r3.01.00 The isoparametric finite element method

Summary:

This document presents the foundations of iso-parametric finite elements introduced in Code_Aster for modeling 2D and 3D continuous media. First, we recall the transition from a strong formulation to a variational formulation, then we detail discretization by finite elements: use of a reference element, calculation of form functions and evaluation of elementary terms. The principle of assembling these terms and imposing boundary conditions is also briefly described, and the matrix resolution methods used are discussed. Finally, the main steps of a finite element calculation as designed and implemented in Code_Aster are presented.

Table of Contents

• 1. Introduction
• 2. Obtaining a variational formulation
  • 2.1. Physical problem modeling — Principles and notation
  • 2.2. System equations
  • 2.3. Weighted residue method — Strong integral formulation
  • 2.4. Weak integral formulation
• 3. Finite element method
  • 3.1. General principles
  • 3.2. Geometry approximation
  • 3.3. Representation of the unknowns
  • 3.4. Existence and uniqueness results
• 4. Ritz method
• 5. Construction of the matrix system
  • 5.1. New notation (Voigt notation)
  • 5.2. Discretized system
  • 5.3. Calculation of elementary terms
• 6. Matrix system resolution
  • 6.1. Imposition of kinematic boundary conditions
  • 6.2. Resolution
• 7. Organization of a finite element calculation in the Code_Aster
  • 7.1. Finite element concept in Code_Aster
  • 7.2. Item initializations
  • 7.3. Calculation of elementary terms
  • 7.4. Global resolution
• 8. Bibliography
• 9. Description of document versions