Meshing, Geometric Modeling and Numerical Simulation 3

Chapter 6Meshes and Finite Element Calculations

It is not entirely evident that the finite element method, although taught in engineering schools and on some university courses, is mastered fully from the purely practical point of view¹. In addition, in books or courses, it is often the triangle of degree 1 that is presented without going any further; in other words, where the difficulties begin. These are a few raisons d'etre of this chapter.

Therefore, in the example of a simple problem with an unknown scalar (the heat equation with temperature as unknown) and an unknown vector (the equation of elasticity with displacements as unknowns), it will be shown how to calculate the matrices and second elementary (thereby, per element) and global (per assembly) members intervening in the resolution by finite elements of the system of partial differential equations associated with the problem being considered.

Synthetically, for a linear problem, a calculation using the finite element method consists, in practice, of:

• – building a mesh of the computational domain, a cover composed of geometric elements;
• – shifting from geometric elements to finite elements;
• – calculating matrices and right-hand sides of every mesh element;
• – building, by assembly, the matrices and second global members;
• – solving the corresponding system (here linear) by taking into account the possible essential boundary conditions;
• – analyzing (drawing) the solution.

This scheme is the simplest that can be imagined. ...