Univerza v Ljubljani Fakulteta za gradbeništvo in geodezijo PODIPLOMSKI ŠTUDIJ GRADBENIŠTVA DOKTORSKI ŠTUDIJ Kandidat: UROŠ BOHINC, univ. dipl. inž. fiz. PRILAGODLJIVO MODELIRANJE PLOSKOVNIH KONSTRUKCIJ ADAPTIVE MODELING OF PLATE STRUCTURES Soglasje k temi doktorske disertacije je dal Senat Univerze v Ljubljani na 3. seji 20. novembra 2005 in za mentorja imenoval doc. dr. Boštjana Branka, za somentorja pa prof. dr. Adnana Ibrahimbegoviča. Na 27. seji 3. decembra 2009 je Komisija za doktorski študij dala soglasje k pisanju disertacije v angleškem jeziku. Doktorska disertacija štev.: 212 Doctoral thesis No.: 212 Univerza v Ljubljani Fakulteta za gradbeništvo in geodezijo I^W.TM h VTTl i 1 i|. iTTl In iih i ili mj Ä11I Komisijo za oceno ustreznosti teme doktorske disertacije v sestavi - doc. dr. Boštjan Brank, - prof. dr. Adnan Ibrahimbegovič, ENS Cachan, - izr. prof. dr. Jože Korelc, - doc. dr. Tomaž Rodič, UL NTF je imenoval Senat Fakultete za gradbeništvo in geodezijo na 19. redni seji dne 20. aprila 2005. Komisijo za oceno doktorske disertacije v sestavi - prof. dr. Ivica Kožar, Sveučilište u Rijeci, Gradevinski fakultet, - prof. dr. Miran Saje, - izr. prof. dr. Marko Kegl, UM FS je imenoval Senat Fakultete za gradbeništvo in geodezijo na 17. redni seji dne 26. januarja 2011. Komisijo za zagovor doktorske disertacije v sestavi - prof. dr. Matjaž Mikoš, dekan UL FGG, predsednik, - prof. dr. Ivica Kožar, Sveučilište u Rijeci, Gradevinski fakultet, - prof. dr. Miran Saje, - izr. prof. dr. Marko Kegl, UM FS, - prof. dr. Boštjan Brank, mentor, - prof. dr. Adnan Ibrahimbegovič, ENS Cachan, somentor. je imenoval Senat Fakultete za gradbeništvo in geodezijo na 20. redni seji dne 20. aprila 2011. Univerza v Ljubljani Fakulteta za gradbeništvo in geodezijo I^W.TM h VTTl i 1 i|. iTTl In iih i ili mj Ä11I IZJAVA O AVTORSTVU Podpisani UROŠ BOHINC, univ. dipl. inž. fiz., izjavljam, da sem avtor doktorske disertacije z naslovom: »PRILAGODLJIVO MODELIRANJE PLOSKOVNIH KONSTRUKCIJ«. Ljubljana, 5. maj 2011 (podpis) BIBLIOGRAPHIC-DOCUMENTARY INFORMATION UDC Author: Supervisor: Co-supervisor: Title: Notes: Key words: 519.61/.64:624.04:624.073(043.3) Uros Bohinc assoc. prof. Boštjan Brank prof. Adnan IbrahimbegoviC Adaptive analysis of plate structures 289 p., 8 tab., 185 fig., 332 eq. structures, plates, plate models, finite element method, discretization error, model error, adaptivity Abstract The thesis deals with adaptive finite element modeling of plate structures. The finite element modeling of plates has grown to a mature research topic, which has contributed significantly to the development of the finite element method for structural analysis due to its complexity and inherently specific issues. At present, several validated plate models and corresponding families of working and efficient finite elements are available, offering a sound basis for an engineer to choose from. In our opinion, the main problems in the finite modeling of plates are nowadays related to the adaptive modeling. Adaptive modeling should reach much beyond standard discretization (finite element mesh) error estimates and related mesh (discretization) adaptivity. It should also include model error estimates and model adaptivity, which should provide the most appropriate mathematical model for a specific region of a structure. Thus in this work we study adaptive modeling for the case of plates. The primary goal of the thesis is to provide some answers to the questions related to the key steps in the process of adaptive modeling of plates. Since the adaptivity depends on reliable error estimates, a large part of the thesis is related to the derivation of computational procedures for discretization error estimates as well as model error estimates. A practical comparison of some of the established discretization error estimates is made. Special attention is paid to what is called equilibrated residuum method, which has a potential to be used both for discretization error and model error estimates. It should be emphasized that the model error estimates are quite hard to obtain, in contrast to the discretization error estimates. The concept of model adaptivity for plates is in this work implemented on the basis of equilibrated residuum method and hierarchic family of plate finite element models. The finite elements used in the thesis range from thin plate finite elements to thick plate finite elements. The latter are based on a newly derived higher order plate theory, which includes through the thickness stretching. The model error is estimated by local element-wise computations. As all the finite elements, representing the chosen plate mathematical models, are re-derived in order to share the same interpolation bases, the difference between the local computations can be attributed mainly to the model error. This choice of finite elements enables effective computation of the model error estimate and improves the robustness of the adaptive modeling. Thus the discretization error can be computed by an independent procedure. Many numerical examples are provided as an illustration of performance of the derived plate elements, the derived discretization error procedures and the derived modeling error procedure. Since the basic goal of modeling in engineering is to produce an effective model, which will produce the most accurate results with the minimum input data, the need for the adaptive modeling will always be present. In this view, the present work is a contribution to the final goal of the finite element modeling of plate structures: a fully automatic adaptive procedure for the construction of an optimal computational model (an optimal finite element mesh and an optimal choice of a plate model for each element of the mesh) for a given plate structure. BIBLIOGRAFSKO-DOKUMENTACIJSKA STRAN UDK Avtor: Mentor: Somentor: Naslov: Obseg in oprema: Ključne besede: 519.61/.64:624.04:624.073(043.3) Uros Bohinc izr. prof. dr. Boštjan Brank prof. dr. Adnan IbrahimbegoviC Prilagodljivo modeliranje ploskovnih konstrukcij 289 str., 8 pregl., 185 sl., 332 en. konstrukcije, metoda konCnih elementov, plosCe, modeli plosC, napaka diskretizacije, modelska napaka, prilagodljivost Izvlecek V disertaciji se ukvarjamo z različnimi vidiki modeliranja ploskovnih konstrukcij s končnimi elementi. Modeliranje plosč je nekoliko specifično in je zaradi kompleksnosti in pojavov, ki jih opisuje, bistveno prispevalo k razvoju same metode končnih elementov. Danes je na voljo več uveljavljenih modelov plosč in pripadajočih končnih elementov, ki uporabniku nudijo siroko mnozičo moznosti, iz katere lahko izbira. Prav siroka moznost izbire predstavlja tudi največjo tezavo, saj je tezje določiti, kateri model je primernejsi in tudi, katera mreza končnih elementov je za dan problem optimalna. Glavni čilj disertačije je raziskati ključne korake v pročesu prilagodljivega modeliranja plosč, ki omogoča samodejno določitev optimalnega modela za dan problem. Ker je prilagodljivo modeliranje odvisno od zanesljivih očen napak, je večji del disertačije posvečen metodam za izračun diskretizačijske in modelske napake. Na praktičnih primerih smo preučili nekaj najbolj uveljavljenih metod za očeno napake. V nasprotju z očenami napake diskretizačije, je modelsko napako mnogo tezje določiti. Posebna pozornost je bila zato namenjena metodi uravnoteženja rezidualov, ki ima potential tudi na področju očene modelske napake. V tem smislu to delo predstavlja pomemben prispevek k področju računanja modelske napake za plosče. Končept prilagodljivega modeliranja ploskovnih konstrukčij je bil preskusen na hierarhični druzini končnih elementov za plosče - od tankih plosč do modelov visjega reda, ki upostevajo deformačije po debelini. Ravno dobro vzpostavljena hierarhija v druzini končnih elementov se je pokazala za ključno pri zanesljivi očeni modelske napake. Prilagodljivo modeliranje ploskovnih konstrukčije je bilo preskuseno na nekaj zahtevnejsih primerih. Območje je bilo najprej modeliranjo z najbolj grobim modelom na sorazmerno redki mrezi. Z uporabo informačije o napaki začetnega izračuna je bil zgrajen nov model. Primerjava izračuna na novem modelu z začetnim računom je pokazala, daje predlagan način prilagodljivega modeliranja sposoben nadzorovati porazdelitev napake, kakor tudi zajeti vse pomembnejse pojave, ki so značilni za modeliranje plosč. INFORMATION BIBLIOGRAPHIQUE-DOCUMENTAIRE CDU Auteur: Directeur de these: Co-directeur de these: 519.61/.64:624.04:624.073(043.3) Titre: Notes: Uros Bohinc prof. Boštjan Brank prof. Adnan IbrahimbegoviC Modelisation adaptives des structures Mots cies: 289 p., 8 tab., 185 fig., 332 eq. structures, plaques, modele de plaques, methode elements finis, erreur de discretisation, erreur de modelisation, adaptativite Resume Le rapport de these traite la modelisation adaptative des plaques par la methode des elements finis. Le domaine de recherche sur la methode des elements finis de plaques a atteint une certaine maturite et a contribue d'une maniere significative au developpement de la methode des elements finis dans un sens plus large, en apportant les reponses a des problemes tres specifiques et complexes. A present, on dispose de nombreux modeles de plaques et des formulations d'elements finis correspondantes efficaces et operationnelles, qui offrent a l'ingenieur une bonne base pour bien choisir la solution adaptee. A notre avis, les principaux problemes ouverts pour la modelisation des plaques par elements finis sont aujourd'hui lies a la modelisation adaptative. Une modelisation adaptative devrait aller au-dela de l'estimation d'erreurs dues a la discretisation standard (maillage elements finis) et de l'adaptivite de maillages. Elle devrait inclure aussi l'estimation d'erreur due au modele et l'adaptativite de modeles, afin de disposer d'un modele le mieux adapte pour chaque sous-domaine de la structure. L'objectif principal de la these est de repondre a des questions liees aux etapes cle d'un processus de l'adaptation de modeles de plaques. Comme l'adaptativite depend des estimateurs d'erreurs fiables, une part importante du rapport est dediee au developpement des methodes numeriques pour les estimateurs d'erreurs aussi bien dues a la discretisation qu'au choix du modele. Une comparaison des estimateurs d'erreurs de discretisation d'un point de vue pratique est presentee. Une attention particuliere est pretee a la methode de residuels equilibres (en anglais, "equilibrated residual method"), laquelle est potentielle-ment applicable aux estimations des deux types d'erreurs, de discretisation et de modele. Il faut souligner que, contrairement aux estimateurs d'erreurs de discretisation, les estimateurs d'erreur de modele sont plus difficiles a elaborer. Le concept de l'adaptativite de modeles pour les plaques est implemente sur la base de la methode de residuels equilibres et de la famille hierararchique des elements finis de plaques. Les elements finis derives dans le cadre de la these, comprennent aussi bien les ele ments de plaques minces et que les elements de plaques epaisses. Ces derniers sont formule s en s'appuyant sur une theorie nouvelle de plaque, integrant aussi les effets d' etirement le long de l' epaisseur. Les erreurs de modele sont estimees via des calculs el ement par el ement. Les erreurs de discre tisation et de modele sont estimees d'une maniere independante, ce qui rend l'approche tres robuste et facile a utiliser. Les methodes developpees sont appliquees sur plusieurs exem-ples numeriques. Les travaux realises dans le cadre de la these representent plusieurs contributions qui visent l'objectif final de la modelisation adaptative, ou une procedure completement automatique permettrait de faire un choix optimal du modele de plaques pour chaque el ement de la structure. ZAHVALA Za nesebično pomoč in vso spodbudo se zahvaljujem mentorjema, prof. dr. Boštjanu Branku in prof. dr. Adnanu Ibrahimbegoviču. Na prehojeno pot se oziram z zadovoljstvom, da sem imel prilošnost in čast delati z vama. Zahvaljujem se izr. prof. dr. Andrašu Legatu, ki me je podpiral ves čas študija, tudi takrat, ko to ni bilo lahko. Vesel sem, da sem pri študiju vztrajal, saj sicer ne bi spoznal takih prijateljev, kot sta Jaka in Damijan. Ceprav se nisva spoznala med študijem, pa si prav takrat pokazal svojo ■srčnost in prijateljstvo. Vlado, hvala ti, ker si znal biti in ostati prijatelj do konca. Brez svojih najblišjih, mojih staršev in staršev moje šene tega dela danes gotovo ne bi bilo. Ob dolgi in zaviti poti sem bil vedno deležen vaše nesebične podpore in pomoši, za kar sem vam iskreno hvalešen. Moja drušina, Polona, Blaš in Tine so sodelovali pri tem delu bolj, kot sem si upal predstavljati na začetku. Neskonšno zaupanje in podporo, ki sem ju dobil od vas, lahko dobiš le od nekoga, ki te ima resnično rad. To delo posvečam tebi, Klemen. Contents 1 Introduction 1 1.1 Motivation........................................................................1 1.1.1 Verification and Validation..............................................2 1.1.2 Discretization............................................................2 1.1.3 Model error. Discretization error. The optimal model..................3 1.1.4 Adaptive modeling. Error estimates and indicators..................5 1.2 Goals of the thesis ..............................................................6 1.3 Outline of the thesis ............................................................7 2 Thin plates: theory and finite element formulations 9 2.1 Introduction......................................................................9 2.2 Theory............................................................................9 2.2.1 Governing equations ....................................................9 2.2.2 Further details of the Kirchhoff model ................................19 2.3 Finite elements..................................................................26 2.3.1 Preliminary considerations..............................................26 2.3.2 An overview of thin plate elements ....................................29 2.3.3 Natural coordinate systems ............................................30 2.3.4 Conforming triangular element ........................................37 2.3.5 Discrete Kirchhoff elements ............................................41 2.4 Examples ........................................................................50 2.4.1 Uniformly loaded simply supported square plate ......................51 2.4.2 Uniformly loaded clamped square plate ................................53 2.4.3 Uniformly loaded clamped circular plate ..............................54 2.4.4 Uniformly loaded hard simply supported skew plate ..................70 2.5 Chapter summary and conclusions ..............................................75 3 Moderately thick plates: theory and finite element formulations 77 3.1 Introduction......................................................................77 3.2 Theory............................................................................78 3.2.1 Governing equations....................................................78 3.2.2 Strong form..............................................................82 3.2.3 Weak form................................................................85 3.2.4 Boundary layer and singularities........................................86 3.3 Finite elements..................................................................89 3.3.1 Elements with cubic interpolation of displacement....................92 3.3.2 Elements with incompatible modes....................................95 3.4 Examples ........................................................................97 3.4.1 Uniformly loaded simply supported square plate ......................97 3.4.2 Uniformly loaded simply supported free square plate ................98 3.4.3 Uniformly loaded simply supported skew plate............100 3.4.4 Uniformly loaded soft simply supported L-shaped plate.......101 3.5 Chapter summary and conclusions.......................119 4 Thick plates: theory and finite element formulations 121 4.1 Introduction...................................121 4.2 Theory ...................................... 122 4.2.1 Governing equations .......................... 122 4.2.2 Principle of virtual work ........................ 124 4.3 Finite elements ................................. 128 4.3.1 Higher-order plate elements ...................... 128 4.4 Hierarchy of derived plate elements ...................... 133 4.5 Examples .................................... 136 4.5.1 Uniformly loaded simply supported square plate ........... 136 4.5.2 Uniformly loaded soft simply supported L-shaped plate ....... 137 4.5.3 Simply supported square plate with load variation .......... 138 4.6 Chapter summary and conclusions ....................... 145 5 Discretization error estimates 147 5.1 Introduction. Classification of error estimates ................147 5.2 Definitions. Linear elasticity as a model problem...............149 5.3 Recovery based error estimates ........................151 5.3.1 Lumped projection...........................153 5.3.2 Superconvergent patch recovery (SPR)................153 5.4 Residual based error estimates.........................155 5.4.1 Explicit .................................155 5.4.2 Implicit .................................158 5.5 Illustration of SPR and EqR methods on 1D problem............169 5.6 Chapter summary and conclusions.......................172 6 Discretization error for DK and RM elements 175 6.1 Introduction...................................175 6.2 Discretization error for DK elements .....................176 6.2.1 Discrete approximation for error estimates..............176 6.2.2 Formulation of local boundary value problem.............179 6.2.3 Enhanced approximation for the local computations.........183 6.2.4 Numerical examples ..........................185 6.2.5 An example of adaptive meshing ...................189 6.3 Discretization error for RM elements.....................201 6.4 Chapter summary and conclusions.......................201 7 Model error concept 203 7.1 Introduction...................................203 7.2 Model error indicator based on local EqR computations ......................205 7.2.1 Definition of model error indicator ..................205 7.2.2 Construction of equilibrated boundary tractions for local problems . 210 7.3 Chapter summary and conclusions.......................216 8 Model error indicator for DK elements 219 8.1 Introduction...................................219 8.2 Model error indicator..............................221 8.2.1 Regularization ............................................................223 8.2.2 The local problems with RM plate element ............................224 8.2.3 Computation of the model error indicator ............................226 8.2.4 Numerical examples ....................................................226 8.3 Chapter summary and conclusions.......................241 9 Conclusions 243 10 Razširjeni povzetek 247 10.1 Motivacija....................................247 10.1.1 Verifikacija in validacija ........................248 10.1.2 Prilagodljivo modeliranje - optimalni model .............248 10.1.3 Napaka diskretizacije, modelska napaka................250 10.2 Cilji .......................................251 10.3 Zgradba naloge ..................................................................252 10.4 Plosce......................................253 10.4.1 Tanke plosce...............................253 10.4.2 Srednje debele plosce..........................261 10.4.3 Debele plosce..............................268 10.5 Ocene napak ......................................................................273 10.5.1 Napaka diskretizacije..........................273 10.5.2 Ocena modelske napake ................................................278 10.6 Zakljucek ....................................281 List of Figures 2.2.1 Mathematical idealization of a plate, • • •....................................10 2.2.2 Stress resultants - shear forces and moments - on a midsurface differential element ........................................................................13 2.2.3 Rotations at the boundary....................................................14 2.2.4 Boundary coordinate system, moments and test rotations ................17 2.2.5 Concentrated corner force at node I..........................................19 2.2.6 To the definition of a corner force in (2.2.47)................................19 2.2.7 Plate boundary traction loading • • • ........................................20 2.2.8 Definition of a polar coordinate system (r, at the corner................24 2.3.1 Problem domain Q and its boundary r (left), finite element discretization Qh, Qh and patch of elements around node i (right) ..................28 2.3.2 Bi-unit element QD in coordinate system (£,n)..............................31 2.3.3 Quadrilateral element with the alternate shading between the coordinate lines of the quadrilateral natural coordinate system........................33 2.3.4 Triangular (area) coordinates................................................33 2.3.5 Triangular element with the alternate shading between the coordinate lines (i = const. of triangular coordinate system.............. 35 2.3.6 To the definition of eccentricity parameter ß12 = 2e1/l12 ......... 37 2.3.7 Triangular Argyris plate element - ARGY................. 38 2.3.8 Hierarchical shape functions......................... 42 2.3.9 Euler-Bernoulli beam element........................ 42 2.3.10 Hermite shape functions........................... 43 2.3.11 Node and side numbering in (2.3.64).................... 45 2.3.12 Triangular DK plate element - DK..................... 46 2.3.13 Hierarchic shape functions of the triangular DK plate element...... 48 2.3.14 Hierarchic shape functions of the quadrilateral DK plate element .... 49 2.4.1 Designation of boundary conditions and loads ............... 50 2.4.2 Problem definition and geometry for the uniformly loaded simply supported square plate (t/a=1/1000)...................... 51 2.4.7 Problem definition and geometry for the uniformly loaded clamped square plate (t/a=1/1000).............................. 53 2.4.12 Clamped circular plate with uniform load (t/R=1/500).......... 55 2.4.3 Reference solution of the uniformly loaded simply supported square plate •••....................................... 57 2.4.4 FE solution of the uniformly loaded simply supported square plate with Argyris plate element ............................ 58 2.4.5 Finite element solution of the uniformly loaded simply supported square plate with DK plate elements ........................ 59 2.4.6 Comparison of the convergence of FE solutions for the uniformly loaded simply supported square plate ....................... 60 2.4.8 Reference solution of the uniformly loaded clamped square plate , ■ ■ ■ . . 61 2.4.9 FE solution of the uniformly loaded clamped square plate with Argyris plate element ................................. 62 2.4.10 FE solution of the uniformly loaded clamped square plate with DK plate elements .................................... 63 2.4.11 Comparison of the convergence of FE solutions for the uniformly loaded clamped square plate ............................ 64 2.4.13 Reference solution of the uniformly loaded clamped circular plate , ■ ■ ■ . 65 2.4.14 The sequence of meshes used for the finite element solution of uniformly loaded clamped circular plate ........................ 66 2.4.15 FE solution of the uniformly loaded clamped circular plate with Argyris plate element ................................. 67 2.4.16 FE solution of the uniformly loaded clamped circular plate with DK plate elements .................................... 68 2.4.17 Comparison of the convergence of FE solutions for the uniformly loaded clamped circular plate ............................ 69 2.4.18 Problem definition and geometry for the uniformly loaded simply supported skew plate (t/a=100)......................... 70 2.4.19 Reference solution of the uniformly loaded simply supported skew plate , ■ ■ ■...................................... 71 2.4.20 FE solution of the uniformly loaded simply supported skew plate with ARGY element, ■ ■ ■ ............................. 72 2.4.21 FE solution of the uniformly loaded simply supported skew plate with DK elements, ■ ■ ■............................... 73 2.4.22 Comparison of the convergence of FE solutions for the uniformly loaded simply supported skew plate ........................ 74 3.3.1 To the computation of the nodal shear ■yh,I from (3.3.18)......... 94 3.3.2 Quadrilateral plate element P3Q with cubic displacement interpolation . 96 3.4.1 Problem definition and geometry for the uniformly loaded hard-soft simply supported square plate (t/a=1/10)................... 97 3.4.6 Problem definition and geometry for the uniformly loaded hard simply supported-free square plate (t/a=10).................... 99 3.4.11 Problem definition and geometry for the uniformly loaded soft simply supported skew plate (t/a=10).......................100 3.4.16 Problem definition and geometry for the uniformly loaded soft simply supported L-shaped plate (t/a=10).....................101 3.4.2 Reference solution of the uniformly loaded hard-soft simply supported square plate with legend also valid for Figures 3.4.4 and 3.4.3 ...... 103 3.4.3 FE solution of the uniformly loaded hard-soft simply supported square plate with PI plate elements ......................... 104 3.4.4 FE solution of the uniformly loaded hard-soft simply supported square plate with P3 plate elements (mesh as in Figure 3.4.3) .......... 105 3.4.5 Comparison of the convergence of FE solutions for the uniformly loaded hard-soft simply supported square plate .................. 106 3.4.7 Reference solution of the uniformly loaded hard simply supported-free square plate with legend also valid for Figures 3.4.9 and 3.4.8 ...... 107 3.4.8 FE solutions of the uniformly loaded hard simply supported-free square plate with PI plate elements ......................... 108 3.4.9 FE solutions of the uniformly loaded hard simply supported-free square plate with P3 plate elements (mesh as in Figure 3.4.8)..........109 3.4.10 Comparison of the convergence of FE solutions for the uniformly loaded hard simply supported-free square plate .................. 110 3.4.12 Reference solution of the uniformly loaded soft simply supported skew plate with legend also valid for Figures 3.4.14 and 3.4.13.........111 3.4.13 FE solution of the uniformly loaded soft simply supported skew plate with PI plate elements ............................ 112 3.4.14 FE solution of the uniformly loaded soft simply supported skew plate with P3 plate elements (mesh as in Figure 3.4.13).............113 3.4.15 Comparison of the convergence of FE solutions for the uniformly loaded soft simply supported skew plate ...................... 114 3.4.17 Reference solution of the uniformly loaded soft simply supported L-shaped plate with legend also valid for Figures 3.4.19 and 3.4.18.....115 3.4.18 FE solution of the uniformly loaded soft simply supported L-shaped plate with PI plate elements ............................ 116 3.4.19 FE solution of the uniformly loaded soft simply supported L-shaped plate with P3 plate elements (mesh as in Figure 3.4.18).............117 3.4.20 Comparison of the convergence of FE solutions for the uniformly loaded soft simply supported L-shaped plate ................... 118 4.2.1 Kinematics of through-the-thickness deformation in the thick plate model 122 4.3.1 Designation of plate surfaces Q+, Q- and r................132 4.5.1 Problem definition and geometry for the uniformly loaded hard-soft simply supported square plate (t/a=1/10)...................136 4.5.4 Problem definition and geometry for the uniformly loaded soft simply supported L shaped plate (t/a=1/10)....................137 4.5.7 Problem definition and geometry for the simply supported square plate with load variation (t/a=1/10 and t/b=1).................138 4.5.2 FE solution of the uniformly loaded simply supported square plate with PZ plate elements .............................. 139 4.5.3 Legend for Figures 4.5.2...........................140 4.5.5 FE solution of the uniformly loaded soft simply supported L shaped plate with PZ plate elements ............................ 141 4.5.6 Legend for Figures 4.5.5...........................142 4.5.8 FE solution of the simply supported square plate with load variation with PZ plate elements ............................ 143 4.5.9 Legend for Figures 4.5.8...........................144 5.1.1 Rough classification of a-posteriori error estimates/indicators ......149 5.3.1 SPR recovery of the enhanced stress a} at node I.............155 5.4.1 Notation....................................159 5.4.2 Relation of tractions tf to their projections rf,r..............163 5.4.3 Element nodal residual Rf and boundary tractions projections rf,ri2 at node I of element e..............................164 5.4.4 Continuity of boundary tractions projections rf,r.............164 5.4.5 Patch p of four elements around node I..................165 5.4.6 Representation of element nodal residuals Rf by the projections rf,r . . 168 5.4.7 Boundary tractions tf replace the action of element projections rf,r . . . 168 5.5.1 One dimensional problem: hanging bar with variable cross section .... 169 5.5.2 Solution of the model problem with linear elements............171 5.5.3 Solution of the model problem with quadratic elements..........171 5.5.4 Computation of enhanced stress a*; comparison of SPR and EqR method 172 6.2.1 Subdivision schemes for the DKT element.................184 6.2.2 Simply supported square plate under uniform loading - global energy error estimates ................................................................186 6.2.3 Simply supported square plate under uniform loading - effectivity index of the global energy error estimates ..........................................187 6.2.4 Simply supported square plate under uniform loading - comparison of relative local error estimates ..................................................192 6.2.5 Clamped square plate under uniform loading - global energy error estimates 193 6.2.6 Clamped square plate under uniform loading - effectivity index of the global energy error estimates ................................................193 6.2.7 Clamped square plate under uniform loading - comparison of relative local error estimates ..........................................................194 6.2.8 Clamped circular plate under uniform loading - global energy error estimates ..........................................................................195 6.2.9 Clamped circular plate under uniform loading - effectivity index of the global energy error estimates ................................................195 6.2.10 Clamped circular plate under uniform loading - comparison of relative local error estimates ..........................................................196 6.2.11 Morley's skew plate under uniform loading - global energy error estimates197 6.2.12 Morley's skew plate under uniform loading - effectivity index of the global energy error estimates ........................................................197 6.2.13 Morley's skew plate under uniform loading - comparison of relative local error estimates ................................................................198 6.2.14 L-shaped plate under uniform loading - global energy error estimates . . 199 6.2.15 L-shaped plate under uniform loading - effectivity index of the global energy error estimates ............................ 199 6.2.16 L-shaped plate under uniform loading - comparison of relative local error estimates ................................... 200 7.2.1 Local computation on a finite element domain Qeh with Neumann boundary conditions tpe on re...........................207 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.2.6 8.2.7 8.2.8 8.2.9 8.2.10 8.2.11 8.2.12 8.2.13 8.2.14 8.2.15 8.2.16 Model error indicator for thick SSSS problem for rectangular meshes Model error indicator for thin SSSS problem for distorted meshes . . Model error indicator for thin SSSS problem for rectangular meshes . Model error indicator for thin SSSS problem for distorted meshes . . Model error indicator for thick SFSF problem for rectangular meshes Model error indicator for thick SFSF problem for distorted meshes Model error indicator for thin SFSF problem for rectangular meshes Model error indicator for thin SFSF problem for distorted meshes LSHP -LSHP -LSHP -LSHP -SKEW SKEW SKEW model error indicator stress resultant mxx . stress resultant mxy . stress resultant qx . . - model error indicator - stress resultant mxx - stress resultant mx xy SKEW - stress resultant qx 228 229 230 231 232 233 234 235 235 236 237 238 238 239 239 240 List of Tables 2.1 Boundary conditions for thin plates. * dn = ds is zero due to w = 0..... 23 2.2 Critical opening angles for the Kirchhoff plate (* - the case reexamined in this section) ................................... 26 2.3 4 point triangular integration scheme for triangular finite elements (DKT) 47 2.4 4 point quadrilateral integration scheme for quadrilateral finite elements (DKQ)...................................... 48 3.1 Boundary conditions for moderately thick plates............... 80 3.2 Leading terms of thickness expansion in (3.2.38)............... 88 3.3 The singularity coefficients, Ai and A2..................... 89 6.1 Comparison of errors of FE computation of the square clamped plate problem with various meshes adaptively constructed from discretization error estimate......................................191 Chapter 1 Introduction 1.1 Motivation Understanding the behavior of structures under various loads has always been of primary interest to man. Throughout the history the knowledge on the structural behavior was predominantly acquired by experiments. But due to the rising complexity of the man made structures it became obvious that it is not possible nor economical to perform the experimental validation of all possible loading situations. It was realized that mathematical models (i.e. equations) describing physical phenomena, which would be able to predict the structural behavior, were needed. The models of physical phenomena have been build upon the experimental observations. The models are therefore only as accurate as the experiments are. The accuracy of the model predictions inherently depend on the accuracy of the experimental results. The accuracy is limited also by the computational ability of the solver, used to solve equations of the model, which affects the level of the detail attainable in the model. In the early days the model equations were solved analytically. Only relatively small set of engineering problems with simple geometry and material laws were suitable to be solved that way. Some numerical approximations were attempted but were seriously limited by the computing power of the solver (usually human at that time). With the advent of the modern computers the numerical approximations of the solution became more elaborate and it became possible to numerically solve complex real life problems. Numerical computations have nowadays become an integral part of engineering design process. Critical design decisions are routinely made on the basis of numerical simulations. It is therefore vitally important to clearly define the reliability and accuracy (i.e. the error) of the computational predictions of the model. A numerical simulation (i.e. the mathematical model plus the numerical methods) of a physical phenomena is accepted only when the predictions match the outcome of the controlled experiments. The model predictions are therefore always made in the form of measurable physical quantities. Since the engineering approach to modeling goes by the paradigm as simple as possible and as complicated as necessary the necessary level of complexity of a physical model is defined by the required accuracy of the model. The difference between the controllably measured physical reality and the model predictions can be attributed to one of the two reasons: (i) the mathematical model is either based upon the wrong assumptions and/or (ii) the equations of the mathematical model can be solved only approximately for a given data. The error of a model prediction, which is caused by the wrong assumptions, is usually referred to as the model error. The approximative nature of the model results is related to the discretization error. 1.1.1 Verification and Validation The total error of the prediction is thus always composed of a model error and a discretization error. The approximation of the exact model equations reflects itself in the model error, while the discretization error is a consequence of the approximation in the solution of the chosen model equations. The concept of verification and validation (V&V) has been established to study the model performance in terms of accuracy. Comparison of the solution of the discretized model with the exact solution of the model is a subject of verification. The validation usually concerns the comparison of the model predictions with the results of the experiments. 