The Finite Element Method

PrefaceAcknowledgmentsChapter 1 The Formulation of Physical Problems 1.1 Introduction 1.2 Classification of Physical Problems 1.3 Classification of the Equations of a System ReferencesChapter 2 Field Problems and Their Approximate Solutions 2.1 Formulation of Field (Continuous) Problems 2.2 Classification of Field Problems 2.3 Equilibrium Field Problems 2.4 Eigenvalue Field Problems 2.5 Propagation Field Problems 2.6 Summary of Governing Equations 2.7 Approximate Solution of Field Problems 2.8 Trial Function Methods in Equilibrium Problems 2.9 Trial Function Methods in Eigenvalue Problems 2.10 Trial Function Methods in Propagation Problems 2.11 Accuracy, Stability, and Convergence 2.12 Approximate Solutions for Nonlinear Problems 2.13 The Extension to Vector Problems 2.14 The Finite Element Method ReferencesChapter 3 The Variational Calculus and Its Application 3.1 Maxima and Minima of Functions 3.2 The Lagrange Multipliers 3.3 Maxima and Minima of Functionals 3.4 Variational Principles in Physical Phenomena ReferencesChapter 4 The Variational Method Based on the Hilbert Space 4.1 The Hilbert Function Space 4.2 Equilibrium and Eigenvalue Problems 4.3 The Variational Solution of the Equilibrium Problem 4.4 Inhomogeneous Boundary Conditions 4.5 Natural Boundary Conditions 4.6 The Variational Solution of the Eigenvalue Problem ReferencesChapter 5 Fundamentals of the Finite Element Approach 5.1 Classification of Finite Element Methods 5.2 The Finite Element Approximation 5.3 Elements and Their Shape Functions 5.4 Variational Finite Element Methods 5.5 Residual Finite Element Methods 5.6 The Direct Finite Element Method 5.7 Significant Features of a Finite Element Method 5.8 The Coefficient Finite Element Method 5.9 The Cell Finite Element Method 5.10 Convergence in the Finite Element Method ReferencesChapter 6 The Ritz Finite Element Method (Classical) 6.1 Statement of the Problem 6.2 The Equivalent Variational Problem 6.3 The Subdivision of the Region 6.4 The Element Shape Function 6.5 The Subdivision of the Functional 6.6 The Minimization Condition 6.7 The Element Matrix Equation 6.8 The System Matrix Equation 6.9 Insertion of the Dirichlet Boundary Condition 6.10 The Finite Element Approximation 6.11 The Two-Dimensional Region 6.12 Structural Formulations of the Finite Element Method ReferencesChapter 7 The Ritz Finite Element Method (Hilbert Space) 7.1 The Ritz Finite Element Method for the Equilibrium Problem 7.2 Rayleigh-Ritz Finite Element Solution for the Eigenvalue Problem ReferencesChapter 8 Finite Element Applications in Solid and Structural Mechanics 8.1 The Solid Mechanics Formulation of the Finite Element Method 8.2 The Structural Formulation of the Finite Element Method ReferencesChapter 9 The Laplace or Potential Field 9.1 The Laplace Equation 9.2 The Variational Formulation for the Laplace Field 9.3 The Ritz Finite Element Solution of the Laplace Field 9.4 Summary ReferencesChapter 10 Laplace and Associated Boundary-Value Problems 10.1 The Potential Flow Field 10.2 The Electrostatic Field 10.3 The Thermal Conduction Field 10.4 Porous Media Flows 10.5 The Quasi-Harmonic Equation 10.6 The Poisson Equation 10.7 Unsteady Potential Fields (Moving Boundaries) 10.8 Lifting Bodies with Appreciable Boundary Displacement ReferencesChapter 11 The Helmholtz and Wave Equations 11.1 Physical Phenomena and the Helmholtz Equation 11.2 Physical Phenomena and the Wave Equation ReferencesChapter 12 The Diffusion Equation 12.1 Forms of the Diffusion Equation 12.2 The Finite Element Solution of the Diffusion Equation ReferencesChapter 13 Finite Element Applications to Viscous Flow 13.1 Oden and Somogyi 13.2 Tong 13.3 Baker 13.4 Leonard 13.5 Atkinson et al. 13.6 Reddi 13.7 Argyris and Scharpf 13.8 Other Formulations ReferencesChapter 14 Finite Element Applications to Compressible Flow 14.1 Leonard 14.2 Gelder 14.3 De Vries, Berard, and Norrie 14.4 Reddi and Chu 14.5 Other Formulations ReferencesChapter 15 Finite Element Applications to More General Fluid Flows 15.1 Skiba 15.2 Oden—I 15.3 Oden—II 15.4 Oden—III 15.5 De Vries and Norrie 15.6 Baker 15.7 Bramlette and Mallett 15.8 Other Formulations ReferencesChapter 16 Other Finite Element Applications 16.1 Solid-Fluid Coupled Vibrations 16.2 Further Finite Element Applications 16.3 Further Developments ReferencesAppendix A Matrix Algebra A.1 Matrix Definitions A.2 Matrix Algebra A.3 Quadratic and Linear Forms ReferencesAppendix B The Differential and Integral Calculus of Matrices B.1 Definition of Differentiation and Integration of Matrices B.2 Differentiation of a Function of a Matrix with Respect to the Matrix B.3 Partial Differentiation of Matrices B.4 Differentiation of Functions of Several Variables ReferencesAppendix C The Transformation Matrix C.1 The Functional Relationship Between Coordinate Systems C.2 The Local Transformation Matrix C.3 The Linear Transformation C.4 The Translation Matrix C.5 The Rotation Matrix C.6 Successive Transformations C.7 Transformation of Matrices C.8 Principal Axes, Diagonalization, and Eigenvalues ReferencesAdditional References A. Mathematical Methods B. Variational Principles and Formulations C. Finite Element Analysis D. Finite Element Solutions of Physical Problems E. Finite Element Computational Procedures F. Input/Output Data Procedures G. Large-Scale Finite Element SystemsAuthor IndexSubject Index