.. _R3.03.01:

**r3.03.01** Dualization of boundary conditions
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**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 :math:`{\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**




.. toctree::
    :hidden:

    self

.. toctree::
    :maxdepth: 2
    :numbered:

    Introduction
    Dualisation_des_conditions_aux_limites_cin_matiques__principe_des_multiplicateurs_de_Lagrange
    Inconv_nients_de_cette_dualisation
    Principe_des__doubles_Lagrange_
    Avantage_suppl_mentaire
    Remarque_sur_le_conditionnement_du_syst_me
    Modes_propres_et_param_tres_de_Lagrange
    Description_des_versions_du_document