1.1.2 Discretization Mathematical model (or just model) is a mathematical idealization of a physical model which is build upon certain assumptions and simplifications. Certain degree of abstraction is used in the model derivation. It usually involves the concept of continuity, which assumes that the solution is known at every point of the problem domain. Such models cannot be solved numerically since the number of points in the problem domain is infinite. In order to obtain a numerically solvable system, the problem has to be discretized with a finite number of variables. A widespread method of the discretization of a given continuous mathematical model is the finite element method (FEM). The discretization introduces a discretization error. The finite element method was first conceived by engineers and mathematicians to solve the problems of linear elasticity. Nowadays, the prevalent use of the finite element method remains in the field of linear structural mechanics. The main focus of this work is the finite element analysis of bending of plate structures, which fits in that framework. 1.1.3 Model error. Discretization error. The optimal model. In order to find the optimal model for the analysis of a structure behavior - in our case elastic plate bending - the criteria have to be set first. If the accuracy is the only criteria, the best possible model is the exact fully three-dimensional model of elastic plate bending. Such model is of course prohibitively complicated for usual plate bending problems. The plate model is therefore proclaimed as optimal in the engineering sense: besides the necessary (desired) accuracy, the model must be as simple as possible. The plates are basic structural elements and several models of their bending behavior exist. Since the third dimension of the plate is significantly smaller than the other two, the plate models are subject to dimensional reduction. The computational advantage of using two-dimensional model instead of fully three-dimensional model is obvious. Although the three-dimensional model should asymptotically converge to two-dimensional plate model (for decreasing plate thickness), not many plate models are strictly build on the principles of asymptotic analysis. Various important plate models are rather based on a-priori assumptions coming from the engineering intuition. Nevertheless, a hierarchy of such plate models can still be established with respect to the convergence to the full three-dimensional model, [Babuska Li, 1990]. It is thus possible to form a family of hierarchically ordered plate models - a series of models, whose solutions converge to the exact three-dimensional solution of plate bending problem. The convergence to the exact three-dimensional solution manifests itself, for example, in the ability to describe the boundary layer effects, which are typical for the three-dimensional solution. Model error The a-priori assumptions of a plate model are usually related to its kinematics. The kinematics of the plate strongly depends on the boundary conditions, concentrated forces and abrupt changes in thickness or material properties. Since the kinematics changes over the plate problem domain, so does the validity of model assumptions. The model is optimal only in the region, where its assumptions are valid. It is therefore clear that the optimal model is location dependant. There is no single model which is optimal for the whole plate domain. The plate domain should therefore be divided into regions, each being modeled with its own optimal model. The main goal of the model adaptivity is to identify the regions and determine the optimal model for each one of them. The region dependent optimal model of the plate structure can be built iteratively. The engineering approach is to start from bottom up. Preferably, we start the analysis with the coarsest possible model over the whole problem domain. Through the postprocessing of the solution, the model error estimates are obtained. Based on the prescribed accuracy (i.e. the prescribed value of the model error), regions, where the starting model is to be enhanced, are identified. A new, mixed model, is built for the plate structure under consideration and a new solution is searched for. The model error is estimated again and the procedure is repeated iteratively until the desired accuracy is achieved in all the regions of the problem domain. The goal of this approach is an effective and accurate estimation of the model error. Ideally, the model error estimate should be effective and as accurate as possible. It is clear, that the above described procedure for choosing the most suitable plate model for each region of a plate structure is possible only when the equations of the chosen models are solved exactly. The discretization error Let us assume that a regional optimal model of a plate structure is somehow chosen. It has to be now discretized in order to obtain a numerically solvable problem. The finite element discretization divides the problem domain into the finite elements. On each of the elements the approximation of the solution is build based on the finite number of degrees of freedom. The quality of the approximation depends on the element size and the relative variation of the solution. The higher the variation of the solution, the smaller elements have to be used to properly describe it. The optimal discretization is the one, for which the discretization error is roughly the same for every element. In order to keep the discretization error uniformly distributed between the elements, the size of the element has to be adapted to the solution. The partition of the problem domain (the mesh) is therefore constructed iteratively. First, a relatively coarse mesh is constructed and an original solution is computed. A discretization error estimate is computed from the original solution. Since the dependance of the discretization on the element size is usually known a-priori and the predefined error tolerance is set, a new element size can be computed. A new mesh is generated, which takes into account the computed element size distribution. The process is iterated until the discretization error is evenly distributed over the problem domain. Total error In the above paragraphs it has been assumed that the model error can be obtained with fixed (preferably zero) discretization error, and that discretization error can be obtained with fixed model error. In reality both errors are connected and can be separated only under certain assumptions. Optimal model The optimal model, in the sense of model and discretization error, has to be adapted to the problem studied. The main benefit of using the adapted model is probably not in the gain in the computational efficiency but rather in the control of the accuracy of the model predictions. The automated procedure for the selection of the optimal model should prevent a designer to oversimplify the model and potentially overlook the possible problematic issues. On the other hand, due to the ever increasing computational power, the temptation to use full three-dimensional models in the whole problem domain is increasing. Although the fine models do capture the physics of the phenomena, they are more difficult to interpret and they obscure the basic phenomena by unnecessary details. 1.1.4 Adaptive modeling. Error estimates and indicators The adaptive modeling is an iterative process which crucially relies on the estimates of both model error and discretization error. At this point, one should clearly distinguish between the error estimates and error indicators. The estimates define the boundaries of the error. They are usually quite conservative and they tend to overestimate the error by several orders of magnitude. Indicators on the other hand do not provide any guarantees on the error. In return they can give quite sharp indication of the error. Which type of model error to use depends on the purpose of the computation of the error. If the main goal is to control the adaptive construction of the optimal model, the indicators are the preferable choice. The computation of reliable and efficient error indicators is a key step in the construction of the optimal model. Although the model error can be orders of magnitudes higher than the discretization error, its evaluation is much harder. Therefore there are not many existing methods for the computation of model error indicator. Their development is mostly on the level of general treatment. In this work one of the existing concepts of model error indicators was applied to the subject of plate modeling. Its development is the main achievement of this thesis. The effective procedure of computation of model error indicator is build upon the following idea. The model error, in principle, measures the difference of the solution of the applied (current) model and the solution of the exact (full three-dimensional) model. The global computation of the true solution - just to serve as the reference to model error computation - is obviously too expensive. An effective compromise is to repeat the computations with the exact model on smaller domains - preferably on elements. The local problems have to replicate the original problem. The interpretation is, that the elements are extracted from the continuum and its actions are replaced by boundary tractions. Therefore the boundary conditions for the local problems are of Neumann type. The primary question of model error estimation now becomes: "How to estimate the boundary conditions for the local problems?". If the exact solution had been known, the boundary conditions for the elements would have been computed from the true stress state using the Cauchy principle. This is, however, not possible since the stresses are determined from the true solution, which is unknown and yet to be computed. An approximation of the boundary stresses must therefore be built based on the single information available: current finite element solution, an approximation to the true solution. The development of the method of the construction of the best possible estimates for the boundary conditions for local problems is thus the central topic of the model error computation. The application of this general principle to the subject of modeling of plates is not very straightforward. The specific issues inherent to plate modeling have to be properly addressed. Additional complication presents the fact, that the formulations of plate elements are mostly incompatible. 1.2 Goals of the thesis The main goal of the thesis is to develop a concept for adaptive modeling of plate structures. Therefore the first goal of the thesis is to derive a hierarchical family (in a model sense) of fine performing triangle and quadrilateral plate elements. The concept of adaptive modeling depends on the reliable error estimates and error indicators. In the following we will not always strictly distinguish in text between the error indicator and error estimate. The second goal of the thesis is related to discretization error estimates. We derived, implemented and tested some of the most established a posteriori discretization error estimate methods. Two most prominent methods of discretization error estimation are superconvergent patch recovery SPR and method of equilibrated residuals EqR [Stein Ohn-imus, 1999]. We implemented and compared the discretization error estimate methods for Discrete Kirchhoff (DK) and Reissner/Mindlin plate elements, which are very popular ana reliable finite elements for analysis of plate structures. Currently, there exist several methods to obtain estimates of discretization error, while there are only a few methods available to obtain the model error estimates. The third goal of the thesis is to explain the idea to use equilibrated residual method (EqR) to derive model error estimate. The final goal of the thesis is to derive a procedure for adaptive modeling of plates. The procedure should go as follows: obtain an initial finite element solution of a chosen plate problem; estimate both model and discretisation error; build better (in terms of model and discretization errors) finite element model of the problem. This adaptive modeling procedure is planned to be build from bottom up: from the relatively coarse model and mesh to the model and mesh capable of capturing all the important phenomena and to control the overall error as well as the error distribution. 1.3 Outline of the thesis The thesis consists of nine chapters. In the first chapter an overview of the subject and the state of the art is presented. Chapters 2 to 4 present an overview of the theory of plates. The theories are presented in the bottom up fashion: from most basic one towards " quasi" three-dimensional theory. The theoretical treatment of plates reveals the phenomena typical for the plates: boundary layers and singularities. The plates theories which can be arranged in the hierarchical order are accompanied by the description of some possible discretizations: finite elements. Some of the most important and established finite elements are presented in these sections. The chapters conclude with the treatment of selected test cases each one illustrating different aspects of plate modeling. Chapter 5 starts with an overview of the most important methods for the estimation of discretization error. The general description of the methods is followed by the more detailed treatment of superconvergent patch recovery method and a method of equilibrated residuals. The implementation of the latter for the case of discrete Kirchhoff element is given a special focus in Chapter 6. The implementation of the method for the non-conforming plate elements is somewhat special and not treated before in the literature. Several numerical test illustrate the differences between the various error estimation methods. Chapters 7 and 8 cover the question of model error. After the general treatment of the model error an approach to estimate the model error based on the method of equilibrated residuals is followed. The implementation of the method for the case of plate bending is covered in detail and tested by several numerical tests in Chapter 8. The tests are selected to exhibit the basic and most important phenomena specific to the plate models. Concluding remarks of the thesis are given in Chapter 9. Chapter 2 Thin plates: theory and finite element formulations 2.1 Introduction In this chapter, the equations describing the linear elastic bending of thin plates - called Kirchhoff thin plate theory or Kirchhoff thin plate model - will be re-derived, and some conforming and nonconforming (discrete) triangular and quadrilateral Kirchhoff thin plate finite element formulations will be presented in detail - in forms suitable for immediate numerical implementation. For the sake of completeness, some specific details of the Kirchhoff thin plate model will be presented as well. 2.2 Theory The first part of this section, related to the the basic equations of Kirchhoff thin plate theory, is organized in such a way that many equations are first derived for moderately thick plates (commonly called Reissner/Mindlin plates), and only further specialized for thin (i.e. Kirchhoff) plates. Moderately thick plates will be further addressed in the next chapter. 2.2.1 Governing equations Kinematic equations Let the position of an undeformed plate in the xyz coordinate system be given by Q x [—1/2, t/2], where t is plate thickness. The region Q at xy plane (at z = 0) defines the midsurface of the plate. We denote the boundary of Q by r. It is assumed that x ft x> Figure 2.2.1: Mathematical idealization of a plate (left), a plate differential element (middle), directions of rotations (right) Slika 2.2.1: MatematiCna idealizacija ploSCe (levo), diferencialni element (sredina), smer rotacij (desno) z y y when external loading is applied to the plate, any midsurface point can displace only in the direction of z coordinate; we will denote this displacement by w. It is further assumed that the deformation of the plate "fiber", which has initial orientation of the midsurface normal nn = [0, 0,1]T, can be described by a small rigid rotation, defined by a rotation vector 0 = [ftx,fty]T, see Figure 2.2.1. In accordance with the above, the kinematic assumption, which defines small displacements of a point P(x,y,z) of a plate is given as: Ux + Zfty zftx; uz w (2.2.1) The displacements (2.2.1) yield the following strains: dux dx dx ' yy duy dy z dftftx dy xy 09y dftx dy dx (2.2.2) £ =1(+B + dw\ ; £ Sxz = 2 I +fty + "TT" I ; e XZ n 1 1 yJy 1 dx J ' '"yz 2 ( ftx + dy ) ) '-ZZ dw\ dy J and the engineering transverse shear strains, defined as £z Y =[7x,7y] = [2^xz, 2£yz] T (2.2.3) The transverse strain £zz is for thin and moderately thick plate significantly smaller than the strains £xx, £yy and £xy, which makes the relation £zz = 0 justified. However, in plate regions near the supports, near the point loads, at sharp changes of thickness, etc., where £zz = 0 no longer holds true, the thin plate theory and the moderatelly thick plate theory are not adequate, and should be, if possible, replaced by a plate theory which better describes such local phenomena. We note, that the assumption £zz = 0 implies plane strain state in a plate. However, a closer approximation to the physics, according to the £ 0 experimental observations, is the plane stress state with ozz = 0, which implies nonzero . As shown further below, the plane stress state is assumed in the thin plate theory and in the moderatelly thick plate theory rather than the plane strain state. The curvatures k = [kxx, Kyy, Kxy]T at the plate midsurface point are defined as Ody Kxx = dS^ Kyy = + ~6y ; d0x 30y Kxy = ix (2.2.4) dx' dy ' dx dy which leads to the following expressions for nonzero in-plane strains e = [exx,£yy,£xy]T ^xx zkxx j £yy ZKyyj 2^xy ZKxy (2.2.5) In the thin plate Kirchhoff model the transverse shear strains are negligible: ,x - 0 (2.2.6) dw , ix = dx + oy = 0j dw 0 t;-- dy which relates the midsurface deflection with the rotations dw dw o =_; o =__ Vx - ^ i Vy - ~ (2.2.7) dy' dx From (2.2.4) it thus follows that the curvatures for the Kirchhoff thin plate model are rd2w d2w „ d2w K = [ , , 2 ] dx2' dy2' dxdy T and, in more compact writing K = CW- C =[ — ,— , 2 ]T ' dx2' dy2' dxdy (2.2.8) (2.2.9) Constitutive equations Let us assume that a plate under consideration is made of isotropic elastic material, characterized by elastic modulus E and Poisson's ratio v. According to the accumulated engineering experience, the thin plate small strain elastic behavior is described well by the plane stress Hook's law: qxx E °yy = 1 - v2 Qxy 1 v 0 v 1 0 0 0 1 (1 - v) £xx £yy 2f ^ xy (2.2.10) Integration of (2.2.10) through the plate thickness leads to the plate in-plane forces, which are zero, and plate bending moments m = [mxx, myy, mxy]T with dimension of moment per unit length (Figure 2.2.2). p+t/2 f+t/2 p+t/2 (2.2.11) l-t/2 r+t/2 '-t/2 z qxx dz- myy I z Qyy dz- "mxy -t/2 Z Q' xy dz Positive moments yield tensile stresses at z < 0. Rotational equilibrium of a plate differential element around z axis implies Jyx = Jxy and myx = mxy. Using (2.2.5) and (2.2.10), we obtain the moment-curvature relationship m Cb k (2.2.12) where CB is 1 v Cb = D v 1 0 0 0 0 (2.2.13) and D = lEt3/(1 — v2) is isotropic plate rigidity constant. The transverse shear forces, with dimension of force per unit length, are defined as f+t/2 '—1/2 where Ox J yz Jxz dz; Ec Qy +t/2 -t/2 ( yz dz 10 Yx 01 . Yy _ (2.2.14) (2.2.15) 2(1 + v) and c is transverse shear correction factor, which will be discussed in the next chapter. The relationship between the constitutive transverse shear forces and the engineering transverse shear strains can be given in compact form as q = CS7, where q = [qx,qy]T and structure of CS can be seen from the above. For the Kirchhoff thin plate theory the transverse shear forces qx and qy, when computed from the constitutive equations, are zero, see (2.2.6). Thus, in the Kirchhoff thin plate theory the shear forces need to be computed from the equilibrium equations, as shown below. We will not strictly use different notation for the "constitutive" and "equilibrium" shear forces to distinguish between them. Equilibrium equations The equilibrium of a midsurface differential element can be deduced by using Figure 2.2.2. The equilibrium of forces in z direction is V. q = dx + = —f dx oy where f is the applied transverse force per unit area (in the direction of Equilibrium of moments around x and y axes gives: 3mx: dx + dm xy dy Qx; dmyy . dm. dy + xy dx qy (2.2.16) and V = [dx, dy]T. L ax' ay J (2.2.17) 2 q x qy + dy -g>— ^ dy + d y: qx ® Qgx + dx + x qy dx mxy + ^^ dy + mxy + ^dx^ dx + Omx: dx -dx + • • • x y y Figure 2.2.2: Stress resultants - shear forces and moments - on a midsurface differential element Slika 2.2.2: Rezultante napetosti - strižne sile in momenti - na diferencialnem elementu nevtralne ploskve or shortly V ■ M = [ V, V ]TM = M mxx mxy myy (2.2.18) A single equilibrium equation is obtained by elimination of the shear forces from (2.2.17) and (2.2.18) d mxx d ^^xy d myy + 2- + f (2.2.19) Ox2 dxdy dy2 or in a more compact notation L ■ m = f (2.2.20) Although the Kirchhoff model assumes constitutive transverse shear forces as zero, the equilibrium transverse shear forces should be non-zero also for the Kirchhoff model, if the equilibrium equations are to be satisfied. Boundary conditions Let us introduce at each boundary point an orthogonal basis (n, s), where n is the boundary normal and s is the boundary tangential vector, see Figure 2.2.1. We also define 0 = [0x, 0y]T = [—9y, 9x]T. We further denote part of the boundary where w is prescribed (w = w) as rw. Similarly, part of the boundary where is prescribed = is denoted as , and part of the boundary where is prescribed = is denoted as r^s. We further define rw U U r^s = rD. The transformation relations between I|A denotes the value of mn at point I on its A side. The term in the sum corresponding to point I is nonzero only if the mn is discontinuous at that point. The difference can be attributed to the corner force FC I = [mn>I]A, which is an internal force, see Figure 2.2.4. 02 Figure 2.2.6: To the definition of a corner force in (2.2.47) Slika 2.2.6: K definiciji vozlisCne sile (2.2.47) 2.2.2 Further details of the Kirchhoff model Effective transverse shear force B. Figure 2.2.5: Concentrated corner force at node I Slika 2.2.5: Koncentrirana sila v vozliscu I The biharmonic equation (2.2.26) is a partial differential equation of the fourth order. It can only have two boundary conditions at each boundary point. In the moderately thick plate model three independent quantities appear naturally at a boundary point: the transverse displacement w, the normal rotation 9n and the tangential rotation 9s. The conjugate quantities are the transverse shear force q, twisting moment mn and tangential (bending) moment ms respectively. Integration per partes in (2.2.43), however, shows that only two pairs of conjugate quantities exist for the Kirchhoff model: (1) displacement w and effective shear qef, and (2) tangential rotation 9s and tangential moment ms. This is the direct consequence of the Kirchhoff assumption (2.2.6) which relates displacement to the rotations. Figure 2.2.7: Boundary loading mn, ms and q for moderately thick plate (left) and Kirchhoff plate (right) Slika 2.2.7: Robna obtežba mn, ms in q za srednje debele plosCe (levo) in Kirchhoffove plosCe (desno) Reaction corner force Let us consider the point lying at the corner of the plate, denoted by I. The corner with opening angle a extends between angles 01 and 02 (a = 02 — 0i), see Figure 2.2.6. jjn If the moment stress resultants at the corner I are [mxx,myy,mxy]T, the twisting moment at the side with angle 0 is mn(0) = 2 (myy — mxx) sin 20 + mxy cos 20 (2.2.46) With this result and trigonometric identities sin a — sin b =2 sin((a — b)/2) cos((a + b)/2) and cos a — cos b = —2 sin((a — b)/2) sin((a + b)/2), the corner force at I is expressed as Fc,i (0,a)= mn (02) — mn (0i ) = 2mn(0) sin a; 0= 01 + 02 + 4 (2.2.47) For a given angle 0 the corner force depends only on the opening angle. The maximum corner force occurs at a = n/2. For the obtuse corner, we have Fci (0, 2n — a) = — Fci(0, a). If the vertical displacement at the corner boundary area of a plate is not restricted, the corner force tends to move the corner in the direction opposite to f. The reaction corner force is present if the displacement at corner boundary area is fixed. The reaction corner force, however, does not appear if the edges meeting at the corner are free or clamped, because the twisting moment mn on both sides of the corner point is zero. Weak form of the problem The weak form of the plate bending problem for the Kirchhoff model is f m ■ ß dQ= f uf dQ + [ (qef u — ms ^) ds (2.2.48) Jn Jn Jr dn The corner forces are omitted from the expression (2.2.48). The constitutive equation and the kinematic equation are used point-wise. Therefore, the vector ß is a function of u only, since ß = Lu and m = CBk = CBLw, where k = Lw. Left hand side of (2.2.48) becomes a bilinear form of w and u aK(w; u) = uLtCbLw dQ (2.2.49) n while the right hand side is a linear functional of u only: r r Qu _ Ik (u) = J uf dQ + J (qef u — ms—) ds + ^ Fcu (2.2.50) The weak form of a thin plate Kirchhoff bending problem can be presented as For given geometry, Dirichlet and Neumann boundary conditions and area loads, find displacement w G V, such that aK(w; u) = lK(u) Vu G V0 (2.2.51) is satisfied for arbitrary test function u from the test space V0 The test space V consist of square integrable functions, which satisfy the boundary condition of the problem, while the test space V0 consists of square integrable functions which are zero at the part of the boundary where the Dirichlet boundary conditions are prescribed. Virtual work The principle of virtual work can be regarded as the weak form of the equilibrium equations. That becomes clear if the test functions are denoted as u = Sw and ß = Sk where S stands for the variation. The variations Sw and SjW = ^iw are zero at the boundary an an J where the displacement w or rotation S are defined, respectively. The virtual work of internal forces for the thin plate is thus Snint = f Sktm dQ (2.2.52) n and the virtual work of external forces: SWxt = f Swf dQ+ f(qef 5w - Ms )ds (2.2.53) Jn Jy dn By using constitutive and kinematic equations m = CBk = CBLw the virtual work equation transforms into the weak problem. The equilibrium displacement w is such that the virtual work of internal forces equals the virtual work of external forces for arbitrary admissible virtual displacement Sw, which conforms to the boundary conditions Snint = aK(w; Sw) = Snext = Ik (Sw) VSw e Vo (2.2.54) Boundary conditions The part of the boundary, where displacement and its normal derivative is prescribed, is called fixed boundary and the boundary conditions are expressed as dw — w = w; 0s = = Os (2.2.55) on Since the test function u and its normal derivative are zero, at that parts nonzero contribution to (2.2.50) is only from the region of boundary where stress resultants are prescribed (and displacements and its normal derivatives are not). The linear functional (2.2.50) suggests that there are two independent loads on the boundary rN: moment ms and shear force qef. The boundary conditions for the traction boundary are: qef = qef; m = Ms (2.2.56) From (2.2.43) we notice that at free edges the shear force qef and the tangential moment ms vanish. However, (2.2.44) shows that vanishing shear force qef = 0 does not necessarily imply that the shear q at free edges also vanish. In fact the shear q and twisting moment mn can both be nonzero at the free edges of the Kirchhoff plate. Another implication is that the support forces are not identical to shear forces if the twisting moment exists. Some common boundary conditions for the thin plate problem are listed in Table 2.1. Notice, that the prescription of the displacement of the boundary implicitly defines also the normal rotation. Since the Kirchhoff constraint along the boundary is js = ds — 0n = 0 Table 2.1: Boundary conditions for thin plates. * 9n = ^w is zero due to w = 0 Tabela 2.1: Robni pogoji za tanke plosce. * 9n = Qww je nic zaradi w = 0. Clamped (fixed) Simply supported Free w = 0 9s = 0 6n = 0 w = 0 ms = 0 9n = 0 qef = 0 's = 0 the normal rotation is defined by 6n = f. In the case of simply supported or clamped boundary the constraint w = 0 implies also 0n = 0. Therefore, with the Kirchhoff model it is possible to model only hard simply supported boundary. The difference between the hard and the soft simple support will be presented in the next chapter. Principal directions of the moments The twisting moment mn and the bending moment ms in a given direction at a point (x,y) G Q are defined by (2.2.36). Let n in (2.2.36) be defined as n = [cos p, sin p]T. From the condition mn = 0, one can compute angle p at which the twisting moment vanishes: tan 2p = 2m_xy (2.2.57) ' xx - ' yy The twisting moment has its extreme value at p ± n/2: ' nmax ' xx - ' yy 2 The maximum and minimum bending moment are: + m2xy (2.2.58) mi ii = mxx + myy ± mmax (2.2.59) Singularities An interesting implication of the Kirchhoff constraint occurs in the case of curved simply supported boundary, which is approximated by a polygonal boundary. The normal rotation at each corner point is defined twice - for each side a normal rotation 9n = dW = Vw ■ s = 0 vanishes. Since the orientations of the sides meeting in corner point do not match, si = s2, this is only possible when Vw = 0 at the corner points 2 Vw ■ s1 = 0; Vw ■ s2 = 0 ^Vw = 0 (2.2.60) Slika 2.2.8: Definicija polarnega koordinatnega sistema (r, 0 (2.2.67) For the displacement w and rotation dr to remain finite at r = 0 the condition A > 0 has been taken into account. The leading term in the expression for the displacement w is of the form rx cos(Ap). From (2.2.63), we see that the leading term for the moments mr and mv is rx-2, while the leading term for the qr, qv and mrv is rA-3. The singularity occurs when the exponent is < 0. The singular behavior of (i) mr and mv is expected when A< 2 or a > n/2. Correspondingly, the singularity in (ii) qr, qv and mrv can occur at angles a > n/3. At a given angle the strongest singularity occurs for the stress components (ii) followed by one order weaker singularity in the components (i). The strength of the singularity A = n/a is inversely proportional to the opening angle a. The angles a = n/4,n/2, 3n/4, 3n/2, 2n/2 are exceptions. In general, the leading term rx in this case is replaced by rx (log r)p where p is an integer. In the neighborhood of these exceptions the singular behavior is not continuous with respect to a. A list of critical angles for Kirchhoff plates for different boundary conditions of intersecting edges is shown in Table 2.2 (after [Melzer Rannacher, 1980]). Table 2.2: Critical opening angles for the Kirchhoff plate (* - the case reexamined in this section) Tabela 2.2: Kriticni koti za Kirchhoffove plosce (* - primer je obravnavan v tem razdelku) Support conditions mr, mv Qr, Qv, m, clamped - clamped 180° 126° clamped - simply supp. 129° 90° clamped - free 95° 52° simply supp. - simply supp.* 90° 60° simply supp. - free 90° 51° free - free 180° 78° 2.3 Finite elements 2.3.1 Preliminary considerations The departure point for the development of finite element method for the thin plate bending problem is the weak form (2.2.51). Let us define the residuum as the difference between the bilinear and linear forms of the weak form (i.e. the difference between the virtual works of internal and external forces), by R(w; u) = aK(w; u) — lK(u) (2.3.1) The solution w satisfies R(w; u) = 0 for arbitrary u G V0. To solve the problem for an approximate solution, the problem domain Q is discretized by the finite element mesh Qh consisting of finite elements Qh and nodes i =1, 2, ■ ■ ■ , Nn (see Figure 2.3.1) Q « Qh = IJ Qh (2.3.2) n i&h Each finite element Qh is defined by a fixed number of nodes and the boundary re. The approximation of the solution of the problem is now limited to a search in a finite dimensional vector test space Vh C V1. It is defined by a finite set of parameters w^, associated to the nodes of the mesh, i = 1, 2, ■ ■ ■ , Nn and shape functions N (N equals 1 Subscript h defines an approximation 1 at node i, and 0 at all other nodes)2 N„ wh = YI NiwWi (2.3.3) i After Bubnov and Galerkin, the optimal approximation of the solution is the one for which the residuum is zero only for the arbitrary test function uh from the test space Vo,h RK; uh) = 0 ^uh e Vo,h (2.3.4) The test space V0^h is spanned over the shape functions Ni. An arbitrary test function uh is a linear combination of shape functions: uh = Niui. The residuum for arbitrary test function uh vanishes, if it vanishes for each shape function Ni R(wh,Ni) = 0 or J] ax (Nj, Ni) Wj = Ik (Ni) VN (2.3.5) j In the matrix form KU = f (2.3.6) where K = [Kij ] = [ax (Ni,Nj)], U = [w^, ••• ,voNn ]T and f = [Ik (N1),Ik (N2), ••• ,1k (Nn„ )]t . The linear system (2.3.6) is finally solved for the unknown parameters U. The approximate solution can be also searched from the spirit of the principle of virtual work. The solution wh = i Niwi must be such that the virtual work of internal forces 5nint equals the virtual work of external forces 5next for arbitrary virtual displacement $wh = i Nj,SWj,, member of the test space V0,h 6Wnt = 5Wxt W5wh e Vo,h (2.3.7) Snint = ax(wh; 6wh); önext = Ik(Swh) From wh = i Niwi and 5wh = i Ni öwjji we have 6nint = ax Niöuji) = ax (wh; N^Svbi = ^ ^ vojax (Nj, N^i i i i j where the linearity of bilinear form aK was exploited. Using the matrix notation we have Snint = 5ut KU; K = [Kij ] = ax (Nj, Ni); Su = [5wu 5vj2, ••• , ötUn]T (2.3.8) 2 For the sake of compactness the shape functions N are here defined over the whole domain $next = lK(Swh) = Y, tWilK(Ni) = SuT f; f = [fi] = [Ik(Ni)] i Inserting the relations (2.3.8) and (2.3.9) into (2.3.7) we obtain SuT(KU - f) = 0 (2.3.9) (2.3.10) Due to the fact that the variations tWi are arbitrary and independent, the equation (2.3.10) is fulfilled only when KU — f = 0 holds. This is equivalent to (2.3.6). Q r__........ \ r / \ i Figure 2.3.1: Problem domain Q and its boundary r (left), finite element discretization Qh, Qh and patch of elements QP around node i (right) Slika 2.3.1: ObmoCje Q in njegov rob r (levo), diskretizacija s konCnimi elementi Qh, Qh in krpa elementov QP okoli vozlisCa i (desno) Shape functions The shape function Ni takes unit value at node i and is zero at all other nodes of the finite element mesh. The shape function Ni is nonzero only on a finite domain . Hence, the adjective 'finite' in the name of the method. Let m denote the variational index of the variational functional (the highest derivative present in the functional) that is to be solved by the finite element method. A basic requirement for the convergence of approximate solution is that the shape functions satisfy the completeness condition: the element shape functions must exactly represent all polynomial terms of order < m in the Cartesian coordinates. In the case of (2.2.51), the variational index is m = 2, therefore the interpolation on ^h should be able to represent any field, which is a linear or quadratic polynomial in x and y; in particular, a constant value. Constant and linear polynomials represent rigid body motions, whereas quadratic polynomials represent constant curvature modes. The C0 continuity requires that the shape functions are continuous across the element edges. The shape functions are continuous only if the displacement over any side of the element is defined only by DOFs on that side. The second derivatives in (2.2.51) indicate that the shape functions must satisfy not only C0 but also C1 continuity - continuity of the slopes of the shape functions. This can only be achieved if the shape functions are C0 continuous and the normal slope ^n over any edge of the element is completely defined only by the DOFs on that edge. The elements based on C1 continuous shape functions are often referred to as conforming or compatible. While obtaining C0 continuity is relatively simple, this is certainly not the case with C1 continuity. It was proven in ([Zienkiewicz Taylor, 2000], [Felippa, 2009]) that, when only w and its slopes are prescribed at the corner nodes, it is impossible to specify simple polynomial expressions for shape functions, which ensure the full compatibility. 