Portela A., Charafi A. Finite Elements Using Maple. A Symbolic Programming Approach

Springer, 2002. — 354 p. — (Engineering Online library). — ISBN: 3-540-42986-7. OCRMaple is a computational environment with symbolic, numerical and graphical programming capabilities that allows a radical change in the way computers are used in education. Effectively, Maple software can be used in the form of non-declarative programming which means that the user tells the system what to do, without telling it how to do. Thus, Maple opens to the student the possibility of investing less time in programming and much more time in the study of the problem under consideration.
This textbook illustrates how Maple can be used in a finite element introductory course. Providing the user with a unique insight into the finite element method, along with symbolic programming that fundamentally changes the way applications can be developed. This book is an essential tool written to be used as a primary text for an undergraduate or early postgraduate course, as well as a reference book for engineers and scientists who want to develop quickly finite-element programs. The book is split into 7 chapters and 1 appendix.
Chapter 1 presents a brief introduction of the computational system Maple, referring only to the aspects considered relevant in programming the finite element method which include mainly symbolic programming and graphic visualization.
Chapter 2 presents an introduction to Computational Mechanics which deals with the mathematical modeling process of physical systems. The chapter begins with the presentation of the essential objective of the whole modeling process, the substitution of the continuous model of the physical system by a discrete model that is represented by a system of algebraic equations. A classification of physical systems, based on the type of the differential equation that defines the respective continuum model, is presented. As a consequence of the difficulty in obtaining analytical solutions of the differential equation that represents the continuous model of the physical system, the discretization process is then introduced to generate discrete models which lead to approximate solutions.
Chapter 3 deals with the formulation of weighted residual approximation methods. The general equation of weighted residuals is presented as the starting point of their formulation. The chapter considers first the case of approximation functions with a global definition and indirect discretization, setting up their admissibility conditions. Domain and boundary models are defined on the basis of the possibility of the approximation function satisfying the boundary conditions. The methods of Galerkin, least squares, moments and collocation, obtained by defining the appropriate weighting functions, are presented. Integration by parts of the general weighted residual equation is used to obtain weaker admissibility conditions for the approximation function, leading to the weak and transposed forms of the weighted residual equation. Approximation functions with a piecewise continuous definition and direct discretization are then considered, as well as their respective admissibility conditions. Finally, the models of finite differences, finite elements and boundary elements are presented, as representative of the direct methods with piece-wisely defined continuous approximation.
Chapter 4 presents some topics of interpolation. The chapter begins with general aspects of interpolation with both global and piecewise functions. The difficulty of spline interpolation is contrasted with the simplicity of finiteelement interpolation. Finite element interpolation functions, defined in terms of generalized coordinates, are first introduced along with the convergence conditions, referred to as conditions of compatibility and completeness. Finite elements with interpolation in terms of shape functions are then considered. Natural coordinates as well as curvilinear coordinates are introduced leading to the formulation of parametric finite elements.
Chapter 5 introduces the finite element method. A steady-state continuous model, with a scalar variable, is considered for two-dimensional problems. Linear triangular isoparametric finite elements are used. The finite element package Cgt.fem, specially developed using Maple, is used to present the basic steps in the application of the finite element method.
Chapter 6 applies the finite element method to problems of Fluid Mechanics. A description of continuous models relative to perfect-fluid flows, free-surface flows and flow through porous media is first presented. Finally, the finite element method is applied to solve steady-state problems, with the Maple package Cgt.fem.
Chapter 7 formulates and applies the finite element method to problems of Solid Mechanics. The presentation, confined to the linear theory, deals with the so-called assumed-displacement formulation. The chapter begins with a summary of the general continuous model that is the three-dimensional theory of elasticity, presenting the concepts of static and kinematic admissibility. The correspondence between the work theorem, specified for a virtual displacement, and the equation of weighed residuals is then presented. The minimum total potential energy is used to show that the finite element model is more rigid than the exact solution. Asymptotic models, derived from the three-dimensional model, are established for both one-dimensional and two-dimensional structural elements. The essential aspects of finite-element meshes are analyzed focusing, in particular, on the respective topology optimization. Maple package Cst_fem, specially developed for the finite element analysis of two-dimensional elasticity problems with linear triangular isoparametric elements, is then presented and applied to the solution of several problems.
Appendix A presents details of the content of the companion CD-ROM. All the application examples of the book are included in the CD-ROM, where the results are presented in colour and with animations.