Introduction to Linear and Nonlinear Finite Element Analysis

"This is an introductory textbook on finite element analysis and practice aimed at students with diverse backgrounds from engineering, technology, physics, geophysics and applied mathematics. The book provides accessibility to all students, with a minimum of mathematical analysis... The last chapter is dedicated to computer programs in Mathematica, Ansys, Matlab and Fortran. There are six appendices, 87 examples and 148 exercises. The book ends with a bibliography and a detailed subject index." -Mathematical Reviews "This book is introductory in the sense of being accessible to students not only of mathematics, but also of the physical and the engineering sciences once they have mastered the introductory mathematical courses. It is also introductory in the sense of not providing the reader with all the theoretical framework of convergence analysis of the FE-method based on Sobolev spaces, etc. Rather it is content with explaining the very basic ideas behind FE. In a different sense it does however lead to relatively advanced topics, namely from the standpoint of applications... Overall, the presentation is quite detailed regarding the needs of the practitioner with many examples to engineering, earth sciences, etc. (among others elasticity, vibrations, heat transfer, fluid flow; also eigenvalue problems), and special but important items not so often covered in other texts, e.g., how to cope with the specific difficulties arising in polar coordinates. Both numerous exercises and codes in Ansys, Fortran, Mathematica and MATLAB direct the reader towards experimentation of his own." -Monatshefte fur Mathematik

• Preface.- Notation.- 1 Introduction.- 1.1 Historical Sketch.- 1.2 Euler-Lagrange Equations.- 1.3 Weak Variational Form.- 1.4 Galerkin Method.- 1.5.- 1.6.- 2 One-Dimensional Shape Functions.- 2.1 Local and Global Linear Shape Functions.- 2.2 Local and Global Quadratic Shape Functions.- 2.3 Parametric Coordinates.- 2.4 Hermite Shape Functions.- 2.5 Exercises.- 3 One-Dimensional Second-Order Equation.- 3.1 Galerkin Finite Element Method.- 3.2 Two Dependent Variables.- 3.3 Exercises.- 4 One-Dimensional Fourth-Order Equation.- 4.1 Euler-Bernoulli Beam Equation.- 4.2 Exercises.- 5 Two-Dimensional Elements.- 5.1 Linear Three-Node Triangular Elements.- 5.2 Bilinear Four-Node Rectangular Elements.- 5.3 Global Shape Functions.- 5.4 Triangular Coordinates.- 5.5 Shape Functions on the Sides of a Triangle.- 5.6 Exercises.- 6 Two-Dimensional Problems.- 6.1 Single Dependent Variable Problems.- 6.2 Exercises.- 7 More Two-Dimensional Problems.- 7.1 Heat Transfer.- 7.2 Torsion.- 7.3 Seepage.- 7.4 Fluid Flows.- 7.5 Exercises.- 8 Axisymmetric Heat Transfer.- 8.1 Radial Symmetry.- 8.2 Linear Elements.- 8.3 Linear Elements for Heat Transfer in Fluids.- 8.4 Nonlinear Heat Transfer.- 8.5 Exercises.- 9 Transient Problems.- 9.1 Classical Methods.- 9.2 One-Dimensional Transient Problems.- 9.3 Time-Dependent Heat Conduction.- 9.4 Two-Dimensional Transient Problems.- 9.5 Exercises.- 10 Single Nonlinear One-Dimensional Problems.- 10.1 Newton’ method.- 10.2 Radiation Heat Transfer.- 10.3 Stress Analysis of Plastic Rods.- 10.4 Power-Law Pressure Driven Flow between Two Plates.- 10.5 Mixing-Length Equation for Turbulent Flow in Pipes.- 10.6 Rayleigh-Ritz and Nonlinear Gradient Methods.- 10.7 Exercises.- 11 Plane Elasticity.- 11.1 Stress-Strain Relations.- 11.2 Constant-Strain Triangular Element.-11.3 Virtual Displacement Finite Element Model.- 11.4 Weak Form Finite Element Model.- 11.5 Stiffness Matrix and Load Vector.- 11.6 Exercises.- 12 Stokes Equations and Penalty Method.- 12.1 Equality-Constrained Programs and Lagrange Multipliers.- 12.2 Penalty Formulation for Linear Stokes Equation.- 12.3 Penalty Linear Triangular Stokes Element.- 12.4 Penalty Bilinear Rectangular Stokes Element.- 12.5 Penalty Linear Triangular Power-law Stokes Element.- 12.6 Solutions by Conjugate Gradient Methods.- 12.7 Exercises.- 13 Vibration Analysis.- 13.1 Hamiltonian Principle.- 13.2 Free Axial Vibrations of an Elastic Rod.- 13.3 Free Vibrations of a Euler Elastic Beam.- 13.4 Free In-Plane Vibrations of an Elastic Plate.- 13.5 Axial Vibrations of a Plastic Rod.- 13.6 Eigenvalue Problems.- 13.7 Exercises.- 14 Computer Codes.- 14.1 Mathematica Codes.- 14.2 Ansys Codes.- 14.3 Matlab Codes.- 14.4 Fortran Codes.- Integration Formulas.- A Special Cases.- B Temporal Approximations.- C Isoparametric Elements.- D Green’ Identities.- E Gaussian Quadrature.- F Gradient-Based Methods.