2.3.2 An overview of thin plate elements Among the first successful conforming plate elements was the rectangular element with 16 DOFs (w, w,x,w,y and wxy at each node) developed by [Bogner et al., 1966]. The element uses Hermitian interpolation functions with bi-cubic shape functions. The triangular elements offer more flexibility in the design of the conforming elements that the rectangular ones. It was shown in [Felippa, 2009] that a complete fifth order polynomial is the first which satisfy the C1 continuity requirements. From the Pascal's triangle it follows that the complete quintic polynomial contains 21 terms. Conforming triangular plate element with 21 DOFs was independently developed by several authors ([Argyris et al., 1968] et al., [Bell, 1969], [Bosshard, 1968], [Visser, 1968]). Conforming quadrilateral elements can be produced from the conforming triangular elements. If the combined element has an additional internal degrees of freedom they are eliminated by static condensation. A direct derivation of conforming quadrilateral element is also possible. The derivation was first proposed by [Sander, 1964] and [De Veubeke, 1968]. Although the conforming plate elements exhibit excellent accuracy and performance they have some important drawbacks. The C1 continuity presents a difficulty when discontinuous variation of material properties occurs. It is also difficult to impose the right boundary conditions since 'nodal forces' as energy conjugates to chosen degrees of freedom can no longer be interpreted intuitively. It is relatively simple to obtain shape functions, which respect C0 continuity but violate the slope continuity between the elements. They can still be used to build plate elements. It is namely possible to found the convergence of such elements if only they pass the "patch test" [Zienkiewicz Taylor, 2000]. They are referred to as non-conforming and can in many cases perform even better than the conforming ones. Additional conditions for the shape functions follow from the patch test. They are formulated for each straight side of the element r: f dw f dw IA(äü)ds = 0; X A(T.)ds = 0- (2311) where A denotes the mismatch along the side of neighboring elements. Notice that, if the function w and its first derivatives are defined at the corner nodes, the satisfaction of the second condition in (2.3.11) is always ensured. An early attempt to build a conforming rectangular element lead to the formulation of the first successful non-conforming quadrilateral plate bending element proposed by [Adini Clough, 1960]. The element has 12 DOFS and it satisfies completeness as well as transverse deflection continuity but normal slope continuity is maintained only at four corner points. One of the first successful non-conforming triangular plate elements is that presented by [Bazeley et al., 1965]. The element has nine DOFs. Since full cubic expansion contains 10 terms, one DOF is eliminated in such a way that the completeness is maintained. Although the element satisfies constant curvature criterion it does not pass the patch test for arbitrary mesh configurations. Despite the shortcomings the element became quite popular in practical applications. A rather successful approach involves a direct interpolation of displacement w and rotation 9. Due to the independent interpolations the continuity requirement is only C0 for both unknowns w and 9. The interpolations are constructed such that the Kirchhoff constrain is satisfied only in certain discrete collocation points. Hence the name discrete Kirchhoff - DK elements. It is possible to impose the Kirchhoff constraint along the sides of the element resulting in superior performance. The most successful elements of this type are discrete Kirchhoff triangle (DKT) and discrete Kirchhoff quadrilateral (DKQ) ([Dhatt, 1970], [Batoz et al., 1980] and [Batoz, 1982]). Many finite element analyses of a plate bending problem are done with DK plate elements. They are widely accepted due to their efficiency and ease of use. One of the reasons for their popularity is also the fact that they employ only displacement and rotations as the nodal degrees of freedom. 2.3.3 Natural coordinate systems We have thought of shape function N in section 2.3.1 as of a function defined over . In practice, the shape functions are defined over a finite element Qeh. The description of the element shape functions in the Cartesian coordinates (x,y) is not very convenient. We therefore seek for coordinates £i, i =1, 2 such that the location of a point P(x, y) inside an element defined by nodes (xj, yi),i = (1, 2, 3)(triangle) or i = (1, 2, 3, 4)(quadrilateral) would be described in a "natural" way; i.e. as x = Ni(£k)xi where Ni is now an element shape function, defined by £i over Qeh. The coordinate system which emerges naturally is therefore called a natural coordinate system. Quadrilateral geometry The natural coordinate system for the element with quadrilateral geometry is shown in Figure 2.3.2 (£i = £,£2 = n). The location of a point P(x,y) is given as 4 4t 14 'x ' +1 QG -1 0 * 3 + 1 4 2 Figure 2.3.2: Bi-unit element QD in coordinate system (£,n) Slika 2.3.2: Enotski element QD v koordinatnem sistemu (£,n) n £ y 1 x = Nixi + N2x2 + N3x3 + N4x4; y = Niyi + N2 y2 + N3y3 + or in matrix form (2.3.12) x y xi x2 x3 x4 yi y2 y3 y4 Ni N2 N3 N4 (2.3.13) where Ni are bilinear Lagrangian functions; defined over QD as: Ni = (1 - £)(1 - n)/4; N2 = (1 + £)(1 - n)/4 (2.3.14) N3 = (1 + £)(1 + n)/4; N4 = (1 - £)(1 + n)/4 The differentials of (x,y) are related to those of (t,n) and vice versa by: dx dV dx dx df dn dy dy df dn dt dn dt dn J dx dV (2.3.15) The Jacobian of the transformation is denoted by J and is computed from (2.3.15), (2.3.12) and (2.3.14): J E dNi x df Xi dNi df Vi dNi i xi dn dAi V. dn Xi X2 X3 X4 Vi V2 V3 V4 dNi dNi df dn dN2 dN3 dN4 an (N n dN:i n dN4 (2.3.16) f n The derivatives of a function f with respect to the global coordinates (x,y) are related to the derivatives of f in the natural coordinate system by (2.3.17) The integration over the element domain is performed in the natural coordinates over the unit element Qa = (t,n),t € [-1, +1],n ^ [-1, +1]: r f i r df dn ' r df i r df i x f = x df dx dn df f = J-T df f - dy . - dy dy - n n fdxdV laP f (t,n) det J dtdn (2.3.18) The expression for the determinant of Jacobian can be simplified into: det J = (Aah + dit + 4n)/4 (2.3.19) di = ((x4 - x3)(V2 - Vi) - (x2 - xi)(v4 - V3))/2 d2 = ((x3 - x2)(V4 - Vi) - (x4 - xi)(V3 - V2))/2 The area of the element is denoted as Aah and the parameters di and d2 define the distortion of the quadrilateral element. Distortion vanishes if the quadrilateral element is parallelogram. In that case det J is constant. Figure 2.3.3 shows the quadrilateral element with the coordinate lines of the natural coordinate system. Alternate shading between the coordinate lines is used to illustrate the variation of values of det J. ah Triangular geometry The coordinates used for the triangular elements are often called area, natural or even barycentric coordinates. The coordinates are denoted by (ti = Zi, t2 = Z2, t3 = (3) with (3 Figure 2.3.3: Quadrilateral element with the alternate shading between the coordinate lines of the quadrilateral natural coordinate system Slika 2.3.3: Štirikotni element z izmeniCnim senCenjem med koordinatnimi Črtami naravnega koordinatnega sistema being function of Z1 and Z2. The coordinate line Zi = const. is parallel to the side opposite to node i. The term area coordinates originates from the definition. Let us consider a point p inside the triangle defined by the nodes i, j, k. The triangle coordinate of the point is Zi = Apjk/Aijk (see Figure 2.3.4). Area of the triangle ijk is denoted by Aj, while Apjk is the area of the triangle defined by the nodes p, j and k. Since Aijk = Apij+Apjk+Apki the sum of the coordinates equals to unity Figure 2.3.4: Triangular (area) coordinates Slika 2.3.4: Trikotniske koordinate 1 = Ci + C2 + Ca; X = Cixi + to + Csx?; y = (ivi + to + to (2.3.20) or in matrix form 1 1 1 1 "Zi " X = Xi X2 Xs Z2 y _ yi y2 ys Zs (2.3.21) By inverting this relation, we get 1 Zi Z2 Zs 2 Ar X2 ys - Xs y2 +y2 - ys -X2 + Xs 1 Xs yi - Xi ys +ys - yi -Xs + Xi X Xi y2 - X2 yi +yi - y2 -Xi + X2 y (2.3.22) where (xi ,yi) are the coordinates of the nodes of the triangle and Ane is its area. The Jacobian of the triangular coordinate transformation is J 111 Xi X2 X3 yi y2 ys (2.3.23) The det J is twice the area of the triangle: det J = 2Ane. The integral over the element domain is performed in the natural coordinates over the unit element : f (Zi,Z2,Zs)dnA Since the unit element is a triangle, we have 1 2 -'nA f dxdy f (Zi,Z2,Zs)det J d^A = An (2.3.24) r ri fi-C 1 / f(Zi,Z2,Zs)dnA = dZi f(Zi,Z2, 1 - Zi - Z2)dZ2 IQA JO JO Derivatives with respect to global coordinates (x, y) are related to the derivatives with respect to the triangular coordinates (Zi,Z2,Z3) via r df i dx df = - dy . dCi dZ2 dCs dx dx dx dZi dzf 0(3 dy dy dy df dZl df d(2 df L öCs (2.3.25) From (2.3.22) we have r df i 1 dx df - dy . = 2Anh y2 - ys ys - yi yi - y2 X3 — X2 Xi — Xs X2 — Xi df dZl df d(2 df L dCs (2.3.26) and from (2.3.21) df dCl df d(2 df d(s J Xi yi X2 y2 Xs ys df dx df dy Figure 2.3.5 shows the triangular element with the coordinate lines of the natural coordinate system. Alternate shading provides the indication of det J. Since the factor det J is constant over the element domain and equals det J = A^/2, the shaded areas are equal in size. Infinitesimal vector in the triangular coordinate system d£ = [dZ1, dZ2, dZs]T is related to the infinitesimal vector dr = [dx, dy]T as dZi dZ2 dZs 1 2Ac +y2 - y3 -X2 + X3 +ys - yi -x3 + xi +yi - y2 -xi + x2 dx dy (2.3.28) For the triangular coordinate system the total differential is Figure 2.3.5: Triangular element with the alternate shading between the coordinate lines Zi = const. of triangular coordinate system Slika 2.3.5: Trikotni element z izmenicnim sencenjem med koordinatnimi crtami trikotniskega koordinatnega sistema df df df df = f dZi + f dZ2 + f dZ3 (2.3.29) dZi dZ2 dZ3 To obtain the derivative along a defined direction, the directional vector d£ has to be expressed in the triangular coordinate system. Next, the derivative along the line will be defined. The position of the point lying on the line defined by the two points is: r = ri + s ri2 s G [0,1] where ri2 = r2 - ri. Inserting dr = [dx, dy]T = ds[(x2 - xi), (y2 - yi)]T into (2.3.28), we obtain dZ = [dZi, d(2, d(3]T = ds[-1, +1, 0]. The derivative along the direction ri2 with respect to the parameter s is given by (2.3.29) I = (f - f) ^ (2.3.30) The definition of the second derivative along the line is straightforward d2f = d (df df d2f 2 df df + d2f f (2 3 31) dS2 = dS(- dZi)) =(šži - 2dZi+ dčf) = f,i22 (2.3.31) To determine the derivative with respect to the normal to the side of the triangle, one has to follow similar steps . First we parameterize the perpendicular line to a side (in this case side 1-2) going through the side middle point: 1 2 (ri + r2) + 2 ds I13 ni2 = ri2 + 2 ds I13 ni2 (2.3.32) where ni2 is a normal vector to the side 1-2 with length /i2, h3 is height of point 3 and ri2 = 2 (ri + r2). The differential vector dr = 2ds h3 ni2 is in the global basis dr dx dy 2 ds h3 +y2 - y1 - x2 + x1 // 12 (2.3.33) and can be written in the triangular basis using (2.3.28) 2h3 dZ dZi d(2 d(3 2Aq -(y3 - y2) + (x3 - x2) +(y3- yi) (x3- xi) -(y2 - yi) + (x2 - xi) +y2 - y1 -x2 + x1 // 12 We further notice that AQe = h3 /i2/2, introduce the notation xij the upper expression as xj - xi, and rewrite dZ dZi d(2 d(3 2 1 i2 y23 y12 - x23 x12 +y13 y12 + x13 x12 72 li2 -/2 71 +1 - 2 ri2 ■ (r3 - ri2>//i2 +1 + 2 r i 2 ■ (r3 - r i 2)// 22 2 (2.3.34) We introduce the parameter ßi 2 = 2ri 2 ■ (r3 - ri 2)// 22 = 2e i//i2, which has a meaning of normalized eccentricity of node 3 with respect to the symmetry line of edge 1-2 (see Figure 2.3.6). Finally, we have d( i = (1 - ßi 2) ds; d(2 = (1 + ßi 2) ds; d^3 = 2ds and the derivative normal to the edge 1-2 df _ndf df Of, , df df dn = 2 - (šži + ) + ßi 2 (- šži) (2.3.35) r Figure 2.3.6: To the definition of eccentricity parameter ^12 = 2e1/112 Slika 2.3.6: K definiciji parametra ekscentriCnosti ^12 = 2ei/1i2 2.3.4 Conforming triangular element The derivation of the element presented below follows the work of [Argyris et al., 1968]. Here, only the outline of the derivation procedure will be presented. The element will be called TUBA6 or Argyris element in what follows. The displacement field is interpolated with complete 5th order polynomial in natural (area) coordinates (Ci,C2,C3). Namely, the complete quintic polynomial, defined with 21 parameters, is the lowest discrete approximation, which satisfies C1 continuity across element boundaries [Zienkiewicz Taylor, 2000], [Argyris et al., 1968]. Hence, the discrete approximation of the displacement field can be written as: wh = (2.3.37) where p = [p1,p2, ■ ■ ■ ,p21]T denotes a vector of nodal interpolation parameters and w the set of shape functions. The latter contains 21 polynomials and can be written as: W =[<1,<25,<3, C1C2 , C2C3 , C3C1 , C2C1 , C3C2 , C1C3 , 3 2 3 2 3 2 3 2 3 2 3 2 S 1S2 , S2S3 , S3 S1,S2S1,S3S2,S 1S3 , C 1C2C3, C 1C2 C3, Ci C2C3, C 2Z2 C3, C1C2C3, C 1 C2 C3]T The element nodal interpolation parameters pertain to 3 vertex nodes and 3 mid-side nodes. Each vertex node has 6 degrees of freedom: displacement, its first derivatives, Figure 2.3.7: Triangular Argyris plate element - ARGY Slika 2.3.7: Trikotni Argyrisov končni element za ploSče - ARGY and its second derivatives with respect to x and y. The mid-side nodes contain the first derivative in the direction of the exterior normal. The complete set of global degrees of freedom can be written as: w = [Wf>!, ww2, wt>3, , W2,x,W2,y, W3,x,W3 y w,rai2 , w,n23 , w,«.3i ] y> ^3 'ly) 1Vi,xx, ww1,xy ,«1,yy, w2,xx,«2,xy ,«2,yy, w3,xx,«3,xy ,«3,yy, T To establish the relation between p and w, all the derivatives are computed at the nodes. As an intermediate step, we introduce an additional set of degrees of freedom wp, where the derivatives are formed with respect to natural (area) coordinates: Wp =[ Wi ,«2,«3, Wi,i2 ,«1,31, «2,23,^2,12, «3,31 ,«3,23, «1 ,122 , Wi,3i2 , WÜ1,232 ,«2,122 , «2,312 , «2,23 2 ,«>3,122 , «3,312 , «3,232 , > > > ]T The relation between p> and wp is defined by the system of equations which are obtained by using results in (2.3.30), (2.3.31) and (2.3.36) at the nodes of the element. At node 1 (2.3.38) (Zi = 1, Z2 = 0, Z3 = 0) we thus have: wi = pi wi,i2 = -5pi + P4 wi,3i = +5pi -P5 wüi,122 = 20pi - 10pip4 + 2pio wi,3i2 = 20pi - 10pip5 + 2pii 11)1,232 = -2P4P5 + 2pio + 2pii and at the midside node on edge 1 - 2 (Z1 = 1/2, Z2 = 1/2, Z3 = 0) we have w,ni2 =(-5(1 + ^i2)pi + 5(-1 + ßi2)p2 - (5 + 3^i2)p4 (2.3.39) + 2p5 + 2p6 - (5 - 3^i2)pz - (5 + ^i2)pio - (5 - ^12)pi3 + 2pi6 + 2pi7 + 2pig)/16 A linear system can be formed by writing down the equations (2.3.38) and (2.3.39) for all the nodes: w p A-1p (2.3.40) where the matrix of the linear system is denoted as A 1. From (2.3.30) we have: W, 12 Xi2 yi2 w,23 = X23 y23 _ w,31 . _ X31 ysi _ w, w y (2.3.41) and from (2.3.31) we get w 122 w 312 = w 232 X12 y22 X31 y31 X23 y23 -Xi2yi2 -X3iy31 X23y23 w, w yy 2w xy (2.3.42) -- T This relations enable to define the transformation between w and w p, which can be written in matrix notation as wp = XTw, where XT is matrix of the linear transformation. We insert this result in (2.3.40) and obtain p = Xw; X = AXt (2.3.43) The discrete approximation of curvature vector is the second derivative of the displacement field wh = wTXw according to (2.2.8): wh ,xx wh,yy 2wh,xy Computing further the inverse of the relation (2.3.42), we obtain: Kh = TKh,p; Kh,p = [wh, 1 22 , wh,232 , wh,312 ]T = wTpw (2.3.45) where w,p = [w,i22, w 232, w3i2]T. Stiffness matrix The element stiffness matrix is obtained from the bilinear form defined for a single element aK,e(wh; uh) = / kt(uh)mh(wh) dQ = / kt(uh) Cb Kh(wh)dQ h = / KTp(uh)TT Cb t Kh,p(wh)dü (2.3.46) •/nh where the interpolation of the displacement is wh = wTXw and of the test function uh = wTXuu where uu is vector of test function nodal parameters in cartesian coordinates. The product G = TT CB T is constant in the case of homogeneous material. In that case it remains only to evaluate the integral 'nh Jnh / Khp(uh)Kh,p(wh) dQ = uu / w,p w,p dQ Uu = uUS w (2.3.47) ^h V^h / where S is obtained by analytic integration with the use of: dQ=(2+^ + r)! <2-3-48» The expression (2.3.46) can be put in a familiar matrix form aK,e(wh; uh) = UU t ^ ] Gij Sij L i,j w = U Kw. (2.3.49) where K = [Kij] = [^ij GijSij] is the stiffness matrix of the element. Load vector The equivalent load vector is computed from the virtual work of external forces (pressure load f and boundary tractions qef ,ms) f f du lK,e(uh)= uh f dQ+ / (uh qef - msdh)ds (2.3.50) Qh r = uTXT( f w f dQ + f (w qef — ms ^^) ds V Jne Jre dn By using (2.3.43) the nodal load vector f is finally written as: f = X^^ w f dQ + ^ (w qef - ms dn) | (2.3.51) The derivatives are evaluated with the use of (2.3.36). The integrals in (2.3.51) can be obtained in a closed form with the aid of the formula (2.3.48). 2.3.5 Discrete Kirchhoff elements The motivation for the Discrete Kirchhoff (DK) plate elements is based on the observation that it is quite challenging to construct a conforming approximation of the plate displacement field. Since the Kirchhoff constraint of zero shear strains is very difficult to satisfy at every point of the domain, a compromise is made: the constraint Y = Vw — 0 = 0 is relaxed and is satisfied only at chosen points and directions. Relaxation of the Kirchhoff constraint implies that the discretization of displacement wh and rotations 0h are no longer tied together but are rather independent. They are related only along the sides of the element. Otherwise independent interpolations for displacement wh and rotations 0h are constructed such that the shear strain Yh along the element boundary (in the direction s) vanishes d Yh ' s = ds — n ' 0h = 0 (2.3.52) Edge interpolation equals beam element interpolation To illustrate the idea, let us consider a discretization of the Euler Bernoulli beam, see Figure 2.3.9. Two independent interpolations of displacement wh and rotation 9h are set: wh = wiNi + w2N2 + w3N3 + VJ4N4 (2.3.53) Oh = O1N1 + d2N2 + (9sNs (2.3.54) where the hierarchic shape functions are (see Figure 2.3.8): Ni = (1 — 0/2; N2 = (1 + 0/2 e e [—1, +1] (2.3.55) N3 = (1 — e2); N4 = e (1 — e2) Due to the requirement for the C1 continuity of the displacement, the order of the interpolation is cubic. Only nodal displacements w1, w2 and rotations O1, O2 are acceptable N, = (1 - 0/2 N3 = (i~e) Figure 2.3.8: Hierarchical shape functions Slika 2.3.8: HierarhiCne interpolacijske funkcije N2 = (1 + 0/2 N4 = a i-e2) wi^Jh_w^ Ö2 1 r " 2 X L X = L (1 + 0 Figure 2.3.9: Euler-Bernoulli beam element Slika 2.3.9: Koncni element za nosilec po Euler-Bernoullijevi teoriji z nodal degrees of freedom. Other parameters vjs,vj4,ds are thus eliminated by enforcing the constraint of vanishing transverse shear strain Yh = ^Wt ~ Öh = 0. One gets from the latter equation and (2.3.53), (2.3.54) T T A A 1 whs = +L (h - Ö2); wh4 = L (- -(h + Ö2)); h = - wh4 (2.3.56) 8 4 L 2 L Using (2.3.56), the interpolations (2.3.53) and (2.3.54) become: wh = wiHi + VJ2H2 + h H3 + O2H4 Öh = 101 2 - - 1) + 1h (3^ + - 1) + 3(w22Lwi)(1 - e2) The functions Hi are the well known Hermitian cubic shape functions (see Figure 2.3.10): Hi = 1(1 - e)2(2 + e) H2 = 1(1 + e)2(2 - e) Hs = +L(1 - e)2(1+e) H4 = -L(1+e)2(1 - e) 8 8 (2.3.58) Figure 2.3.10: Hermite shape functions Slika 2.3.10: Hermite-ove interpolacijske funkcije Interpolation for Discrete Kirchhoff (DK) plate elements Similar approach as for Euler-Bernoulli beam is taken also in the case of the Discrete Kirchhoff plate element: the side of the element is treated as Euler-Bernoulli beam. The interpolation on the boundary equals the interpolation (2.3.57). The rotation 9 is now the normal rotation 9n = d • n. The interpolation of the displacement is defined as a sum of nodal (I) and side contributions (I J): wh = Wi Ni + Y, (Ws,ij NIJ + w4,ij MI J ) (2.3.59) i ij where Ni are nodal shape functions while Nij and Mij are shape functions related to element side IJ. This nonconventional notation is beneficial since it holds generally. It enables that both triangular as well as quadrilateral DK element are treated in exactly the same way. The parameters W31 j and W41 j are eliminated from the interpolation by enforcing zero transverse shear strain along the side IJ. It turns out that w3,ij = -^r nu • (Oi - Oj ); o Lij wj - wi 1 - .. wA,IJ =—T(—t--Ö nIJ • (0I + °J)) 4 -IJ 2 (2.3.60) where 0I = [9xI, 9yI] and LIJ is length of side between nodes I and J, and nIJ is normal to the side between nodes I and J. Note the similarity between (2.3.60) and (2.3.56). The interpolation of the rotation is defined as 3 Oh 9x,h 9y,h Ni + J2 93,IJnIJNij (2.3.61) i=i i j where, due to zero transverse shear strain along the side IJ (2.3.56) 6 93,IJ L W4,IJ (2.3.62) The above interpolation leads to an element with only vertex nodes with degrees of freedom denoted by wI = [wI,9xI,9yI]T. The interpolations (2.3.59) and (2.3.61) can also be expressed as: wh = NwW = Nw,iWi; Oh = NW = Ne,iWi; II Kh = BkW = Bk,iWi (2.3.63) where W = [W 1, W2, • • • , Wnen] and nen is the number of element nodes. Explicit forms of Nw I and NÖI = [Nöx I, Nöy I]T are obtained from (2.3.59)-(2.3.62): N T w,I N T dxj N T ' Ni - 1 + 8 0 0 0 Ni 3 + 4 0 0 0 3 + 4 . Ni _ 2 (Mki-Mij) Lij (Nij -Mi j )nu,x -Lki ( Nki + Mki )nKi,-. Lij (Nij - Mu )nu,y - Lki (Nki + Mki )nKi,c 2 (NkinKi,x/LKi - Nijnij,x/Lij) -(Nki nKix + Nu n?^) -(Nki nKi ,xnKI,x + Nij nij, xnIJ,y ) 2 (NkinKi,y/Lki - Nijnu,y/Lij) -(Nki nKi,xnKi,y + Nu nu^nu^) _ (NKI nKiyy + NIJ n?j,y ) (2.3.64) (2.3.65) where the node ordering and side numbering is presented in Figure 2.3.11. Oy i K Figure 2.3.11: Node and side numbering in (2.3.64) Slika 2.3.11: Oznacevanje vozlisc in stranic v (2.3.64) The curvature vector Kh is computed from (2.2.4): K = r dÖy,h ddx,h dÖx,h 9dy,h ] (2 3 67) dx ' dy ' dx dy ' ' Since the shape functions are defined in the natural coordinate system £i, i = 1, 2, their derivatives are related to those in the Cartesian coordinate system (x,y) by dx = ^i df where Jx^ = is the term of the Jacobian matrix of transformation. It depends on the geometry ((2.3.16), or (2.3.23)). The sum runs over the natural coordinates ( which are ^ = £ and £2 = n in case of quadrilateral geometry and ^ = £2 = (2, £3 = 1 - Z1 - Z2 for the triangles). Using these results and the results from (2.3.65), (2.3.66), we arrive at BK>I = [BKxx>1, BKyy,i, BKxy,i]T: B-, = -^ = - M^) (2.3.68) = + dNoxj = +^( ,-i dNoxi* KyyJ + dy + 2-^(Jyti ' B = + ONoxi dNey ,I = 1 ONoxi 1 ÖNöy ^ BKxy,I = +--HI---ČT = + / , ( Jx£i E7---Jy dx dy ^v x?i d£i yii d£i Stiffness matrix The stiffness matrix K of the element is obtained by the discretization of the weak form (2.2.51). It is written in a block matrix form with K = [KIJ] aK,e(wh; uh)= kt(uh)m(wh) dQ = (2.3.69) Jni = £ uT kijwj; kij = f ^ BtkJcbbk,j dQ The stiffness matrix depends entirely on the interpolation of rotations. Notice, that the interpolation of displacement wh is needed only for the computation of the consistent load Figure 2.3.12: Triangular DK plate element - DK Slika 2.3.12: Trikotni DK končni element za plošče - DK vector f. The explicit form of the displacement interpolation for the DK element was first obtained in [IbrahimbegoviC, 1993]. Load vector From (2.2.50) we have (2.3.70) I f uhdQ + (&/ uh + ms 6h(uh) ■ s)ds + [mnuh,r] j I The consistent nodal load vector is fI = f/;I + ftiI. Triangular DK plate element - DKT The DKT element has three vertex nodes with nodal degrees of freedom UUI = [wI, 9xI, 9yI]. The shape functions are formulated in the area coordinates: Ni = Zi ; Nij = 4I and Kyy,i are set to zero. In the mid-side nodes lying at the supported edges the nodal normal slope w,n\I is left free. Figures 2.4.4 and 2.4.5 show FE solutions. The comparison of FE solutions obtained with Argyris and DK elements shows good agreement with the reference solution. Since the range (and the color coding) in Figures 2.4.3, 2.4.4 and 2.4.5 is the same, it is possible to visually asses the differences of the FE solutions. In the current case, the differences between the solutions are, however, minor. More quantitative comparison of element performance is shown in Figure (2.4.6), where a study of element performance versus mesh density is presented. The plots show the loglog plot of relative error of several selected quantities versus the total number of degrees of freedom (both constrained and unconstrained) which is inversely proportional to the average element size. The a-apriori estimate of the convergence rate is hp, where h is the average size of the element and p the interpolation order. In the log-log plot, the convergence lines are straight lines. The slope of the lines in the log-log plot indicates the rate of convergence. The comparison of Argyris and DK elements clearly shows the superior convergence rate of the Argyris element in all of the studied quantities: displacement w and moment mxx at the center of the plate as well as the total deformation energy of the plate, defined as Wint = e fQe ktCbKdQ. The integrand is evaluated at h integration points and numerically integrated. Convergence of Wmt is not monotone for the case of ARGY element which explains a kink in the convergence plot. The effect can be attributed to the choice of the numerical integration scheme. The superconvergent points of the ARGY element are the element nodes, while the integration points of the triangular integration scheme (2.3) lie in the interior of the element domain. The convergence rate for the ARGY element is roughly twice as higher as the convergence rate of DK elements. 2.4.2 Uniformly loaded clamped square plate A rectangular plate of side length a = 10 and thickness t = 0.01 under uniform loading f = 1 is analyzed. All sides of the plates are clamped. The material is linear elastic and isotropic, with Young's modulus E = 10.92 x 1010 and Poisson's ratio v = 0.3. a Figure 2.4.7: Problem definition and geometry for the uniformly loaded clamped square plate (t/a=1/1000) Slika 2.4.7: Definicija in geometrija enakomerno obremenjene vpete kvadratne plosce (t/a=1/1000) a y Reference solution The reference solution is the analytical solution in the form of series expansion. Since the series converges quite slowly, a special treatment is necessary to be able to efficiently compute the reference result (see e.g. [Taylor Govindjee, 2002]). The main results are given at the center of the plate: fa4 w = 1.26531908710-3^d ; mxx = -2.29050835210-2/a? (2.4.2) -p 2 4 The total deformation energy is Wint = 3.89120077510-4^. In this work we use the above results with the material and geometry data as given above. Finite element solutions FE solutions are shown in Figure 2.4.9 and Figure 2.4.10. The boundary conditions for DK elements were w = 0, 9x>I = 0 and 9y>I = 0 at the clamped sides of the plate. The boundary conditions for the ARGY elements were w = 0,w,x = 0,w,y = 0 at the vertex nodes and w,n = 0 at the midside nodes. The qualitative comparison does not reveal any significant differences in the performance of the elements. However, the difference becomes clearly visible, when the convergence is observed (see Figure 2.4.11). The convergence of the ARGY element is one order higher that that of DK elements. Notice, however, that for the very coarse mesh, the results of both type of elements are quite comparable. 2.4.3 Uniformly loaded clamped circular plate A clamped circular plate of radius R = 5 and thickness t = 0.01 under uniform loading / = 1 is considered. The material is linear elastic and isotropic, with Young's modulus E = 6.825 x 109 and Poisson's ratio v = 0.3. y Figure 2.4.12: Clamped circular plate with uniform load (t/R=1/500) Slika 2.4.12: Enakomerno obremenjena vpeta krožna plosCa (t/R=1/500) Reference solution By taking into account the symmetry, it is possible to obtain the closed form solution of the problem (see [Timoshenko Vojnovski-Kriger, 1959]). The displacement of uniformly loaded clamped circular plate is defined by: w = f (R2 - r2)2/64/D (2.4.3) The moment resultants at the center of the plate are: mxx = - -6 fR2/D(1 + v); myy = - -6 fR2(1 + v); mxy = 0 (2.4.4) In current case, where we have D = R4 = 625, the displacement at the enter is w = 1/64 = 0.015625 and moment resultant mxx = -25/16 * 1.3 = -2.03125. Finite element solutions A quarter of of the plate is modeled in FE analyses. The symmetry boundary conditions are taken into account at x = 0 and y = 0. The boundary conditions are listed in (2.4.5). DK wI @x,I Qy,I ARGY wI Wi,x Wl,y Wl,xx wI,yy wI,xy Wi -J,n x=0 - - 0 x=0 - - 0 - -0 0 y=0 - 0 - y=0 - 0 -- -0 0 r=R 0 0 0 r=R 0 0 0- -- 0 (2.4.5) The meshes were constructed so, that the distortion of the elements was minimized. In the case of quadrilateral elements, the distortion of the elements is, however, still significant. The sequence of meshes used in the convergence analysis is shown in Figure 2.4.14. Notice that the area of the FE problem domain Qh converges to the area Q as the mesh is refined. As shown in Figure 2.4.15, the FE solution with ARGY element shows very good qualitative agreement with the reference solution. Results of the DK elements are being influenced by the mesh distortion (quadrilaterals) and mesh orientation (triangles). The influence is, however, noticeable only when the comparison of the equilibrium shear forces with the reference solution is made. One can thus conclude that the mesh distortion (quadrilaterals) and mesh orientation (triangles) has the greatest influence on the computation of the transverse shear forces. The comparison of the convergence, shown in Figure 2.4.17, reveals an interesting fact. The convergence of the ARGY element is no longer superior to that of DK elements, but is rather of the same order. 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'°g ^ JS [S CD >CD aj >QQ H £ Ti cu 2 8 'Ö CD -Ö «u «u ft ft > B a cö Ö CD O ö CD Ti ^ ö CÖ CD £ £ S § fH fH O CD S o CÖ Ö CD CD O Ö ">00 O qj fH cö "O ^ ® § H 2 IC Tf T-H Q) C^ 3 J fn 55 g s H J D Z m D J m ^ Z o 5 z 3 l-H K M D D ^ o ü S m H > h P ffi rt ^ Q C < rt o H D _ O Ol Q d' g S o m Qx,e Qx,e lxy FE mesh lxy FE mesh (DKT element) (DKQ element) Figure 2.4.16: FE solution of the uniformly loaded clamped circular plate with DK plate elements Slika 2.4.16: Računska rešitev enakomerno obremenjene vpete krožne plošče z DK končnimi elementi oo co (Displacement w at center) -0.5 -1 -1.5 -2 -2.5 -3 -3.5 • ARGY ■ DKT DKQ -1.5 -2 -2.5 -3 -3.5 1.5 2 2.5 3 3.5 log (■ndof) (Moment resultant mxx at center) • ARGY ■ DKT DKQ 1.5 2 2.5 3 3.5 4 log (ndof) (Energy norm) Figure 2.4.17: Comparison of the convergence of FE solutions for the uniformly loaded clamped circular plate Slika 2.4.17: Primerjava konvergence racunske resitve problema enakomerno obremenjene vpete krožne plosce 2.4.4 Uniformly loaded hard simply supported skew plate The analysis of Morley's a = 30° skew plate under uniform loading f =1 with thickness t = 0.1 and side length a = 10 is considered. The plate is hard simply supported on all sides. The Young's modulus is E = 10.92 x 107 and Poisson's ratio is v = 0.3. Figure 2.4.18: Problem definition and geometry for the uniformly loaded simply supported skew plate (t/a=100) Slika 2.4.18: Definicija in geometrija problema enakomerno obremenjene prosto podprte rom-boidne plosce (t/a=1/100) Reference solution The most interesting feature of the solution concerns two singular points at the two obtuse corners of the plate, which strongly influence the quality of the computed results (e.g. see [Morley, 1963]). The strength of the singularity is A = 6/5 = 1.2 see (2.2.66). In the vicinity of the corner, the moment resultants mxx, myy vary as rA-2, while the moments resultant mxy and shear forces qx and qy vary as rA-3, where r is the distance from the corner. As the analytic solution is not available, a finite element computation with Argyris element on a fine mesh (the elements with side h = 0.1) is taken as the reference solution. The reference solution confirms the existence of the singularity in the obtuse corners. Finite element solutions The boundary conditions used for the hard simply supported edge at y = const. are: Wj = 0, 9x>j = 0 for DK elements and Wj = 0, Wj>x = 0 for the ARGY element. At the hard simply supported sloped edge y/x = tan a the rotation 9n is set to zero, which results in the boundary condition: 9x j — tan a 9yj = 0 for DK elements and wj>y + tan a wj>x = 0. The finite element solutions with ARGY element is shown in Figure 2.4.20. A serious deterioration of the accuracy is observed at the obtuse corners. In order to capture the a w 0x 6y m xx myy mxy qx 1y n ■ 0. -0.0003 -0.000145 -1.5 -2.2 -0.5 -1.5 -3. 0.00005 -0.00025 -0.000121 -1.38 -1.98 -0.417 -1.25 -2.5 ■ 0.00005 -0.00025 -0.000121 -1.38 -1.98 -0.417 -1.25 -2.5 0.0001 -0.0002 -0.0000967 -1.25 -1.75 -0.333 -1. -2. ■ 0.0001 -0.0002 -0.0000967 -1.25 -1.75 -0.333 -1. -2. 0.00015 -0.00015 -0.0000725 -1.13 -1.53 -0.25 -0.75 -1.5 ■ 0.00015 -0.00015 -0.0000725 -1.13 -1.53 -0.25 -0.75 -1.5 0.0002 -0.0001 -0.0000483 -1. -1.3 -0.167 -0.5 -1. ■ 0.0002 -0.0001 -0.0000483 -1. -1.3 -0.167 -0.5 -1. 0.00025 -0.00005 -0.0000242 -0.875 -1.08 -0.0833 -0.25 -0.5 □ 0.00025 -0.00005 -0.0000242 -0.875 -1.08 -0.0833 -0.25 -0.5 0.0003 0. 0. -0.75 -0.85 0. 0. 0. □ 0.0003 0. 0. -0.75 -0.85 0. 0. 0. 0.00035 0.00005 0.0000242 -0.625 -0.625 0.0833 0.25 0.5 □ 0.00035 0.00005 0.0000242 -0.625 -0.625 0.0833 0.25 0.5 0.0004 0.0001 0.0000483 -0.5 -0.4 0.167 0.5 1. □ 0.0004 0.0001 0.0000483 -0.5 -0.4 0.167 0.5 1. 0.00045 0.00015 0.0000725 -0.375 -0.175 0.25 0.75 1.5 □ 0.00045 0.00015 0.0000725 -0.375 -0.175 0.25 0.75 1.5 0.0005 0.0002 0.0000967 -0.25 0.05 0.333 1. 2. □ 0.0005 0.0002 0.0000967 -0.25 0.05 0.333 1. 2. 0.00055 0.00025 0.000121 -0.125 0.275 0.417 1.25 2.5 ■ 0.00055 0.00025 0.000121 -0.125 0.275 0.417 1.25 2.5 0.0006 0.0003 0.000145 0. 