r3.03.01 Dualization of boundary conditions

Summary:

We explain the principle of Lagrange multipliers to solve linear systems under affine constraints resulting from the imposition of kinematic boundary conditions. Since the stiffness matrix obtained is no longer positive, certain resolution algorithms therefore become unusable. We are therefore looking for a technique to be able to continue using the factorization algorithm \({\mathit{LDL}}^{T}\) without permutation and without elimination. The technique proposed is that of « double Lagrange » (used in the Castem2000 code). It is shown that this technique is effective. Some indications are given on the conditioning of matrices obtained by this technique.

The problem of finding the natural modes of constrained systems is then examined. It is shown that a possible solution is to add the dualized boundary conditions to the « stiffness » matrix and not to touch the « mass » matrix.

Table of Contents

• 1. Introduction
• 2. Dualization of kinematic boundary conditions, principle of Lagrange multipliers
• 3. Disadvantages of this dualization
• 4. Principle of « double Lagrange »
• 5. Additional benefit
• 6. Note on system conditioning
• 7. Eigenmodes and Lagrange parameters
  • 7.1. Introduction
  • 7.2. Mechanical problem to be solved
  • 7.3. Reduced system
  • 7.4. Dualized system
  • 7.5. example
  • 7.6. Conclusions
• 8. Description of document versions