An Introduction to Linear and Nonlinear Finite Element Analysis: A Computational Approach

Modern finite element analysis has grown into a basic mathematical tool for almost every field of engineering and the applied sciences. This introductory textbook fills a gap in the literature, offering a concise, integrated presentation of methods, applications, software tools, and hands-on projects. Included are numerous exercises, problems, and Mathematica/Matlab-based programming projects. The emphasis is on interdisciplinary applications to serve a broad audience of advanced undergraduate/graduate students with different backgrounds in applied mathematics, engineering, physics/geophysics. The work may also serve as a self-study reference for researchers and practitioners seeking a quick introduction to the subject for their research.

Table of Contents:
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.