0.5 0.5 1.5 3. Figure 2.4.19: Reference solution of the uniformly loaded simply supported skew plate with legend also valid for Figures 2.4.20 and 2.4.21 Slika 2.4.19: Referencna resitev enakomerno obremenjene prosto podprte romboidne plosce z legendo, ki velja tudi za Sliki 2.4.20 in 2.4.21 singularity better, the density of the mesh has to be increased at the obtuse corners. The finite element solution with DK elements qualitatively matches the reference solution in problem domain except at the obtuse corners. The quality of the finite element solution is influenced by the mesh distortion (see Figure 2.4.22). The performance of triangular element DKT does not the reach the performance of the quadrilateral element DKQ. The comparison of the converge rates shows, surprisingly, that the superior accuracy of the ARGY element, which was observed in the previous tests, is now lost. In Figures 2.4.20 and 2.4.21, the singularity can be observed at the obtuse corners in moments and mxy and in equilibrium shear forces qx and qy. The contour plots are, however, not very suitable to study the singularities due to the fixed finite values of contour lines. At the obtuse corners (singularities) the gradients of the solution are infinite and in the limit, the optimal size of the elements is zero. With the decreasing mesh size, the singularity is captured increasingly better. The singularity thus manifests itself the most in the convergence rates (see Figure 2.4.22). The DK elements and ARGY element exhibit similar convergence rates. The monotonic convergence is observed at DK elements, while the convergence of the ARGY element is not monotone. Additionally the difference between the convergence of the triangular and quadrilateral elements is observed: the later converge slightly faster. This fact can be contributed to the singularity and the effect of distortion on the element performance. W m/:x Qx,e (ARGY element) Figure 2.4.20: FE solution of the uniformly loaded simply supported skew plate with Argyris plate element Slika 2.4.20: Računska resitev enakomerno obremenjene prosto podprte romboidne plosče z Argyrisovim končnim elementom (DKT element) (DKQ element) Figure 2.4.21: FE solution of the uniformly loaded simply supported skew plate with DK plate elements Slika 2.4.21: Računska resitev enakomerno obremenjene prosto podprte romboidne plosče z DK končnimi elementi b0 O -0.5 -1 -1.5 ■ • ARGY ■ DKT DKQ 1.5 (Displacement w at center) 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1.2 1.5 2.5 log (ndof) (Moment m^x at čenter) 2.5 3 3.5 log (■ndof) 3.5 ✓ • ARGY ■ DKT DKQ v / V (Energy norm) Figure 2.4.22: Comparison of the convergence of FE solutions for the uniformly loaded simply supported skew plate Slika 2.4.22: Primerjava konvergence racunske resitve problema enakomerno obremenjene prosto podprte romboidne plosce 2.5 Chapter summary and conclusions In this chapter, we studied and revisited the theory of the (thin) Kirchhoff plate model and derived several finite element formulations based on that model. The motivation to dedicate some effort to study in more detail the well known topic - the Kirchhoff plate theory - is strongly connected to the main goals of this thesis. The main goals of the thesis are namely related to the derivation of procedures for discretization error estimation and model error estimation in the process of the finite element computation of plate-like structure. As it will be explained in the following chapters, both discretization error estimation and model error estimation can be performed by a comparison of numerical results that are obtained by using two different level of complexities in the computations. In the model error estimation, the complexity is associated with the type of the used hierarchical plate model (i.e. thin, moderately thick, thick). Those models are covered in the first three chapters. On the other hand, in the discretization error estimation, the complexity is associated with the type of finite element formulation (less rigorous, more rigorous) within the framework of a single plate model. Due to the above reasons, the thin plate model (theory) was studied in some detail in the first part of this chapter. The topis, like strong form of the boundary value problem, and singularities of solution, which may not be very familiar to those dealing with finite element computations of plates, were considered in order to prepare ourselves for better understanding of problems that might pop-up when dealing with discretization and model error of plates. In the second part of this chapter, several finite elements were derived. First, a conforming triangular element with 21 dofs was presented in detail. This element has not been used by the engineers, although, due to its completeness, it is relatively popular by mathematicians dealing with the plate problems (e.g. [Bernadou, 1996]). This element is very rigorous with respect of taking into account assumptions of Kirchhoff plate model. Next, triangular and quadrilateral discrete Kirchhoff plate elements were rederived. Those elements are more relaxed with respect to the Kirchhoff plate theory assumptions than the conforming one. The elements perform very well in general and may be found in many commercial finite element codes (e.g. [Wilson, 1997]). The chapter concludes with the numerical examples, where the results obtained by the three above mentioned elements are compared by each other and by reference analytical solutions. As expected, the conforming element shows superiority to the DK elements. In summary, this chapter deals with the thin plate theory and some corresponding finite element formulations that are of our interest for the later purpose. The main novelty of this chapter can be related to the nonconventional notation in the derivation of the DK elements, which is such, that both triangular and quadrilateral element formulations can be covered by using a single notation. Chapter 3 Moderately thick plates: theory and finite element formulations 3.1 Introduction The theory of moderately thick plates, in contrast to the Kirchhoff plate theory, takes into account transverse shear deformations. The Kirchhoff constraint is relaxed. Therefore, an additional freedom exists for the modes of deformation. A relaxation of the assumptions of the Kirchhoff theory was first made by [Reissner, 1945] and later, in a slightly different manner, by [Mindlin, 1951]. Since Reissner and Mindlin theories are not very different, it is common to denote a theory for plates with moderate thickness as a theory of Reiss-ner/Mindlin type. An analogy exists between the Euler-Bernoulli and the Timoshenko beam theories and Kirchhoff and Reissner/Mindlin plate theories, respectively. In the Kirchhoff theory, the rotation of the plate normal is completely defined by the displacement of the midsurface, whereas in the Reissner/Mindlin plate theory it is independent of displacement. The independence of rotations and transverse displacement has several consequences. The continuity of the rotations is no longer required and Reiss-ner/Mindlin plate loaded with a line load, can have kinks with abrupt change of slope. On the other hand, a point load, which produces in Kirchhoff plate finite displacement exhibits singularity in the Reissner/Mindlin plate. Perhaps the most interesting manifestation of the additional freedom, gained by the relaxation of Kirchhoff constraint, is the existence of a boundary layers - a region of a plate close to the boundary, where quick and significant changes in twisting moment and shear forces occur. The boundary effect, however, is not only a consequence of the assumptions of the Reissner/Mindlin plate theory, but rather a real physical phenomenon. In Reissner/Mindlin treatment of plate bending the basic assumptions are: (i) the material normal to the original reference surface remains straight, although no longer normal to the deformed reference surface, (ii) the thickness of the plate does not change during deformation and (iii) transverse normal stresses are negligible. The latter two assumptions are contradictory, which can easily be verified. Reissner and Mindlin plate theories differ from each other in a way how they solve this problem. While Reissner assumed a cubic variation in thickness direction of transverse normal stress, Mindlin manipulated the material law in order to comply with both assumptions [Reissner, 1945], [Mindlin, 1951]. Although conceptionally different, both plate theories deliver practically the same results for transverse displacements, shear forces and bending moments in actual structural problems. The basic consequence of the above assumptions is that the deformation of the moderately thick plate can no longer be defined by the transverse displacement w and its derivatives only. The rotation of material normal 0 is introduced as an independent variable. 3.2 Theory 3.2.1 Governing equations Kinematics In the following, the rotation 0 will be employed instead of the conventional rotation 0, see Figure 2.2.1. Recall that the rotations are related as 0 = [0x,0y]T = [—9y,9x]T. According to (2.2.3) and (2.2.4), the kinematic relations, which relate the displacement w and rotation 0 to the curvatures k = [Kxx,Kyy, Kxy]T and transverse shear strains Y = [Yx,Yy]T, are k = [o^x d^L (d^y + d^x)]T. y = [dw — 0 dw — (321) dx ' dy ' dx dy ' dx x' dy y In a compact notation we have Y = Vw — 0. k = D0 (3.2.2) where the differential operator D is V ^ 0 dx 0 dy JL JL dy dx (3.2.3) Equilibrium equations The equilibrium equations were derived already in the previous chapter. We recall the main result: V ■ qeq = -f (3.2.4) where the shear forces obtained from the equilibrium equations are denoted in this chapter as qeq = [qx,eq, qy,eq]T. We also recall from (2.2.18): V ■ M = -qeq; M mxx mxy mxy myy (3.2.5) Constitutive relations To obtain the moment-curvature relationship, we recall from (2.2.12) and (2.2.13): the moments are related to curvatures as m = Cb k (3.2.6) The kinematic shear strains y computed from the kinematic assumptions (3.2.1) are constant through the thickness, which is approximation to the actual variation, even for a homogeneous cross-section. The equilibrium equations show, that the variation of the shear stresses through the thickness must exist. For the homogeneous cross-sections, the shear strain distribution is commonly accepted to be a parabolic function of z. Strains following from the kinematic assumptions are in contrast with the equilibrium of differential plate element. To overcome this difficulty, two theories were presented ([Reissner, 1945] and [Mindlin, 1951]), both resulting in the introduction of the shear correction factor c which enables the definition of the following simplified constitutive relation, where the constitutive shear forces are denoted by qc: cEt qc = CsY; Cs = cGtl = 1 = C/t2D1; 1 10 01 (3.2.7) where G = E/2(1 + v) is shear modulus and C = 6c(1 — v). The shear correction factor c is usually set to 5/6, although also different values can be taken. Boundary conditions The Reissner/Mindlin plate theory provides a wider range of possibilities for the boundary conditions than the Kirchhoff theory. Some of them are listed in the Table 3.1. Notice Table 3.1: Boundary conditions for moderately thick plates Tabela 3.1: Robni pogoji za srednje debele ploSCe Clamped - hard w = 0 0s =0 0n =0 Clamped - soft w = 0 0s =0 mn =0 Simply supported - hard w= 0 0n =0 ms =0 Simply supported - soft w= 0 ms =0 mn =0 Free q = 0 ms =0 mn =0 the difference between hard and soft simply supported plate. In many cases more realistic is soft simple support with both twisting and normal moments set to zero. In the case of thin plates, however, (Table 2.1) it is possible to model only hard simple support. Shear correction factor The shear correction factor approximates, on an average basis, the transverse shear strain energy. The transverse shear strain deformation energy density equals w(0) = Yxz °xz + Yyz (7yz = (^Xz + rfz )/G (3.2.8) where jx = jxz = axz/G and jy = Yyz = ayz/G, was employed. The superscript (0) denotes that the the stress level constitutive relations were used. Integration of w(0) through the thickness gives the transverse shear strain deformation energy per unit area of a plate: dW (0) r t/2 r t/2 ^Wf- = W0)dz = (axz + < )/G dz (3.2.9) di' J-t/2 J-t/2 The transverse shear deformation energy per unit area can, on the other hand, be computed also with the shear strains 7 and the constitutive shear forces qc = [qx,c,qy,c]T as dW(1) = YxQx,c + Yy Qy,c = (ql,c + qy,c)2/(cGt) (3.2.10) where the (3.2.7) was employed. The superscript (1) denotes that the stress resultant level of constitutive relations was used. The shear correction factor c can be computed from the requirement that the shear deformation energies (3.2.10) and (3.2.9) are equal. Let us now write the expression (3.2.9) with equilibrium stress resultants. We start with the two stress equilibrium equations, which are related to the transverse shear stresses: + 9aXy + doxx _ 0, dayz + dayy + 9&xy _ q (32 n) dz dy dx ' dz dy dx The distribution of transverse shear stresses across the thickness at fixed (x,y) can be computed by integration of (3.2.11): r r Oxz(z) I (^xx,x + @xy,y)dZ; @yz(z) I (&yy,y + ^xy,x)dz (3.2.12) J-t/2 J-t/2 We know, that the stresses axx,ayy and axy share the same (i.e. linear) through the thickness dependence, which we denote by g(z). These stresses can be written as: axx(x,y,z) = ^xxü(x, y) g(z); Gyy(x,y,z) _ Oyy0(x,y) g(z); Gxy(x,y,z) _ ^(x,y) g(z) (3.2.13) After inserting (3.2.13) into (3.2.12) and integrating through the thickness we can write (3.2.9) as dW(0) 1 f+t/2 frz \2 d^ _ G [(axxü,x + Vxy0,y )2 + (^yyü,y + &xyü,x )2] J J g(z)dzj dz (3.2.14) Next, we write the equilibrium shear forces (2.2.14) in terms of: dmxx dmxy ft/2 . ft/2 Qx,eq _ —R--+ ^- _ <7xx0,x zg(z)dz + <7xy0,y zg(z)dz (3.2.15) dx dy J-t/2 J-t/2 r t/2 Qx ,eq (^xx0,x + &xy0, y) / zg(z)dz -t/2 dm dmx ft/2 ft/2 Qy,eq _ + _ ^0 y J zg(z)dz + axy0,x J zg(z)dz r t/2 Qy,eq (^yy0,y + @xy0,x ) zg(z)dz -t/2 Using (3.2.15) in (3.2.14), we finally have (3.2.9) written in terms of equilibrium stress resultants dw(1) - - ' r+t/2 dQ -x,ey ■ , y j-t/2 v , , S (Qx,eq + qLq Vel/ P 2(z)dz|/R2 (3.2.16) where the following definition was used: r+t/2 PZ R _ / zg(z) dz; P (z) _ g (z) dz (3.2.17) -t/2 -t/2 The demand that the transverse shear energies defined in (3.2.16) and (3.2.10) must match, leads to the expression for the shear correction factor: R2 c = —F72--(3.2.18) tf% P2(z) dz For the case of homogeneous isotropic plate, where g(z) = 2z/t, we have R = t2/6 and P (z) = (t/4)((2z/t)2 — 1). Since f-^ P2(z) dz = t3/30, we thus obtain c = 5/6. 3.2.2 Strong form From the constitutive relations (3.2.7) and kinematic relations (3.2.2), we have t2 Y = Vw — 0 =— q (3.2.19) Dc Taking a divergence of (3.2.19) and using the definitions (2.2.21) V-0 = m/D and (2.2.16) V ■ q = —f, we arrive at △w=m—Di <3-2-2°) By applying the A operator to both sides and using (2.2.22) Am = f, the modified biharmonic equation for the displacement of the thick plate is obtained: △Aw = + D — tC f (3.2.21) Notice, that in the limit as t ^ 0, the standard biharmonic equation (2.2.26) is recovered. This is also the case, when Af = 0. From the definition Yx = dW — 0x, Yy = dr — 0y and by noticing that d-dw = , ix dx ' 'y dy ry j o dy dx dx dy " we have &Yx dYy _ d0y 80x dy dx dx dy where we introduced the function —Q(0) , (3.2.22) Q(0) = 0x,y — 0y,x (3.2.23) which can be interpreted as the local transverse twist. Notice that, in the thin plate limit Q = 0 or 0x y = 0y x. The vector field 0 is in that case potential, which implies, that there exists such a scalar field (displacement w in this case) whose gradient equals the vector field 0 (0 = Vw or y = Vw — 0 = 0). A useful relation is obtained if we express transverse shear strain 7 with the rotations 0. We start with (3.2.7), where we take into account the equilibrium equations (3.2.5). The constitutive relation (3.2.6) relates moments to curvatures which are defined by the rotations through (3.2.2). Finally we have 7x = 2C(( +v) s? — (1 +v»äxsy— 2' - =t2 ((—1 + v) — (1 + v) — 2 ^) 2C dx2 dxdy dy2 dx dy From the results above, we now compute Q = ^ — ^ ' ± dx dy Q = 12_ (+ dVx — d30y — ) (3 2 24) 12c dy3 dy dx2 dx dy2 dx3 By taking AQ = + |yr and using the definition (3.2.23), we observe that the following relation holds: t2 Q = — AQ (3.2.25) Additional result relates shear forces to the Marcus moment (see (2.2.21) for the definition of m): qx = —mx + 1 D(1 — v)Qy (3.2.26) qy = —my — 1 D(1 — v)Q,x Note the similarity to (2.2.24), which relates Marcus moment to the equilibrium shear forces for the Kirchhoff theory. The behavior of the plate according to Reissner/Mindlin theory is governed by the system of differential equations for displacement w and transverse twist Q: t2 t2 AAw = f/D — —A f; Q — 12c AQ = 0 (3.2.27) with the appropriate boundary conditions specified for the transverse displacement, rotations or stress resultants (expressed in terms of w and Q). The two equations in (3.2.27) are coupled through the boundary conditions. With the decreasing thickness, the transverse twist plays an increasingly smaller role in the interior part of the plate. In thin plate situations Q is substantially different from zero only near the boundary. An alternative formulation of the Reissner/Mindlin theory, which uses the displacement w and rotations 0 is also possible. Differentiating (3.2.25) and again using Am = f, we obtain the relations AA0x - (AQ),y = f,x/D (3.2.28) AA^y + (AQ),x = f,y/D Inserting AH = H into (3.2.28), we obtain together with (3.2.21) the following system of fourth-order differential equations for the displacement w and rotations 0 t2 AAw = f/D - — Af (3.2.29) t2 ■ 12C AA0x - (0x,y - 0y,x),y = f,x/D t2 + 12C AA0y + (0x,y - 0y,x),x = f,y/D Strong form of the Reissner/Mindlin problem can now be formulated as: by knowing f, D, t and c, find the solution of (3.2.29) for transverse displacement w and rotations 0 that satisfies the corresponding boundary conditions. Kirchhoff versus Reissner/Mindlin solutions The Reissner/Mindlin plate theory is a hierarchic extension of the Kirchhoff theory. It is thus possible to express the solutions wM, 0M (the superscript M refers to the Reiss-ner/Mindlin theory) of the equations (3.2.21) and (3.2.29) as the hierarchic extension of the Kirchhoff solution wK (the superscript K refers to the Kirchhoff theory) (see [Lee et al., 2002]). To that end, we introduce the scalar functions $ and tf with the following properties: AA$ = 0; Atf = 0 AmK = AmM = f; mM = mK + DA$; mK = DAwK (3.2.30) From (3.2.30), (3.2.20) and (3.2.26), the relations between Reissner/Mindlin and Kirchhoff solutions are obtained as t2 t2 wM = wK - — mK + tf + $ = wK - -AwK + tf + $ Dc c d /t2 \ t2 0m = w£ + ^(j a$ + * + $j + — n,y = 0k + a2 d /12 \ t2 0m = w,k + (jA$ + tf + $) - ^= 0k + a (3.2.31) To obtain the rotations 0M, in addition to the Kirchhoff solution the functions Q and $ have to be computed from the equations AA$ = 0; A^ = 0: Q =-AQ 12c (3.2.32) The equations above are solved with some given boundary conditions (note, that $, ^ and Q depend both on displacement/rotation boundary conditions and loading). All the Mindlin stress resultants can be computed from (3.2.31) and are related to the Kirchhoff stress resultants. The Kirchhoff shear forces are computed from the equilibrium, using (2.2.24). This relations are: (3.2.33): y — 0y>x can be interpreted as the local transverse twist. From inspection of Q = AQ, we conclude that the term on the right decreases as t ^ 0 and in the limit the Kirchhoff assumption is met: x. The inspection shows also that for small t the functions Q exhibits a strong boundary layer within a distance of the order t/V 12c from the boundary. According to the relation (3.2.21), the transverse displacement w has no boundary layer. Transverse twist Q alone governs the edge-zone behavior of rotations and stress resultants. For that reason [Haggblad Bathe, 1990], see also [Arnold Falk, 1989], assumed that in the neighborhood of the domain boundary it is possible to make an asymptotic expansion of displacement w and rotation 0 in powers of the plate thickness t: X X / < \ w tw 0 = ^210i + S K] t%) (3.2.38) i=0 i=0 V i=0 / The terms w^ and 0j are smooth functions independent of thickness. The first two terms in (3.2.38) also satisfy Vw0 = 00 and Vwp = 0p. Function S is a cut-off function, which equals unity when the point in question is close to boundary and is zero otherwise. The functions $ are exponential decay functions of the form where c is the shear correction factor and Fi are smooth functions independent of t. The boundary fitted coordinate system (p, p) defines the position of the point relative to the boundary: the coordinate p is the coordinate along the boundary curve and p is the distance from the point under consideration to the nearest point on the boundary. Due to the presence of the decay function and the cut-off function S, all the boundary effects die out quickly and are negligibly small outside a layer of width of order t adjacent to the boundary. In [Arnold Falk, 1989] it is shown that depending on type of support condition certain terms in the expansion of w and 0 vanish. Since the boundary effects are determined by the existence of these terms, the boundary layer strength is defined by the power of t corresponding to first non-vanishing term of the boundary layer expansion. In the finite element analysis of the plates the stress resultants are of primary importance. The boundary effect is more pronounced for the shear forces since they are derived both from rotations 0 and displacement w. The bending moment for instance depends on the first derivatives of 0 and it has a boundary layer one order higher than that of 0. The shear force vector is defined by q = AD/t2(Vw - 0) and can be expanded as According to Table 3.2 ([Selman et al., 1990]) a strong boundary layer effect is expected in the case of soft simply supported or free boundary. The boundary layer dominates the shear force vector. The strength of the effect is inversely proportional to thickness. Since the boundary layer is limited to distances of order t from the boundary, the shear force as well as the twisting moment have very steep gradients. Notice, however, that despite the steep gradients the shear force and twisting moment will always be bounded for any thickness of the plate. Since all the terms in the expansion (3.2.38) are differentiable with respect to p and t, the shear forces and moments are continuous everywhere along the edges. The boundary layer does not introduce any singular point into the problem domain. The boundary layer effect of order one is expected for the shear force vector in the case of hard simply supported or clamped (fixed) edge. The soft clamped edge experiences the weakest boundary layer. On the other hand, the soft simply supported plate and $ = e-VT2Cp/tF(p/t,p) (3.2.39) q = AD[(Vw2 - 02) + t(Vwa - 0s) + ■ ■ ■ ] - S(t-1$i + $2) (3.2.40) Table 3.2: Leading terms of thickness expansion in (3.2.38) Tabela 3.2: Vodilni cleni v izrazu (3.2.38) On Os mn ms mns Qn Qs Clamped - soft t4 t3 t3 t3 t2 t2 t Clamped - fixed t3 t2 t2 t2 t t 1 Simply supported - hard t3 t2 t2 t2 t t 1 Simply supported - soft t2 t t t 1 1 1/t Free t2 t t t 1 1 1/t free plate have the strongest boundary layers. It is interesting to notice (see [Arnold Falk, 1989], [Brank Bohinc, 2006], [Brank et al., 2008]) that the boundary layer effect disappears altogether in the case of hard simply supported or soft clamped edge if the boundary is straight. Singularities The response of the Reissner/Mindlin plate to the point load is singular - the displacement under load is infinite. Although the finite element approximation predicts finite displacement at point load application, the magnitude of the displacement increases without limit as mesh is refined near the loads. As already discussed in the case of Kirchhoff plate theory certain configurations of the boundary conditions can cause the singularities in the stress resultants. Such singular effects are most commonly induced at a corner of the plate when the internal angle is greater than the right angle. In the vicinity of the plate corners the stress resultants can be expressed in polar coordinates (r, as (see [Babuska Li, 1992]) C1 rXl-1^ns(w) + smoother terms C2 rA2-1^s(^) + smoother terms min(Ai,A2) (3.2.41) where the functions ns and are analytical functions of The C1(z), C2(z) are smooth functions of z . The powers A1, A2 define the strength of the singularity. If A < 1 the moments and shear forces will become unbounded at the corner point. Furthermore, if A is not an integer and 1 < A < 2 the moments and the shear forces will be discontinuous at the singular points. The parameter A depends on the opening angle of the plate a as well mn Qs A Table 3.3: The coefficients Ai and A2 for the hard (AH, A^) and soft (Af, Af) simple support Tabela 3.3: Koeficienta A1 in A2 za togo (AH, AH) in mehko (Af, Af) prosto podporo a AH AH Af Af 30° 3.4846 6 8.0630+i4.2028 6 45° 2.4129 4 5.3905+i2.7204 4 90° 1.4208 2 2.7396+i1.1190 2 120° 1.2048 3/2 2.0941+i0.6046 3/2 135° 1.1368 4/3 1.8853+i0.3606 4/3 150° 1.0832 6/5 1.5339 6/5 225° 0.7263 4/5 0.6736 4/5 270° 0.5951 2/3 0.5445 2/3 315° 0.5330 4/7 0.5050 4/7 360° 0.500 1/2 1/2 1/2 as on the support conditions in the neighborhood of the corner point but is independent of plate thickness t and Poisson's ratio v. Moments and shear force will be unbounded or discontinuous only at the singular point. The singular behavior is confined to the immediate neighborhood of the point in question. The Table (3.3) lists the parameters A1, A2 for the hard (H) and soft (S) simply supported boundary conditions ([Babuska Li, 1990]) with respect to the opening angle of the corner a. It can be seen, that the strength of singularity for the shear forces is the same for both support condition. This is, however, not the case for the moments. Coefficients A can also be complex. It such a case, there is always a pair of conjugate coefficients, since the quantities in (3.2.41) have to be real. 3.3 Finite elements Reissner/Mindlin plate theory allows greater flexibility in the element design than the Kirchhoff theory. The Kirchhoff constraint is no longer required nor is the need to build the C1 continuous approximation of the displacement. Since in the Reissner/Mindlin model the shear deformation is allowed, the rotation 0 is no longer defined by the displacement w. Independent approximations are built for the displacement w and the rotations 0 = [OX, Oy]T. Principle of virtual work The displacement based finite element discretization is built upon the weak form of the equilibrium (3.2.4), which has already been presented in (2.2.41) and is repeated here for the convenience in the form of principle of virtual work Snmt = Snext: Snint(w, 0; Sw,S0)= [ (m(0) • ök(ö0) + q(w, 0) • Sy(Sw,S0))dQ (3.3.1) JQ Snext(Sw,S0)= / f Sw dQ+ / qn Sw ds + / 50 • m ds n 'rq where f is the surface loading. The boundary is a union of Dirichlet and Neumann boundary. The tractions prescribed at Neumann boundary have three independent components; shear force line load q and two components of moment line load m = \mx,my]T. Interpolation The displacement w and the rotation 0 are independent and can be interpolated over a single element in a usual fashion as wh = NwUu; 0h = N U (3.3.2) where the element degrees of freedom are collected in U and Nw and N^ are matrices of interpolation functions. When wh depends on both wbj and 0/, and 0h depend on both wbj and 0/ we talk about linked interpolation. The matrices Nw and N^ are specified for the considered elements. The discretized curvatures and shear strains are: Kh = BkuU; Yh = B7 UU (3.3.3) where BK, BY are matrices of derivatives of interpolation functions. Stiffness matrix The stiffness matrix K of the element is revealed by the discretization of the weak form (3.3.1) Q ~ (Je Qh and by using the interpolation (3.3.3): ae(uh; Suh) = Snent(uh; Suh) = (3.3.4) [ Skt(S0h)m(0h)dQ+ / Syt(Swh,S0h)q(wh; 0h) dQ = SUTKU J Qe J Qe Q r m where uh = (wh; 0h and Suh = (Swh, S9h). The stiffness matrix of the element K has two parts: bending stiffness and shear stiffness K = KB + KS (3.3.5) where Kb = BT CB BKdQ; KS = B^ CS B7 dQ (3.3.6) The matrices BK and BY are specified for considered elements, by using Nw and Nö. Load vector Following from the discretization of the (3.3.1) we have le(5uh) = ön:xt(öuh) = £SUTfi = (3.3.7) i [ SUT f Nw dQ + / SUT (gNw + mTNe) ds = SUT f where the consistent nodal load vector is f = ff + ff ff = / f NwdQ; ff = f (qNw + mTNe) ds ./ne ./rNe Shear locking The leading term of the bending stiffness is t3 while the leading term of the shear stiffness is t. In the limit as t ^ 0, the shear strain, due to the high shear stiffness, vanishes and the bending prevails. Bending is most favorable deformation mode in the thin plate limit. The problem occurs if the discretization is unable to describe accurately pure bending of thin plates. To do so, the discretization must be able to satisfy the Kirchhoff constraint Vwh = 0h, where 0h = [—9y>h, 6x,h]T. If the discretization fails to eliminate shear strains in the thin plate limit, the total deformation energy is dominated by the shear and the minimization of total potential energy de facto equals the minimization of the shear deformation energy. In the thin plate limit, the shear stiffness becomes very large relative to bending stiffness. Unless the finite element is able to describe pure bending, and thus completely avoid the shear deformation, the displacements of the plate vanish. The finite element solution "locks". The phenomena is known by the name shear locking. In order to build a successful Reissner/Mindlin element which avoids shear locking the discretization of displacement wh and rotation 0h must be carefully balanced. 3.3.1 Elements with cubic interpolation of displacement In the following, a family of Reissner/Mindlin plate elements will be presented. The derived elements follow the work of [Ibrahimbegovic, 1992] and [Ibrahimbegovic, 1993]. They are extensions of DK elements presented in previous chapter, therefore linked interpolation is used. Edge interpolation equals Timoshenko beam interpolation Again, the basis of the element construction is an analogy with beams. Recall that in the development of DK elements the shear free interpolation of the Euler Bernoulli beam, (2.3.53)-(2.3.56) and Figure 2.3.9, was employed. We modify that interpolation so that the constraint of zero shear Yh = — Oh = 0 is replaced by the constraint of constant shear Yh = — Oh = Y0 = const. In that case the Euler-Bernoulli parameters (2.3.56) transform into the Timoshenko beam parameters w = + L(O 1 — O2); w4 = L(W2_w — 1(O 1 + O2)) — L^0; O3 = Lw4 (3.3.8) The Timoshenko (TM) beam interpolation can be presented as the hierarchical extension of the Euler Bernoulli (EB) interpolation wTM = wEB — L Y0N4; oTm = oEb + 2 (3.3.9) where EB interpolation is given in (2.3.57)-(2.3.58). P3T and P3Q elements Let us express the interpolation of considered plate elements as a hierarchic extension of the DK interpolation (2.3.63): wh = wDK — £ 4 Lij Yij Mi j ; (3.3.10) IJ 4 3 Oh = 0DK + J] -niJYiJNu IJ The shear along the side IJ is denoted by jIJ, LIJ is length of the side IJ, and nIJ is normal to the side IJ, see Figure 2.3.11. Note the similarities between (3.3.9) and (3.3.10). The P3Q/P3T elements have the same vertex degrees of freedoms uI = [WI, Ox>I, OViI]T as DK elements. However, they have additional midside degrees of freedom uIJ = [YIJ]. The displacement and rotation of an element point is designated by uh = [wh, 6Xyh, 0y,h]T• The elements have nen vertex nodes and nen midside nodes (nen=4 for the quadrilateral, and nen = 3 for triangular elements). The vertex nodes are numbered as I = 1,..., nen, the midside nodes as IJ = (nen + 1),..., 2 nen. Element degrees of freedom are arranged into u = [uT, ■ ■ ■ , 'Unen, ünen+1, ' ' ' , ü2nenV. nen+1> In the matrix notation, the interpolations can be written as Li Wh = Nwü = J] NwJÜ! + Y, NW,IJÜU; Nw,u = —jMiJ (3.3.11) i i J 3 Oh = N ü = J] Ngj ÜI + Neu ü i j ; Ne,ij = +2 nij Nu i i j where the Nw i = N^j, Ne i = NDJ are the matrices introduced in DK elements, see (2.3.63)-(2.3.66). Notice, that the additional terms in the interpolations (3.3.11) and (3.3.12) depend only on midside degrees of freedom collected in UiJ. The curvature Kh is computed directly from the rotations Oh and is expressed as Kh BkÜ = J] BK,I Üi + J] Bkj üij (3.3.12) iJ DK k,i . The matrix Bk i equals to the matrix of the DK element defined in (2.3.68): Bk i = B The curvature interpolation can therefore be written as: Kh = kdk + Kh, where Kh is the part of the curvature, depending entirely on UiJ: Kh = ^2 BK,IJüIJ; (3.3.13) iJ Bk,IJ = - [-nij,yNij,x, +nij,xNij,y, nj^Nj — njNij,y]T (3.3.14) 2 [—nr 7 y Nt 7 x, +nr 7 xNT 7 y , n,7 7 xNT 7 x — n,7 7 y NT 7 y]T The P3T/P3Q element stiffness matrix is a sum of bending stiffness matrix KB and shear stiffness matrix KS. The stiffness matrix is expressed as a block matrix K = [Kj], where blocks K j correspond to the interaction of degrees of freedom of nodes i and j. The indexes referring to the vertex node are denoted by I and J. The indexes IJ and K L are used to denote the midside nodes. The stiffness matrix is then organized as: [Ki,j] [KI,KL] [KUJ] [KUKL] K (3.3.15) Bending stiffness The bending stiffness matrix is obtained as KB =[KIBj]= t BTCbBKdQ; Kg = / B^CbBkjdQ (3.3.16) JQe JQe where BK = [BM, ••• , BK,nen, BK,nen+1, ••• , BKMen\T with BK,/ defined in (2.3.68) and Bk,/j defined in (3.3.14) and Cb defined in (2.2.12). Notice, that the block matrix [KJ equals the stiffness matrix of the corresponding DK element Kb'dk, defined in (2.3.69). Shear stiffness Although the interpolations wh and 0h were constructed such that the shear strain is constant along the sides of the element, i.e. equals Y/J along side IJ, these interpolations still fail to describe the pure bending deformation without the spurious shear strains. Therefore, the shear locking is still present. A robust and efficient remedy for the shear locking is the use of the assumed shear strain approach. The shear strains may be interpolated in the form of bilinear distribution where Yh/ are transverse shear nodal parameters at vertex nodes and N/ are standard Lagrangian function. The Yh,/ are chosen to be consistent with the values of y/j, which define the constant shear along the side IJ. For the node I, where the sides IJ and IK meet, the nodal parameters ■yh,/ are chosen so, that the projections of ■yh,/ to the sides IJ and IK match the values y/j and y/k respectively (see Figure 3.3.1): Yh = N/Yh,/ = ^2 By,/JU/J = B7U (3.3.17) / /J Yh,/ ■ s/j = Y/J; Yh,/ ■ s/k = Y/k (3.3.18) Figure 3.3.1: To the computation of the nodal shear Yh,/ from (3.3.18) Slika 3.3.1: K izraCunu vozliSCnega vektorja strižne sile Yh,/ iz izraza (3.3.18) The solution of the linear system is: Yh,/ Y/Kn/j - Y/Jn/K n/j ■ s/k (3.3.19) Notice, that the nodal shear strains 7h;I depend only on the parameters /yIJ. Consequently the discretization 7h depends only on UIJ: 7h = ^ B7)iju u IJ (3.3.20) The element shear stiffness matrix KS is K1 [Kfj] [K?kl] [J [jl] (3.3.21) where K/,J = °3X3; KI,KL = 03xi; K/J,J = 01x3 (3.3.22) and K KIJ,KL B7,IJ CSBT,KLd^ (3.3.23) The explicit form of matrix B7)IJ, related to the side IJ, is: n t B7)IJ = ( Nj -nuK- - Ni -IS!-) fca (3.3.24) V -JK ■ Sij -IJ ■ sm) KJ / AW The above developments apply for both quadrilateral and triangular elements. We named such elements as P3Q and P3T, respectively. Since the P3Q/P3T elements are hierarchical extensions of DKQ/DKT elements, they share the same shape functions, given in (2.3.71) and (2.3.74). 3.3.2 Elements with incompatible modes The midside parameters UIJ complicate the practical usage of the P3Q/P3T element. It is possible to reduce that midside degrees of freedom collected UIJ by elimination made by means of static condensation. To do that we follow the steps of method of incompatible modes presented in [Ibrahimbegovic, 1992]. Not only that the parameters UIJ define the 7h but they also affect Kh, since Kh = Bk U = Bk iUi + Bk,ij-iJYij; (3.3.25) I IJ The BK iJ—IJ can be seen as additional deformation modes. The parameters YIJ can thus be interpreted as the amplitudes of the additional deformation modes. These modes are Figure 3.3.2: Quadrilateral plate element P3Q with cubic displacement interpolation Slika 3.3.2: Štirikotni konCni element za ploSCe P3Q s kubiCno interpolacijo pomika treated as incompatible modes and are subject to static condensation. To ensure that the element passes the patch test (see [Ibrahimbegovic, 1992]), the matrix Rkjj must be modified in the following way BkJJ ^ BkJJ - f Bjn (3.3.26) Si ./Qe The stiffness matrix of the element is reduced by standard static condensation procedure ([Zienkiewicz Taylor, 2000]). The resulting elements will be denoted as PIT and PIQ. 3.4 Examples 3.4.1 Uniformly loaded simply supported square plate The square plate of side length a =10 and thickness t =1 under uniform loading f = 1 is analyzed. The sides of the plate at x = 0, a are hard simply supported, while the sides at y = 0, a are soft simply supported. The difference between soft and hard simply supported boundary condition is defined in Table 3.1. The material is linear elastic and isotropic, with Young's modulus E = 10.92 x 104 and Poisson's ratio v = 0.3. a a y x Figure 3.4.1: Problem definition and geometry for the uniformly loaded hard-soft simply supported square plate (t/a=1/10) Slika 3.4.1: Definicija in geometrija problema enakomerno obremenjene togo-mehko podprte kvadratne plosCe (t/a=1/10) Reference solution The reference solution is an analytical solution in the form of series expansion (see Section 3.2.2). Using the approach suggested in [Lee et al., 2002], it is possible to express the solution of the Reissner/Mindlin model as a hierarchic extension of the Kirchhoff solution (see [Brank, 2008]). While the differences between the solutions are small in the interior of the plate, they are significant along the boundary. The characteristic width of the boundary layer is proportional to the plate thickness. The local transverse twist Q = I and 9y>I. The PI elements are derived from the P3 formulation, where the shear deformation is treated as the incompatible deformation mode. The boundary conditions for the P3 and PI elements are: UI = 0 at soft simply supported sides at y = 0, a and UI = 0, 9y = 0 at hard simply supported sides at x = 0, a. All other degrees of freedom are unconstrained. The FE solution on a regular mesh is shown in Figure 3.4.3 and Figure 3.4.4. One immediately notices that the PI elements completely fail to capture the boundary layer. In the interior of the domain the PI solution is close to the reference solution. The P3 elements on the other hand successfully capture the boundary layer effect and show otherwise satisfactory performance. The performance of the elements is further studied in the convergence plots, see Figure 3.4.5. The PI solutions apparently do not converge to the reference solution. Additionally one notices the differences in the performance of the triangular and quadrilateral P3/PI elements. Slower convergence of triangular elements compared to the convergence of quadrilateral elements is partly due to the fact that the quadrilateral elements are not distorted in the regular mesh. 3.4.2 Uniformly loaded simply supported free square plate The problem of uniformly loaded (f = 1) square plate of side length a =10 and thickness t = 1 is considered. The sides of the plate at x = 0, a are hard simply supported, while the sides at y = 0,a are free. The material is linear elastic and isotropic, with Young's modulus E = 10.92 x 104 and Poisson's ratio v = 0.3. a y* x Figure 3.4.6: Problem definition and geometry for the uniformly loaded hard simply supported-free square plate (t/a=10) Slika 3.4.6: Definicija in geometrija problema enakomerno togo prosto podprte - proste kvadratne plosCe (t/a=1/10) Reference solution The reference solution is the analytical Mindlin model solution in the form of series expansion, expressed as the hierarchic expansion of Kirchhoff model according to [Lee et al., 2002]. The problem is solved with 40 terms as shown in [Brank, 2008]. The solution, contrary to intuition, but according to Table 3.2, exhibits a strong boundary layer along the free sides. The boundary layer is again most pronounced for the stress components mxy and qx, qy. The local transverse twist shows strong resemblance with that in Figure 3.4.2. Reference equilibrium transverse shear forces are equal to the reference constitutive transverse shear forces. Finite element solutions The comparison of the P3 and PI finite element solutions shows that the PI elements are incapable of detecting the boundary layers. Qualitative comparison of P3 solution with reference solution does not reveal any significant discrepancies. The convergence plots shown in Figure 3.4.10 confirm that the PI elements do not converge to the reference solution. The P3 finite element solution converges to the reference solution. The relative error (in total deformation energy, center displacement and center moment resultant mxx) is inversely proportional to the number of global degrees of freedom (convergence rate is 1). 