Finite Elements Using Maple

Almost all physical phenomena can be mathematically described in terms of differential equations. The finite element method is a tool for the appro- mate solution of differential equations. However, despite the extensive use of the finite element method by engineers in the industry, understanding the principles involved in its formulation is often lacking in the common user. As an approximation process, the finite ele~ent method can be for- lated with the general technique of weighted residuals. This technique has the advantage of enhancing the essential unity of all processes of approxi- tion used in the solution of differential equations, such as finite differences, finite elements and boundary elements. The mathematics used in this text, though reasonably rigorous, is easily understood by the user with only a basic knowledge of Calculus. A common problem to the courses of Engineering is to decide about the best form to incorporate the use of computers in education. Traditional c- pilers, and even integrated programming environments such as Turbo Pascal, are not the most appropriate, since the student has to invest much time in developing an executable program that, in the best of cases, will be able to solve only one definitive type of problems. Moreover, the student ends up learning more about programming than about the problem that he/she wants to solve with the developed executable program.

Introduces a new, more systematic approach to Finite Element Analysis using Symbolic Computation Can be a primary text for an undergraduate or early postgraduate course, as well as a reference book for engineers and scientists

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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 for undergraduate or early postgraduate course, as well as a reference book for engineers and scientists who want to develop quickly finite-element programs. The use of symbolic computation in Maple system delivers new benefits in the analysis and understanding of The finite element method.

Contenu

1. Introduction to Maple.- 1.1 Basics.- 1.2 Entering Commands.- 1.3 Fundamental Data Types.- 1.4 Mathematical Functions.- 1.5 Names.- 1.6 Basic Types of Maple Objects.- 1.7 Evaluation Rules.- 1.8 Algebraic Equations.- 1.9 Differentiation and Integration.- 1.10 Solving Differential Equations.- 1.11 Expression Manipulation.- 1.12 Basic Programming Constructs.- 1.13 Functions, Procedures and Modules.- 1.14 Maple's Organization.- 1.15 Linear Algebra Computations.- 1.16 Graphics.- 1.17 Plotter: Package for Finite Element Graphics.- 2. Computational Mechanics.- 2.1 Introduction.- 2.2 Mathematical Modelling of Physical Systems.- 2.3 Continuous Models.- 2.4 Mathematical Analysis.- 2.5 Approximation Methods.- 2.6 Discrete Models.- 2.7 Structural Models.- 3. Approximation Methods.- 3.1 Introduction.- 3.2 Residuals.- 3.3 Weighted-Residual Equation.- 3.4 Approximation Functions.- 3.5 Admissibility Conditions.- 3.6 Global Indirect Discretization.- 3.7 Integration by Parts.- 3.8 Local Direct Discretization.- 4. Interpolation.- 4.1 Introduction.- 4.2 Globally Defined Functions.- 4.3 Piecewisely Defined Functions.- 4.4 Finite Element Generalized Coordinates.- 4.5 Finite Element Shape Functions.- 4.6 Parametric Finite Elements.- 4.7 Isoparametric Finite Elements.- 4.8 Linear Triangular Isoparametric Element.- 5. The Finite Element Method.- 5.1 Introduction.- 5.2 Steady-State Models with Scalar Variable.- 5.3 Finite Element Mesh.- 5.4 Local Finite Element Equations.- 5.5 Global Finite Element Equations.- 5.6 Exact Boundary Conditions.- 5.7 Solution of the System of Equations.- 5.8 Computation of Derivatives.- 5.9 Finite Element Pre- and Post- Processing.- 5.10 Cgt-fem: Package for Finite Element Analysis.- 5.11 Example.- 5.12 Example.- 5.13 Example.- 5.14 Example.- 6. Fluid MechanicsApplications.- 6.1 Introduction.- 6.2 Continuous Models of Fluid Flow.- 6.3 Confined Flows.- 6.4 Unconfined Flows.- 6.5 Groundwater Flows.- 6.6 Example.- 6.7 Example.- 7. Solid Mechanics Applications.- 7.1 Introduction.- 7.2 Continuous Models.- 7.3 Fundamental Continuous Model: Elasticity Theory.- 7.4 Finite Element Model.- 7.5 Mesh Topology.- 7.6 Constrained Displacements.- 7.7 Application of the Finite Element Model.- 7.8 Three-Dimensional Equilibrium States.- 7.9 Two-Dimensional Equilibrium States.- 7.10 One-Dimensional Equilibrium States.- 7.11 Further Study.