e9780750664318v1 -- Front Cover -- The Finite Element Method: Its Basis and Fundamentals -- Copyright Page -- Contents -- Preface -- Chapter 1. The standard discrete system and origins of the finite element method -- 1.1 Introduction -- 1.2 The structural element and the structural system -- 1.3 Assembly and analysis of a structure -- 1.4 The boundary conditions -- 1.5 Electrical and fluid networks -- 1.6 The general pattern -- 1.7 The standard discrete system -- 1.8 Transformation of coordinates -- 1.9 Problems -- Chapter 2. A direct physical approach to problems in elasticity: plane stress -- 2.1 Introduction -- 2.2 Direct formulation of finite element characteristics -- 2.3 Generalization to the whole region- internal nodal force concept abandoned -- 2.4 Displacement approach as a minimization of total potential energy -- 2.5 Convergence criteria -- 2.6 Discretization error and convergence rate -- 2.7 Displacement functions with discontinuity between elements - non-conforming elements and the patch test -- 2.8 Finite element solution process -- 2.9 Numerical examples -- 2.10 Concluding remarks -- 2.11 Problems -- Chapter 3. Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches -- 3.1 Introduction -- 3.2 Integral or 'weak' statements equivalent to the differential equations -- 3.3 Approximation to integral formulations: the weighted residual-Galerkin method -- 3.4 Virtual work as the 'weak form' of equilibrium equations for analysis of solids or fluids -- 3.5 Partial discretization -- 3.6 Convergence -- 3.7 What are 'variational principles'? -- 3.8 'Natural' variational principles and their relation to governing differential equations -- 3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations -- 3.10 Maximum, minimum, or a saddle point?.

3.11 Constrained variational principles. Lagrange multipliers -- 3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods -- 3.13 Least squares approximations -- 3.14 Concluding remarks - finite difference and boundary methods -- 3.15 Problems -- Chapter 4. 'Standard' and 'hierarchical' element shape functions: some general families of C0 continuity -- 4.1 Introduction -- 4.2 Standard and hierarchical concepts -- 4.3 Rectangular elements- some preliminary considerations -- 4.4 Completeness of polynomials -- 4.5 Rectangular elements- Lagrange family -- 4.6 Rectangular elements- 'serendipity' family -- 4.7 Triangular element family -- 4.8 Line elements -- 4.9 Rectangular prisms - Lagrange family -- 4.10 Rectangular prisms - 'serendipity' family -- 4.11 Tetrahedral elements -- 4.12 Other simple three-dimensional elements -- 4.13 Hierarchic polynomials in one dimension -- 4.14 Two- and three-dimensional, hierarchical elements of the 'rectangle' or 'brick' type -- 4.15 Triangle and tetrahedron family -- 4.16 Improvement of conditioning with hierarchical forms -- 4.17 Global and local finite element approximation -- 4.18 Elimination of internal parameters before assembly - substructures -- 4.19 Concluding remarks -- 4.20 Problems -- Chapter 5. Mapped elements and numerical integration- 'infinite' and 'singularity elements' -- 5.1 Introduction -- 5.2 Use of 'shape functions' in the establishment of coordinate transformations -- 5.3 Geometrical conformity of elements -- 5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements -- 5.5 Evaluation of element matrices. Transformation in ξ,η,ζ coordinates -- 5.6 Evaluation of element matrices. Transformation in area and volume coordinates -- 5.7 Order of convergence for mapped elements -- 5.8 Shape functions by degeneration.

5.9 Numerical integration- one dimensional -- 5.10 Numerical integration- rectangular (2D) or brick regions (3D) -- 5.11 Numerical integration - triangular or tetrahedral regions -- 5.12 Required order of numerical integration -- 5.13 Generation of finite element meshes by mapping. Blending functions -- 5.14 Infinite domains and infinite elements -- 5.15 Singular elements by mapping - use in fracture mechanics, etc. -- 5.16 Computational advantage of numerically integrated finite elements -- 5.17 Problems -- Chapter 6. Problems in linear elasticity -- 6.1 Introduction -- 6.2 Governing equations -- 6.3 Finite element approximation -- 6.4 Reporting of results: displacements, strains and stresses -- 6.5 Numerical examples -- 6.6 Problems -- Chapter 7. Field problems - heat conduction, electric and magnetic potential and fluid flow -- 7.1 Introduction -- 7.2 General quasi-harmonic equation -- 7.3 Finite element solution process -- 7.4 Partial discretization- transient problems -- 7.5 Numerical examples- an assessment of accuracy -- 7.6 Concluding remarks -- 7.7 Problems -- Chapter 8. Automatic mesh generation -- 8.1 Introduction -- 8.2 Two-dimensional mesh generation- advancing front method -- 8.3 Surface mesh generation -- 8.4 Three-dimensional mesh generation- Delaunay triangulation -- 8.5 Concluding remarks -- 8.6 Problems -- Chapter 9. The patch test, reduced integration, and non-conforming elements -- 9.1 Introduction -- 9.2 Convergence requirements -- 9.3 The simple patch test (tests A and B) - a necessary condition for convergence -- 9.4 Generalized patch test (test C) and the single-element test -- 9.5 The generality of a numerical patch test -- 9.6 Higher order patch tests -- 9.7 Application of the patch test to plane elasticity elements with 'standard' and 'reduced' quadrature -- 9.8 Application of the patch test to an incompatible element.