3.4.3 Uniformly loaded simply supported skew plate Uniformly loaded (f = 1) skew plate (a = 30°) with side length a =10 and thickness t = 1 is analyzed. The sides of the plate are soft simply supported. The material is linear elastic and isotropic, with Young's modulus E = 10.92 x 104 and Poisson's ratio v = 0.3. Figure 3.4.11: Problem definition and geometry for the uniformly loaded soft simply supported skew plate (t/a=10) Slika 3.4.11: Definicija in geometrija enakomerno obremenjene mehko podprte romboidne plosce (t/a=10) Reference solution As the analytical solution of the problem is unavailable, a finite element computation with P3T elements on a dense mesh (the elements with side h = 0.1) is taken as the reference solution. The Figure 3.4.12 shows that a boundary layer develops at the sides of the plate. The width of the boundary layer is of the order of plate thickness. Notice, that, compared to the Kirchhoff solution of the hard simply supported skew plate (see Figure 2.4.19), the singularity vanishes at the obtuse corners (see Table 3.3). Finite element solutions At the soft simply supported sides of the plate only displacement is constrained WI = 0. The boundary condition is the same for both P3 and PI elements. The additional degree of freedom jIJ is unconstrained at the supports. Since the reference solution is dominated by the boundary layer effect, the performance of P3 elements is clearly superior to that of PI elements. The comparison of the convergence rates in Figure 3.4.15 confirms that the PI elements do not converge to the reference solution. From Figure 3.4.13 it is also clear, that the PI elements are not able to distinguish between soft simple and hard simple support. 3.4.4 Uniformly loaded soft simply supported L-shaped plate Uniformly loaded soft simply supported L-shaped plate with the geometry shown in Figure 3.4.16 is analyzed. The side length is a =10 and thickness of the plate is t = 1. The material is linear elastic and isotropic, with Young's modulus E = 10.92 x 104 and Poisson's ratio v = 0.3. a/2 y, i-------1 E ,v,t,f a/2 a a Figure 3.4.16: Problem definition and geometry for the uniformly loaded soft simply supported L-shaped plate (t/a=10) Slika 3.4.16: Definicija in geometrija problema enakomerno obremenjene mehko podprte L plosče (t/a=1/10) Reference solution The reference solution is taken to be the finite element solution with the P3Q elements on a dense mesh (the elements with side h = 0.1). The solution shown in Figure 3.4.17 exhibits a singularity at the reentrant corner as well as the boundary layers along all the sides of the problem domain. According to the Table 3.3, the strength of the singularity is Ai = 0.5445 for the moment resultants and A2 = 2/3 for the moment resultant mxy and shear forces qx, qy. Finite element solutions At the soft simply supported sides of the plate only displacement is constrained wI = 0. The comparison of finite element solutions (see Figures 3.4.19 and 3.4.18) shows that both P3 and PI elements qualitatively capture the reference solution. The boundary layer is, however, captured only by the P3 elements. The finite element solution with PI elements does not converge to the reference solution. Additionally, the Figure 3.4.20 shows that due to the boundary layers, the convergence rate of P3 elements is deteriorated. The singularity at the re-entrant corner exists for the moments mxx, myy, mxy and shear forces qx, qy. i 2 co CO o CO (M O O O o o >8 828 8?8 8 2 o 828 o o o 1 o o 1 1 o o o 1 d d o 1 1 1 -1.17 —0?583 -0.583 o! CO . , 00 o 2io d CO K d lO lO t- s. " t 00 j CO CO K 1 1 1 1 d2S d 0.58 1 1 1 -1.17 —CL583 -0.583 o! o 2 »o d m 2 ^ d iS S no lO § o «o 1 CO 2 CO 1 1 T i •72er Pi^H 1 1 lO lO CO CO S CM lO 00 o ss 1 CO C CO 1 1 co 2oi T 7 P 1? lO in 10 iO to ° 1 § O O s 10 ?§ 82° §28 8?d d 28 o 1 6 ? 1 1 ° i o 1 o d 0 . d ° t2 o 12 "3 M S d 8 26 • i o o 2 o S "3 N d 1 1 1 ° i d 1 o d 0 . d ° o O CO s s S o o CO CO S >o o o 2 o o o o 2 o o o o 2 o o o o 2 o o o 0 2 0 0 0 o o o o o o o o o d 0 J □ □ ■ a □ S 2§ 8?° ■ 2 8 28 ■ ■ S'8 □ CD 1ö CD n >0 >co CD 0 f-i CÖ p CD rH co ■ CÖ TJ CD f-l f-l CÖ O > ft ft CD P ^ CO f-i PP Ti O H ft O ^ '00 fH M—1 O CD CB 1 S '"O £ O bi O i-M '"O CD CD Ö CD CÖ * CD Ö CD a a f-l Q o: ^ CD f-i ^ O co O Ö "rH CD a C^ a Q O co CÖ rH rH co CD O CD > • i-H f-l p bi CD O CO E >co CD CD f-i 0 Q CÖ rH Ö CD f-l rH CD 1 0 CD fH CD x,y - 4>y,x) (P3Q element) 1y ^ = {x,y - 4>y,x) Figure 3.4.4: FE solution of the uniformly loaded hard-soft simply supported square plate with P3 plate elements (mesh as in Figure 3.4.3) Slika 3.4.4: Računska rešitev enakomerno obremenjene togo-mehko podprte kvadratne plošče s P3 končnimi elementi (mreža, kot je prikazana na Sliki 3.4.3) o ® H S O z v a ISi - > F H ^ Š > t l-H 0 r o a t < > o z ^ > s o c: g /h M E %% G H H TJ H f C? ° > £2 > š O g > a w m o h z H S < S o 3 o -0.5 ^ -1 £ 3 ^ -1.5 c -2 -2.5 . P3T ■ P3Q PIT » PIQ 1.5 2 2.5 3 3.5 log (■ndof) (Displacement w at center) -0.5 -1 -1.5 -2 -2.5 -3 *—*— -A A A A A s*-*—♦ ♦ ♦ ♦ ♦ ♦♦ . P3T ■ P3Q PIT » PIQ 1.5 2 2.5 3 3.5 log (ndof) (Moment resultant mxx at center) (Energy norm) Figure 3.4.5: Comparison of the convergence of FE solutions for the uniformly loaded hard-soft simply supported square plate Slika 3.4.5: Primerjava konvergence racunske resitve problema enakomerno obremenjene togo-mehko podprte kvadratne plosce qx Lyy qy ^ (ßy^y H- w 0x 6y 7T1-XX myy mxy qx qy ■ \ ~ -0.001 -0.005 15. -3. -1.5 -6. -1.5 0.00125 -0.000833 -0.00417 13.8 -2.75 -1.25 -5. -1.25 ■ 0.00125 -0.000833 -0.00417 -13.8 -2.75 -1.25 -5. -1.25 0.0025 -0.000667 -0.00333 -12.5 -2.5 -1. -4. -1. ■ 0.0025 -0.000667 -0.00333 -12.5 -2.5 -1. -4. -1. 0.00375 -0.0005 -0.0025 -11.3 -2.25 -0.75 -3. -0.75 ■ 0.00375 -0.0005 -0.0025 -11.3 -2.25 -0.75 -3. -0.75 0.005 -0.000333 -0.00167 -10. -2. -0.5 -2. -0.5 ■ 0.005 -0.000333 -0.00167 -10. -2. -0.5 -2. -0.5 0.00625 -0.000167 -0.000833 -8.75 -1.75 -0.25 -1. -0.25 □ 0.00625 -0.000167 -0.000833 -8.75 -1.75 -0.25 -1. -0.25 0.0075 0. 0. -7.5 -1.5 0. 0. 0. □ 0.0075 0. 0. -7.5 -1.5 0. 0. 0. 0.00875 0.000167 0.000833 -6.25 -1.25 0.25 1. 0.25 □ 0.00875 0.000167 0.000833 -6.25 -1.25 0.25 1. 0.25 01)1 0.000333 0.00167 -5. -1. 0.5 2. 0.5 □ 0.01 0.000333 0.00167 -5. -1. 0.5 2. 0.5 0.0113 0.0005 0.0025 -3.75 -0.75 0.75 3. 0.75 □ 0.0113 0.0005 0.0025 -3.75 -0.75 0.75 3. 0.75 0.0125 0.000667 0.00333 -2.5 -0.5 1. 4. 1. □ 0.0125 0.000667 0.00333 -2.5 -0.5 1. 4. 1. 0.0138 0.000833 0.00417 -1.25 -0.25 1.25 5. 1.25 ■ 0.0138 0.000833 0.00417 -1.25 -0.25 1.25 5. 1.25 0.015 0.001 0.005 0. 0. 1.5 6. 1.5 Figure 3.4.7: Reference solution of the uniformly loaded hard simply supported-free square plate with legend also valid for Figures 3.4.9 and 3.4.8 Slika 3.4.7: Referenčna rešitev enakomerno togo prosto podprte - proste kvadratne plošče z legendo, ki velja tudi za Sliki 3.4.9 in 3.4.8 w o tß K f-H > hH 0 - S Cd E < O > > C t"1 V K % H N O < O S > K o S M O u-i H K O O o -vT qx ///AW yy qy FE mesh (PIT element) M M yy /g (PIQ element) Figure 3.4.8: FE solutions of the uniformly loaded hard simply supported-free square plate with PI plate elements Slika 3.4.8: Računska rešitev enakomerno togo prosto podprte - proste kvadratne plošče s PI končnimi elementi qx qx (P3T element) qy K> r ' " \ \ V F B ^ Š > t l-H 0 r o a t < > o z ^ > s o c: g E %% G H H 13 B f C? ° > £2 O < o s > m a w dO o b z ISi H H S < S o 3 o o (Displacement w at center) -1 -1.5 -2 -2.5 . P3T ■ P3Q PIT » PIQ (Moment resultant mxx at center) -1.5 -2 -2.5 bO —3 —3.5 -4 1.5 2 2.5 3 3.5 log (■ndof) . P3T ■ P3Q PIT » PIQ 1.5 2 2.5 3 3.5 log (■ndof) (Energy norm) Figure 3.4.10: Comparison of the convergence of FE solutions for the uniformly loaded hard simply supported-free square plate Slika 3.4.10: Primerjava konvergence racunske resitve problema enakomerno togo prosto podprte - proste kvadratne plosce w ex 6y 772 x x myy mxy qx Qy Ü ■ 0. -0.0003 -0.000145 —1.5 -2.2 -0.5 -1.5 -3. -0.00011 0.00005 -0.00025 -0.000121 — L 38 -K 98 —0^417 -L25 -2.5 -0.0000917 □ 0.00005 0.0001 -0.00025 -0.0002 -0.000121 -0.0000967 -1.38 -L25 -1.98 -K 75 -0.417 —0?333 -1.25 -1. -2.5 —2. -0.0000917 -0.0000733 ■ 0.0001 0.00015 -0.0002 -0.00015 -0.0000967 -0.0000725 -1.25 -L13 -1.75 -K 53 -0.333 -a 25 -1. -(175 -2. -1.5 -0.0000733 -0.000055 ■ 0.00015 -0.00015 -0.0000725 -1.13 -1.53 -0.25 -0.75 -1.5 -0.000055 0.0002 -0.0001 -0.0000483 -1. —T. 3 -0J67 -0.5 -1. -0.0000367 ■ 0.0002 -0.0001 -0.0000483 -1. -1.3 -0.167 -0.5 -1. -0.0000367 0.00025 -0.00005 -0.0000242 —0J375 -K08 -0.0833 -(125 -0.5 -0.0000183 □ 0.00025 -0.00005 -0.0000242 -0.875 -1.08 -0.0833 -0.25 -0.5 -0.0000183 0.0003 0. 0. -a 75 -a 85 0. a 0. 0. □ 0.0003 0. 0. -0.75 -0.85 0. 0. 0. 0. 0.00035 0.00005 0.0000242 —0?625 —0?625 0.0833 0^25 (15 0.0000183 □ 0.00035 0.00005 0.0000242 -0.625 -0.625 0.0833 0.25 0.5 0.0000183 0.0004 0.0001 0.0000483 -0.5 -0.4 0.167 0^5 1. 0.0000367 □ 0.0004 0.0001 0.0000483 -0.5 -0.4 0.167 0.5 1. 0.0000367 0.00045 0.00015 0.0000725 -0^375 -0T75 0^25 0^75 L5 0.000055 □ 0.00045 0.00015 0.0000725 -0.375 -0.175 0.25 0.75 1.5 0.000055 0.0005 0.0002 0.0000967 -a 25 om 0.333 L 2. 0.0000733 □ 0.0005 0.0002 0.0000967 -0.25 0.05 0.333 1. 2. 0.0000733 0.00055 0.00025 0.000121 -0T25 0.275 0.417 1^25 2.Ö 0.0000917 ■ 0.00055 0.00025 0.000121 -0.125 0.275 0.417 1.25 2.5 0.0000917 0.0006 0.0003 0.000145 0. (15 CL5 K5 .1 0.00011 Figure 3.4.12: Reference solution of the uniformly loaded soft simply supported skew plate with legend also valid for Figures 3.4.14 and 3.4.13 Slika 3.4.12: Referencna resitev enakomerno obremenjene mehko podprte romboidne plosCe z legendo, ki velja tudi za Sliki 3.4.14 in 3.4.13 (PIT element) (PIQ element) Figure 3.4.13: FE solution of the uniformly loaded soft simply supported skew plate with PI plate elements Slika 3.4.13: Racunska resitev enakomerno obremenjene togo podprte romboidne plosce s PI koncnimi elementi (P3T element) (P3Q element) Figure 3.4.14: FE solution of the uniformly loaded soft simply supported skew plate with P3 plate elements (mesh as in Figure 3.4.13) Slika 3.4.14: Racunska resitev enakomerno obremenjene togo podprte romboidne plosce s P3 koncnimi elementi (mreža, kot je prikazana na Sliki 3.4.13) 1.5 2 2.5 3 3.5 4 log (■ndof) (Displacement w at center) 1.5 2 2.5 3 3.5 4 log (■ndof) (Moment resultant mxx at center) 1.5 2 2.5 3 3.5 4 log ('ridof) (Energy norm) Figure 3.4.15: Comparison of the convergence of FE solutions for the uniformly loaded soft simply supported skew plate Slika 3.4.15: Primerjava konvergence racunske resitve problema enakomerno obremenjene mehko podprte romboidne plosce CD n CD & l-J CD >0 >co CD O fH H O a l-J a CD co fH ft o ft .d o CO ^ CD m a '"O CD CD rH CD CÖ o CD d oc CD H T-1 fH ^ , O o M—1 'd d co Ü o (H fH CD CD i-d H o ^ Q ^ CÖ (H d o P co SH CD co CD fH d > CD .""K Q m bi '>03 CD & E fH O CÖ d Q CD ^H >0 fn & d CD & i fH x,y ~ ' (P3Q element) qx ^ — 0x,y ~ Figure 3.4.19: FE solution of the uniformly loaded soft simply supported L-shaped plate with P3 plate elements (mesh as in Figure 3.4.18) Slika 3.4.19: Računska rešitev enakomerno obremenjene mehko podprte L plošče s P3 končnimi elementi (mreža, kot je prikazana na Sliki 3.4.18) o ® H S O z v a ISi - * (-1 > F > l-H ° r o a t < > o > s o C g M G H H TJ B f C? ° > £2 O < o s > m a w m o b z o -a -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 : . P3T ■ P3Q PIT » PIQ 2.5 3 3.5 log (ndof) (Displacement w at center) -0.75 F : 2 2.5 3 log (ndof) (Moment resultant mxx at center) -0.5 -1 -1.5 -2 ■ . P3T ■ P3Q PIT » PIQ 2 2.5 3 3.5 4 log (ndof) (Energy norm) Figure 3.4.20: Comparison of the convergence of FE solutions for the uniformly loaded soft simply supported L-shaped plate Slika 3.4.20: Primerjava konvergence racunske resitve problema enakomerno obremenjene mehko podprte L plosce 3.5 Chapter summary and conclusions A Reissner/Mindlin plate theory, which covers the behavior of moderately thick plates is presented. The Kirchhoff assumption is abandoned as the theory takes into account the transverse shear: the midsurface deflection of the plate is independent of rotation of normal fibers. In the first section the basic equations are derived. The shear correction factor is treated alongside for the sake of completeness. There are three coupled differential equations of sixth order for midsurface deflection w and two components of rotation 0x, which govern the deformation of the moderately thick plates. Since the hierarchic relation of the Reissner/Mindlin theory to the Kirchhoff theory is followed throughout the derivation, one of the equations can be presented as the modified biharmonic equation. This hierarchical treatment is advantageous since it enables us to clearly identify the novelties. The Reissner/Mindlin theory is able to describe a larger set of boundary conditions. There are three independent components of boundary traction loading: normal shear, twisting and bending moment. However, being closer to the full three-dimensional theory, and thus able to describe more flexible plate structures, the Reissner/Mindlin plates do not handle concentrated loads - contrary to the Kirchhoff plates. The introduction of transverse shear triggers some interesting phenomena. Most notable is the boundary layer effect, which concern the areas near the plate boundary, where steep gradients of stress resultants exist. The influence of boundary layer effects is limited to the distance of the order of plate thickness into the plate interior. The existence and strength of boundary layers is related to the type of boundary conditions imposed. Maybe the most counter-intuitive fact concerning the boundary layers is, that it is the most prominent along the free boundary of the plate. The singularities, already encountered in the Kirchhoff theory of thin plates, exist also in the Reissner/Mindlin theory. The second section deals with the development of the finite element based on Reiss-ner/Mindlin theory. An alternative notation enables the presentation of the finite elements as the hierarchical extension of the DK thin plate elements. Additional degrees of freedom at the element mid-sides are directly related to the projection of the transverse shear on the direction of the element side. An analysis of the stiffness matrix of the element shows that the part related to the shear strain depends only on the mid-side degrees of freedom. Equivalent formulation of triangular and quadrilateral elements is possible due to the alternative element notation. A variation of the elements is also presented. Here the additional degrees of freedom are related to incompatible deformation modes, and are thus subject to elimination by means of static condensation. The chapter concludes with several typical plate problems, which illustrate the specific issues of moderately thick plate modeling including singularities and boundary layers. The performance of the presented finite elements is assessed in terms of the convergence rates of various selected quantities. The functionality of element implementation is tested through the comparison to the reference solutions. An interesting observation is made, regarding the performance of the elements with the incompatible modes. Although the elements are able to capture the transverse shear, they fail to detect the boundary layers. Chapter 4 Thick plates: theory and finite element formulations 4.1 Introduction A logical successor of the Reissner/Mindlin model in the hierarchy of the models for bending of elastic isotropic plates is a higher-order plate model which includes through the thickness stretching. The higher-order plate model, considered in this chapter is a special case of a shell model presented in [Brank, 2005] and [Brank et al., 2008]. It treats a plate as a 2d surface in a full 3d stress state. This is advantageous, since the usual modification of 3d constitutive equations due to the plane stress assumption (see chapters 2 and 3) is omitted, resulting in a simpler implementation of complex material models for example hiperelasticity or elasto-plasticity. Also the boundary layer effect, which is full 3d effect, is much better represented by the higher-order plate models as shown below. The considered higher-order plate model fits between the Reisnner/Mindlin model and the 3d solid model, which makes it an ideal choice for the model adaptivity of plates. In this chapter we will thus present a new higher-order plate theory and a new higherorder plate elements based on that theory. The elements will be a direct refinement of the Reissner/Mindlin finite elements with cubic interpolation of displacement and linked interpolation of displacement and rotations, denoted in previous chapter as P3T and P3Q. t = (1 + £{zJ)t t w -Kzz (t/2)2 z Figure 4.2.1: Kinematics of through-the-thickness deformation in the thick plate model Slika 4.2.1: Kinematika deformacije vzdolž debeline ploSCe v modelu debelih ploSC 4.2 Theory We start by abandoning the assumption of zero through the thickness stress: ozz = 0. In the case of 3d linear elasticity we thus have: Ozz = (A + 2ß)£zz + A(£xx + £yy) = 0 (4.2.1) with A = (1+vV(E-2v), ß = 2(i+v) as Lame coefficients. Nevertheless in the limit t ^ 0 the plane stress assumption ozz = 0 must be satisfied resulting in the thickness stretching £zz: = - A ( ) £zz a + 2ß (£xx + £yy) This kind of limit behavior can be achieved only if the strains £xx, £yy, £zz are of the same order with respect to z coordinate. In such a case they can cancel each other in (4.2.1). If for instance £xx, £yy are linear with respect to z coordinate, so must be £zz. The linear variation of £zz with z also implies that the transverse displacement uz should be at least quadratic function of z since £zz = ^^. 4.2.1 Governing equations Kinematic equations The following kinematics of plate displacement is therefore assumed: u = uRM + uS where uRM denotes kinematics of Reissner/Mindlin model uRM = wnn - z0; nn = [0,0,1]T and uS defines the displacements that produce the through-the-thickness stretch (see Figure 4.2.1) uS = ((t/t - 1)z - Kzzz2/2) nn = {e^z - £^z2/t) nn (4.2.2) The uS is a quadratic function of z coordinate; i is the thickness of the deformed plate, t is the thickness of the plate prior to deformation, kzz = £(z 2/t is the parameter of the stretch and nn the normal to the midsurface of the undeformed plate. The uS direction is limited to the direction of nn. We denote the relative thickness change by eZZ^ = i/t - 1. The components of displacement w = [ux ,uy,uz]T are: ux = -z0x; uy = -z0y; uz = w + eZZ^ z - eZZ^z2/t (4.2.3) and the resulting strains are £xx zKxx; £yy = -zKyy; £zz ezz - 2zezz^/t £zz - zKzz; where the definitions from chapters 2 and 3 were employed Yx = w,x - 0x (4.2.5) Yy = w,y - 0y Additionally, using the analogy with the deformation terms due to bending, which vary with z, the curvature kzz was introduced in (4.2.4): 2£xy zKxy (4.2.4) 2£ = Y + £(0) z - £(1) z2 /t 2£xz fx + £zz,xz £zz,xz 2£ = Y + £(0) z - £(1) z2 /t 2tyz = fy + £zz,yz tzz,yz Kxx 0x,x; Kyy = 0y,y; K xy x,y 0x,y + 0 y,x Kzz = £ziz)/(t/2) (4.2.6) Constitutive equations The Hooke's relations for the 3d linear elastic material are: Gxx (A + 2ß)e xx + A^yy + A^zz; Gyy = (A + 2ß)eyy + Aexx + A^zz; G zz = (A + 2ß)£zz + A^xx + A^yy; through nzz = t(A + 2ß)£z°z) = Cz £z°i; Cz = t(A + 2ß) In matrix notation the constitutive matrix is (4.2.13) C B " (A + 2ß) A 0 A t3 A (A + 2ß) 0 A 12 0 0 ß 0 A A 0 (A + 2ß) (4.2.14) which allows for the relations (4.2.12) to be expressed in a compact form: m = C B K (4.2.15) In order to make the the theory more comparable to the Reissner/Mindlin one, we decide to make the following decomposition of kzz : A , -v Kz Kzz + K^zz ; kKzz = A + 2ß (kxx + Kyy) 1 - V (kxx + Kyy) (4.2.16) n Since we have (0) -zKzz £zz = ^z) zKzz = ^z) ^ + 2^ (£xx + £yy) zkKz:z (4.2.17) £ziz) = (t/2)Kzz =(t/2)(Kz°z) + kz1z)) the decomposition leads to plane stress assumption azz = 0, if £z!) = 0 and kH) = 0. According to (4.2.3), the kinematics of stretch becomes: uz = w + £z°z)z + 2(Kxx + Kyy)z2 - 2K^z2 (4.2.18) By using (4.2.18), eq. (4.2.11) can be simplified to: otf = / ÖKTmdn = [ öKTCbKdQ = önf'rm + önKf'A (4.2.19) Jn Jn önint,RM = ökCb K dft; ön^ = ökziz)Cb,zz K^dn nn where k = [kxx, Kyy, Kxy]T and CB is the constitutive matrix as defined in (2.2.13). Additionally we have Cb,zz = i2(Jf+3v()1--)2v) = D (I^Ž), where D = . Next, the expanded shear strain vector is introduced, see (4.2.4) 7=[Yx,Yy ,Yxi),Yyi),Yx2),Yy2)]T (4.2.20) where [2£xz, 2£yz]T = 7 + z7(1) + y7(2); t = z/(t/2) (4.2.21) [fx, Yy]T = 7 = Vw - 0 [f^T = 7(1) = (t/2)V£z0i bf^T = 7(2) = - (t/2) V£zz) The components of the shear force vector q = [qx, qy, qy1), qr2), qy2)]T are obtained from (4.2.9) as: q = cpt (7 + 7(2)/6) ; q(1) = cpt (7(1)/3) ; q(2) = cpt (7/6 + 7(2)/20) (4.2.22) where axz = cp 2£xz; ayz = cp 2£yz (4.2.23) was used and the shear correction factor is denoted by c (which should have different value than 5/6, derived in chapter 3). They can also be expressed in a compact form: q = C s Y (4.2.24) where the block matrix CS is defined as C S = Cßt ^2x2 0 11 0 61 3-2x2 6 -2x2 0 L 612x2 - 20 0 2012X2 (4.2.25) and 12x2 denotes the 2 x 2 identity matrix. The term contributing to the virtual work (second term on the rhs of (4.2.11)) is thus: sn int Y sqT dn = snnt'RM + čn;nt'A (4.2.26) where and SR int,RM Y syt cs y dft; sny SYT C S 7dn (4.2.27) SY = [S7® j S7y] ; S7=[S7x,S7y ,S7X1),S7yi),S7X2),S7y2)] T (4.2.28) (4.2.29) C S = cßt 0 0 11 0 61 3—2x2 g —2x2 0 L 612x2 - 20 0 2012X2 (4.2.30) With the notation introduced above, the virtual work of internal forces (4.2.11) is expressed hierarchically as: snint =snr + snYnt + snzn = snint>RM + snnt'A + snnt'A + sn pnt int int _ XTlint.RM jint, A Tint, A Y int zz (4.2.31) where sn int.RM (skt c b k + s7T cs 7) d^ = snnt'RM + snynt'rm (4.2.32) and SnYnt'A is given in (4.2.28), SnKnt'A is given in (4.2.8) and SnZf is given in (4.2.11). Y n n n n Virtual work of external forces The external forces (boundary tractions) act on the boundary surface(s) of the plate: the upper and bottom surfaces of the plate (commonly indicated by and Q-) and the side surface of the plate (indicated by r): snext = 8next>n + 8next>r (4.2.33) The tractions acting on the side of the plate t are assumed to be constant over the plate thickness, therefore the traction load can be integrated through the thickness and presented in form of normal moment mn, twisting moment ms and shear force qn: , ^/2 8next'r(8w,80) = 8uTt dz ds = (4.2.34) Jy J-t/2 J (8wqn + 89nmn + 89sms)ds = J (8wqn + 80 m) ds Since the top (+) and bottom (-) of the plate are distinguished in the higher-order model, the distributed surface load can in general act on one or both surfaces (note that , (0) i 2\ uz = w + eZz z - 2 Kzzz2) 8next'n(8w,84°z),8Kzz )=/ f+8u+ dQ+ / f-8u- dQ+= (4.2.35) Jn+ Jn- t „ ^ t2 „ s ^ f . t „ ^ t2 j f+(8w + 28ez°z - -8kzz)dQ + / f-(8w - 28ez°) - -8nzz)dQ 2 zz 8 zz n- 2 zz 8 4.3 Finite elements 4.3.1 Higher-order plate elements Since our aim is to build a hierarchic family of plate finite elements (in the model sense), the higher-order finite elements are built on the basis of P3 elements derived in chapter 3. They are designated by PZ. Discretization The transverse displacement w and the rotations 0 are interpolated using the same discretization as in the case of P3 elements. Additionally, the fields ez°z) and are to be discretized. A bilinear discretization of both fields is assumed: J0) zz,h E / N r(o) • N/ £zz,/• K (1) zz,h E / Ni KZZ^i (4.3.1) where eZZ,/ and &ZZ, are the additional nodal degrees of freedom. The vertex degrees of freedom of the PZ element are thus ii/ = [w/, , ,eZ'Z)/, kZZ)/]T and the midside degrees of freedom are ii/j = [y/j]. The elements have nen vertex nodes and nen midside nodes (nen=4 for the quadrilateral, and nen = 3 for triangular elements). The vertex nodes are numbered as I = 1,..., nen, the midside nodes as IJ = (nen + 1),..., 2 nen. Element degrees of freedom are arranged into u = [uf, ■ ■ ■ , ULn+D ''' , UTnen]T. The interpolations can be expressed as: Kh = bk,/u/ + b K,/J11/J • i ij 4°z!h = 5] b£(o; ,/u / • i KZZ),h = bkM ,/u/• i where Yh = B7,/Ju/j• (4.3.2) /j Yh1) = ^ B7(D ,//• / Yh2) = BY(2) ,// + S BY(2) ,/juu / /j Kh = [k h] • Yh = [Yx,h,Yy,h] lT Yh1) dp(0) dp1 (0) T (2) Yh ^zz^h dezz,h .(1) T (4.3.3) t The following definitions apply: bk,i Bk,ij B (i) j k«« ,I By1 ,I BY(2) i [B Kxx,I , BKyy,I, B KXy ,1^ = [BP,/, 03x2] (4.3.4) Bp}j; bY,IJ = [0, 0, 0, 0, Ni]; Be«,) I 000 (t/2) >P 3 [0, 0, 0, NI, 0] 0 0 0 Nix 0 0 0 0 N^y 0 B(°) + B( B7(2),i + B7(2),I »(1) B (°) /(2) ,I B (1) /(2) ,I 1 - v -(t/2)2 (t/2)2 dx (B^xx,i ++BB dy (B Kxx,i + Kyy j1) 0 0 0 0 Nix 0 0 0 0 Niy by(2),ij 1v (t/2)2 dx (BKxx,ij+B dy (BKxx,iJ + B Kyy,IJ ) Kyy,IJ ) Stiffness matrix The stiffness matrix K of PZ element is obtained by inserting the discretization (4.3.2) into (4.2.11) and taking into account the constitutive relations (4.2.13), (4.2.15) and (4.2.24). It consists of three parts, which correspond to bending stiffness, shear stiffness and the through-the-thickness stretching stiffness: K = Kb + KS + KZ (4.3.5) The stiffness matrix is expressed as a block matrix K = [Kj], where blocks Kij correspond to the interaction of degrees of freedom of nodes i and j. The stiffness matrix is expressed as a block matrix K = [Kj], where blocks Kij correspond to the interaction of degrees of freedom of nodes i and j. The indexes referring to the vertex node are denoted by I and J. The indexes IJ and K L are used to denote the midside nodes. The stiffness matrix is then organized as: K [Ki,j] [kikl] [Kijjj ] [kijkl] (4.3.6) ) v v According to the (4.2.19), the bending stiffness KB is a sum of two components KB = Kb'° + KB' j j 1 j where Kj = J B^CbBäj. dQ; K? = J ^m^zzB^, dQ (4.3.7) Notice that, since BK / = [B0, 03x2], the matrix KJ corresponds to the bending stiffness matrix of P3 elements. K B,° bk,/cb bk,j dq kb,p3 0 K/;J 03x2 02 3 02 2 (4.3.8) Since the interpolation of k^*1 is independent of all degrees of freedom other than k^^ the deformation mode kÜ) is independent of bending. The block matrix K^fJ is thus: K B,A 1,J -»4x4 ■Mxl 01x4 Km (4.3.9) where K (iu j = / BT(i) CbzzB (i) , dQ KzZ , I , J Jn , I B ' Kzz ,J Cb , zz N Nj dQ (4.3.10) Similarly, from (4.2.26), (4.2.27), we can decompose the shear part of the stiffness matrix into KS = KS'° + Ks'a where (according to (4.2.26), (4.3.2)) K j = BT,iCSBYJ dQ (4.3.11) and KSj'A =3 Jn BT(i)'iCsBY(i)'j dQ+ 20 i BT(2)'iCsBY(2)'j dQ+ 1 f ^ „ _ _ 1 9 I BTiCsB,®, dQ + - I BT(2) iCsB7 j dQ 6 n 6 n The part KS'A corresponds to the contribution of the through the thickness stretching to the shear deformation. n t n Notice, that the block matrix KsJ0kl corresponds to the shear stiffness matrix of P3 elements (4.3.2) K S,0 IJ,KL B7,IJ Cs BY,KLd0 B7,IJ CSB7,KLdO — KJ,KL (4.3.12) and Ks,J — °5x5; kS'K L — °5xi; ksJ0J — 0lx5 The part of the stiffness matrix, which directly corresponds to the through the thickness stretching KZ is a block matrix K BT(0) rczb (0) d0; kZj,j — 0ix5; kZkL — °5xi; kZJ,KL — Olxl ■I,J ~ I " (0) (0) J ' Jn £zz J zz 'J Load vector Figure 4.3.1: Designation of plate surfaces Q+, Q and r Slika 4.3.1: Oznaka povrSin Q+, Q- in r n n e t Z The load vector f is revealed by inserting the discretization (4.3.2) into (4.2.34) and (4.2.35). Since we distinguish between the top and bottom surface (see Figure 4.3.1) and , the load vector consists of three parts f — ff + + ff- + f where f ++2 ff± — f ± (Bw ± -Be(0) + -Bkzz)dO; (4.3.13) Jne 2 £zz 8 f — f (i + 6>J)) Using the given kinematic assumptions, the interpolation of curvature is computed directly through _[_9dy,h ddx,h ddx,h 9dy,h ] (4 4 2) dx ' dy ' dx dy ' ' From the interpolation of rotation 0DK, we have kdk = £ BDK ni; BDK (4 . 4 . 3) I The transverse shear is neglected; bending is the only deformation mode. P3 plate element In addition to the nen vertex nodes with the degrees of freedom üI = [wI, dx>I, 0yj]T, the element has nen midside nodes with the degrees of freedom nIJ = [7IJ]T. The interpolations are 1 wP3 = -L/j 1/jM/j• (4.4.4) /j 4 #3 = + £ 3n/j 1/jN/j 2 /j The curvature interpolation Kp3 is defined by the rotation 6t P3 h KP 3 = k£k + K h Kh = £ B k,/j u/j • B «,/j (4.4.5) /j The interpolation of the transverse shear is: yP3 = £ N/Yh,/ (4.4.6) / where the nodal values Yh)/ depend on the midside degrees of freedom Yj according to Yh/ = n/J - 1/Jn/K (4.4.7) n/j ■ s/K Note, that the interpolation of the transverse shear conform to the assumed kinematics only along the sides of the element. At the element interior, an assumption of the variation of the transverse shear strain is made. PZ plate element Element has nen vertex nodes with the degrees of freedom it/ = [wW/, , , , lZZ),/]T, and nen midside nodes with the degrees of freedom ii/J = [Yj]t. The kinematics of the material point is interpolated as = <3 + uf (4.4.8) u P3 = wP3nn - z^ uf = (42* - 4*iz2/0 nn nn = [0, 0,1]T• 0P3 = h||E l|eh||E = J] ||ee,h||| (5.2.10) e 5.3 Recovery based error estimates The recovery based error estimates are based on a rather heuristic basis. The main idea originates from the fact, that the stresses of the finite element solution are in general discontinuous across the interelement boundaries. The stresses of the exact solution are, however, continuous. By constructing smooth stress fields from the original finite element solution (by post-processing) a better estimate of the true stresses can be obtained. An estimate for the true error is obtained by comparing the post-processed stresses and the non post-processed stresses of the approximation. Let us denote the stresses obtained from the original finite element solution as ah and the exact stresses as a. The error (5.2.9) can then be expressed in terms of stresses only |eh||| = / (a - ah)TC-1(a - ah) dQ (5.3.1) JQh where the relations a(u — uh) = a(u) — a(uh), a = Cs and ah = Csh are taken into account, with C being the constitutive matrix. The exact stresses u are not known. Their approximation is tried to be reconstructed from uh by a postprocessing procedure. The recovered values, denoted by u*, are continuous stresses over the element U = £ u* N (5.3.2) i where i runs over the nodes of the element and, N* are shape functions defined on the patch pertaining to node i, denoted by P*. Nodal stresses are denoted as u*. The recovery based approximation of the error can thus be written as ||e*||| = / (u* - Uh)TC-1(u* - Uh)dQ (5.3.3) Jnh The integral in (5.3.3) can be decomposed into element contributions as l|e*||E = £ ne2; n2 = (u* - ufe)TC-1(u* - uh)dfi (5.3.4) e e jHh where the indicator of the global error ||e*||E is expressed as the sum of the local error indicators ne2. The primary goal of the recovery based error estimates becomes the determination of the enhanced stresses u* or rather their nodal values ui*. They can be determined by using the following argumentation: continuous approximation of the stress field u* should be as close as possible to the finite element approximation uh, therefore we seek for the function u* which minimize the functional 1 f (u* - ufc)2dft ^ min (5.3.5) 2 J H The minimization of (5.3.5) with respect to u* leads to f (u* - uh)N*dQ = 0 (5.3.6) Jn-p. ' i Finally, we obtain a system of equations for the u* in the form: Y,(l NNd^)u* = f uhNdn (5.3.7) j JHPi JHPi This problem is global, therefore the solution is as computationally expensive as the solution of the original problem. This is of course not a desired feature of the effective error estimation. 5.3.1 Lumped projection A more effective procedure for the computation of nodal values a* can be obtained by the diagonalization of (5.3.5). The left hand side of (5.3.7) is simplified to * a Nj)dHI = ahNidH (5.3.8) 'np. j / Jnp. by assuming a* = a*. Note, that the shape functions N are complete, therefore we have N = 1 and the nodal value a* can be explicitly computed from a* = — [ ahNidH (5.3.9) miJ np. ' i where m, = fn N,dH. Due to its simplicity, this procedure is implemented in various finite element codes to smooth the stress field prior to plotting. The procedure (5.3.8)-(5.3.9) is denoted as "lumped projection" in Figure 5.1.1. 