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Geometric Discretization Schemes and Differential Complexes for Elasticity

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Geometric Discretization Schemes and Differential Complexes for Elasticity GEOMETRIC DISCRETIZATION SCHEMES AND DIFFERENTIAL COMPLEXES FOR ELASTICITY A Thesis Presented to The Academic Faculty by Arzhang Angoshtari In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Civil and Environmental Engineering Georgia Institute of Technology August 2013 Copyright © 2013 by Arzhang Angoshtari GEOMETRIC DISCRETIZATION SCHEMES AND DIFFERENTIAL COMPLEXES FOR ELASTICITY Approved by: Professor Arash Yavari, Advisor Professor Hamid Garmestani School of Civil and Environmental School of Materials Science and Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Reginald DesRoches Professor Naresh Thadhani School of Civil and Environmental School of Materials Science and Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Wilfrid Gangbo Date Approved: 13 May 2013 School of Mathematics Georgia Institute of Technology ACKNOWLEDGEMENTS I want to express my gratitude to my advisor Prof. Arash Yavari for his support and for letting me be creative in my research. I am grateful to my committee members Prof. Wilfrid Gangbo, Prof. Hamid Garmestani, Prof. Reginald DesRoches, and Prof. Naresh Thadhani, and also my former advisor Prof. Mir Abbas Jalali for their support. I should also thank Prof. Mohammad Ghomi, Prof. Andreas Cap,ˇ and Prof. Marino Arroyo for valuable discussions that were helpful in the development of this research. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................. iii LIST OF TABLES .................................. vi LIST OF FIGURES ................................. vii SUMMARY ....................................... x I INTRODUCTION ................................ 1 1.1 MainContributions.............................. 2 II A GEOMETRIC NUMERICAL SCHEME FOR INCOMPRESS- IBLE ELASTICITY ............................... 6 2.1 IncompressibleElasticity . .. 11 2.1.1 IncompressibleFiniteElasticity . .. 11 2.1.2 Incompressible Linearized Elasticity . ... 21 2.2 DiscreteExteriorCalculus. .. 28 2.2.1 PrimalMeshes ............................ 28 2.2.2 DualMeshes ............................. 31 2.2.3 DiscreteVectorFields........................ 34 2.2.4 PrimalandDualDiscreteForms . 35 2.2.5 DiscreteOperators.......................... 37 2.2.6 AffineInterpolation ......................... 42 2.3 Discrete Configuration Manifold of Incompressible Linearized Elasticity 43 2.4 DiscreteGoverningEquations . 56 2.4.1 KineticEnergy ............................ 57 2.4.2 ElasticStoredEnergy ........................ 58 2.4.3 Discrete Euler-Lagrange Equations . 64 2.4.4 DiscretePressureField ....................... 69 2.5 NumericalExamples ............................. 75 iv III COMPLEXES OF LINEAR AND NONLINEAR ELASTOSTAT- ICS .......................................... 79 3.1 AlgebraicandGeometricPreliminaries . ... 80 3.1.1 CategoriesandFunctors . 80 3.1.2 TensorProductandExteriorPower . 82 3.1.3 LieAlgebrasandLieGroups. 86 3.1.4 FiberBundles............................. 93 3.1.5 DerivativesonVectorBundles . 108 3.1.6 Connections.............................. 113 3.2 Differential Operators of Elastostatics. ...... 123 3.2.1 ProjectiveDifferentialGeometry . 124 3.2.2 TheKillingOperator ........................ 126 3.2.3 The Curvature Operator and the Compatibility Equations. 128 3.2.4 The Bianchi Operator and Stress Functions . 143 3.3 Complexes in Linear and Nonlinear Elastostatics . ..... 150 3.3.1 ResolutionsofSheaves . 151 3.3.2 LinearElastostaticsComplexes . 154 3.3.3 NonlinearElastostaticsComplexes . 157 3.4 Linear Elastostatics Complexes and HomogeneousSpaces............................. 164 3.4.1 SemisimpleLieAlgebras . 164 3.4.2 Irreducible Representations of SL Cn .............. 171 ( ) 3.4.3 ParabolicGeometries . 176 3.4.4 Associated Representations of Homogeneous Bundles and In- variantDifferentialOperators . 183 3.4.5 The Linear Elastostatics Complex as a BGG Resolution . .. 187 3.4.6 The twisted de Rham Complex of Linear Elastostatics . .. 193 IV CONCLUDING REMARKS ........................ 197 REFERENCES ..................................... 200 v LIST OF TABLES 1 CommonLiegroupsandtheirLiealgebras. 87 vi LIST OF FIGURES 1 A primal 2-dimensional simplical complex (solid lines) and its dual (dashed lines). The primal vertices are denoted by and the dual vertices by . The circumcenter c i,j,k is denoted● by cijk, etc. The highlighted○ areas denote the support([ volumes]) of the corresponding simplices, for example, i, k is the support volume of the 1-simplex i, k . Note that the support[ ] volume of the primal vertex j coincides with[ ] its dual j and support volume of the primal 2-simplex l,m,n coincideswithitself.[ ] .............................