9.9 Higher order patch test- assessment of robustness -- 9.10 Concluding remarks -- 9.11 Problems -- Chapter 10. Mixed formulation and constraints- complete field methods -- 10.1 Introduction -- 10.2 Discretization of mixed forms - some general remarks -- 10.3 Stability of mixed approximation. The patch test -- 10.4 Two-field mixed formulation in elasticity -- 10.5 Three-field mixed formulations in elasticity -- 10.6 Complementary forms with direct constraint -- 10.7 Concluding remarks - mixed formulation or a test of element 'robustness' -- 10.8 Problems -- Chapter 11. Incompressible problems, mixed methods and other procedures of solution -- 11.1 Introduction -- 11.2 Deviatoric stress and strain, pressure and volume change -- 11.3 Two-field incompressible elasticity (u-p form) -- 11.4 Three-field nearly incompressible elasticity (u-p-εv form) -- 11.5 Reduced and selective integration and its equivalence to penalized mixed problems -- 11.6 A simple iterative solution process for mixed problems: Uzawa method -- 11.7 Stabilized methods for some mixed elements failing the incompressibility patch test -- 11.8 Concluding remarks -- 11.9 Problems -- Chapter 12. Multidomain mixed approximations- domain decomposition and 'frame' methods -- 12.1 Introduction -- 12.2 Linking of two or more subdomains by Lagrange multipliers -- 12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods -- 12.4 Interface displacement 'frame' -- 12.5 Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements -- 12.6 Subdomains with 'standard' elements and global functions -- 12.7 Concluding remarks -- 12.8 Problems -- Chapter 13. Errors, recovery processes and error estimates -- 13.1 Definition of errors -- 13.2 Superconvergence and optimal sampling points -- 13.3 Recovery of gradients and stresses.

13.4 Superconvergent patch recovery - SPR -- 13.5 Recovery by equilibration of patches - REP -- 13.6 Error estimates by recovery -- 13.7 Residual-based methods -- 13.8 Asymptotic behaviour and robustness of error estimators - the Babuška patch test -- 13.9 Bounds on quantities of interest -- 13.10 Which errors should concern us? -- 13.11 Problems -- Chapter 14. Adaptive finite element refinement -- 14.1 Introduction -- 14.2 Adaptive h-refinement -- 14.3 p-refinement and hp-refinement -- 14.4 Concluding remarks -- 14.5 Problems -- Chapter 15. Point-based and partition of unity approximations. Extended finite element methods -- 15.1 Introduction -- 15.2 Function approximation -- 15.3 Moving least squares approximations - restoration of continuity of approximation -- 15.4 Hierarchical enhancement of moving least squares expansions -- 15.5 Point collocation- finite point methods -- 15.6 Galerkin weighting and finite volume methods -- 15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement -- 15.8 Concluding remarks -- 15.9 Problems -- Chapter 16. The time dimension- semi-discretization of field and dynamic problems and analytical solution procedures -- 16.1 Introduction -- 16.2 Direct formulation of time-dependent problems with spatial finite element subdivision -- 16.3 General classification -- 16.4 Free response - eigenvalues for second-order problems and dynamic vibration -- 16.5 Free response - eigenvalues for first-order problems and heat conduction, etc. -- 16.6 Free response- damped dynamic eigenvalues -- 16.7 Forced periodic response -- 16.8 Transient response by analytical procedures -- 16.9 Symmetry and repeatability -- 16.10 Problems -- Chapter 17. The time dimension- discrete approximation in time -- 17.1 Introduction.

17.2 Simple time-step algorithms for the first-order equation.