5.3.2 Superconvergent patch recovery (SPR) The SPR procedure takes advantage of the superconvergence property of the finite element solution (e.g. see [Zienkiewicz Taylor, 2000], [Ibrahimbegovic, 2009]). Namely, the solution error in displacements is locally the smallest at the nodes of the element, whereas the error in displacement gradients (or stresses) is smallest not at nodes but at the points inside the element usually coinciding with the Gauss points. At such points the order of convergence of the gradients is at least one order higher than the one that would be anticipated from the approximation polynomial. This is known as the superconvergence property. The values of stresses sampled in Gauss points (or other optimal sampling points) are denoted by ah(£gp). These values serve as the basis for the reconstruction of the recovered stress field a*, which will, due to the superconvergent property of sampling points, provide the best approximation of exact solution a. In the neighborhood of a particular node i each component of a* is interpolated by a* = afcp (5.3.10) where, for example for a 2D problem p = = [1,x,y, ••• ,yp]T (5.3.11) afc = [afcl) afc2, • • • j afcm] and x = x — xi,y = y — yi with xi = (xi,yi) denoting the coordinates of node i. The nodal parameters c* are taken simply as the value of C * at the node i c* = C *(Xi) (5.3.12) The parameters ak are determined using the least square fit of C * to the values ch(xgp) sampled in the ngp superconvergent points xgp = (xgp,ygp) over the patch of elements surrounding node i. For each stress component k we have 1 ngp 2 YI (xgp) — ^fc(xgp))2 1-► (5.3.13) gp=1 This leads to a small set of linear algebraic equations for the coefficients ak Mak = b; M = [Mj ]; b = (5.3.14) Mij = YPi(xgp) Pj(xgp); bi = °"h,k(xgp)pi(xgp) gp gp The approximation c* is superconvergent in each Gauss integration point in a case of regular mesh with rectangular elements. For the elements with distorted geometry the superconvergence is no longer exactly satisfied at the Gauss points. However, the procedure gives reasonably good approximation even in such cases. An overview of the SPR procedure is given in Box 1. Box 1 : The superconvergent patch recovery - SPR Algoritem 1: SPR metoda 1. Obtain original finite element solution uh of the problem (5.2.4) and obtain ch(xgp) 2. Chose interpolation (5.3.10) of each a* in the neighborhood of node i 3. Evaluate [Mj] and [bi] from (5.3.14) 4. Solve (5.3.14) for ak 5. Define c* from (5.3.12) 6. Use the interpolation (5.3.2) to get c* Figure 5.3.1: SPR recovery of the enhanced stress a*j at node I Slika 5.3.1: IzraCun izboljSanih napetosti a*j v vozliSCih I po metodi SPR 5.4 Residual based error estimates 5.4.1 Explicit One can find an estimate for the global error in the energy norm (5.2.9) by explicit residual based error estimates. They are based on the the direct computation of the interior element residuals and the jumps at the element boundaries. The basic idea of residual based error estimates comes from the observation that the residual R(uh, v) = a(uh, v) — l(v) is directly related to the error eh. From (5.2.1) and (5.2.7) if follows that the error satisfies the equation a(eh, v) = l(v) — a(uh, v); vv g vo (5.4.1) where v belongs to the test space and it holds true for an arbitrarily chosen test function v G V0. The integrals in (5.4.1) are split into element contributions: a(eh, v) = £|/e(v) - ae(uh, v)}; Vv G V0 (5.4.2) e The forms le and ae are: ae(uh, v) = f [a(uh)] : [e(v)]dH (5.4.3) 'ne = ([a(uh)]n) • v dr - (V^ [a(uh)]) • v dH Jre Jne le(v) = f v • t dr+ / f • v dH (5.4.4) JreN Jne where the Gauss statement was used to perform integration per partes in (5.4.3). The divergence of stress tensor [a] is denoted by V • [a] and the normal to the element side is denoted by n. The sides with Neumann boundary conditions are denoted by (re = rN U rD). Using the definitions (5.4.3) and (5.4.4), the expression (5.4.2) transforms to a (eh, v) = V f (f + V • [a(uh)])dH (5.4.5) e Jne + V/ (t - [a(uh)]n) • v dH e JrN + W ([a(uh)]n) • v dr e J^b With the rearrangement of the summation it is possible to split the residual (5.4.5) into element and side contributions: (eh, v) = £/ Rne (uh) • v dH + £ /^K) • v dr (5.4.6) op d ne ti j r a(eh, v) = ^ Rne(uh) • v dH + ^ / Rr(uh) • v ne d ne r ^ r |[a(uh)]n] vr G re Rne = f + V • [a(uh)]; Rr ={ t - [a(uh)]n Vr G rew (5.4.7) 0 vr g rD The jump term, denoted by [• ], is related to the edge r, adjacent to elements e, and ej, and is defined as [a(uh) • n]r = [ae; (uh)] • ne; + [ae^ (uh)] • n, (5.4.8) Note that nei = — nej = n, therefore the jump term equals to Muh)] = ([0-ei (uh)] — K, (uh)]) ■ n The Clement operator nh maps the test functions v from V to Vh and builds all possible test functions from Vh: nh = V ^ Vh with nhv G Vh and vh = nhv (5.4.9) Due to Galerkin orthogonality (5.2.8), we may write (5.4.6) a(eh, v) = £ f Rne(uh) ■ (v — nhv) dQ + ^ I Rr(uh) ■ (v — ^v) dr (5.4.10) ^e J r ^ r After applying the Cauchy-Schwartz inequality element-wise for the integrals in (5.4.10), the residual is bounded by a (eh, v) h denote the solution of the weak form of the local boundary value problem for the error computation over the element e: öe(^e,h, v) = le(v) - «e(uh, v) +/ tf ■ v ds; Vv G Vo (5.4.21) Jre Note, that the local element problems (5.4.21) for unknown $e>h are of Neumann type, since only tractions tr are prescribed on element edges. Moreover, the boundary tractions tr are unknown and are yet to be constructed from the finite element solution uh, and in agreement with (5.4.20). In this manner, the global error computation (5.4.1) has been split into element contributions a(eh, v) = J] ae(tfe,h, v) (5.4.22) e With the use of the Cauchy-Schwartz inequality, the relation between global error eh and local solutions $e>h, the upper bound of a global error can be estimated Kill Uphill = £n2 (5.4.23) ee where the local error indicator ne is defined as ne2 = uphill (5.4.24) In the case when element boundary tractions tr coincide with the boundary tractions obtained from the exact solution for the right choice of v G V0, the estimated error is exact, since the exact solution to (5.4.18) can be recovered. It is thus obvious that the quality of approximation of element boundary tractions is critical for the effective error estimate. In the following, an efficient method for building the approximation of true boundary tractions will be presented. We first see that the Galerkin orthogonality property (5.2.8) of the finite element solution uh through the use of (5.4.22) implies ae($e>h, vh) = 0. Then, by using (5.4.21), an additional constraint for the unknown boundary tractions tf can be constructed: / tr ■ vh ds = ae(uh, vh) - le(vh) (5.4.25) Jre The meaning of the equation (5.4.21) becomes more evident if $e>h is defined as the difference between element approximation to the exact solution üe>h, and the FE solution ue,h #e,h = ue,h - ue,h Due to linearity of bilinear form a, the local problem (5.4.21) for ue then becomes ae(ue,h, v) = 1e(v) + / tr ■ v ds; Vv0 G V (5.4.26) J re The result ae($e,h, vh) = 0 implies ae(ue,h, vh) = ae(ue,h, vh). It is thus clear that ue,h = ue,h, when the subspace used to obtain a local solution of (5.4.26) is identical to the original subspace Vh, which was used to obtain the global finite element solution. Therefore, to be able to get an error estimate from (5.4.26), an enhanced test space should be used with vh+ G V. The enhanced space deformation modes should also include rigid body modes va, for which it holds ae(uh, vA) = 0. In this case, the result in (5.4.25) will reduce to / tr ■ va ds + 1e(vA) = 0 (5.4.27) Jre By closer inspection of the expression (5.4.27), the equilibrium of the boundary tractions with the external loading can be deduced in the sense of virtual work principle. An estimate of the discretization error can then be obtained through the solution of local problems (5.4.26) of Neumann type where V is replaced by its best possible approximation Vh+ that must be sufficiently enhanced with respect to Vh ae(ue,h+, vh+ ) = 1e(vh+)+/ tr ■ vh+ ds; vh+ G Vh+ (5.4.28) Jre The loading of element problems consists of the sum of the external loading and the boundary tractions tf. They are referred to as equilibrated boundary tractions, due to the property (5.4.27) which implies that they are in equilibrium with the external loading. In a sense, the equilibrated boundary tractions are introduced to represent the action of surrounding elements on the selected element. Introduced boundary tractions are continuous across element edges. In summary, the main steps involved in the error estimation procedure are given in Box 2. Computation of equilibrated boundary tractions for EqR The performance of element residual method depends crucially on the estimation of equilibrated boundary tractions (see Box 2). In the following, a general procedure for the computation will be presented, which follows the work of [Ladeveze Leguillon, 1983]. Box 2 : The equilibrated residual method - EqR Algoritem 2: EqR metoda 1. Obtain original finite element solution uh of the problem (5.2.4) 2. Compute equilibrated boundary tractions tf for each edge r C re of each element e (see Box 3) 3. Approximately solve (5.4.28) for Ue,h+ by using test space Vh+ as enhanced approximation to Vh 4. Compute local error estimates n^ from (5.4.24) and $e>h = Ue,h+ — ue,h Recall that j-th component of the test function vh is a linear combination of shape functions Nj,k and nodal degrees of freedom Vk. The interpolation of j-th component of vh along an edge r is thus (vh|r)j = £ Nj,kvk = £ NK • Vk (5.4.29) K The alternative notation (NK; VK) refers to a set of the shape functions and interpolation parameters which pertain in any manner to node K. Specifically, we have VK = [Vi)K]T and NjK = [NK]t with i =1... nndofK, where nndofK is number of degrees of freedom at node K. The virtual work of boundary tractions on edge r, spanning between nodes I and J, can then be reduced to1 : J tr • Vh ds = re,r • V/ + rJr • Vj, (5.4.30) where r/ r denotes the "projection" (in a virtual work sense) of boundary traction tf to node I. The components of r'e r are directly related to the components of V/. Thus, the number of components in re r matches the number of nodal degrees of freedom at node I. Let us assume that the boundary tractions tr variations are described with some chosen interpolation. Regarding edge r, boundary tractions tr, can thus always be recovered from the projections re,r, rJ,r: re,r, rJ,r ^ tr (5.4.31) 1We reduce attention to the edge with two vertex nodes, although the edge with more nodes could be treated in the same manner /v» e /v» e ' /,r\ tr ------ J,r r=r u Figure 5.4.2: Relation between the tractions tr and their projections rf r Slika 5.4.2: Relacija med robno obtežbo tr in projekcijami r/,r We therefore decide to replace in what follows the unknown tractions tr with their projections rf,r. This will enable the patch-wise computation of boundary tractions tr. By using Rf to denote the element nodal residuals2, the right hand side of (5.4.25) can be rewritten as ae(uh, vh) - le(vh) = Rf ■ v/ (5.4.32) Moreover, the continuity condition for boundary tractions (5.4.20) can be rewritten as, see Figure 5.4.3: rf,r + rf,r = 0 (5.4.33) From (5.4.25), (5.4.30) and (5.4.32) we have (with a slight change in notation3), see Figure 5.4.4 rf.n + rf,r2 = Rf (5.4.34) where r and r2 are edges of the same element e meeting at node I. Gathering (5.4.33) and (5.4.34), we obtain a linear system of algebraic equations with the projections rf,r as unknowns. These systems are formed patchwise for the projections rf,r of the node I. To illustrate the patchwise formation of a local system of this kind we inspect an example with a patch of four 2D elements surrounding node I of the FE mesh (see Figure 5.4.5). Equations (5.4.34) and (5.4.33) for this case read (with a slight abuse of notation): Rf1 = rf1r4 + rfk ^ + rffFl = 0 (5.4.35) Rf2 = rf2ri + rf2r2 ^ + ^ = 0 Re3 = ^3 I ^3 ^3 I ^4 =0 = '/,r2 + '/,r3 '/,r3 + '/,r3 = 0 = '/,r3 + '/,r4 '/,r4 + '/,r4 = 0 2Element nodal residuals can also be interpreted as the element nodal reactions i.e. the forces acting on the nodes of the element. 3The edge r has sometimes subscript related to the index of the edge; e.g. ri, r2, ■ ■ ■ i,r2 Re r i,ri Figure 5.4.3: Element nodal residual and boundary tractions projections r| r12 at node I of element e Slika 5.4.3: VozliSCne reakcije elementa in projekcije robnih obtežb r|r12 v vozliSCu I elementa e e r /,r i,r e e e e Figure 5.4.4: Continuity of boundary tractions projections r|r Slika 5.4.4: Zveznost projekcij robnih obtežb r| r Note, that the equations (5.4.35) also imply the node equilibrium Rp = 0. Combining the last set of equations, we obtain the following system of equations with the only remaining unknowns refcrfc (5.4.36) Unfortunately, the local system in (5.4.36) is singular and its solution is not unique. To obtain the unique solution, a regularization has to be used. In the following, the regularization introduced in ([Stein Ohnimus, 1999], [Ainsworth Oden, 2000]) will be used. ■ RD ■ " +1 0 0 -1 re1 Re2 -1 +1 0 0 re2 Re3 0 -1 +1 0 re3 _ RD . 0 0 -1 +1 re4 r2 r ri r4 Figure 5.4.5: Patch P/ of four elements around node I Slika 5.4.5: Krpa P/ Štirih elementov okoli vozliSCa I Regularization of (5.4.33), (5.4.34) First, an alternative estimate for the boundary tractions which will be denoted as tp has to be constructed. Next, we use (5.4.30) to compute the projections ff,r: J tr ■ Vh ds = ff,r ■ V/ + f Jr ■ Vj, (5.4.37) The basic idea of the regularization is to seek for the projections rf,r which are as close as possible, in a least squares sense, to the projections ff,r. Hence, beside imposing the constraints (5.4.33) and (5.4.34), we will seek for the projections r|,r that minimize the objective function over a patch P/ around node I (a slight change in notation is introduced4) 2 E E far,* - re,pIK)2 min. (5.4.38) eePi Tjk The alternative boundary tractions ff are usually computed from the stresses oh(uh). Since the finite element stresses oh are discontinuous over the edges of the elements their continuity is first enforced. Simple averaging can be performed o |r = (oeh |r + oh |r)/2 (5.4.39) The alternative boundary tractions are then f = [o |r]nf (5.4.40) The procedure for the determination of the equilibrated boundary tractions is summarized in Box 3: Box 3 : Computation of equilibrated boundary tractions Algoritem 3: Uravnoteženje robnih obtežb 1. Compute element nodal residuals R from uh using (5.4.32) 2. Construct f from (5.4.40) and (5.4.39) 3. Compute alternative projections rf,r from (5.4.37) 4. Form patchwise systems of equations (5.4.35) and solve them for r|,r by (5.4.38) 5. Choose the interpolation —k, —J for boundary tractions tf and solve for nodal values of boundary tractions p^,/ from (5.4.43) 6. Compute boundary tractions by (5.4.40) Let us now elaborate on Step 5 in Box 2 and introduce parametrization of k-th component of the boundary traction tf (let r be element edge between element nodes I and J) as (tf )k = -k Pk,/ + -Jpk,j (5.4.41) where —k are interpolation functions (yet to be chosen) and pk>/ are the nodal values of (tf)k. The variation of virtual displacements along the edge r can be written as nndof, nndofj (vh,r)k = £ N&v + £ NJiVi,j (5.4.42) i=1 i=1 4The edge r (see Figure 5.4.2) has sometimes subscript related to the indices of vertex nodes of the edge: e.g. ru, , • • • where {j is related to the j-th degree of freedom at node I, and N^ are the corresponding shape functions. In other words, the shape function N^ interpolates the k-th component of virtual displacements vh,r and is weighted by nodal degree of freedom {i)/ in the linear interpolation (5.4.42). We can now insert the above parameterizations into (5.4.30) and express residuals at nodes I and J with respect to the boundary traction on edge r (that spans between I and J). The result can be written in a compact form as: r/ re e /,r e J,r = ■ Mf1 _ Mr1 MJ " MJ _ " Pr,/ " _ Pr,j _ MJ = [Mb ]; mj = [M/^m (5.4.43) where pr, [Pfc.j]T, k 1,... ndim, and ndim is a dimension of tr. The block matrices MpJ have nrado/I rows and ndim columns with the elements M /,J r,nm N1 /J ds m ,n r m (5.4.44) The index m is denotes the virtual displacement component which is conjugated (in the sense of the virtual work principle) to the traction component m. By solving (5.4.43) for every element edge r, we obtain parameters pr>/ and pr>J, which completely define element boundary traction tp7j through (5.4.41). The choice of the shape functions /, which describe the variation of the boundary tractions along the edge, is somewhat arbitrary. It is suggested in the literature (e.g. [Ainsworth Oden, 2000], [Oden Cho, 1997]) that shape functions /J and / should be orthogonal to the shape functions describing the variation of the test function In this case the terms MfJ and Mr1 vanish and the matrix M becomes block diagonal. The relation (5.4.43) simplifies to rf,r Mr pr (5.4.45) which significantly reduces the computation cost. Note, however, that the specific choice of parametrization (5.4.45) does not affect the local patch-wise computation of boundary tractions. r R R R 3,rsi 4 r3,r23 J3 Ri Ri /J]T is the unit exterior normal and ■ ■ ■ , 04 are shape functions. Similarly, variation of rotation vector along the same edge can be written as 0h,rij = 0/ pi + 0j ^2 + n/jA0/j ^3 (6.2.3) The shape functions ^ in (6.2.2) and (6.2.3) are defined as, see (2.3.55) Pi = (1 — 0/2 P2 = (1+ £ )/2 (6.2.4) P3 = (1 — £2)/2 = £(1 — £2)/2, where £ G [—1,1]. We compute the boundary traction projections rf,riJ, which pertain to the boundary tractions tpij = [qe/ , ms,mn] through (5.4.30). In order to evaluate the traction projections, the following integral has to be evaluated: (qe/,riJwh,riJ + riJ(s ■ 0h,rij^ ds + [mnwh,rij]j = rf,p^ ■ u/ + rJ,p^ ■ uj (6.2.5) •/rij Comparison of the corresponding terms on both sides of the equation (6.2.5) will provide the components of the projections rf,r = [rw r;rjxr>r/r]T, where the notation r = r/J is employed. These components are energy conjugated to the plate element degrees of freedom and the chosen discretization. The boundary traction tf consists of three components: shear qr and moments ms,r, mn;r. For the given interpolation for tf, the components can be reconstructed from the projections rf,r from (7.2.25). We chose to interpolate each of the components of tf with the same interpolation: a linear combination of the shape functions V/ and Vj: qr = qf V/ + qr Vj (6.2.6) ms,r = mS, rV/ + mjr Vj mra;p = m£ rV/ + m^p Vj Note, that q^r in (6.2.5) is defined as qr — d™,S'r, where s is a boundary coordinate, see Figure (2.2.7), with normal [nx,ny]T. The shape functions -0/ are not yet specified here, except that — k)/ = —/ for k G [q,ms,mn]. The relation between the projections of rf,r and the parameters [qf, mS,r, m^p] can then be obtained according to (compare to more general formulation (5.4.43)): f f Aq J qJ + Am mS,r m' + Am mn,r m (6.2.7) where Am Ak (—ny + nxAmy); A" A Afc A// A/J A A AJ/ AJJ. - A/ J = 100000 0 0 0 1 0 0 T N/Mx N/- (6.2.8) —j ds ; k G [q,mx,my]; k G [w,6x,6y] The shape functions employed in (6.2.8) can be written explicitly as: = Pi — ^4/2 NJ,w- = ^2 + ^4/2 (6.2.9) N —Lnx (—^3 + <^)/2 NJ —Lnx(+^3 + P4)/2 = —Lny (—^3 + ^4)/2 = —Lny (+^3 + ^4)/2 s n s /a NU = +(3nx/L) ^3 = ^ + 3^3/2 = + 3nx ny <^3/2 Ni,,® = — (3nx/L) ^3 NlA = ^2 + 3^3/2 N/x>- = + 3nx ny ^3/2 N/ N0y N/ + (3ny/L) ^3 : +3nxny ^3/2 : ^i + 3n;;^3/2 Nj,, = —(3ny/L) ^3 Nj= +3nxny ^3/2 Nj- = ^ + 3n2 ^3/2 where L is the length of the edge r/J and nx, ny are components of the normal to that edge. They are obtained from (6.2.2) and (6.2.3) according to (5.4.42). According to (5.4.41) the following notation is used for edge r: Pq,/ = q£; Pms,/ = mS,r; Pmn,/ m —/ = <^i; —j = ^2 (6.2.10) (6.2.11) we can explicitly compute the inverse of the system (6.2.7) and obtain the final result for the traction components: «r = - § (n^r + 3r%) + ny (2r?yr + 3j)) (6.2.12) 2 mir = +T(ny(-2r?V + rj>) + n*(+2r& - r%)) m, «X I „«I \ I „ l I Orr"y ^ y n,r - + L(nx(r«:r + 7rJxr) + ny(r?yr + 7r$V)) - rj»r J = + L2 {nx(2r% + 3r«:r) + ny(2 J + 3r%0) mJr = + L(ny(-2rJ> + r«^) + ^(+2^ - r^)) mJ = _1 " L mn,r = -T (nj + 7r«*r) + ny (J + 7r£r)) + J By inserting (6.2.12) into (6.2.6) the boundary tractions = [qr, ms>r, mra>r]T at the element edge r/j can be obtained. Thus, the procedure, described in this section, represents step 4 of Box 3 for DKT plate element for the case of interpolation (6.2.6) and (6.2.11). 6.2.2 Formulation of local boundary value problem The discretization error is estimated by solving local problems (5.4.28). In the case of thin plates, the weak formulation of local problem is defined as: find the displacement we h+, which satisfies / m(we,h+) ■ K(vh+) dQ = fvh+ dQ (6.2.13) Jne Jne + W [qe/,rjj vh+ + mSip;j9.s(vh+)] ds + [m^ruVh+ ]J}; Vvh+ G Vo,h+ T"1 _ _ JVjl P/J where Vh+ is the best possible approximation of the solution space V that must be sufficiently enhanced with respect to the finite element space Vh constructed by DK plate element. The equilibrated boundary tractions tf,JJ = [qe/,rJJ,ms>rIJ,mn,rIJ]T are those obtained from the original finite element solution using the equilibration procedure (see Box 3). On the other hand, the finite element solution we,h satisfies the following weak form defined on the element domain: m(we,h) • K(vh) dQ fvh dQ + y] [qef,ru Vh + mSlr,j 0S (vh)] ds + [mn,p„Vh]J The equivalent matrix form of (6.2.14) can be written as Kene = ffe + £ [re,r„ + r J>r„] rjj Vvh eVc,h (6.2.14) (6.2.15) where rI,rlJ are the projections defined by (6.2.5). Isolating the equations regarding the node I we arrive at (compare to (5.4.34)): R = V KJUJ - f = rf;ru + rehVlK J (6.2.16) Local patchwise systems with the unknowns r1TlJ are formed using the equilibration procedure described in Box 3. Solution of the local system gives projections r1TlJ for all nodes I and edges rIJ. The boundary traction tpij at the edge r1J is recovered from projections r1VlJ and r J,rij using (6.2.12). t t Regularization To form the optimization function (5.4.38), the projections are required which are computed from the improved approximation of the boundary traction tf = [qef ,fhs,m,n] using the inverse of (6.2.12). The boundary traction approximation is computed from the current finite element solution using the following simple technique: edge boundary traction tp is taken as an average of the boundary tractions tf, and tp computed from the stress resultants of the two elements e and e' adjacent to the edge r: tr = (tr - tr )/2 (6.2.17) Boundary tractions t]T are evaluated at side r from element stress resultants from the Cauchy principle. First, the stress resultants are evaluated at the integration points and are denoted as mh(£gp) = CBKh(^gp). A linear interpolation of the stress resultants over the element m =Y mi Ni (6.2.18) i is constructed using the least squares fit to mh(£gp) 1 2 Y12(m(igp) — mh(igp)) ^ min. gp At the element side r defined by unit exterior normal n = [nx, ny]T we thus have: mn = (fhyy — rnxx) n^ny + rnXy(nX — n^) (6.2.19) Tfls (m XX nx + m yy ny + 2mmxy nxny ) The shear component normal to the edge is obtained from q = r]T given in (6.2.6). 6.2.4 Numerical examples The selected numerical examples are presented in this section in order to illustrate different aspects of the two error estimates: the SPR and the EqR methods. The error estimate for the plate problems is computed according to (5.3.3): IKHE2 = ^ nf = (m* - mh)TC-1(m* - mfc) dH (6.2.23) e e ^-w-- The energy norm of the true error of the FE solution, denoted by ||eh||E, was also computed by using the reference "exact" solution. For a number of simple plate structures, the reference Kirchhoff solution can be obtained analytically. In the case where analytical solution is not available, the reference solution is computed using the FE approximation with very fine mesh of Argyris TUBA6 (ARGY) element. The effectivity index of the error estimate is defined as the ratio between estimated and the true error: 8 = ^^ (6.2.24) In the process of adaptive meshing, the new mesh is generated using the distribution of the error as the input. In order to asses the performance of the error estimate in the context of adaptivity, the local error estimate is introduced. The local error estimate is *_ M 11 * ne = ||ehHE,e and the global error estimate 22 n =2^ ne The relative local error estimate n*,r is defined as n*,r = (6.2.25) || || E,e where the energy norm of the solution ||uh||E is defined as ||uh|E = y2 |K||Ee; |K||Ee = mhC-1mh ds (6.2.26) e ' The relative local error estimate is normally expressed in percents. The relative global error estimate is defined as: ME nr* = TTii (6.2.27) ||Uh||E Simply supported square plate under uniform loading In the first problem we consider simply supported square plate of side length a =10 and thickness t = 0.01 under uniform loading f =1. The material behavior is linear elastic and isotropic, with Young's modulus E = 10.92 x 1010 and Poisson's ratio v = 0.3. The reference solution is the analytic solution in the form of series expansion; see [Timoshenko Vojnovski-Kriger, 1959]. The solution is regular over the whole domain and does not exhibit any singularities. In Figure 6.2.2 we give the results for energy norm computations showing the convergence of the FE solution with respect to the number of elements expressed as 1/h, with h indicating the typical element size. The typical element size is computed from the average area of the element h2 = Q/nelem. The global error estimates are compared to the true global error. The comparison shows that the "EqR-SDKT" estimate is lower than the true error, while other error estimation techniques overestimate the error. The error estimation procedure EqR-ARGY performs best. The effectivity index converges towards 1 as the number of elements is increased (see Figure 6.2.3). The Figure 6.2.4 shows the comparison of the relative local error estimates . The overall distribution of the error is captured quite well by all the error estimation procedures, with only minor differences between them. The error estimates based upon "EqR-ARGY" again performs best. 100 101 1/h Figure 6.2.2: Simply supported square plate under uniform loading - global energy error estimates 100 101 1/h Figure 6.2.3: Simply supported square plate under uniform loading - effectivity index of the global energy error estimates Slika 6.2.3: Enakomerno obremenjena prosto podprta kvadratna plosCa - indeks učinkovitosti ocene Clamped square plate under uniform loading The second example is a clamped square plate of side length a =10 and thickness t = 0.01 under uniform loading f =1. The material behavior is linear elastic and isotropic, with Young's modulus E = 10.92 x 1010 and Poisson's ratio v = 0.3. The reference solution is the analytical solution in the form of series expansion; see [Taylor Govindjee, 2002]. The solution is regular over the whole domain and does not exhibit any singularities. In Figure 6.2.5 we show the convergence of the FE solution in the energy norm with respect to the number of elements. The comparison of the global error estimates shows that the "EqR-SDKT" estimate is again lower than the true error, while other error estimation techniques overestimate the error. The effectivity index (see Figure 6.2.6) is close to 1 for the SPR procedure. The effectivity index of the EqR-CDKT and EqR-ARGY procedures remains close to 1. However, it increases as the mesh density is increased. The performance of both procedures is comparable. Comparison of the relative local error estimates ne,r, shown in Figure 6.2.7, reveals an interesting detail. While the overall distribution of the error is estimated quite similarly by all the error estimation procedures, the error in the corners is severely underestimated by both SPR and EqR-SDKT procedures. The local error estimates is again favorable to EqR-ARGY procedure. Clamped circular plate under uniform loading The clamped circular plate of radius a =1 and thickness t = 0.001 under uniform loading f = 1 is considered in this example. The material is linear elastic and isotropic, with Young's modulus E = 1.70625 x 108 and Poisson's ratio v = 0.3. A quarter of the plate is modeled only, with symmetry enforcing boundary conditions. The reference solution is the analytic solution; see [Timoshenko Vojnovski-Kriger, 1959]. In the convergence study the mesh was refined in each step by factor 2. The mesh refinement was done carefully to minimize the effects of Babuska paradox [Babuska Pitkaranta, 1990]. Namely, since the curved boundary cannot be captured exactly, the discretization of the domain introduces spurious singularities in the obtuse corners. With the mesh refinement, the curved boundary is better represented, but on the other hand the number of obtuse corners, where the singularities are expected, also increases. The convergence of the FE solution is shown in Figure 6.2.8 in terms of the energy norm for increasing number of elements. All the error estimates overestimate the error in this particular case, due to high solution regularity (see Figure 6.2.9). The comparison of the relative local error estimates shown in Figure 6.2.10 again reveals some interesting discrepancies. The error is overestimated in the corner elements of the discretized domain for the EqR error estimates, while this is not the case for the SPR error estimate. Main reason seems to be deficiency of the equilibration procedure in the local patch systems for corner nodes, which are overly constrained. Enakomerno obremenjena vpeta kroZna plosca - globalna ocena napake Morley's skew plate under uniform loading The analysis of Morley's 30° skew plate (see [Morley, 1963]) with thickness t =1, side length a = 10, simply supported on all sides, and loaded with unit uniform pressure is examined. The plate material is linear elastic and isotropic, with Young's modulus E = 10.92 and Poisson's ratio v = 0.3. The most interesting feature of the solution to this problem concerns two singular points at the two obtuse corners of the plate, which strongly influence the quality of the computed results (e.g. see [Ibrahimbegovic, 1993]). In Figure 6.2.11, the convergence of the solution is shown along with the error estimates. Both SPR and EqR-SDKT procedure underestimate the error, while EqR-ARGY and EqR-CDKT procedures overestimate the error by approximately the same amount. It is interesting to note that the convergence of the solution is not monotonic. The effectivity index of all the error estimates is relatively close to 1 despite the singularities (see Figure 6.2.12). The comparison of the relative local error estimates given in Figure 6.2.13 clearly shows that EqR-SDKT is completely incapable of capturing the singularities. The SPR procedure performs better but it clearly underestimates the error at the obtuse corners. The EqR-CDKT and EqR-ARGY both capture the singularities. L-shaped plate under uniform loading The analysis of L-shaped plate with thickness t = 0.01, side length a = 10, simply supported on all sides, and loaded with unit uniform pressure is considered in this example. The plate is of linear elastic isotropic material, with Young's modulus E = 10.92 x 109 and Poisson's ratio v = 0.3. The solution exhibits a singularity in stress resultant components at the re-entrant corner. The singularity for the components mxx and myy is defined by the leading term rA-2 and for the component mxy by rA-3, where r is the distance from the singular point and A the exponent which depends on the opening angle a: A = n/a. In the case of the opening angle a = 3n/2, we have A = 2/3. The problem is solved with a non-uniform mesh which is denser around the singular point. The distortion of the element (measured by the eccentricity) is significant. The Figure 6.2.14 shows the non-monotonic convergence of the true error of the solution although all the error estimates indicate otherwise. The effectivity index of SPR and EqR error estimates is close to 1.0 which indicates that the mesh distortion does not deteriorate the effectivity of error estimates significantly. 6.2.5 An example of adaptive meshing The discretization error estimate is usually employed to control the adaptive mesh generation. The main goal of the adaptive meshing is to generate a mesh where the local element error ne will be approximately constant over the domain i.e. equal for every element ne ~ ne. By knowing the desired local element error ne and the local error estimate n* it is possible to deduce the element size from the a-priori estimate of the local element error ne = Chp (6.2.28) where the element size is denoted by h, C is a constant and p the polynomial order of finite element interpolation. The estimated element size h is deduced from ne* = Chp; ne = Chp ^ h = h(ne/ne*)i/p (6.2.29) The software gmsh [Geuzaine Remacle, 2009] was employed to generate the mesh. It enables the generation of the mesh according the input scalar field ('background mesh') which defines the element size over the domain. In the following numerical test the comparison of different error estimation procedures with respect to the adaptive meshing is presented. A problem of uniformly loaded clamped square plate is chosen as the model problem. The analysis was performed as follows: after the first FE solution was obtained on a uniform mesh, a model problem was recomputed on several meshes, which were generated according to different error estimates. The true error of each solution was computed using the reference solution. Finally a comparison of local element errors ne was made. The number of the elements in the generated meshes was controlled to approximately match the number of the elements in the original mesh. A value p = 1.1, determined from Figure 6.2.5, was used in (6.2.29). The results of the computation are summarized in Table 6.1. Along the meshes, it contains also the distributions of the local element error ne and their histograms. In the first row of the table, the errors of original solution made with a uniform mesh, are presented. The histogram of the local element error is relatively coarse, as expected. In the second row the results of the solution computed on the mesh, based on the true local element errors of the original solution, are presented. The histogram is noticeably narrower but it is not just a single peak. This can be accounted to the limited capability of the mesh generation algorithm and the lack of ability of the a-priori error estimate to predict the effect of element distortion. Other rows depict the results of error computation on the FE solutions obtained on several other meshes which were based on different error estimates. The results (local element errors) are quite comparable and do not expose any specific method. The difference of the generated meshes is however apparent. The visual assessment of the overall distribution of the local element errors clearly reveals the difference between the original solution and the solution obtained on the 'adapted' meshes. Table 6.1: Comparison of errors of FE computation of the square clamped plate problem with various meshes adaptively constructed from discretization error estimate. Tabela 6.1: Primerjava napak resitev s KE za problem enakomerno obremenjene vpete plosce za razlicne mreZe, konstruirane na osnovi ocene napake diskretizacije. n i i i i 7L_ Local element error Ve Histogram of local element error ne Original mesh 1156 elements, nr: 6.2% -6 -5 -4 logio ne Mesh based on true error 1171 elements, nr: 5.4% % -5 -4 logio ne Mesh based on SPR error estimate 1170 elements, nr: 5.7% 300 200 100 0 -6 -5 -4 logio ne Mesh based on EqR-SDKT error estimate 1168 elements, nr: 5.5% 300 200 100 0 L -6 -5 -4 logio ne Mesh based on EqR-CDKT error estimate 1158 elements, nr: 5.4% 300 200 100 0 j -6 -5 -4 logio ne Mesh based on EqR-Argyris error estimate 1261 elements, nr: 5.4% \ 300 200 100 0 -6 -5 -4 logl0 ne (a) (m) Figure 6.2.4: Simply supported square plate under uniform loading Comparison of relative local error estimates infer in [%] on the mesh (m): (a) True error, (b) SPR, (c) EqR-SDKT, (d) EqR-CDKT, (e) EqR-ARGY Slika 6.2.4: Enakomerno obremenjena prosto podprta kvadratna plosča Primerjava relativnih lokalnih ocen napake nI r v [%] na mreži (m): (a) Dejanska napaka, (b) SPR, (c) EqR-SDKT, (d) EqR-CDKT, (e) EqR-ARGY V 10-1 1/h Figure 6.2.5: Clamped square plate under uniform loading - global energy error estimates Slika 6.2.5: Enakomerno obremenjena vpeta kvadratna plosča - globalna ocena napake 100 101 1/h Figure 6.2.6: Clamped square plate under uniform loading - effectivity index of the global energy error estimates Slika 6.2.6: Enakomerno obremenjena vpeta kvadratna plosča - indeks učinkovitosti ocene (a) I (b) L > (d) r i Ll I TU < J 1 ■ (m) (c) (e) :J L T E ■ Figure 6.2.7: Clamped square plate under uniform loading Comparison of relative local error estimates n* r in [%] on the mesh (m): (a) True error, (b) SPR, (c) EqR-SDKT, (d) EqR-CDKT, (e) EqR-ARGY Slika 6.2.7: Enakomerno obremenjena vpeta kvadratna plosCa 0 100 1/h Figure 6.2.8: Clamped circular plate under uniform loading - global energy error estimates Slika 6.2.8: Enakomerno obremenjena vpeta krožna plosca - globalna ocena napake 100 1/h Figure 6.2.9: Clamped circular plate under uniform loading - effectivity index of the global energy error estimates Slika 6.2.9: Enakomerno obremenjena vpeta krožna plosca - indeks ucinkovitosti ocene (a) (m) (b) (c) (d) (e) 10 9 8 7 6 5 4 3 2 1 0 Figure 6.2.10: Clamped circular plate under uniform loading Comparison of relative local error estimates nI r in [%] on the mesh (m): (a) True error, (b) SPR, (c) EqR-SDKT, (d) EqR-CDKT, (e) EqR-ARGY Slika 6.2.10: Enakomerno obremenjena vpeta krožna plosca 101 100 100 1/h 101 Figure 6.2.11: Morley's skew plate under uniform loading - global energy error estimates Slika 6.2.11: Enakomerno obremenjena prosto podprta romboidna plosca - globalna ocena napake ® -2 True error SPR EqR-SDKT ♦ EqR-CDKT EqR-ARGY NIMM! i : ! 10 -1 100 101 1/h 8 6 4 2 0 Figure 6.2.12: Morley's skew plate under uniform loading - effectivity index of the global energy error estimates Slika 6.2.12: Enakomerno obremenjena prosto podprta romboidna plosca - indeks ucinkovitosti napake (a) (m) (b) (c) i100 90 80 70 (d) (e) 60 50 40 30 20 10 Figure 6.2.13: Morley's skew plate under uniform loading Comparison of relative local error estimates nI r in [%] on the mesh (m): (a) True error, (b) SPR, (c) EqR-SDKT, (d) EqR-CDKT, (e) EqR-ARGY Slika 6.2.13: Enakomerno obremenjena prosto podprta romboidna plosca 0 100 101 1/h Figure 6.2.14: L-shaped plate under uniform loading - global energy error estimates Slika 6.2.14: Enakomerno obremenjena prosto podprta L plosca - globalna ocena napake 100 101 1/h Figure 6.2.15: L-shaped plate under uniform loading - effectivity index of the global energy error estimates Slika 6.2.15: Enakomerno obremenjena prosto podprta L plosca - indeks ucinkovitosti ocene (a) (m) C (b) C (d) \/ /\ (c) (e) 100 90 80 70 60 50 40 30 20 10 0 Figure 6.2.16: L-shaped plate under uniform loading Comparison of relative local error estimates nI r in [%] on the mesh (m): (a) True error, (b) SPR, (c) EqR-SDKT, (d) EqR-CDKT, (e) EqR-ARGY Slika 6.2.16: Enakomerno obremenjena prosto podprta L plosca 6.3 Discretization error for RM elements RM plate elements, specifically the P3 elements introduced in Chapter 3, conform to the same principles given above for the DK elements. The definition (5.3.4) of the local error indicator is modified so, that it includes, besides the bending, also the shear deformation energy: ne2, RM = £ / (m*-mh)TCB1(mI-mh)dQ+^/ (q*-qh)TC-V-q^dQ (6.3.1) e e SPR is method is applied in exactly the same manner as for the DK elements (practial issues are discussed in [Lee Hobbs, 1998] and [Kettil Wiberg, 1999]). The application of the EqR method to the P3 elements follows the same same steps as for the DK elements. The reason for the similar treatment lies in the close similarity of the interpolations since they are build on a hierarchic basis. Conceptually, the only novelty is the treatment of the residuals to the midside degrees of freedom in P3 elements in the equilibration procedure. It is apparent, that due to the simple topology, equilibration of the midside DOF is not needed. Thus the equilibration procedures for DK and P3 elements are effectively equal. The enhanced interpolations for the local computations are, however, limited to the subdivision schemes, since there does not exist an enhanced P3 element (equivalent to the ARGY element in the case of DK elements). 