The Finite Element Method: Its Basis and Fundamentals

Chapter 1: The standard discrete system and origins of the finite element method1.1 Introduction1.2 The structural element and the structural system1.3 Assembly and analysis of a structure1.4 The boundary conditions1.5 Electrical and fluid networks1.6 The general pattern1.7 The standard discrete system 1.8 Transformation of coordinates1.9 ProblemsChapter 2: A direct physical approach to problems in elasticity: plane stress2.1 Introduction2.2 Direct formulation of finite element characteristics2.3 Generalization to the whole region ¨C internal nodal force concept abandoned2.4 Displacement approach as a Minimization of total potential energy2.5 Convergence criteria2.6 Discretization error and convergence rate2.7 Displacement functions with discontinuity between elements ¨C non-conforming elements and the patch test2.8 Finite element solution process2.9 Numerical examples2.10 Concluding remarks2.11 ProblemsChapter 3: Generalization of finite element concepts3.1 Introduction3.2 Integral or ¡®weak¡¯ statements equivalent to the differential equations3.3 Approximation to integral formulations: the weighted residual-Galerkin method3.4 Virtual work as the ¡®weak form¡¯ of equilibrium equations for analysis of solids or fluids3.5 Partial discretization3.6 Convergence3.7 What are ¡®variational principles¡¯?3.8 ¡®Natural¡¯ variational principles and their relation to governing differential equations3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations3.10 Maximum, minimum, or a saddle point?3.11 Constrained variational principles. Lagrange multipliers3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods3.13 Least squares approximations3.14 Concluding remarks ¨C finite difference and boundary methods3.15 ProblemsChapter 4: Element shape functions4.1 Introduction4.2 Standard and hierarchical concepts4.3 Rectangular elements ¨C some preliminary considerations4.4 Completeness of polynomials4.5 Rectangular elements ¨C Lagrange family4.6 Rectangular elements ¨C ¡®serendipity¡¯ family4.7 Triangular element family4.8 Line elements4.9 Rectangular prisms ¨C Lagrange family4.10 Rectangular prisms ¨C ¡®serendipity¡¯ family4.11 Tetrahedral elements4.12 Other simple three-dimensional elements4.13 Hierarchic polynomials in one dimension4.14 Two- and three-dimensional, hierarchical elements of the ¡®rectangle¡¯ or ¡®brick¡¯ type4.15 Triangle and tetrahedron family4.16 Improvement of conditioning with hierarchical forms4.17 Global and local finite element approximation4.18 Elimination of internal parameters before assembly ¨C substructures4.19 Concluding remarks4.20 ProblemsChapter 5: Mapped elements and numerical integration5.1 Introduction5.2 Use of ¡®shape functions¡¯ in the establishment of coordinate transformations5.3 Geometrical conformity of elements5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirementsContents ix5.5 Evaluation of element matrices. Transformation in ¦Î, ¦Â, ¦Æ coordinates5.6 Evaluation of element matrices. Transformation in area and volume coordinates5.7 Order of convergence for mapped elements5.8 Shape functions by degeneration5.9 Numerical integration ¨C rectangular (2D) or brick regions (3D)5.10 Numerical integration ¨C triangular or tetrahedral regions5.11 Generation of finite element meshes by mapping. Blending functions5.12 Required order of numerical integration5.13 Meshes by blending functions5.14 Infinite domains and infinite elements5.15 Singular elements by mapping ¨C use in fracture mechanics, etc.5.16 Computational advantage of numerically integrated finite elements5.17 ProblemsChapter 6: Linear elasticity6.1 Introduction6.2 Governing equations6.3 Finite element approximation6.4 Reporting of results: displacements, strains and stresses6.5 Numerical examples6.6 ProblemsChapter 7: Field problems7.1 Introduction7.2 General quasi-harmonic equation7.3 Finite element solution process7.4 Partial discretization ¨C transient problems7.5 Numerical examples ¨C an assessment of accuracy7.6 Concluding remarks7.7 ProblemsChapter 8: Automatic mesh generation8.1 Introduction8.2 Two-dimensional mesh generation ¨C advancing front method8.3 Surface mesh generation8.4 Three-dimensional mesh generation ¨C Delaunay triangulation8.5 Concluding remarks8.6 ProblemsChapter 9: The patch test and reduced integration9.1 Introduction9.2 Convergence requirements9.3 The simple patch test (tests A and B) ¨C a necessary condition for convergence9.4 Generalized patch test (test C) and the single-element test9.5 The generality of a numerical patch test9.6 Higher