The Finite Element Method
Fundamentals and Applications
1st Edition
Published on 10. May 2014
336 pages
978-1-4832-1891-5 (ISBN)
Description
The Finite Element Method: Fundamentals and Applications demonstrates the generality of the finite element method by providing a unified treatment of fundamentals and a broad coverage of applications. Topics covered include field problems and their approximate solutions; the variational method based on the Hilbert space; and the Ritz finite element method. Finite element applications in solid and structural mechanics are also discussed. Comprised of 16 chapters, this book begins with an introduction to the formulation and classification of physical problems, followed by a review of field or continuum problems and their approximate solutions by the method of trial functions. It is shown that the finite element method is a subclass of the method of trial functions and that a finite element formulation can, in principle, be developed for most trial function procedures. Variational and residual trial function methods are considered in some detail and their convergence is examined. After discussing the calculus of variations, both in classical and Hilbert space form, the fundamentals of the finite element method are analyzed. The variational approach is illustrated by outlining the Ritz finite element method. The application of the finite element method to solid and structural mechanics is also considered. This monograph will appeal to undergraduate and graduate students, engineers, scientists, and applied mathematicians.
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Content
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.