6.4 Chapter summary and conclusions The focus of the chapter is the discretization error for the case of finite element Discrete Kirchhoff plate model, which is widely accepted as finite element for the analysis of bending of thin plates. Of particular challenge are the non-conventional finite element interpolations for Discrete Kirchhoff plate bending elements. Hence, we can only provide the error indicators (rather that rigorous error bounds), and even those require quite ingenious procedures for constructing enhanced finite element space for error estimates. The SPR method is nowadays widely accepted and found its way in many finite element codes. The implementation for the Discrete Kirchhoff plate elements follows the general principles of the method and it does not take into account the specific nature of the problem. The EqR method, however, is significantly more difficult to implement, but it has the potential of capturing specific issues related to modeling of plate bending. In this work a special attention was given to the development of EqR method for this case. In particular, we have tested the equilibrated residual method with enhanced space constructed by either subdividing the original element into 6 smaller elements (subdivision into 3 smaller elements proved to be of no interest) or by using the Argyris plate element. Although each of these enhanced elements possesses 21 degrees of freedom, they differ in terms of providing the enhanced space. The former uses the patch like construction with third order polynomials, whereas the latter uses the complete fifth order polynomial. Thus, although both enhanced elements provide useful estimates, the latter, or the Argyris element, furnishes superior results quality. Chapter 7 Model error concept 7.1 Introduction For the case of linear elasticity, which is the subject of this work, a complete and validated three-dimensional theory exists, which captures all the physics of the deformation. In the practical work in solid and structural mechanics, however, the theory is rarely applied. The reason is, that for a given accuracy, the computational effort to obtain a solution is exceptionally high. Simplified models were therefore developed which are valid only under certain assumptions. The accuracy of this models is satisfactory for the most engineering computations as long as the assumptions are satisfied. Similar situation exists also in other areas. The motion of free falling body in the air, for instance, is effectively computed using a Newton theory, rather than using quantum theory, which is known to be more accurate. If the result of the computation is simply a trajectory of the ball with a given accuracy, the Newton theory is obviously quite adequate. For the practical use, the model must be computationally efficient while the accuracy must match the prescribed one. Engineering approach to modeling requires that the models are only as complex as necessary. In the structural mechanics the need to develop the computationally effective models is of particular importance. Several models of the typical structural members were developed and tested which give accurate measurable predictions. They are based on certain assumptions, which are typically satisfied in regular regions of a structure. In the disturbed regions such as boundary layers, thickness jumps, stiffeners and regions with concentrated loads enhanced models have to be used. Ultimately in the regions where none of the assumptions apply a full three-dimensional theory of elasticity have to be employed. Adaptive modeling in structural analysis has the goal to produce the most suitable (i.e. optimal) model of a structure. In order to construct a mixed - regionally adapted -model, a sequence of hierarchically reduced, i.e. simplified, mathematical models derived from an appropriate master model is required. The choice of particular model depends on estimate of the model error. This information is the key to successful control of the adaptive construction of the optimal model. The model error is related to the suitability of model itself. One should clearly distinguish between the model error estimates and model error indicators. The estimates define the boundaries of the error. They are usually quite conservative and they tend to overestimate the error by several orders of magnitude. Indicators on the other hand do not provide any guarantees on the error. In return they can give quite sharp indication of the error. Which type of model error to use depends on the purpose of the computation of the model error. If the goal is to control the adaptive construction of the optimal regional dependent model, the model indicators are the preferable choice. To locate regions of the domain where the chosen mathematical model (usually the simplest one in a set of available hierarchic models) no longer performs well, one has to provide an estimate of model performance. Ideally, such an estimate for the chosen model should follow from the comparison with the best possible mathematical model (which is often the 2D/3D solid model). However, the best model estimate is in general not feasible, since it remains prohibitively expensive or simply inaccessible. Thus, for the practical model error estimation, it is sufficient to compare the chosen model with the one which is known to perform better; the latter will be called the enhanced model. In principle, two global computations would be required to compare two different mathematical models: the chosen against the enhanced one. However, the computations with the enhanced model is usually simplified in trying to estimate the true stress state. This is made possible by using the EqR method originaly developed for discretization error estimation (see section 5.4.2). The procedure starts by extracting a portion of the domain, reducing the computation to a single finite element based on the enhanced model, applying the loading on its edges according to the true stress state estimate, computing the local enhanced solution and comparing it to the original solution obtained by the chosen model. This is of course possible only in principle, since the true stress state is unknown. However, its approximation can be obtained by improving the FE solution obtained with the chosen model to be best-possible approximation of the true stress state. This is done as follows: the FE solution for any stress component (which is discontinuous between the elements) is improved to be continuous in order to approximate the true stress state, which is continuous (unless there is thickness, loading or material discontinuity). In accordance with the best-possible approximation of the true stress state, the edge loading (so-called boundary traction) for each finite element of the mesh is computed. That loading is further used in computation of local (element-wise) Neumann problems based on the enhanced model. Comparison of two mathematical models can be thus achieved by one global and a set of local computations. The effective procedure of computation od model error indicator is build upon the following idea. The model error, in principle, measures the difference of the solutions of the current and full three-dimensional model. Based on the assumption that the family of models is hierarchically ordered, the model solution does not need to be compared to the solution of the exact model. Due to the hierarchy, it suffices to compare the current solution with the solution of the next model from the hierarchic family of models. The global computation of the enhanced solution - just to serve as the reference to model error computation - is too expensive. An effective compromise is to repeat the computations with the enhanced model on smaller domains - preferably on elements. The local problems have to replicate the original problem. The interpretation is, that the elements are extracted from the continuum and its actions are replaced by boundary tractions. The boundary conditions for the local problems are therefore of Neumann type. 7.2 Model error indicator based on local EqR computations 7.2.1 Definition of model error indicator The primary question of model error computation based on EqR method becomes: "How to estimate the boundary conditions for the local problems?". If the enhanced solution is known, the boundary conditions for the elements are computed from the stress state using the Cauchy principle. This is, however, not possible since the stresses are computed from the enhanced solution, which is unknown and yet to be computed. An approximation of the boundary stresses must therefore be built based on the single information available: current finite element solution. The development of the method of the construction of the best possible estimates for the boundary conditions for local problems is thus the central topic of the model error computation based on EqR method. The enhanced solution, to which the original solution is compared to, is thus ap- proximated twice: (i) the boundary conditions for the local problems are computed only approximately and (ii) the local problems are not solved exactly but rather in the weak sense (finite element method) with the enhanced model. Such double approximation is however still acceptable since the goal of the model error indicator is not to provide the error bounds but to drive the model adaptivity procedure. The starting point for the computation of the model error indicator is the finite element solution of the problem using the coarse model - model 1. The finite element solution is ui, h. Let us suppose that the finite element solution u2 ,h using the enhanced model - model 2 - is available. Since the model error is of primary interest, let us assume also that the discretization error is negligible or at least much smaller than the model error. In this case the difference between the solutions can be entirely contributed to the model error. Since the difference u2 ,h — u1, h is mathematically inconsistent, the difference of the solutions is computed as: eu,mod = u2,h — P«,1^2u1,h where PU;1^2uh;1 denotes the kinematic transformation of the solution uh)1 to the solution space of model 2. Several measures of the difference of the solutions u2)h,uh)1 exist for the definition of the model error indicator. The most straightforward is probably to use the energy norm of eu,mod as the model error indicator Vu,mod = ||eu,mod||E = ||u2,h — P«,1^2uh,11| E This definition of model error indicator is, however, prone to strong overestimation. Another option is to measure the model error in terms of the energy difference nu,mod = |u2,h|E — |uh,1|E In most cases investigated this estimate is not sensitive enough, since the global energies of the models 1 and 2 usually do not differ enough. In order to avoid the problems mentioned, a model error is rather measured in terms of the stresses, i.e.: ea,mod = 02,h — The transformation transforms the stresses from model 1 to model 2 using the constitutive relation of model 2. Note, that eu,mod and ea,mod are in different solution spaces. The model error indicator is computed as the energy norm of ea,mod: Va,mod = \ea,mod He = — Pa,1^20^1^ Figure 7.2.1: Local computation on a finite element domain Qh with Neumann boundary conditions tpe on re Slika 7.2.1: Lokalni izracun s koncnimi elementi na obmocju z Neumannovimi robnimi pogoji t^e on re The complementary energy norm || • ||E is related to the model 2: I^IIe = : C- : ^dQ n Construction of local problems The model error can be estimated only via the comparison of the current solution with the solution of the enhanced model. The global computation of the enhanced solution u2,h is, however, not feasible. An alternative is the decomposition of the global problem into smaller local problems. The local computations - preferably on elements - should ideally give the same results as the global one. By a suitable choice of boundary conditions it is possible in principle to define local problems with this property. The finite element solution of the enhanced model 21, denoted by u2 ,h, satisfies the weak form: «2(U2,h, V2,h) = l2(V2,h); Vv2 ,h € V2 ,h (7.2.1) The linear forms a2 and l2 refer to the model 2. The finite element solution space is denoted by V2 ,h. The global problem can be replaced by a set of local problems, formed 1The subscript 2 will be related to the enhanced model. a2,e(U2,e,h, «2,h) = l2,e(V2,h)+/ t^e ' V2,hds; Vv2,h G V2,h (7.2.2) where the boundary conditions are of Neumann type. Along the boundary of the element, denoted by re = 3Vle, the element is loaded by boundary tractions t|,re. Boundary tractions t2,rjj are undefined and are yet to be determined. Additional condition for their construction comes from the observation, that the solution of the local problems should match the global finite element solution: U2,e,h = U2,e,h (7.2.3) The element solution u2,e,h = u2,h|ne is the restriction of the global finite element solution to the element domain Assuming that the global finite element solution u2,h is known, the additional condition for the boundary tractions is: / t2,re ■ V2,hds = a2,e(U2,e,h, V2,h) - l2,e(V2,h); V^2,h G V2,h (7.2.4) Jre The symbolic form of the discretization of the j-th component of the test function v2,h is (v2,h)j = j v2,j, where Nj,j are the shape functions. Employing this notation, we have for the integral over side rJJ: Tjj t2,rjj ■ V2,h du = re,j,rJJ ■ + r2,j,rJJ ■ V2,j (7.2.5) where the r|,j,rjj is the 'projection' of the boundary traction t2,rjj to the node I in the finite element test space V2,h. We can interpret r2,j rjj as residual forces at node I of element e due to the boundary traction applied on edge rJJ, i.e. awjj^rjj) öv2 r2,i,rjj = ^ 'ijj/; Wrejj(^) = / tS,r„ ■ (7.2.6) 2,j Jru From (7.2.4) the finite element equilibrium equation for a single element (7.2.4) can be written also as R2 = Fmt,2 - Fext,2 (7.2.7) where F^2, R and F^2 are nodal values of external, residual and internal forces, respectively. The later are proportional to the stiffness matrix K2 and the nodal displacements so that (7.2.7) becomes: K2U = F22xt,e + R Residual forces R represent the action of the surrounding elements. The goal is to find such boundary traction forces that exactly replace the residual forces and are continuous across element boundaries. We thus seek for boundary traction tfe acting on an edge re, which will replace (in an energy manner) the action of residual nodal forces and reflect the continuity of the stress field. We can write Y^ R2,/ ■ «2.I = tfe ■ V2,re;h ds = Y trjj ' V2.ru.hds (7.2.8) t «'T6 r> ^rrj where v2.re = v2|re are edge virtual displacements, and v2)/ are nodal virtual displacements. The relation follows directly from (7.2.4): a2,e(U2,e,h, Vž.fc) - k.e^.h) = R2.I ' «2,/ (7.2.9) I where R|,I is the nodal residual on node I of the element e. The vector of degrees of freedom pertaining to the node I is denoted by V2)I. From (7.2.5) we further express the nodal value of the element residual force Ä|,I in (7.2.8) as the sum (with slight abuse of the notation) R2.I = r2,i,ri + r2,i,r2 (7.2.10) where r and r2 are two edges of element e that both include node I, see Figure 5.4.4. Since the boundary tractions reflect the continuity of stress field, they conform to, see Figure 5.4.3: t2,r„ + e2 i,ri „e3 r>e2 i,r2 ri,r + re3r3 r>e4 i,r3 + r e4 i,r4 The demand for continuity further leads to r i,ri + re r r e2 I re3 i,r2 + ' i e3 + re4 i,r3 r e4 + re1 0 0 0 0 e4 ei ra -0- es e2 i,ri e3 i,r2 e4 i,r3 ri1,r4 Combination of the last two sets of equations leads to the following system ri Figure 7.2.2: A patch of four elements surrounding node I ■ Rf1 ■ " +1 0 0 -1 Rie2 -1 +1 0 0 Rie3 0 -1 +1 0 Rie4 0 0 -1 +1 i,r1 e2 i,r2 e3 i,r3 e4 i,r4 (7.2.21) for the unknowns r(^zrk. Since all the unknowns refer only to node I of the FE mesh, the system is independent from other nodes. One thus obtains a local (patch-wise) problem; a global system is actually a composition of independent patch-wise systems. If local system (7.2.21) is solved for each node of the FE mesh, we obtain rffcr, for all elements of the mesh (ek = e = 1,..., ne1), all element edges (r = 1,... nr) and all element nodes (I = 1,...nen). Recovery of boundary tractions from the projections Result of the local patchwise computations are the projections re1)i riJ for all nodes and edges riJ. Two projections exist for every edge riJ: re1)iriJ and re1)jriJ. After a suitable discretization of t2,rjj is chosen, the boundary tractions are reconstructed using the relation (7.2.18). The boundary tractions t2,rjj are consistent with model 2 and caution should be taken when solving for t2,rjj. The relation (7.2.18) namely assumes that the boundary tractions conform to the kinematics of model 1. Evaluation of boundary traction parameters from (7.2.18) If rf,r are known2, the boundary traction can be computed for each element as follows. We introduce parametrization of k-th component of the boundary traction tf (let r be element edge between element nodes I and J) as (tep)fc = pk ik + pJ iJ + pK iK (7.2.22) where ik are shape functions (yet to be chosen) and pk are nodal values of (if )k. Here we assumed, the edge has vertex nodes I and J and a midside node K. The variation of virtual displacements along the same edge r can then be written as (vr,h)k = NkmVm + Nfcjmvm + Nfcy>m (7.2.23) where v^ is the m-th degree of freedom at node I, and N[m are the corresponding shape functions. In other words, the shape function N^ interpolates the k-th component of virtual displacements vh,r and is weighted by nodal degree of freedom vi;/ in the linear interpolation (7.2.23). We can now insert the above parameterizations into (7.2.18) and express residuals at nodes I and J with respect to the boundary traction on edge r (that spans between I and J). The result can be written in a compact form as: r/,r = rJ,r _ ■ Mf7 _ M77 MfJ M7J MfK 1 MrK J pI pJ pK ; (7.2.24) MfJ = [MfJ ]; Mf'J = [Mflim] where pf = [pk, k = 1,... udim]T, and udim is vector dimension. The block matrices Mf'J have Undofj rows and udim columns with the elements M J = J NinijJ ds (7.2.25) The index m is denotes the virtual displacement component which is conjugated (in the sense of the virtual work principle) to the traction component m. By solving (7.2.25) for every element edge r, we obtain parameters pf, which define element boundary traction tf through (7.2.22). It is suggested in the literature (e.g. [Ainsworth Oden, 2000], [Oden Cho, 1997]) that shape functions iJ and ik should be such that MfJ and MjT7 vanish and the matrix M 2The subscript related to the model is dropped. becomes block diagonal. In such a case ( ) simplifies to r}r = Mr pr (7.2.26) This makes the bookkeeping for the computations much easier. However, our experience is that the form of boundary traction obtained in this way is very specific and it does not represent well the stress field. We also note that the specific choice of parametriza-tion (7.2.26) is not necessary, since local patch-wise computation can be performed, see (7.2.21). Regularization of local system (7.2.21) The solution of local system of equations, e.g. (7.2.21) for 2D problem, is not unique, unless we impose an adequate additional regularization. The latter can be derived in accordance with the estimate of the true stress state with the continuous boundary traction across the element interfaces. However, the true stress state is not known, but only its (discontinuous) element-wise approximations obtained by model 1. Continuous stresses at the edge r, denoted as |r, follow from the smooth stress recovery which may be coupled with previous mesh adaptivity procedure. Boundary traction forces resulting from a such stresses ^°|r are obtained by the use of the Cauchy principle tp° = ^o |rne. It is now possible to calculate the effect of this edge loading on neighboring nodes by using3 (7.2.6) rfr = ^ (7.2.27) We want r}r to be as close as possible to this result in the least square sense. We construct the following constrained patch-wise minimization problem: L{r},r; Ar; ae} = 2 £ £ eePj r 1 2 / ^ / ^ vri;r- ri,r + £ a}(R} r},r) eePj r + E Ar(r},r + r},r) (7.2.28) kr(re , re' r where the constraints (7.2.11) and (7.2.10) are also introduced by means of the corre t} and Ay sponding Lagrange multipliers a} and Ay, and Pi is patch of elements around node I of the FE mesh. 3The subscript related to the model is dropped again. The Kuhn-Tucker optimality conditions for stationary point are then given by differentiating the Lagrangian with respect to unknowns rf r: rf ,r - rf,0r - of + A,r = 0 The same equation also holds for the residual force on the neighboring element rf r: rf'r - rf'r - of' + A,r = 0 If we sum up the last two equations and take into account the continuity condition vr rf r + rf r = 0, we can express the Lagrange multiplier Ay as A j / f i f , 6,0 , f',0\ (Of + Of + rf,r + rfr) The unknowns rf r can then be expressed in terms of the multiplier of 1 e e e,0 e ,0 (7.2.29) The condition rf = rf,ri + rf,r2 is rewritten as (rf,r1 - rflrO) + >f,r - ^) + >f - of1 + of - of2) ,61,0 - e,0 62,0 > 2 2 2 By exploiting the above results, the local system for the patch of four elements can be rewritten as (7.2.30) Rf1 " 2 -1 0 -1 rof1] RRf2 1 -1 2 -1 0 oje2 RRf3 = 2 0 -1 2 -1 of3 RRf4 . -1 0 -1 2 . of4 . where we introduced notation RRf = Rf - £ red^i e,0 < rf,r >; e,0 < rfr >= The notation < ■ > implies averaging. The < rf,r > thus represent averaged boundary traction on r (evaluated from FE solution) 'projected' to node I. To summarize, the element-wise boundary traction are computed from FE solution (based on chosen mathematical model) in two steps: (i) by solutions of patch-wise problems in (7.2.30) to obtain rf,r; (ii) by solutions of element-wise problems (7.2.25) to compute nodal values pk of boundary tractions (7.2.22). 1 e,0 e',0 (rfr - rib) 2 i' 1 e r j Local problems as floating structures The local problems (7.2.2) are of Neumann type. An element with only Neumman boundary conditions is essentially a floating structure, and one thus ought to eliminate the rigid body modes to get a unique result. This can be done simply by using the element geometry, as shown in [Park Felippa, 1998]. A rigid body displacement dR of an element is a linear combination of all its rigid body modes dR = Dea (7.2.31) where a is a vector of amplitudes that correspond to rigid body modes, and De is a matrix containing the rigid body modes of the element (arranged column-wise). Since the element stiffness matrix Ke is singular due to the rigid body modes contribution, one can form a modified nonsingular stiffness matrix Ke by adding a product DeDeT to Ke Ke = Ke + DeDeT The inversion of K is possible and solution d ('rigid-body-polluted' element nodal displacements/rotations) can be obtained. Since the matrix DeDeT projects onto the space spanned by rigid body modes, we have d = de + d R where de are real nodal displacements/rotations. As we are interested only in stress resultants, such a 'pollution' is not critical due to the following property KedR = 0. 7.3 Chapter summary and conclusions A general discussion of the model error estimation is given and a concept of hierarchically ordered models is introduced. Hierarchy of models enables computation of a model estimate on the basis of the comparison of the results of two model computations: original (model 1) and the enhanced one (model 2). Comparison of the results of two global computations is clearly not rational. It is therefore desirable if the computation with the enhanced model is done only locally - preferably on finite elements. The local problems are of Neumann type - the boundaries are loaded with boundary tractions representing the action of the surrounding continua. The central question of model error estimation is: how to make an educated guess of the boundary tractions for the local problems? The boundary tractions of the local problems are optimally computed from the true stress state using the Cauchy principle. However, since the true stress state is unknown, we seek for an alternative method for the computation of boundary tractions. The alternative identified as useful is the method of equilibrated residuals (EqR). The EqR method produces the boundary tractions, which exactly represent (in the sense of principle of virtual work) the action of the element residual forces of the original model (model 1). The basic principle of EqR method therefore coincides with the concept of model error estimation. The second part of the chapter is thus devoted to a general discussion of the computational aspects of EqR method related to the model error computation. Chapter 8 Model error indicator for DK elements 8.1 Introduction Analysis of plate structure with complex shape, loading and boundary conditions is one of the most frequently encountered problems in structural engineering practice. A problem of selecting the most suitable model for a particular plate structure, which is the topic of this chapter, has therefore interesting practical aspects. If successfully solved, it can lead to an efficient and accurate plate analysis, which is of great practical interest. Since plate structures are often combined with frame and other skeletal structures, for which one can develop by far the most efficient finite element analysis by exploiting one dimensional form of the governing model and the superconvergence properties of the corresponding finite element method, the solution of the above mentioned problem would clearly have a very practical value. Plates are basic structural elements and historically much attention has been given to the derivation of different models. The derivation of the models has often been based on various mechanical considerations and principles with no rigorous proof of the relationship between the three-dimensional solution and the plate model. This kind of derivation has also been used in this work (see chapters 2-4). A sequence of plate models can be derived in an hierarchical way from three-dimensional model. Using a model reduction a hierarchical sequence of elastic 2D plate models can be be produced from 3D theory ([Babuska et al., 1983], [Szabo Sahrmann, 1988]). The models are controlled by two parameters p and q (p denotes the polynomial degree in terms of standard plate coordinates, and q the polynomial degree in normal direction to the mid-surface). Computable upper-bound error estimates are given by [Babuska Schwab, 1993], presuming that the resulting 2D PDEs (by reduction) are solved with prescribed error tolerances. The plate modeling naturally can thus fit into the framework of hierarchic modeling. The hierarchic sequence of plate models can also be constructed with plate models, based on the engineering insight as those presented in Chapters 2-4. Those models are widely accepted in the practical computations since they are computationaly robust and fast. In our work, we choose three plate models, which form a hierarchic family: the Kirchhoff plate model (chapter 2), the Reissner/Mindlin plate model (chapter 3) and the (1,1,2) plate model with through the thickness stretching (chapter 4). An adaptive structural analysis of a plate structure as organized in our work starts with a simplest possible plate model (Kirchhoff model). The model error with respect to Reisnner/Mindlin model is computed and the regions where more refined model should be used can be identified. The procedure can be iterated until the prescribed accuracy is achieved throughout the problem domain by using the available models. An effective computation of the model error is thus the key of the concept of adaptive modeling. The above described procedure implies the need for a hierarchic family of plate models. The model error indicator procedure described in chapter 7 will be specialized for plates in this chapter. To construct the boundary stress resultants (which are plate counterparts of solid boundary traction) we follow the procedure outlined e.g in [Ladeveze Leguillon, 1983], [Ladeveze Maunder, 1996], [Stein Ohnimus, 1997], [Oden Cho, 1997] for 3D elasticity and briefly presented in chapter 7. The local problem that needs to be solved (to get a model error indicator/estimate) deals with single plate element, which is floating structure, loaded on the surface and along its edges in such a way that the loads are self-equilibrated. We can thus speak about the model adaptivity procedure for plates that is based on equilibrated boundary stress resultants1. Due to the hybrid construction of the DK plate elements, the standard computation of equilibrated boundary tractions is modified. The detailed procedure for the computation of the boundary tractions of the local problems will be given in the following (see also [Bohinc et al., 2006], [Bohinc et al., 2009]). 1We will not distinguish between boundary stress resultants and boundary tractions in what follows 8.2 Model error indicator In order to find equilibrated stress resultants for the DK plate elements, we start with interpolations for displacement and rotations. The displacement along the element boundary riJ, spanned between the element nodes I and J, is defined (according to (2.3.59)) by: w>h,rfj = Wi + Wj + LiJnij ■ ((i - (j) ^3 + ^A0j 4 3 (8.2.1) where A0iJ 2 Lj (Wj — wt)i) — 4 niJ ■ (j,r mra/,r pJ = 0 p^s = msJ,r K pq = qK,r J& =0 „K — (8.2.8) (8.2.9) (8.2.10) (8.2.11) To get the nodal parameters of (8.2.6) from (7.2.25) we have to evaluate integrals M /j /r ds, elements of block matrix MfJ MJ L 2 1 ( N w r q N/s w r m, N/ /n T mn N1 - iJ Ox ^q NOS - iJ Ox^ ms On - iJ O-x mn I N7 \ w, Oy iq NOS Oy^ mS On - iJ O-y mn de (8.2.12) where the following transformation was used: No,,- = -ny Nw + nxNoy, = +nxNox,^ + ny Ngy, (8.2.13) with (■) standing for wW, 0x and 0y. 2 The parametrization is arbitrary, however, the total number of free parameters must not exceed the number of components of input data r i, r J = mnJ,r n e The boundary tractions projections rf,r = [rwr, r^, r7yr] are computed from the patch-wise local systems of equations (7.2.30). The parameters for (8.2.6) are computed form the inverse of the system (7.2.25). We obtain: qK ry^WJ _I_ 'W ' i,r + ' j, r L (8.2.14) mS,r 2ny (+rJ;T - 2r^) + 2ng(+2rjyr - J) L mJ,r 2ny (+r£r - 2J) + 2nx(+2rJ;T - rjyr) L mn,r ny (+33rpT + 21rJyr) + nx(+33r0r + 21rJyr) 2L r I, J = ny (+33rJyr + 21rpT) + n*(+33rJr + 21rpT) -K 2L +r j, mn,r = 15(nx(rfxr + J) + ny (A, + J)) L 8.2.1 Regularization To form the minimization function (7.2.28), the projections rf^ are required. They are computed from the improved approximation of the boundary traction tp0 = [q0, , m® ]. If the improved boundary tractions are expressed in the form (7.2.22), the projections are computed using the equation (7.2.25). The boundary traction approximation tp0 is computed from the current finite element solution using the following technique: edge boundary traction t0 is taken as an average of the boundary tractions tp0 and t^0 computed from the stress resultants of the two elements e and e' adjacent to the edge r: tr = (tf,0 - tp,0)/2 (8.2.15) Boundary tractions tp0 are evaluated at side r from element stress resultants from the Cauchy principle. First, the stress resultants at the integration points mh(£flp) = CBKh(^flp) are evaluated. A linear interpolation of the stress resultants m0 = £ m0Nj (8.2.16) j is constructed using the least squares fit to mh(£gp). At the element side r, defined by unit exterior normal n = [nx, ny]T, we thus have: m°n = (m0y - mXx) nxny + mXy(nX - nl) (8.2.17) ms (mxxnx + myy ny + 2mxy nxny) The shear component normal to the edge is obtained from qy = q0xnx + q0ny where the components qx,qy are computed from equilibrium equations: s (dm°xx , 9m°x^ y dm0 dw^y ,0010, qx = —+ ly-); qy = —+ nr) (8.2.18) The effective shear qf however is computed from qf = qy — ^mp and the derivative ^mp is evaluated from (8.2.17) using the interpolation (8.2.16). Model error indicator for the DK plate elements is obtained by comparing its results with the ones obtained by an element based on more refined plate model RM. As shown above, the later results can be obtained by element-wise computation with the DK boundary stress resultants applied as external edge loading. Thus, a Neumann-type problem has to be solved for each element of the mesh. 8.2.2 The local problems with RM plate element The local problems are written as Kyuu y = Fyxt'e + Ry (8.2.19) where Ky is the RM element stiffness matrix, Fyxt'2 is the external load vector contributed to surface load and Ry are the element nodal residuals due to external edge load. The external load vector Ry = [Ry,I, ■ ■ ■ , Ry)IJ, ■ ■ ■ ]T corresponding to the external edge loading is computed from £ Ry,i ■ v2e,i + 5] R,IJ ■ ve2,ij = f tepe ■ ds = Y f thj ■ V2,pu,hds (8.2.20) i IJ Jve rIU Jvu where V2 = [v2)I, ■ ■ ■ , V2,IJ, ■ ■ ■ ]T. The element edge load ty)rIJ = [?rIU, mS)rIJ, mn>rIJ]T is defined by (8.2.14) and (8.2.6). We have r r e e where the virtual displacements vf,h = [vRM, „RM,„R,M]T are those of RM element. The interpolation of the RM plate elements is an hierarchic extension of the interpolation of the DK element: L vRM = vDK - £ 4 Y/J M/J /J / + 6>j)) 9 6 . /J 93,/J = L W—4,/J Taka interpolacija vsebuje le vozlisca v ogliscih elementa, kjer se nahajajo prostostne stopnje wu/ = [w/, 9x/, 9y/]T. Interpolacije zapisemo v kompaktni obliki kot Wh = NwW = Nw,/wW/; 6h = N w = Ne>/wW/; // Kh = BKwu = B«,/W/ / kjer je = [wW 1, 2, ■ ■ ■ , w-nen] ter nen stevilo vozlisc elementa. Eksplicitna oblika Nw>/ ter Ng)/ je podana v (2.3.64), (2.3.65) ter (2.3.66), kjer je stevilcenje predstavljeno na Sliki 2.3.11. Vektor ukrivljenosti Kh dolocimo iz Kh = [— d9y,h d9x,h d9x,h — d9y,h ] öx ' öy ' öx öy kjer je BK,/ = [BW , BKyy,/, BKxy ,/]T: B = _ ÖN^/ ; B =+ dN^z; B =+ ÖN.x,/ dNöy ,/ r\ ) Kyy,i I r\ 5 Kxy 1 O r\ dx dy dx dy Togostno matriko K DK elementa izracunamo s pomocjo diskretizacije sibke oblike (2.2.51). Zapisana je v obliki blocne matrike K = [K/J]: aK,e(wh; uh) = / kt (uh)m(wh)dQ = Jni = Y UT K/JwJ; K/J =f BT,/CbBk)j dQ /,j Prav tako iz sibke oblike dolocimo konsistentni obtezni vektor f = [f/]T, kjer je f/ = f/,/ + ft,/: lK,e(Uh) = UT // = UT/ = / = /uhdn + J] (qe/ Uh + ms 0h(«h) ' s) ds + [m„«h,r]/ •/nh rj L u //,/ =1 / Nr,/dn; /t,/ = (&/Nr,/ + ms sT Ne>/)ds Trikotni DK element ima tri vozli s c a v ogli s c ih trikotnika s prostostnimi stopnjami Uu/ = w,0x/,0y/]. Interpolacijske funkcije so definirane v trikotni skem koordinatnem sistemu N/ = Z/; N/J = 4Z/ Zj; M// = 4Z/ Zj (Zj — Z/) Interpolacijske funkcije (2.3.71) se na robovih ujemajo s hierarhi cnimi funkcijami (2.3.55). Za integracijo se uporablja 4 to ckovna integracijska shema (Tabela 2.3). Stirikotni DK element ima s tiri vozlisca v ogli s cih stirikotnika s prostostnimi stopnjami Uu/ = w,0x/,0y/]. Interpolacijske funkcije so definirane v koordinatnem sistemu (£,n) G [—1, +1] X [—1, +1]: I 1 2 3 4 IJ 12 23 34 41 r e n e n s n e n e Pr +1 +1 -1 -1 Ps +1 -1 -1 +1 N/ = (1 — pr r)(1 — Pss)/4; N/j = (1 — r2)(1 — Pss)/2; M/j = pr r(1 — r2)(1 — pss)/2 Koordinate in ute z i Gaussove integracijske sheme 2 X 2, ki se uporablja za izra cun togostne matrike in obteznega vektorja, so navedene v Tabeli 2.4. V razdelku 2.4 so predstavljeni izbrani racunski primeri, ki prikazujejo ucinkovitost razvitih koncnih elementov za opis glavnih znacilnosti resitev upogiba tankih plo s c . 