Finite Elements Using Maple

A Symbolic Programming Approach

By: Artur Portela, A. Charafi

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Hardcover | 22 April 2003

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346 Pages

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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.

Industry Reviews

From the reviews:

"The purpose of this book is to illustrate how the computational environment of Maple can be used in a finite element ... . It gives to the reader a very unique insight into the finite element method together with symbolic programming approach ... . In summary, this book is an excellent introductory book for finite element method and I strongly recommend to all undergraduate and postgraduate students and also to engineers and scientists who wish to program easily the finite element method." (T.E. Simos, Journal of Computational Methods in Sciences and Engineering, Vol. 3 (2), 2003)

"This book is an essential tool written to be used as the 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." (European Journal of Mechanical and Environmental Engineering, Vol. 47 (2), 2002)

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Introduction to Maple	p. 1
Basics	p. 1
Entering Commands	p. 1
Fundamental Data Types	p. 3
Mathematical Functions	p. 3
Names	p. 4
Basic Types of Maple Objects	p. 5
Sequences	p. 5
Lists	p. 6
Sets	p. 6
Arrays	p. 7
Tables	p. 7
Strings	p. 8
Evaluation Rules	p. 8
Levels of Evaluation	p. 8
Last-Name Evaluation	p. 9
One-Level Evaluation	p. 9
Special Evaluation Rules	p. 10
Delayed Evaluation	p. 10
Algebraic Equations	p. 11
Differentiation and Integration	p. 12
Solving Differential Equations	p. 14
Expression Manipulation	p. 15
Basic Programming Constructs	p. 16
Functions, Procedures and Modules	p. 16
Maple's Organization	p. 19
Linear Algebra Computations	p. 20
Graphics	p. 31
Plotter: Package for Finite Element Graphics	p. 34
Example	p. 39
Example	p. 41
Example	p. 42
Computational Mechanics	p. 45
Introduction	p. 45
Mathematical Modelling of Physical Systems	p. 45
Continuous Models	p. 47
Equilibrium	p. 47
Propagation	p. 49
Diffusion	p. 51
Mathematical Analysis	p. 52
Approximation Methods	p. 52
Discrete Models	p. 55
Structural Models	p. 56
Approximation Methods	p. 59
Introduction	p. 59
Residuals	p. 60
Weighted-Residual Equation	p. 61
Example	p. 61
Approximation Functions	p. 62
Admissibility Conditions	p. 62
Example	p. 63
Global Indirect Discretization	p. 64
Satisfaction of Boundary Conditions	p. 65
Domain Methods of Approximation	p. 66
Galerkin Method	p. 66
Least Squares Method	p. 67
Moments Method	p. 67
Collocation Method	p. 68
Example	p. 70
Example	p. 82
Integration by Parts	p. 84
Strong, Weak and Transposed Forms	p. 84
One-Dimensional Case	p. 85
Example	p. 85
Higher-Dimensional Cases	p. 86
Example	p. 88
Local Direct Discretization	p. 88
Nodes and Local Regions	p. 89
Satisfaction of Boundary Conditions	p. 89
Finite Difference Method	p. 90
Finite Element Method	p. 93
Boundary Element Method	p. 96
Example	p. 98
Example	p. 113
Example	p. 132
Interpolation	p. 135
Introduction	p. 135
Globally Defined Functions	p. 136
Polynomial Bases	p. 136
Example	p. 137
Example	p. 138
Conclusions	p. 142
Piecewisely Defined Functions	p. 143
Spline Interpolation	p. 143
Finite Element Interpolation	p. 144
Finite Element Generalized Coordinates	p. 145
Convergence Conditions	p. 145
Geometric Isotropy	p. 146
Finite Element Families	p. 146
Finite Element Shape Functions	p. 148
Natural Coordinates	p. 150
Curvilinear Coordinates	p. 156
Example	p. 157
Parametric Finite Elements	p. 161
Isoparametric Finite Elements	p. 162
Convergence Conditions	p. 162
Evaluation of Element Equations	p. 164
Numerical Integration	p. 166
Linear Triangular Isoparametric Element	p. 168
Example	p. 169
Example	p. 171
Example	p. 174
Example	p. 176
The Finite Element Method	p. 179
Introduction	p. 179
Steady-State Models with Scalar Variable	p. 179
Continuous Model	p. 180
Weighted Residual Galerkin Approximation	p. 183
Discrete Model	p. 185
Finite Element Mesh	p. 186
Linear Triangular Isoparametric Element	p. 187
Total Potential Energy	p. 188
Internal Potential Energy Density	p. 188
Mesh Topology	p. 189
Local Finite Element Equations	p. 190
Global Finite Element Equations	p. 192
Exact Boundary Conditions	p. 193
Solution of the System of Equations	p. 194
Computation of Derivatives	p. 194
Finite Element Pre- and Post- Processing	p. 196
Cgt_fem: Package for Finite Element Analysis	p. 197
Data Preparation	p. 197
Example	p. 198
Example	p. 208
Example	p. 213
Example	p. 217
Fluid Mechanics Applications	p. 223
Introduction	p. 223
Continuous Models of Fluid Flow	p. 223
Incompressible Fluids	p. 223
Inviscid Fluids	p. 224
Irrotational Flows	p. 224
Steady-State Flows	p. 224
Bernoulli's Energy Conservation	p. 225
Velocity Potential	p. 226
Stream Function	p. 226
Confined Flows	p. 227
Unconfined Flows	p. 228
Groundwater Flows	p. 229
Darcy's Hypothesis	p. 229
Dupuit's Hypothesis	p. 231
Example	p. 232
Flow Under a Dam	p. 232
Problem's Solution	p. 233
Example	p. 240
Flow in an Unconfined Aquifer	p. 240
Problem's Solution	p. 241
Solid Mechanics Applications	p. 251
Introduction	p. 251
Continuous Models	p. 251
Fundamental Continuous Model: Elasticity Theory	p. 252
Strain-Displacement Equations	p. 253
Equilibrium Equations	p. 253
Stress-Strain Equations	p. 254
Boundary Conditions	p. 254
Elastic Fields	p. 255
The Work Theorem	p. 256
Theorem of Virtual Displacements	p. 256
Theorem of Total Potential Energy	p. 256
Finite Element Model	p. 257
Weighted Residual Equation	p. 257
Theorem of Work	p. 258
Theorem of Virtual Displacements	p. 259
Discretization	p. 259
Mesh Topology	p. 261
Total Strain Energy	p. 261
Distribution of the Strain Energy Density	p. 262
Constrained Displacements	p. 262
Application of the Finite Element Model	p. 264
Three-Dimensional Equilibrium States	p. 265
Constant-Strain Tetrahedron Element	p. 265
Two-Dimensional Equilibrium States	p. 267
Plane Stress and Plane Strain	p. 267
Asymptotic Model: Plane Elasticity	p. 269
Constant-Strain Triangular Isoparametric Element	p. 270
Cst_fem: Package for Finite Element Analysis	p. 273
Data Preparation	p. 274
Example	p. 275
Example	p. 281
Example	p. 284
Example	p. 292
One-Dimensional Equilibrium States	p. 302
Asymptotic Model: Theory of Bars	p. 303
Truss Element	p. 312
Skew Elements	p. 314
Beam Element	p. 315
Further Study	p. 318
The Companion CD-ROM	p. 319
References	p. 321
Index	p. 323
Table of Contents provided by Syndetics. All Rights Reserved.

Finite Elements Using Maple

A Symbolic Programming Approach

At a Glance

Hardcover

Industry Reviews

Shipping

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Defective items

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