Finite Element Methods in CAD

ISBN-13: 9781461598817 / Angielski / Miękka / 2012 / 194 str.

Jean-Claude Sabonnadiere

Opis

The finite element method (FEM) has been understood, at least in principle, for more than 50 years. The integral formulation on which it is based has been known for a longer time (thanks to the work of Galerkin, Ritz, Courant and Hilbert,1,4 to mention the most important). However, the method could not be applied in a practical way since it involved the solution of a large number of linear or non-linear algebraic equations. Today it is quite common, with the aid of computers, to solve non-linear algebraic problems of several thousand equations. The necessary numerical methods and programming techniques are now an integral part of the teaching curriculum in most engineering schools. Mechanical engineers, confronted with very complicated structural problems, were the first to take advantage of advanced computational methods and high level languages (FORTRAN) to transform the mechanical models into algebraic equations (1956). In recent times (1960), the FEM has been studied by applied mathematicians and, having received rigorous treatment, has become a part of the more general study of partial differential equations, gradually replacing the finite difference method which had been considered the universal tool to solve these types of problems.

Spis treści

1 Principles of the finite element method.- 1.1. Principle equations of mathematical physics.- 1.2. The idea of a well posed problem.- 1.3. Integral formulation.- 1.4. Approximation of the unknown functions.- 1.5. Minimization of the functional with the aid of approximation functions.- 1.6. The finite element mesh and the approximation functions.- 2 From one dimension… to three.- 2.1. Elementary principles of the finite element method.- 2.2. Example 1: one dimension.- 2.3. Example 2: two dimensions.- 2.4. Example 3: three dimensions.- 2.5. Example 4: problems in the time domain.- 3 Finite elements and approximation functions.- 3.1. Introduction.- 3.2. One-dimensional elements.- 3.3. Two-dimensional elements.- 3.4. Three-dimensional elements.- 3.5. Conclusion.- 4 Numerical methods.- 4.1. Methods for solving systems of linear equations.- 4.2. Non-linear systems: the Newton-Raphson method.- 4.3. Numerical methods for calculating definite integrals.- 4.4. Differential equations with initial conditions.- 4.5. Conclusion.- 5 General theory of second order isoparametric elements.- 5.1. Introduction.- 5.2. Setting up the equations.- 5.3. Application of the Newton-Raphson method.- 5.4. Construction of the matrix H and the vectors F and R.- 5.5. The finite elements.- 5.6. Application.- 6 General architecture of CAD systems based on the finite element method.- 6.1. General structure.- 6.2. The data entry module.- 6.3. The solver.- 6.4. Postprocessors.- 6.5. Architecture of finite element software.- 6.6. Communication between programs.- 6.7. Interdisciplinary software.- 6.8. Conclusion.- 7 Geometry, mesh generation and physical properties.- 7.1. Description of the geometry.- 7.2. Discretization of the domain.- 7.3. Description of physical characteristics.- 7.4. Conclusion.- 8 Postprocessing.- 8.1. Objectives of postprocessing.- 8.2. Extraction of information.- 8.3. Visualization of the information.- 8.4. Conclusion.- 9 Applications.- 9.1. The FLUX2D package.- 9.2. FLUX3D.- 9.3. Microcomputer software.- 9.4. The super programs.- Conclusion.- Index 191.