order patch tests9.7 Application of the patch test to plane elasticity elements with ¡®standard¡¯ and ¡®reduced¡¯ quadrature9.8 Application of the patch test to an incompatible element9.9 Higher order patch test ¨C assessment of robustness9.10 Conclusion9.11 ProblemsChapter 10: Mixed formulation and constraints10.1 Introduction10.2 Discretization of mixed forms ¨C some general remarks10.3 Stability of mixed approximation. The patch test10.4 Two-field mixed formulation in elasticity10.5 Three-field mixed formulations in elasticity10.6 Complementary forms with direct constraint10.7 Concluding remarks ¨C mixed formulation or a test of element ¡®robustness¡¯10.8 ProblemsChapter 11: Incompressible problems, mixed methods and other procedures of solution11.1 Introduction11.2 Deviatoric stress and strain, pressure and volume change11.3 Two-field incompressible elasticity (u¨Cp form)11.4 Three-field nearly incompressible elasticity (u¨Cp¨C¦Åv form)11.5 Reduced and selective integration and its equivalence to penalized mixed problems11.6 A simple iterative solution process for mixed problems: Uzawa method11.7 Stabilized methods for some mixed elements failing the incompressibility patch test11.8 Concluding remarks11.9 ExercisesChapter 12 Multidomain mixed approximations ¨C domain decomposition and ¡®frame¡¯ methods12.1 Introduction12.2 Linking of two or more subdomains by Lagrange multipliers12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods12.4 Interface displacement ¡®frame¡¯12.5 Linking of boundary (or Trefftz)-type solution by the ¡®frame¡¯ of specified displacements12.6 Subdomains with ¡®standard¡¯ elements and global functions12.7 Concluding remarks12.8 ProblemsChapter 13: Errors, recovery processes and error estimates13.1 Definition of errors13.2 Superconvergence and optimal sampling points13.3 Recovery of gradients and stresses13.4 Superconvergent patch recovery ¨C SPR13.5 Recovery by equilibration of patches ¨C REP13.6 Error estimates by recovery13.7 Residual-based methods13.8 Asymptotic behaviour and robustness of error estimators ¨C the Babu¡¦ska patch test13.9 Bounds on quantities of interest13.10 Which errors should concern us?13.11 ProblemsChapter 14: Adaptive finite element refinement14.1 Introduction14.2 Adaptive h-refinement14.3 p-refinement and hp-refinement14.4 Concluding remarks14.5 Problems Chapter 15: Point-based and partition of unity approximations15.1 Introduction15.2 Function approximation15.3 Moving least squares approximations ¨C restoration of continuity of approximation15.4 Hierarchical enhancement of moving least squares expansions15.5 Point collocation ¨C finite point methods15.6 Galerkin weighting and finite volume methods15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement15.8 Closure15.9 ProblemsChapter 16: Semi-discretization and analytical solution16.1 Introduction16.2 Direct formulation of time-dependent problems with spatial finite element subdivision16.3 General classification16.4 Free response ¨C eigenvalues for second-order problems and dynamic vibration16.5 Free response ¨C eigenvalues for first-order problems and heat conduction, etc.16.6 Free response ¨C damped dynamic eigenvalues16.7 Forced periodic response16.8 Transient response by analytical procedures16.9 Symmetry and repeatability16.10 ProblemsChapter 17: Discrete approximation in time17.1 Introduction17.2 Simple time-step algorithms for the first-order equation17.3 General single-step algorithms for first and second order equations17.4 Stability of general algorithms17.5 Multistep recurrence algorithms17.6 Some remarks on general performance of numerical algorithms17.7 Time discontinuous Galerkin approximation17.8 Concluding remarks17.9 Problems Chapter 18: Coupled systems18.1 Coupled problems ¨C definition and classification18.2 Fluid¨Cstructure interaction (Class I problem)18.3 Soil¨Cpore fluid interaction (Class II problems)18.4 Partitioned single-phase systems ¨C implicit¨Cexplicit partitions (Class I problems)18.5 Staggered solution processes18.6 Concluding remarks Chapter 19: Computer procedures for finite element analysis19.1 Introduction19.2 Pre-processing module: mesh creation19.3 Solution module19.4 Post-processor module19.5 User modulesAppendix A: Matrix algebraAppendix B: Tensor-indicial notation in elasticityAppendix C: Solution of linear algebraic equationsAppendix D: Integration formulae for a triangleAppendix E: Integration formulae for a tetrahedronAppendix F: Some vector algebraAppendix G: Integration by partsAppendix H: Solutions exact at nodesAppendix I: Matrix diagonalization or lumping