[ ] 30 2 Oriented meshes: (a) primal mesh and (b) associated circumcentric dual mesh. The primal vertices are denoted by and the dual vertices by . The circumcenter c 1, 2, 3 is denoted by● c123,etc. ....... 33 ○ ([ ]) 3 A primal vector field (arrows on primal vertices ) and a dual vector field (arrows on dual vertices ) on a 2-dimensional● mesh. The solid and dashed lines denote the primal○ and dual meshes, respectively. .. 35 4 Examples of forms on a 2-dimensional primal mesh (solid lines) and its dual mesh (dashed lines): (a) primal and dual 0-forms, which are real numbers on primal and dual vertices, (b) primal and dual 1-forms, which are real numbers on primal and dual 1-simplices, and (c) pri- mal and dual 2-forms, which are real numbers on primal and dual 2-simplices,respectively. 35 5 The discrete 1-forms df obtained from (a) primal and (b) dual 0-form f. The sets f 1,f 2,f 3,f 4 and f 123,f 243 are the sets of values of primal and dual{ 0-forms, respectively.} { Note} that f 123 is the value of f at c 1, 2, 3 ,etc. ............................... 39 ([ ]) 6 A discrete primal vector field, see Fig. 2 for the numbering of the simplices and orientation of the primal and dual meshes. The vector i31,123 is the unit vector with the same orientation as c31,c123 ,etc. 43 [ ] 7 A subset of a primal mesh and its associated dual mesh. The vector ijlk,lqk is the unit vector with the same orientation as c j,l,k ,c l,q,k , etc. ........................................[ ([ ]) ([ ])] 47 8 Two possible ways for adding a triangle to a 2-dimensional shellable mesh, either (a) the new triangle introduces a new primal vertex or (b) the new triangle does not introduce any new primal vertices. ..... 49 9 A 2-dimensional primal mesh with its associated dual mesh and the associated unit vectors. The vector i12,132 is the unit vector with the same orientation as c 1, 2 ,c 1, 3, 2 ,etc. .............. 53 [ ([ ]) ([ ])] vii 10 Dual cells that are used for defining the kinetic energy. Primal and dual vertices are denoted by and , respectively. The solid lines de- note the boundary of the primal● 2-cells○ and the colored regions denote the dual of each primal vertex. The material properties, displacements, and velocities are considered to be constant on each dual 2-cell. For 0 example, consider the primal vertex i (σi ). The velocity at the corre- sponding dual cell is assumed to be equal to the velocity at vertex i, which is denoted by U˙ i............................. 57 11 Regions that are used for calculating the elastic stored energy. Primal and dual vertices are denoted by and , respectively. The dotted lines denote the primal one-simplices.● Displacement○ is interpolated using affine functions in each of the colored triangles which are the intersection of a support volume of a primal 1-simplex with a dual 2-cell. The elastic body is assumed to be homogeneous in each dual 2-cell. The region bounded by the solid lines denotes the dual of the primal vertex i. The stored energy at this dual cell is obtained by summing the internal energy of the corresponding 6 smaller triangles. 59 12 Discrete solution spaces: (a) P0 over primal meshes for the pressure field, (b) P1 over primal meshes for the displacement field in the in- compressibility constraint, (c) P0 over dual meshes for the displacement field for approximating the kinetic energy, and (d) P1 over support vol- umes for the displacement field for approximating the elasticenergy. 67 13 Part of a primal mesh and its associated dual mesh. The 2-simplex j,l,k liesontheboundary. ......................... 72 [ ] 14 Cantilever beam: (a) Geometry, boundary conditions, and loading, (b) a well-centered primal mesh with denoting the circumcenter of each primal2-cell. ..................................○ 73 15 Cantilever beam: (a) Convergence of the normalized displacement of the tip point A (ratio of the numerically-calculated and exact displace- ments). N is the number of primal 2-cells of the mesh. (b) Pressure of the dual vertices that correspond to the primal 2-cells that are on the bottomofthebeam............................... 73 16 The pressure field for the beam problem for meshes with (a) N = 64, (b) N = 156, (c) N = 494, where N is the number of primal 2-cells of themesh. .................................... 75 17 Cook’s membrane: (a) Geometry, boundary conditions, and loading, (b) a well-centered mesh with N = 123 primal 2-cells with denoting thecircumcenterofeachprimal2-cell. .○ . .. 75 18 Convergence of the normalized displacement of the tip point A of the Cook’s membrane. N is the number of primal 2-cells of the mesh. 76 viii 19 The pressure field for the Cook’s membrane for meshes with (a) N = 123, (b) N = 530, (c) N = 955, where N is the number of primal 2-cells ofthemesh...................................
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