Half Title -- Author Biography -- Title Page -- Copyright -- Dedication -- Contents -- List of Figures -- List of Tables -- Preface -- 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 -- References -- 2 Problems in Linear Elasticity and Fields -- 2.1 Introduction -- 2.2 Elasticity equations -- 2.2.1 Displacement function -- 2.2.2 Strain matrix -- 2.2.2.1 Strain-displacement matrix -- 2.2.2.2 Volume change and deviatoric strain -- 2.2.3 Stress matrix -- 2.2.3.1 Mean stress and deviatoric stress -- 2.2.4 Equilibrium equations -- 2.2.4.1 Plane stress and plane strain problems -- 2.2.4.2 Axisymmetric problems -- 2.2.5 Boundary conditions -- 2.2.5.1 Boundary conditions on inclined coordinates -- 2.2.5.2 Normal pressure loading -- 2.2.5.3 Symmetry and repeatability -- 2.2.6 Initial conditions -- 2.2.7 Transformation of stress and strain -- 2.2.7.1 Energy -- 2.2.8 Stress-strain relations: Elasticity matrix -- 2.2.8.1 Isotropic materials -- 2.2.8.2 Deviatoric and pressure-volume relations -- 2.2.8.3 Anisotropic materials -- 2.2.8.4 Initial strain-thermal effects -- 2.3 General quasi-harmonic equation -- 2.3.1 Governing equations: Flux and continuity -- 2.3.2 Boundary conditions -- 2.3.3 Initial condition -- 2.3.4 Constitutive behavior -- 2.3.5 Irreducible form in φ -- 2.3.6 Anisotropic and isotropic forms for k: Transformations -- 2.3.7 Two-dimensional problems -- 2.4 Concluding remarks -- 2.5 Problems -- References -- 3 Weak Forms and Finite Element Approximation: 1-D Problems -- 3.1 Weak forms.

3.2 One-dimensional form of elasticity -- 3.2.1 Weak form of equilibrium equation -- 3.2.1.1 Adjoint forms -- 3.3 Approximation to integral and weak forms: The weighted residual (Galerkin) method -- 3.3.1 Galerkin solution of elasticity equation -- 3.4 Finite element solution -- 3.4.1 Requirements for finite element approximations -- 3.5 Isoparametric form -- 3.5.1 Higher order elements: Lagrange interpolation -- 3.5.1.1 Linear shape functions -- 3.5.1.2 Quadratic shape functions -- 3.5.2 Integrals on the parent element: Numerical integration -- 3.6 Hierarchical interpolation -- 3.7 Axisymmetric one-dimensional problem -- 3.7.1 Weak form for axisymmetric problem -- 3.7.2 A variational notation -- 3.7.3 Irreducible form for axisymmetric problem -- 3.7.4 Finite element solution -- 3.8 Transient problems -- 3.8.1 Discrete time methods -- 3.8.1.1 Stability and dissipation -- 3.8.2 Semi-discretization of the problem -- 3.8.2.1 Stability of modes -- 3.9 Weak form for one-dimensional quasi-harmonic equation -- 3.9.1 Weak form -- 3.9.2 Finite element solution of quasi-harmonic problem -- 3.9.3 Transient problems -- 3.9.3.1 Stability -- 3.10 Concluding remarks -- 3.11 Problems -- References -- 4 Variational Forms and Finite Element Approximation: 1-D Problems -- 4.1 Variational principles -- 4.2 "Natural" variational principles and their relation to governing differential equations -- 4.2.1 Euler equations -- 4.3 Establishment of natural variational principles for linear, self-adjoint differential equations -- 4.4 Maximum, minimum, or a saddle point? -- 4.5 Constrained variational principles -- 4.5.1 Lagrange multipliers -- 4.5.2 Identification of Lagrange multipliers: Forced boundary conditions and modified variational principles -- 4.6 Constrained variational principles: Penalty function and perturbed Lagrangian methods -- 4.6.1 Penalty functions.

4.6.2 Perturbed Lagrangian -- 4.7 Least squares approximations -- 4.8 Concluding remarks: Finite difference and boundary methods -- 4.9 Problems -- References -- 5 Field Problems: A Multidimensional Finite Element Method -- 5.1 Field problems: Quasi-harmonic equation -- 5.1.1 Irreducible form -- 5.1.2 Finite element discretization -- 5.1.2.1 Two-dimensional plane and axisymmetric problem -- 5.1.2.2 Element matrices -- 5.1.3 Shape functions for triangle, rectangle, and tetrahedron -- 5.1.3.1 Triangle with three nodes -- 5.1.3.2 Rectangle with four nodes -- 5.1.3.3 Tetrahedron with four nodes -- 5.2 Partial Discretization: Transient Problems -- 5.3 Numerical examples: An assessment of accuracy -- 5.3.1 Torsion of prismatic bars -- 5.3.1.1 Torsion of rectangular shaft -- 5.3.1.2 Torsion of hollow bimetallic shaft -- 5.3.2 Transient heat conduction -- 5.3.2.1 Transient heat conduction of a rectangular bar -- 5.3.3 Anisotropic seepage -- 5.3.4 Electrostatic and magnetostatic problems -- 5.3.5 Lubrication problems -- 5.3.6 Irrotational and free surface flows -- 5.4 Problems -- References -- 6 Shape Functions, Derivatives, and Integration -- 6.1 Introduction -- 6.2 Two-dimensional shape functions -- 6.2.1 Shape functions for triangles -- 6.2.1.1 Triangle with three nodes -- 6.2.1.2 Higher order triangular elements -- 6.2.1.3 Quadratic triangle (Fig. 6.2b) -- 6.2.1.4 Cubic triangle (Fig. 6.2c) -- 6.2.2 Shape functions for quadrilaterals -- 6.2.2.1 Quadrilateral with four nodes -- 6.2.2.2 Lagrangian family of quadrilaterals -- 6.2.2.3 ``Quadratic'' element (Fig. 6.6b) -- 6.2.2.4 Serendipity family of quadrilaterals -- 6.2.2.5 ``Quadratic'' element (Fig. 6.8b) -- 6.2.2.6 ``Cubic'' element (Fig. 6.8c) -- 6.3 Three-dimensional shape functions -- 6.3.1 Tetrahedral elements -- 6.3.1.1 Tetrahedron with four nodes -- 6.3.1.2 Higher order tetrahedrons.

