Finite Elements Using Maple A Symbolic Programming Approach

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.

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.- 1.6.1 Sequences.- 1.6.2 Lists.- 1.6.3 Sets.- 1.6.4 Arrays.- 1.6.5 Tables.- 1.6.6 Strings.- 1.7 Evaluation Rules.- 1.7.1 Levels of Evaluation.- 1.7.2 Last-Name Evaluation.- 1.7.3 One-Level Evaluation.- 1.7.4 Special Evaluation Rules.- 1.7.5 Delayed Evaluation.- 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.- 1.17.1 Example.- 1.17.2 Example.- 1.17.3 Example.- 2. Computational Mechanics.- 2.1 Introduction.- 2.2 Mathematical Modelling of Physical Systems.- 2.3 Continuous Models.- 2.3.1 Equilibrium.- 2.3.2 Propagation.- 2.3.3 Diffusion.- 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.- 3.3.1 Example.- 3.4 Approximation Functions.- 3.5 Admissibility Conditions.- 3.5.1 Example.- 3.6 Global Indirect Discretization.- 3.6.1 Satisfaction of Boundary Conditions.- 3.6.2 Domain Methods of Approximation.- 3.6.3 Galerkin Method.- 3.6.4 Least Squares Method.- 3.6.5 Moments Method.- 3.6.6 Collocation Method.- 3.6.7 Example.- 3.6.8 Example.- 3.7 Integration by Parts.- 3.7.1 Strong, Weak and Transposed Forms.- 3.7.2 One-Dimensional Case.- 3.7.3 Example.- 3.7.4 Higher-Dimensional Cases.- 3.7.5 Example.- 3.8 Local Direct Discretization.- 3.8.1 Nodes and Local Regions.- 3.8.2 Satisfaction of Boundary Conditions.- 3.8.3 Finite Difference Method.- 3.8.4 Finite Element Method.- 3.8.5 Boundary Element Method.- 3.8.6 Example.- 3.8.7 Example.- 3.8.8 Example.- 4. Interpolation.- 4.1 Introduction.- 4.2 Globally Defined Functions.- 4.2.1 Polynomial Bases.- 4.2.2 Example.- 4.2.3 Example.- 4.2.4 Conclusions.- 4.3 Piecewisely Defined Functions.- 4.3.1 Spline Interpolation.- 4.3.2 Finite Element Interpolation.- 4.4 Finite Element Generalized Coordinates.- 4.4.1 Convergence Conditions.- 4.4.2 Geometric Isotropy.- 4.4.3 Finite Element Families.- 4.5 Finite Element Shape Functions.- 4.5.1 Natural Coordinates.- 4.5.2 Curvilinear Coordinates.- 4.5.3 Example.- 4.6 Parametric Finite Elements.- 4.7 Isoparametric Finite Elements.- 4.7.1 Convergence Conditions.- 4.7.2 Evaluation of Element Equations.- 4.7.3 Numerical Integration.- 4.8 Linear Triangular Isoparametric Element.- 4.8.1 Example.- 4.8.2 Example.- 4.8.3 Example.- 4.8.4 Example.- 5. The Finite Element Method.- 5.1 Introduction.- 5.2 Steady-State Models with Scalar Variable.- 5.2.1 Continuous Model.- 5.2.2 Weighted Residual Galerkin Approximation.- 5.2.3 Discrete Model.- 5.3 Finite Element Mesh.- 5.3.1 Linear Triangular Isoparametric Element.- 5.3.2 Total Potential Energy.- 5.3.3 Internal Potential Energy Density.- 5.3.4 Mesh Topology.- 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.- 5.10.1 Data Preparation.- 5.11 Example.- 5.12 Example.- 5.13 Example.- 5.14 Example.- 6. Fluid Mechanics Applications.- 6.1 Introduction.- 6.2 Continuous Models of Fluid Flow.- 6.2.1 Incompressible Fluids.- 6.2.2 Inviscid Fluids.- 6.2.3 Irrotational Flows.- 6.2.4 Steady-State Flows.- 6.2.5 Bernoulli’s Energy Conservation.- 6.2.6 Velocity Potential.- 6.2.7 Stream Function.- 6.3 Confined Flows.- 6.4 Unconfined Flows.- 6.5 Groundwater Flows.- 6.5.1 Darcy’s Hypothesis.- 6.5.2 Dupuit’s Hypothesis.- 6.6 Example.- 6.6.1 Flow Under a Dam.- 6.6.2 Problem’s Solution.- 6.7 Example.- 6.7.1 Flow in an Unconfined Aquifer.- 6.7.2 Problem’s Solution.- 7. Solid Mechanics Applications.- 7.1 Introduction.- 7.2 Continuous Models.- 7.3 Fundamental Continuous Model: Elasticity Theory.- 7.3.1 Strain-Displacement Equations.- 7.3.2 Equilibrium Equations.- 7.3.3 Stress-Strain Equations.- 7.3.4 Boundary Conditions.- 7.3.5 Elastic Fields.- 7.3.6 The Work Theorem.- 7.3.7 Theorem of Virtual Displacements.- 7.3.8 Theorem of Total Potential Energy.- 7.4 Finite Element Model.- 7.4.1 Weighted Residual Equation.- 7.4.2 Theorem of Work.- 7.4.3 Theorem of Virtual Displacements.- 7.4.4 Discretization.- 7.5 Mesh Topology.- 7.5.1 Total Strain Energy.- 7.5.2 Distribution of the Strain Energy Density.- 7.6 Constrained Displacements.- 7.7 Application of the Finite Element Model.- 7.8 Three-Dimensional Equilibrium States.- 7.8.1 Constant-Strain Tetrahedron Element.- 7.9 Two-Dimensional Equilibrium States.- 7.9.1 Plane Stress and Plane Strain.- 7.9.2 Asymptotic Model: Plane Elasticity.- 7.9.3 Constant-Strain Triangular Isoparametric Element.- 7.9.4 Cst_fem: Package for Finite Element Analysis.- 7.9.5 Data Preparation.- 7.9.6 Example.- 7.9.7 Example.- 7.9.8 Example.- 7.9.9 Example.- 7.10 One-Dimensional Equilibrium States.- 7.10.1 Asymptotic Model: Theory of Bars.- 7.10.2 Truss Element.- 7.10.3 Skew Elements.- 7.10.4 Beam Element.- 7.11 Further Study.