10.4.2 Srednje debele plošče Teorija upogiba srednje debelih plosc v nasprotju s teorijo, ki velja za tanke plosce, ne gradi na predpostavki, daje precna stri zna deformacija plos c zanemarljiva. V tem smislu dopusca dodatne deformacijske nacine. Sprostitev predpostavk Kirchhoffove teorije sta prva izvedla [Reissner, 1945] in [Mindlin, 1951]. Teorijo srednje debelih plos c zato imenujemo Reissner/Mindlinova teorija. Pri tem obstaja analogija med Euler-Bernoullijevo in Timo s enkovo teorijo nosilcev ter Kirchhoffovo in Reissner/Mindlinovo teorijo plo s c . Najpomembnejsa razlika je, da zasuki niso vec popolnoma doloceni s pomikom nevtralne ploskve, temvec so od pomika neodvisni. Ena od posledic je, da ni vec zahtevana zveznost zasukov. Mogoce najzanimivejsa posledica te dodatne svobode je obstoj t.i. robnih obmocij, podrocij blizu roba plosce, kjer prihaja do hitrih in velikih sprememb rezultante momenta mn in striznih sil. Robna obmocja pa niso le posledica predpostavk Reissner/Mindlinove teorije plosc, temvec so dejanski fizikalni pojav. Glavne predpostavke Reissner/Mindlinove teorije so: (i) normala na prvotno ref-erencno nevtralno ploskev ostaja ravna tudi po deformaciji, ceprav ni vec nujno pravokotna na deformirano nevtralno ploskev, (ii) debelina plosce se med deformacijo ne spremeni in (iii) normalne napetosti so zanemarljive. Zadnji dve predpostavki se izkljucujeta, kar je lahko preveriti. Da bi obsla to nasprotje, sta Reissner in Mindlin vsak na svoj nacin prilagodila teorijo. Ceprav se teoriji konceptualno razlikujeta, pa obe dajeta prakticno enake rezultate za precne pomike, strizne sile in rezultante momentov v vseh realnih konstrukcijskih problemih. Osnovna posledica zgornjih predpostavk je dejstvo, da deformacije srednje debelih plosc ni mogoce opisati le s precnim pomikom w in njegovimi odvodi. Kot neodvisno spremenljivko moramo vpeljati tudi zasuk materialnega vlakna 0. Model V nadaljevanju namesto zasukov 0 vpeljemo rotacijo 0 (glej Sliko 2.2.1), ki je definirana kot 0 = [0x, 0y]T = [—0y, 0x]T. Kinematicne relacije, ki povezujejo pomik w in zasuk 0 z ukrivljenostjo k in precnimi striznimi deformacijami y so: K = [k k k ]T =[(d0y + )]T K = [Kxx , Kyy , Kxy] = [ dx> dy '( dx + dy )] Y =[7x,7y]T =[ ^ — 0x,|y — 0y]T Ravnotezna enacba je bila izpeljana ze v prejsnjem poglavju in tu samo ponovimo glavni rezultat: V ■ Qeq = —f kjer so strizne sile izpeljane iz ravnotezja oznacene s bmqeq = [qx,eq, qy,eq]T. Hkrati velja (glej (2.2.18)): V ■ M = —qeq; M = Konstitutivna zveza med ukrivljenostjo k in momentom m je m = CB k mxx mxy mxy myy cEt q = CsY; Cs = cGtl = 1 = c/t2D1; 1 Strizne deformacije, ki jih izracunamo iz kinematicnih predpostavk so konstantne po debelini. Ravnotezne enacbe po drugi strani kazejo, da variacija striznih napetosti po debelini mora obstajati. Pri homogenem preseku je variacija po debelini parabolicna funkcija z. Strizne deformacije, ki izhajajo iz kinematike, so torej v nasprotju z ravnotezjem diferencialnega elementa. Za preseganje tega nasprotja se uvede strizni korekcijski faktor c, ki konstitutivni zakon, ki veze strizne sile in deformacije spremeni tako, da je zagotovljen korekten izracun strizne deformacijske energije: " 1 0 01 kjer je G = E/2(1 + v) strizni modul in c = 6c(1 — v). Za strizni korekcijski faktor c obicajno vzamemo vrednost 5/6. Reissner/Mindlinova teorija omogoca bogatejso mnozico robnih pogojev, kot pri Kirch-hoffovi teoriji. Najpomembnejsa razlika je, da lahko na vsakem robu definiramo tri neodvisne komponente robne obtezbe (precno strizno silo q ter rezultanti momenta mn in ms). Nekatere od moznosti so navedene v Tabeli 2.1. Deformacija srednje debelih plosc je dolocena s sistemom sklopljenih diferencialnih enacb t2 AAw = f/D — — Af J' Dc t2 ■ 12C AA0x — (0x,y — 0y,x),y = f,x/D t2 + 12C AA^y + (^x,y — 0y,x),x = f,y/D Poznavajoc f, D, t in c in ustrezne robne pogoje, lahko iz zgornjega sistema izracunamo precni pomik w in rotaciji 0x ter 0y. Z uvedbo Marcusovega momenta m = D(^^^r + l|f), za katerega velja Am = f ter zvitja, ki ga definiramo kot 0 = ^ — lif in za katerega lahko pokazemo, da velja Q = ^ — ^ = iL0 dobimo alternativni sistem diferencialnih enacb za w, 0 ter m öx öy 12c ' ' — AAw = f/D — ^Af; n = f— ^lyx ^ Am = f ki je sklopljen preko robnih pogojev. Resitev Reissner/Mindlinove teorije lahko prikazemo kot razsiritev resitve Kirchhoffove teorije. Povezavo obravnavamo v razdelku 3.2.2. Princip virtualnega dela za srednje debele plosce je izpeljan iz sistema diferencialnih enacb. V ravnovesju velja (w, 0; 5w, 50) = 5next (5w, 50) kjer je (w, 0; $w,$0)= / ($ktm + $7Tq) dQ In in Text $next ($w,$0)= / dQ+ / quds + / ds + / ds Robna območja Robna območja so območja ob robu plošče, kjer se zasuki in rezultante napetosti hitro spreminjajo. Pojav je bil v literaturi obsirno raziskan, zato na tem mestu povzemamo le glavne rezultate. Bistvena razlika med Reissner/Mindlinovo in Kirchhoff ovo teorijo je, da je resitev slednje neodvisna od debeline plosče (razen seveda za konstantni faktor). Odvisnost resitve Reissner/Mindlinove teorije od debeline plosče je kompleksna, vendar jo lahko obravnavamo kot perturbirano Kirčhhoffovo resitev. Iz relačij, zapisanih v razdelku 3.2.2, hitro prepoznamo, da je vir hitrih sprememb v resitvi lahko le funkčija Q = — 0y>x. Preostali funkčiji ^ in $, ki prispevata k razliki, sta namreč harmonični (A^ = 0, A$ = 0). Iz enačbe Q = it2;AQ je razvidno, da člen na desni konvergira proti 0, ko gre t ^ 0. V limiti velja Kirčhhoffova predpostavka = 0y>x. Hkrati je razvidno, daje sirina robnega območja sormazmerna s Prečni pomik w kljub prisotnosti robnih območij ne izkazuje hitrih sprememb, saj ni neposredno odvisen od funkčije Q, ki je edini izvor za hitre spremembe v robnem območju. Resitev za prečni pomik w ter zasuka 0 lahko razvijemo v potenčno vrsto po debelini (3.2.38). Vodilni členi v razvoju so prikazani v Tabeli 3.2 in so odvisni od robnega pogoja. Mogoče nekoliko presenetljivo se pokaze, da je robno območje najizrazitejse pri prostem in mehko prosto podprtem robu. Druga skrajnost je vpeti rob, pri katerem robnega območja ni. Klub velikim gradientom pri tankih plosčah pa so vrednosti rezultant napetosti vedno končne in robno območje ni vzrok dodatnim singularnostim. Podobno kakor pri tankih plosčah, tudi pri srednje debelih plosčah v določenih primerih pride do singularnosti v rezultantah napetosti. Običajno do singularnosti pride v vogalih plosče, ko je kot večji od nekega mejnega kota, ki je odvisen od robnih pogojev, ki veljajo v opazovanem vogalu. Mejni kot je sičer neodvisen od debeline plosče. Singularnost je omejena na neposredno okoličo vogalne točke. Tabela 3.3 prikazuje eksponenta Ai in A2 za različne kote vogala za trdo in mehko prosto podprto plosčo. Končni elementi Koncne elemente, ki ustrezajo Reissner/Mindlinovi teoriji, je bistveno lazje konstruirati, kot tiste, ki ustrezajo Kirchhoffovi teoriji in morajo ustrezati zahtevi po zvezni odvedljivosti interpolacije pomika. Bistvena razlika je tudi v dejstvu, da sta pomik w in zasuk 0 interpolirana neodvisno. Neodvisnost obeh interpolacij znatno olaj sa gradnjo koncnega elementa. Hkrati morata biti interpolaciji izbrani zelo skrbno, da ne pride do pojava, ki mu obi cajno recemo blokiranje ("locking"). Interpolacije, ki niso usklajene, namrec ne zmorejo opisati stanja z ni celno strizno deformacijo, kar povzroci, da v limiti tankih plo s c, stri z ni deformacijski nac in prevlada nad upogibnim. Stri z na deformacijska energija je namrec premosorazmerna debelini plosce, medtem ko je upogibna deformacijska energija sorazmerna tretji potenci debeline. V limiti tankih plosc je zato strizna deformacijska energija bistveno vecja od upogibne. Prevladujoc del minimizacije potencialne energije je zato zmanj sanje strizne deformacijske energije, kar v limiti neskoncno tanke plo s ce privede do pretogega odziva. V nadaljevanju sta predstavljena trikotni in stirikotni element za plos ce z oznako P3T in P3Q, ki sta grajena kot hierarhi cno nadaljevanje DK elementov. Elementa sta bila prvi c predstavljena v [Ibrahimbegovic, 1992] and [Ibrahimbegovic, 1993]. V tem delu sta elementa izpeljana na novo, pri cemer je posebna pozornost namenjena skrbi, da se P3 elementa predstavi kot hierarhi cno nadgradnjo DK elementov. Interpolacijo za pomik in rotacijo zapi semo po zgledu interpolacij za DK elemente. Spet se opremo na analogijo z nosilci. Tokrat namesto zahteve po ni celni stri zni deformaciji vzdolz nosilca, predpi semo konstantno vrednost Yh = — = 7o = const. To nas pripelje do naslednjih vrednosti za parametre interpolacije 3.3.8: L « « ^ Lw2 — w1 1- « L ^ - 6 ^ w3 = +8— w4 = 4(—L— 2( + ^— 4-o; - = Lw4 Interpolacijo pomika in zasuka za Timo senkov (TM) nosilec lahko predstavimo kot hierarhi cno raz siritev interpolacije za Euler Bernoullijev (EB) nosilec: TM _ „,,EB L - At . nTM _ nEB , 3 - AT wh = wh — 4 7oN4; = Ph + 2 Y0N3 kjer je interpolacija za EB podana v (2.3.57)-(2.3.58). Po analogiji zapi semo interpolacijo za P3 elemente kot hierarhi cno raz siritev interpo- lacije za DK elemente (2.3.63): 1 wh = - Y 4 LiJÜijMiJ; ij 3 Oh = 0DK + E 2 niJYiJNiJ iJ 2 Strig vzdolz stranice IJ je oznaCen z üIJ, LiJ je dolžina stranice IJ in niJ je normala na stranico IJ, glej Sliko 2.3.11. Podobnost med (3.3.9) in (3.3.10) je oCitna. P3 elementi imajo poleg vozlisc v ogliscih, kjer so definirane prostostne stopnje enake tistim za DK elemente üI = [WI, 0xj, 9y j]T, tudi vozlisca na sredini stranic, kjer so definirane prostostne stopnje üIJ = [üIJ]. Prostostna stopnja YiJ je enaka vrednosti strizne deformacije vzdolz stranice IJ. Elementi imajo nen vozlisc v ogliscih in nen vozlisc na sredini stranic (nen=4 za stirikotnike, in nen = 3 za trikotne elemente). Oglisca so ostevilcena z I = 1,..., nen, vozlisca na sredini stranic pa od IJ = (nen + 1),..., 2 nen. Prostostne stopnje elementa so zbrane v uu = [uuT, ■ ■ ■ , UU^, üTen+1, ■ ■ ■ , ü2nen]T. V matricni notaciji interpolacije P3 elementov zapisemo z Wh = Nwü = Y Nw,iüi + Y NwjjUij; Nw,ij = -L4JMu i iJ 3 Oh = Neü = J] Ne,iüi + Y Ne,ijüij; Nejj = nuNJ i iJ 2 kjer so NW)i = NDK, Ngj = NDk matrike, ki so bile definirane pri obravnavi DK elementov, glej (2.3.63)-(2.3.66). Dodatni cleni interpolacij (3.3.11) in (3.3.12) so odvisni le od prostostnih stopenj na sredini stranic, ki so zbrane v üIJ. Vektor ukrivljenosti Kh izracunamo direktno iz Oh: Kh = BKÜ = BK,I üi + ^2 BK,IJ ü i J i iJ DK k,i . Matrika Bki je enaka matriki DK elementa, ki je definirana v (2.3.68): Bki = B Interpolacijo ukrivljenosti tako lahko zapisemo: Kh = kdk + Kh, kjer je Kh del, ki je odvisen le od üIJ: k h = ^2 Bk,ij ü i j ; B„, .jU.JJJ......J - j J Ceprav so bile interpolacije za wh in Oh konstruirane tako, da je strizna deformacija vzdolz stranic konstantna in kot taka lahko tudi enaka nic, kljub temu ni sposobna opisa čiste upogibne deformačije brez dodatnih motenj v obliki strizne deformačije. Zato je blokiranje elementa se vedno prisotno. Robustno in učinkovito zdravilo za blokiranje elementa je pristop s predpostavljenim poljem stri zne deformačije (assumed shear strain approačh). Stri z no deformačijo interpoliramo kot bilinearno interpolačijo Yh = Y N/Yh,i = Y by,ij«i J = B7u i /j kjer so Yh;I vozli s čni parametri interpolačije pre čne striz ne deformačije in N/ standardne Lagrangeove interpolačijske funkčije. Parametri Yh)I so izbrani tako, da so v skladu z vrednostmi 7IJ, ki določajo konstantno vrednost strizne deformačije vzdolz straniče /J. V vozlisču /, kjer se stikata straniči /J in /K, so vozli sčni parametri YhI izbrani tako, da projekčije Yh)I na straniči /J in /K ustrezajo vrednostim 7IJ in yik zaporedoma (glej Sliko 3.3.1): Yh,i ■ «u = Uij; Yh,i ■ Sik = Y/K Re sitev linearnega sistema je: Tik n/j — Uij n/K Yh,i =- n/j ■ Sik Vozli s čne vrednosti striz ne deformačije Yh)I so odvisne le od vrednosti YIJ. Posledi čno je interpolačija Yh odvisna le od üIJ: Yh = ^ B7)/jU/j /j Stri z na togostna matrika KS je definirana v 3.3.21 in 3.3.23. Ekspličitna oblika matrike B7)IJ, ki se nanasa na straničo /J, je: B7)ij = I nj—— — n/—r}:el " j n JK ■ s/j n/j ■ sh^ Ky Zgornja izvajanja veljajo tako za stirikotne, kot tudi trikotne elemente. Interpolačijske funkčije N/, NIJ ter MIJ so enake tistim, ki so uporabljene pri DK elementih. Parametri Yiv, ki so definirani v vozli s čih na sredini robov, otezujejo prakti čno uporabo P3 elementov. Zato jih obravnavamo kot neodvisne koli čine in jih odstranimo s pomočjo stati čne kondenzačije. Parametri Yiv pa ne vplivajo le na interpolačijo Yh, temveč tudi na Kh, saj velja Kh = B« U = Y B«,/ U i + Y Bk,/j n/j Yij ; i /j Prepoznamo, da so dodatni deformacijski nacini definirani z BK,/jn/j in tedaj Y/j lahko interpretiramo kot njihove amplitude. Dodatne deformacijske nacine obravnavmo kot nekompatibilne in jih izlocimo s staticno kondenzacijo. Da bi zagotovili, da element opravi "patch" test (glej [Ibrahimbegovic, 1992]), moramo matriko BK)/J spremeniti na naslednji nacin bk,/j ^ bk,/j — — / BK,/Jd0 O Togostno matriko elementa reduciramo s standardnim postopkom staticne kondenzacije. Tako konstruirani elementi so oznaceni s PIT/PIQ. Razdelek 3.4 obravnava vrsto razlicnih racunskih primerov, ki prikazujejo ucinkovitost izpeljanih koncnih elementov P3 in PI. Posebna pozornost je namenjena racunskim primerom, kjer je opazen razvoj robnih obmocij. 10.4.3 Debele plošče Logicni naslednik Reissner/Mindlinovega modela v hierarhiji modelov za upogib elasticnih izotropnih plosc, je model visjega reda, ki uposteva tudi deformacije vzdolz debeline. Model, ki je predstavljen v nadaljevanju, je poseben primer modela lupine, ki je predstavljen v [Brank, 2005] in [Brank et al., 2008]. Plosco obravnava kot 2d ploskev v polnem trirazseznem napetostnem stanju. To je ugodno, ker ni vec potrebno uporabljati poenostavljenih 3d konstituvnih relacij, ki izhajajo iz predpostavke ravninskega napetostnega stanja. Prav tako so robna obmocja, ki so polni 3d pojav, bistveno bolje obravnavana z modeli visjega reda. Model za debele plosce je umescen med Reissner/Mindlinov model in 3d model, zaradi cesar je primeren za uporabo v postopku prilagodljivega modeliranja. V nadaljevanju je predstavljen model visjega reda za opis deformacij debelih plosc in koncni element, ki temelji na razvitem modelu. Model Gradnjo interpolacije zacnemo z opustitvijo predpostavke o nicelni normalni napetosti: ^zz = 0. V primeru 3d linearne elasticnosti tako velja: CTzz = (A + 2^)^zz + A(£xx + £yy) = 0 z A = (1+vV(E-2v), ß = 21+vy kot Lame-jevimi koeficienti. Kakorkoli, v limiti t ^ 0 velja predpostavka o ravninskem napetostnem stanju A A + 2ß Tovrstno limitno obnas anje je mogo ce le, c e so vse komponente deformacije eyy, ezz enakega reda glede na koordinato z. Le v tem primeru se lahko v zgornjem izrazu za azz c leni medsebojno od s tejejo. Ce sta npr. eyy linearni funkciji koordinate z, mora biti taka tudi komponenta ezz. Ce naj bo ezz linearna funkcija z, potem mora biti pomik v smeri uz najmanj kvadrati cna funkcija z, ker velja ezz = . Kinematiko pomikov debele plo s ce zapi semo z u = uRM + kjer uRM ozna cuje kinematiko Reissner/Mindlinovega modela uRM = wnn — z0; nn = [0, 0,1]T in oznacuje pomike, ki povzrocijo deformacije po debelini (glej Sliko 4.2.1) ((t/t — 1)z — Kzz z2/2) nn = (42 z — ^ZZ}z2/t) nn Komponenta je kvadrati cna funkcija z koordinate: t je debelina deformirane plos ce, t je debelina plos ce pred deformacijo in kzz = eZZ)2/t je parameter deformacije, kjer je nn normala na nevtralno ploskev pred deformacijo. Smer ustreza smeri nn. Relativna sprememba debeline je ozna cena z = t/t — 1. Komponente pomika w = [ux ,uy,uz]T so tako: ux = —; uy = —z0y; uz = w + eZ°Z) z — 4*? z2/t in deformacije, ki iz njih izhajajo (4.2.3) so zKxx; 2exy zKxy eyy = — zKyy; 2exz = 7® + 4z)xz — 4z)xz /t ezz = eZz? — 2zeZz?/t = ezz? — zKzz; 2eyz = 7y + 4z)yz — 4z)yz /t kjer so uporabljene definicije iz poglavij o tankih in srednje debelih plos c ah: Kyy = 0y,y; 7y = w,y — 0y Kxy 0x,y + 0y,x Po analogiji z upogibnimi deformacijskimi cleni, ki so sorazmerni s koordinato z, smo v (4.2.4) uvedli "ukrivljenost" kzz: Kzz = 4*z7(t/2) Hookove relacije za linearno elasticni material so: xx (A + 2ß) ^xx + A,/U/; ezz,h Yh Yh1) EB ,/J U /J; /j > B7(i),/ii/; / K(1) = Kzz,h = B (1) rU/; '22 (2) ^B7(2)_/U/ + E B, (2) ,/JU/J /J kjer je Kh = [K h] ; Yh Yh1) zz,h dfczz,h .(o) öx [7x,h,7y,h]T T (2) de(1) zz,h dfczz,h (1) öx T Pri uporabimo definicije iz razdelka 4.3.1. Togostno matriko K PZ elementa dobimo z vstavljanjem diskretizacije (4.3.2) v (4.2.11), pri cemer upostevamo konstitutivne relacije (4.2.13), (4.2.15) in (4.2.24). Sestavljena je iz treh delov, ki ustreza upogibni, strizni togosti in togosti pri deformaciji vzdoz debeline: Togostno matriko izrazimo kot blocno matriko, kjer K = [Kj], kjer bloki Kj ustrezajo interakciji prostostnih stopenj vozlisc i in j. Indeksi, ki se nanasajo na vozlisca v ogliscih, so oznacena z I in J. Z indeksoma IJ in KL so oznacena vozlisca na sredini stranic. Togostna matrika je organizirana kot: V razdelku 4.5 so obravnavani nekateri razlicni racunski primeri, ki prikazujejo ucinkovitost formulacije koncnega elementa PZ za izracun odziva debelih plosc na zunanjo obtezbo. 10.5 Ocene napak 10.5.1 Napaka diskretizacije Metoda koncnih elementov je racunska metoda, ki daje le priblizno resitev danega robnega problema ([Zienkiewicz Taylor, 2000], [Ibrahimbegovic, 2009]). Priblizek resitve uh, ki ga dobimo z metodo koncnih elementov, v nadaljevanju imenujemo računska resitev. Zelja po oceni kvalitete racunske resitve v primerjavi s tocno resitvijo je zato dokaj naravna. Glavna razloga za oceno, kako blizu tocni resitvi je racunska resitev, sta: (i) nadzor nad natancnostjo racunske resitve in (ii) vodenje prilagodljive gradnje mreze, ki za dano stevilo koncnih elementov zagotovi optimalno resitev po celotnem obmocju (glej [Ainsworth Oden, 2000], [Ladeveze Pelle, 2006], [Stein Ramm, 2003]). Nadzor na natacnostjo potrebujemo, da uspemo zagotoviti, da je racunska resitev znotraj vnaprej dolocenih meja napake. V kolikor zelimo dolociti stroge meje napake, za katere lahko jamcimo, govorimo o ocenah napake. Ceprav je poznavanje tocnih meja napake zelo K = Kb + KS + K Z K = r [KJ [Kikl] - [ [Kiju] [Kijkl] _ K Obtezni vektor je sestavljen iz treh delov f = ff+ + ff- + ft, kjer je zazeljeno, so po drugi strani take ocene obicajno zelo konzervativne in lahko precenijo napako za vec redov velikosti. Zelo dobro oceno velikosti napake dobimo s tako imenovanimi indikatorji napake ([Babuska Rheinboldt, 1978]), ki pa ne jamcijo meja. Ne le, da je izracun indikatorjev napake veliko lazji od izracuna strogih meja, se indikatorji napake bistveno bolje priblizajo dejanski napaki. Zato predstavljajo ucinkovito in zanesljivo osnovo za prilagodljivo gradnjo mreze. Z drugimi besedami: indikatorje napake uporabljamo za ucinkovit nadzor nad algoritmi za prilagodljivo mrezenje, da dobimo optimalno natacnost racunske resitve z minimalnim stevilom prostostnih stopenj. V nadaljevanju se ocene napake vedno nanasajo le na indikatorje napake in ne razlikujemo striktno med ocenami meja napake in indikatorji napake. Druga klasifikacija ocen napake diskretizacije zadeva nacin ocene: a priori ocene napake proti a posteriori nacinom ocene napake. A priori error ocene se ukvarjajo z asimp-toticno oceno obnasanja napake glede na velikost koncnih elementov h in stopnjo poli-nomske aproksimacije p. Ne zagotavljajo informacije od dejanski napaki racunske resitve. A posteriori ocene se za oceno napake zanasajo na postprocesiranje racunske resitve V osnovi locimo dve veliki skupini a posteriori ocen, ki temeljijo na (i) izracunu izboljsanih gradientov racunske resitve ter (ii) na izracunu residuala racunske resitve. Ocene, ki temeljijo na izracunu izboljsanih gradientov racunske resitve so veliko pop-ularnejse, predvsem na racun lazje implementacije. Posebej razsirjena je metoda SPR (Superconvergent Patch Recovery), ki se zanasa na obstoj t.i. superkonvergencnih tock, to je tock, kjer je konvergenca k tocni resitvi hitrejsa kot sicer. Iz vrednosti racunske resitve, vzorcenih v teh tockah, se zgradi izboljsano (zvezno) polje gradientov resitve. Tako izboljsano polje sluzi kot priblizek polju gradientov prave resitve. Kot indikator napake sluzi energijska norma razlike med izboljsanimi gradienti in gradienti racunske resitve. Metode, ki temeljijo na izracunu residuala racunske resitve so bodisi eksplicitne ali implicitne ([Ainsworth Oden, 2000], [Ladeveze Pelle, 2006], [Ladeveze Leguillon, 1983], [Stein Ramm, 2003], [Becher Rannacher, 2001]). Eksplicitne metode ocenijo napako z neposrednim izracunom norme residualov v notranjosti elementov ali z oceno gradientnih skokov na robovih elementov. Implicitne metode vkljucujejo tudi formulacijo ter izracun pomoznih robnih problemov, katerih resitev je priblizek dejanske napake. Pomozni problemi so lokalni v smislu, da so definirani na koncnih elementih ali krpah koncnih elementov. Kljucna pri ocenah te vrste je zagotovitev ustreznih robnih pogojev (Neumannovega tipa) za lokalne probleme. Klasifikacija različnih a posteriori metod za oceno napake diskretizacije je prikazana na Sliki 5.1.1. Zgoraj omenjene napake se vse ukvarjajo z globalno energijsko normo napake. Pri analizah s končnimi elementi je ve čkrat slučaj, da nas posebej zanima dolo Cena koli čina (npr. napetost v dolo C eni to Cki). Za oceno napake teh koli čin so bili razvite metode za ciljno-orientirane ocene napake [Ainsworth Oden, 2000]. Ključna za oceno napake ciljne koli čine je definicija pomoznega problema, ki je dualen primarnemu problemu. Pomoz ni problem na nek način filtrira potrebno informacijo za točen izračun ocene napake ciljne koli čine. Kljub nedavnemu razmahu teh metod, ta metoda ocene napake ni obravnavana v tem delu. Napaka diskretizacije je definirana kot razlika med točno resitvijo u ter računsko re s itvijo uh eh = u - Uh Ta definicija je lokalna, saj se nanas a na določeno to čko območja. Da bi dobili globalno oceno napake, izračunamo energijsko normo napake: l|eh||E = a(eh, eh) Integral po območju Qh razdelimo v prispevke po elementih II II2 — II II2 H eh H E = |ee>hÜE e SPR in EqR metodi za oceno napake diskretizacije V nadaljevanju je podan kratek povzetek dveh metod za oceno napake diskretizacije, ki ju podrobneje obravnavamo v nadaljevanju. Napako v obeh primerih ocenimo iz ocene za gradiente to čne re sitve a* (napetosti) po ||elE = / (a* - ah)TC-1(a* - ah)dQ JQh kjer so a gradienti točne resitve in matrika C predstavlja splosno konstitutivno matriko. Integral v (5.3.3) razbijemo v prispevke po elementih -*||E = Y, ne2; ne2 = W (a* - ah)TC-1(a*-„hi ||e*||E = £ne2; ne2 = £/ (a* - ah)TC-1(a* - ah)dQ e e jQh kjer je ||e* |E indikator globalne napake, izrazen kot vsota lokalnih ocen napake ne2 Bistvena znacilnost metode SPR je, da zgradi izboljsano resitev za gradiente resitve na podlagi vrednosti, ki jih iz racunske resitve odcita v t.i. superkonvergencnih tockah cth(£gp). V okolici vsakega vozlisca po metodi najmanjsih kvadratov dolocimo polinomsko aproksimacijo, ki je najblizje izracunanim vrednostim v superkonvergencnih tockah, ki so izbranemu vozliscu najblizje. Izboljsano vrednost gradientov v vozliscu ct* odcitamo iz tako zgrajene polinomske aproksimacije. Zvezno interpolacijo gradientov ct* v poljubni tocki dolocimo tako, da na vsakem elementu zgradimo linearno interpolacijo, katerih parametri so izboljsane vrednosti ct* v vozliscih ct* = i a*Ni. Povzetek izracuna ocene napake diskretizacije po metodi SPR je podan v Algoritmu 1: 1. Doloci racunsko resitev uh problema (5.2.4) ter izracunaj CTh(xgp) v superkonvergencnih tockah xgp 2. Izberi interpolacijo (5.3.10) vsake komponente a* v okolici vozlisca i 3. Izracunaj [Mij] ter [bi] iz (5.3.14) 4. Resi (5.3.14) za ak 5. Doloci ct* iz (5.3.12) 6. Iz interpolacije (5.3.2) doloci ct* Pri EqR metodi izboljsano resitev dolocimo na podlagi resitev lokalnih robnih problemov. Lokalne robne probleme obicajno formuliramo na koncnih elementih, ki so po robu obremenjeni z Neumannovimi robnimi pogoji. Kljucni element EqR metode je dolocitev robnih pogojev za posamezne koncne elemente. Dolocimo jih tako, da je lokalna resitev na izbranem elementu z danimi robnimi pogoji enaka originalni racunski resitvi na istem elementu, ce racunamo z istim koncnim elementom. S to zahtevo hkrati poskrbimo za to, da so robni pogoji zvezni na vsakem robu in uravnotezeni z delovanjem zunanje obtezbe. Preostane se zadnji korak: izracun resitev lokalnih robnih problemov z izboljsano diskretizacijo. Glavni koraki metode EqR so definirani v Algoritmu 2: 1. Doloci racunsko resitev uh problema (5.2.4) 2. Izracunaj uravnotezeno robno obtezbo lokalnih problemov tp za vsak rob r C re vsakega elementa e (glej Algoritem 3) 3. Priblizno resi lokalni problem (5.4.28) za Ue,h+ z uporabo testnega prostora Vh+ kot izboljsanega priblizka k testnemu prostoru Vh 4. Izracunaj lokalni indikator napake n2 iz (5.4.24) in = Ue,h+ — ue,h Izracun uravnotezene robne obtezbe za lokalne robne probleme na elementih poteka posamicno po vseh krpah elementov okrog vseh vozlisc mreze. Najprej izracunamo vo-zliscne residuale elementov (vozliscne reakcije elementa) R iz racunske resitve Na podlagi preproste ocene dolocimo prvi priblizek obtezbe po robovih elementov tf, ki jo integriramo po principu virtualnega dela v projekcije f r. Izberemo vozlisce I in iz R ter projekcij f r za vse elemente krpe elementov, ki se stikajo v vozliscu I, zgradimo linearni sistem za neznane projekcije r7r. Ko resimo vse linearne sisteme za vsa vozlisca mreze, dobimo za vsak rob (med vozliscema I in J) projekciji r7r te rJ,r. Projekciji izhajata iz se neznane robne obtezbe tJ iz principa virtualnega dela zunanjih sil. Ko izberemo primerno diskretizacijo tJ, izracunamo projekcije obtezbe, ki jih izrazimo s parametri diskretizacije. Ker projekciji poznamo, lahko izracunamo parametre diskretizacije robne obtezbe in s tem dolocimo iskano robno obtezbo. Glavni koraki postopka uravnotezenja robnih obtezb so povzeti v (3): 1. Izracunaj R iz in (5.4.32) 2. Konstruiraj ff iz (5.4.40) ter (5.4.39) 3. Izracunaj projekcije rf,r obtezbe ff iz (5.4.37) 4. Oblikuj sisteme enacb za krpe elementov okrog vsakega vozlisca (5.4.35) in jih resi za neznanke r|,r s pomocjo (5.4.38) 5. Izberi interpolacijski funkciji ik, iJ za robno obtezbo tf ter doloci vozliscne parametre interpolacije pf,/ iz (5.4.43) 6. Izracunaj robne obtezbe iz (5.4.40) V nadaljevanju poglavja 5 sta SPR in EqR metodi ilustrirani na preprostem 1d modelnem problemu visece palice. Poglavje 6 obravnava implementacijo EqR metode za DK in RM elemente. Podrobno so obravnavani vsi koraki EqR metode. Posebna pozornost je namenjena primerni izbiri diskretizacije robne obtezbe ter resevanju lokalnih robnih problemov. Pomembno spoznanje je, da mora biti diskretizacija, ki jo uporabimo pri resevanju lokalnih problemov, izboljsana glede na diskretizacijo, ki je bila uporabljena za dolocitev prvotne racunske resitve. Izboljsavo, oz. razsiritev diskretizacije lahko izvedemo tako, da razdelimo element na manjse elemente, kjer uporabimo prvotno diskretizacijo. Tak pristop z razlicnimi delitvami je bil preiskusen na primeru DK elementov (glej razdelek 6.2.3). Alternativno lahko za izbolj sano diskretizacijo uporabimo tudi razlicen element, katerega stopnja poli-nomske aproksimacije je visja od prvotno uporabljenega. Tak pristop je bil preskusen s trikotnim elementom ARGY. Element uporablja polinomsko aproksimacijo petega reda z 21 prostostnimi stopnjami, kar je znatno vec od 9 prostostnih stopenj pri DKT elementu, ki je bil uporabljen za originalni izra cun. Racunski primeri prikazujejo izracun ocene napake po SPR in EqR metodi za izbrane probleme. Pri tem je analiziranih ve c razli cic metode EqR. Prikazana je primerjava ocen napake s pravo napako, ki jo izracunamo na podlagi primerjave racunske resitve s tocno re sitvijo. Analiza pokaze, da so metode po ucinkovitosti primerljive med seboj in nobena bistveno ne odstopa. Indeks u cinkovitosti je blizu 1 za vse obravnavane metode. Poglavje se zakljuci z racunskim primerom, ki za izbran problem prikazuje prilagodljivo generiranje mrez e, katerega osnova so razli cne ocene napake. Ucinkovitost razli cnih metod za oceno napake pri kontroliranju procesa prilagodljivega mrezenja primerjamo z razporeditvijo napake po elementih. V idealnem primeru bi bila razporeditev napake po elementih enakomerna. Rezultati analize pokazejo, da razli cne metode ocene napak med seboj prakticno nerazlocljive, saj so histogrami razporeditve napak po elementih po prilagodljivem mrez enju med seboj, vsaj na kvalitativni ravni, zelo podobni. 10.5.2 Ocena modelske napake Za modeliranje izbranega fizikalnega pojava obicajno ne uporabimo najpopolnej se ob-stojece teorije, ceprav bi bila s stali s c a natancnosti najprimernej s a. Vodilo pri izbiri modela je ucinkovitost glede na zahtevano natancnost. Preden zgradimo model, moramo poznati zahteve za natancnost njegovih napovedi. Te namrec preverjamo z preskusi, katerih natancnost je vedno omejena. Izbira primernega modela je tesno povezana z inz enirskim pristopom, ki zagovarja, da mora biti izbrani model preprost, kot je le mogoce in le tako zapleten, kot je potrebno. Ker vsak model sloni na dolocenih predpostavkah, moramo zagotoviti, da so njegove predpostavke izpolnjene z vnaprej predpisano natan c nostjo. Veljavnost predpostavk je funkcija kraja, kar pomeni, da je veljavnost modela krajevno omejena. Cilj prilagodljivega modeliranja je identifikacija obmocij veljavnosti posameznega modela in dolocitev najprimernej sega modela (glede na predpisano natancnost) za vsako obmo cje. Ideja za postopno gradnjo optimalnega modela temelji na hierarhi cno urejeni druzini modelov. ceprav modeli niso nujno zgrajeni na podlagi asimptoti cnih principov, temvec so ve čkrat skonstruirani na podlagi inz enirskega uvida, Vsi modeli izhajajo iz iste teorije. Urediti jih je mogo če glede na stevilo predpostavk, ki jih uporabljajo in posledi čno glede na ujemanje njihovih rezultatov z rezultati, ki jih napove teorija. Pri gradnji optimalnega modela začnemo od spodaj navzgor z najpreprostej sim modelom ter poisčemo njegovo računsko resitev. Iz resitve na vsakem elementu ocenimo veljavnost njegovih predpostavk in s tem ocenimo modelsko napako. Na območju, kjer je napaka večja od predpisane napake, model zamenjamo z naslednjim modelom iz druzine hierarhi čno urejenih modelov. Postopek ponavljamo, dokler v celotnem obmo čju modelska napaka ne pade pod predpisano mejo. Ključen element opisane strategije za prilagodljivo modeliranje je ocena modelske napake. Ocena modelske napake mora biti čim natančnej s a in učinkovita. Modelsko napako najbolj neposredno in natančno ocenimo tako, da primerjamo izračun izbranega modela z izra čunom referen čnega modela/teorije. Ker je re s evanje referen čnega modela drago, se zadovoljimo s primerjavo z naslednjim bolj sim modelom iz urejene druz ine modelov. Tak pristop je sprejemljiv, ker se zadovoljimo z oceno napake. Kljub temu je predlagan nač in ocene modelske napake neu č inkovit, ker predvideva dvakratno resevanje celotnega problema. Namesto tega račun z izbolj sanim modelom ponovimo le na posameznih manj s ih območjih - elementih. Pri tem moramo poskrbeti, da je resitev lokalnih robnih problemov na elementih čim blizje globalni re s itvi. Robni pogoji za lokalne probleme so Neumannovega tipa in predstavljajo obtezbo po robu elementa. Osrednje vprasanje takega pristopa postane, kako določiti robno obtezbo lokalnih problemov na način, da bodo ti čim natančneje replicirali rezultate globalnega izračuna. Idejo za konstrukcijo robnih obtezb lokalnih problemov najdemo pri EqR metodi, ki smo jo obravnavali v okviru ocene diskretizacijske napake. Spomnimo, da EqR metoda zgradi robne obtez be za elemente tako, da re s itev lokalnih problemov natanko ustreza globalni re s itvi, če so za izračun lokalnih resitev uporabljeni isti končni elementi, ki so bili uporabljeni za globalni izračun resitve. Tak pristop uporabimo tudi za račun modelske napake, le da lokalni problem, namesto z izbolj s ano diskretizacijo, re sujemo z izbolj s anim modelom. V 7. poglavju je natančno opisan predstavljen koncept ocene modelske napake. Splo sno so obravnavani vsi koraki konstruiranja lokalnih problemov po zgledu EqR metode. Bistvena je jasna ločitev med projekcijami obtez be ter dejansko robno obtezbo, ki omogoč a bolj se razumevanje omejitev in morebitnih potencialov predlaganega pristopa. V 8. poglavju je splo sni koncept izračuna modelske napake implementiran za primer ocene modelske napake za Kirchhoffov model. Ker je koncni element DK, ki temelji na Kirchhoffovi teoriji, nekompatibilen, je dobro razumevanje postopka izracuna robnih obtezb za lokalne probleme kljucno. Poglavje 8.2 je namenjeno predstavitvi celotnega postopka izracuna indikatorja modelske napake za DK elemente. Zacne se z obravnavo interpolacij DK elementa po robovih elementa, ki so pomembne za izracun projekcij robnih obtezb. S parametrizacijo robne obtezbe (8.2.6) lahko zapisemo bistveno relacijo, ki preslikuje robno obtezbo (oz. njene parametre) tf v projekcije v vozliscih (8.2.14). Da bi dobili robno obtezbo, ki je cim blizje dejanski robni obtezbi, sistem enacb za izra cun projekcij robne obtez be (7.2.20) regulariziramo po postopku, ki je opisan v 8.2.1. Lokalne probleme resujemo z RM modelom, s P3 elementi. Ker so elementi DK in P3 delijo iste interpolacijske funkcije, obtezni vektor lokalnih problemov posebej preprost (8.2.24). Izkaze se, da obtezba v vozli s cih, ki se nahajajo v ogli s c ih elementov, ostaja enaka obtezbi DK elementov, dodatna je le obtezba v sredini robov, ki je definirana v (8.2.25). Ko izracunamo novo izbolj s ano re sitev na elementu, izracunamo deformacijsko energijo v skladu z ee ERM'e = mRM'T Cb-lmRM dü+ / qRM'T Cs-lqRM dü J n Jn = EB + ES (10.5.1) kjer je mRM = [mxx,m,yy,mXy]T in qRM = [qx,Qy]T. Tako DK kot tudi P3 elementa dajeta zelo podobne napovedi za rezultante momentov. Bistvena razlika med njima je v oceni stri znih sil, saj DK element striga ne vsebuje. Zato lahko strizno deformacijsko energijo ERRM' uporabimo za oceno veljavnosti predpostavke Kirchhoffovega modela. Indikator modelske napake v obliki nM = ERM,e /ERM,e Ve = ES /E , ki meri delez strizne deformacijske energije v celotni deformacijski energiji, je zato smiseln, saj dobro povzema bistvene razlike med modeloma. V nadaljevanju so predstavljeni nekateri racunski primeri (razdelek 8.2.4, 8.2.4), ki obravnavajo nekaj problemov upogiba plosc , pri cemer je glavna pozornost namenjena izracunu indikatorja modelske napake. Primeri prikazujejo robustnost predlagane metode glede na razli cne mreze in debeline plo s ce. Izkaze se, da indikator modelske napake, ceprav sicer kaz e dobro ujemanje z dejansko modelsko napako, ne zmore zaznati robnih obmocij, kjer pride do hitrih sprememb rezultant napetosti. Zadnji računski primeri (razdelek razdelek 8.2.4, 8.2.4) prikazujejo primer prilagodljivega modeliranja, kjer je bil na podlagi ocene modelske napake zgrajen model, ki vsebuje tako DK, kot tudi P3 elemente. Območja, kjer so uporabljeni P3 elementi, so tista, kjer je presez en postavljen kriterij modelske napake. Z mes anim modelom je bila izračunana nova re sitev, ki smo jo primerjali s prvotno resitvijo z DK elementi in re sitvijo, dobljeno s P3 elementi. Pri gradnji me s anega modela je bilo potrebno ustrezno obravnavati prostostne stopnje na stiku DK in P3 elementov. 10.6 ZakljuCek Naloga se ukvarja z vprasanjem prilagodljivega modeliranja plosč . Ceprav je področje modeliranja plos č s končnimi elementi v svoji zreli fazi, razmislek pokaze, da so se vedno odprta fundamentalna vprasanja točnosti in zanesljivosti. Pred obravnavo prilagodljivega modeliranja smo v poglavjih 2 in 3 najprej obravnavali dve osnovni teoriji plosč . V nadaljevanju je v poglavju 4 predstavljena nova teorija upogiba debelih plosč, ki je bila razvita z namenom gradnje hierarhi čno urejene druzine modelov za plosče. Posebej so obravnavani pomembnej si pojavi, ki so značilni za upogib plos č . Na podlagi predstavljenih modelov je bila na novo razvita druz ina končnih elementov. Diskretni Kirchhoffov element (DK), ki je med najpopularnej simi elementi za analizo tankih plosč, smo reformulirali z uporabo alternativne formulacije, ki omogoča splosno obravnavo tako trikotnih kot tudi stirikonih elementov. Dodatno predstavljena formulacija DK elementov omogoča hierarhično (v modelskem smislu) konstrukcijo končnih elementov, ki so zasnovani na Reissner/Mindlinovi teoriji. Elementi, označeni s P3, so zgrajeni ko hierarhi č na raz s iritev DK elementov. Kon čni elementi, ki so zasnovani na teoriji vi sjega reda in dopus č ajo deformacije po debelini plos č, pa so bili razviti na osnovi P3 elementa. Hierarhi čna konstrukcija omogoča, da se kljub narasčajoči kompleksnosti modelov, ohrani fizikalni vpogled v mehanizem deformacije. Predstavljena sta tudi konformni trikotni element za plo s če, kot tudi razli čica P3 elementa za Reissner/Mindlinov model, ki stri zne deformacijske načine obravnava kot nekom-patibilne načine. Vse predstavljene elemente smo implementirali v ve č računalni skih programih in jih tudi testirali. Z več računskimi primeri je bila opravljena sistemati čna primerjava u č inkovitosti elementov. Ze sama primerjava tako predstavlja doprinos k področju modeliranja plosč s končnimi elementi. Podrocje raziskave na podrocju ocene napake diskretizacije je aktivno in dandanes je na voljo veliko rezultatov, ki se ukvarjajo z napako na splosni ravni. V tem delu smo nekaj teh rezultatov uporabili na primeru modeliranja plosc s koncnimi elementi. Pri uporabi dimenzijsko reduciranih modelov, ki so znacilni za plosce, se pojavijo specificni numericni problemi (blokiranje). Pojavi, znacilni za plosce, zahtevajo globlje razumevanje teorije ocene napak. Uporaba obstojecih metod za oceno napak v problemih upogiba plosc zato ni brez tezav. Velik del raziskovalnega dela, ki je predstavljen v tem delu, se zato ukvarja z uspesno implementacijo metode EqR za DK koncni element za plosce. Implementacija je zahtevna, med drugim tudi na povsem tehnicni ravni: racunski postopek vsebuje vec korakov: (i) izracun uravnotezene robne obtezbe iz prvotne racunske resitve, (ii) definicija lokalnih robnih problemov na elementih, ki so obtezeni po robu, (iii) resitev lokalnih problemov z izboljsano diskretizacijo in (iv) izracun napake diskretizacije iz lokalnih resitev. Implementiranih je bilo vec razlicic EqR metode: lokalni izracuni so bili izvedeni z vec razlicnimi diskretizacijami. Izboljsanje diskretizacije z delitvijo prvotnega elementa se je pokazala kot ucinkovita alternativa bolj zapletenemu izracunu, kjer se za izboljsano diskretizacijo uporablja konformni element. Taka primerjava doslej se ni bila izvedena in predstavlja prispevek k podrocju ocene napake diskretizacije za plosce. Primerjava ucinkovitosti EqR metode s splosno uveljavljeno SPR metodo je predstavljena na vec racunskih primerih. Prikazan je tudi postopek prilagodljivega mrezenja, ki se opira na oceno napake diskretizacije. Obe metodi imata svoje prednosti: EqR metod zagotavlja vecjo lokalno natancnost ocene napake, po drugi strani je implementacija SPR metode znatno lazja. Rac unski primeri so pokazali, da EqR metoda predstavlja u c inkovito alternativo SPR metodi. Modelska napaka je obravnavana v poglavjih 7-8. Po splosni obravnavi principov ocene modelske napake, je obravnavana metoda, ki sloni na EqR metodi. Specificna vpras anja, ki so povezana s konstrukcijo lokalnih problemov za koncne elemente za plo s ce, so obravnavana v poglavju 8. Ve c rac unskih primerov prikazuje u cinkovitost obravnavane metode. Primeri pokazejo, da je metoda sposobna zaznati podrocja s pove cano modelsko napako. To delo prinas a se en pomemben sklep: hierarhija koncnih elementov, ki jih uporabljamo v proceduri za oceno modelske napake je kljucna za uspe sno oceno modelske napake. Le hierarhija med koncnimi elementi, ki vsi izhajajo iz iste interpolacijske baze, omogoca uspe sno implementacijo ocene napake na osnovi EqR metode. Prikazan je primer prilagodljivega modeliranja, kjer za izbiro modela uporabljamo oceno modelske napake. Primeri kazejo, da je predlagan postopek ocene modelske napake sposoben voditi postopek prilagodljivega modeliranja. Ukvarjanje z oceno modelske napake je pripeljajo do naslednjega sklepa: maksima 'model gradimo od najbolj grobega proti bolj kompleksnemu' ima svoje meje. Pokazali smo namrec, da je samo iz rezultatov grobega modela, nemogo ce zaznati robna obmocja. Od predlaganega postopka ocene modelske napake ni realno pri c akovati, da bo zaznal pojave, ki so v svojem bistvu odraz kompleksnosti modela. Grobi model preprosto ne vsebuje dovolj informacije, da bi bil sposoben zaznati prisotnost morebitnih pojavov vi sjega reda. 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