6.3.1.3 ``Quadratic'' tetrahedron (Fig. 6.12b) -- 6.3.1.4 ``Cubic'' tetrahedron (Fig. 6.12c) -- 6.3.2 Hexagon elements: Brick family -- 6.3.2.1 Hexagon with eight nodes -- 6.3.2.2 Brick elements: Lagrangian family -- 6.3.2.3 ``Linear'' element (eight nodes) (Fig. 6.15a) -- 6.3.2.4 Brick elements: ``Serendipity'' family -- 6.3.2.5 ``Linear'' element (eight nodes) (Fig. 6.15b) -- 6.3.2.6 ``Quadratic'' element (20 nodes) (Fig. 6.15b) -- 6.3.2.7 ``Cubic'' elements (32 nodes) (Fig. 6.15b) -- 6.4 Other simple three-dimensional elements -- 6.4.1 ``Serendipity'' quadratic -- 6.5 Mapping: Parametric forms -- 6.6 Order of convergence for mapped elements -- 6.7 Computation of global derivatives -- 6.7.1 Placement of element coordinates -- 6.8 Numerical integration -- 6.8.1 Quadrilateral elements -- 6.8.2 Brick elements -- 6.8.3 Triangular elements -- 6.8.4 Tetrahedral elements -- 6.8.5 Required order of numerical integration -- 6.8.5.1 Minimum order of integration for convergence -- 6.8.5.2 Order of integration for no loss of convergence rate -- 6.8.6 Matrix singularity due to numerical integration -- 6.9 Shape functions by degeneration -- 6.9.1 Higher order degenerate elements -- 6.10 Generation of finite element meshes by mapping -- 6.10.1 Blending functions -- 6.10.2 Bèzier functions -- 6.11 Computational Advantage of Numerically Integrated Finite Elements -- 6.12 Problems -- References -- 7 Elasticity: Two- and Three-Dimensional Finite Elements -- 7.1 Introduction -- 7.2 Elasticity problems: Weak form for equilibrium -- 7.2.1 Displacement method: Irreducible form -- 7.3 Finite element approximation by the Galerkin method -- 7.4 Boundary conditions -- 7.5 Numerical integration and alternate forms -- 7.6 Infinite domains and infinite elements -- 7.6.1 The mapping function -- 7.7 Singular elements by mapping: Use in fracture mechanics, etc.

7.8 Reporting results: Displacements, strains, and stresses -- 7.9 Discretization error and convergence rate -- 7.10 Minimization of total potential energy -- 7.10.1 Bound on strain energy in a displacement formulation -- 7.10.2 Direct minimization -- 7.11 Finite element solution process -- 7.12 Numerical examples -- 7.12.1 Practical examples -- 7.13 Concluding remarks -- 7.14 Problems -- References -- 8 The Patch Test, Reduced Integration, and Nonconforming Elements -- 8.1 Introduction -- 8.2 Convergence requirements -- 8.3 The simple patch test (Tests A and B): A necessary condition for convergence -- 8.4 Generalized patch test (Test C) and the single-element test -- 8.5 The generality of a numerical patch test -- 8.6 Higher order patch tests -- 8.7 Application of the patch test to plane elasticity elements with "standard" and "reduced" quadrature -- 8.8 Application of the patch test to an incompatible element -- 8.9 Higher order patch test: Assessment of robustness -- 8.10 Concluding Remarks -- 8.11 Problems -- References -- 9 Mixed Formulation and Constraints: Complete Field Methods -- 9.1 Introduction -- 9.2 Mixed form discretization: General remarks -- 9.3 Stability of mixed approximation: The patch test -- 9.3.1 Solvability requirement -- 9.3.2 Locking -- 9.3.3 The mixed patch test -- 9.4 Two-field mixed formulation in elasticity -- 9.4.1 General -- 9.4.2 The u-σ mixed form -- 9.4.3 Stability of two-field approximation in elasticity (u-σ) -- 9.5 Three-field mixed formulations in elasticity -- 9.5.1 The u-σ-ε mixed form -- 9.5.2 Stability condition of three-field approximation (u-σ-ε) -- 9.5.3 The u-σ-εen form: Enhanced strain formulation -- 9.6 Complementary forms with direct constraint -- 9.6.1 General forms -- 9.6.1.1 The complementary heat transfer problem -- 9.6.1.2 The complementary elastic energy principle.

9.6.2 Solution using auxiliary functions.