Finite Element Method for Boundary Value Problems

Karan S. Surana attended undergraduate school at Birla Institute of Technology and Science (BITS), Pilani, India and received a B.E. in mechanical engineering in 1965. He then attended the University of Wisconsin, Madison where he obtained M.S. and Ph.D. in mechanical engineering in 1967 and 1970. He joined The University of Kansas, Department of Mechanical Engineering faculty where he is currently serving as Deane E. Ackers University Distinguished Professor of Mechanical Engineering. His areas of interest and expertise are computational mathematics, computational mechanics, and continuum mechanics. He is the author of over 350 research reports, conference papers, and journal papers. J. N. Reddy is a Distinguished Professor, Regents’ Professor, and inaugural holder of the Oscar S. Wyatt Endowed Chair in Mechanical Engineering at Texas A&M University, College Station, Texas. Dr. Reddy earned a Ph.D. in Engineering Mechanics in 1974 from University of Alabama in Huntsville. He worked as a Post-Doctoral Fellow in Texas Institute for Computational Mechanics (TICOM) at the University of Texas at Austin, Research Scientist for Lockheed Missiles and Space Company, Huntsville, during l974-75, and taught at the University of Oklahoma from 1975 to 1980, Virginia Polytechnic Institute & State University from 1980 to 1992, and at Texas A&M University from 1992. Professor Reddy also played active roles in professional societies as the President of USACM, founding member of the General Council of IACM, Secretary of Fellows of AAM, member of the Board of Governors of SES, Chair of the Engineering Mechanics Executive Committee, among several others.

"This book is written by notable experts in the field, and its content has been verified and used in university courses for thirty years. It is self-contained, and it includes a balance of mathematical background/derivations and applications to general problems (rather than restriction to solid mechanics, for example), and this it will be of high interest to students in applied mathematics, applied physics, as well as all branches of engineering mechanics."--John D. Clayton, University of Maryland, College Park, USA

• 1 Introduction General Comments and Basic Philosophy Basic Concepts of the Finite Element Method Summary Concepts from Functional Analysis General Comments Sets, Spaces, Functions, Functions Spaces, and Operators Elements of Calculus of Variations Examples of Differential Operators and their Properties 2.5 Summary Classical Methods of Approximation Introduction Basic Steps in Classical Methods of Approximation based on Integral Forms Integral forms using the Fundamental Lemma of the Calculus of Variations Approximation Spaces for Various Methods of Approximation Integral Formulations of BVPs using the Classical Methods of Approximations Numerical Examples Summary The Finite Element Method Introduction Basic steps in the finite element method Summary Self-Adjoint Differential Operators Introduction One-dimensional BVPs in a single dependent variable 5.3 Two-dimensional boundary value problems5.4 Three-dimensional boundary value problems5.5 Summary6 Non-Self-Adjoint Differential Operators6.1 Introduction6.2 1D convection-diffusion equation6.3 2D convection-diffusion equation6.4 Summary7 Non-Linear Differential Operators7.1 Introduction7.2 One dimensional Burgers equation7.3 Fully developed ow of Giesekus Fluid between parallel plates (polymer flow)7.4 2D steady-state Navier-Stokes equations7.5 2D compressible Newtonian fluid Flow7.6 Summary8 Basic Elements of Mapping and Interpolation Theory8.1 Mapping in one dimension8.2 Elements of interpolation theory over 8.4 Local approximation over : quadrilateral elements8.5 2D p-version local approximations8.6 2D approximations for quadrilateral elements8.10 Serendipity family of interpolations8.11 Interpolation functions for 3D elements8.12 Summary9 Linear Elasticity using the Principle of Minimum Total Potential Energy9.1 Introduction9.2 New notation9.3 Approach9.4 Element equations9.5 Finite element formulation for 2D linear elasticity9.6 Summary10 Linear and Nonlinear Solid Mechanics using the Principle of Virtual Displacements10.1 Introduction10.2 Principle of virtual displacements10.3 Virtual work statements10.4 Solution method10.5 Finite element formulation for 2D solid continua10.6 Finite element formulation for 3D solid continua10.7 Axisymmetric solid finite elements10.8 Summary11 Additional Topics in Linear Structural Mechanics11.1 Introduction11.2 1D axial spar or rod element in R1 (1D space)11.3 1D axial spar or rod element in R2 11.4 1D axial spar or rod element in R3 (3D space)11.5 The Euler-Bernoulli beam element11.6 Euler-Bernoulli frame elements in R211.7 The Timoshenko beam elements11.8 Finite element formulations in R2 and R311.9 Summary12 Convergence, Error Estimation, and Adaptivity12.1 Introduction12.2 h-, p-, k-versions of FEM and their convergence12.3 Convergence and convergence rate12.4 Error estimation and error computation12.5 A priori error estimation12.6 Model problems12.7 A posteriori error estimation and computation12.8 Adaptive processes in finite element computations12.9 SummaryAppendix A: Numerical Integration using Gauss QuadratureA.1 Gauss quadrature in R1, R2 and R3A.2 Gauss quadrature over triangular domains