Introduction
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The consideration of boundary conditions in *Code_Aster* can be achieved either by a direct elimination technique in simple cases (AFFE_CHAR_CINE), or through a dualization technique (AFFE_CHAR_MECA), and the introduction of Lagrange multipliers, for the most general cases. This last approach is very general and makes it possible to treat all types of boundary conditions under the same formalism.

However, this approach has two drawbacks. On the one hand, the addition of Lagrange multipliers increases the number of degrees of freedom of the global problem, and therefore the size of the system to be solved. This point can become a disadvantage as the number of boundary conditions increases (numerous interfaces, kinematic links, rigid motion impositions, etc.). On the other hand, the particular dualization technique leads to the loss of the positive semi-definite character of the stiffness matrix. The search for the solution is then no longer a search for a minimum, but a search for a point saddle []. This change in nature can lead to robustness defects, or even to the failure of resolution in the case of iterative solvers.

This document presents an alternative method to the introduction of Lagrange multipliers to take account of affine boundary conditions. This approach does not replace the introduction of Lagrange multipliers, especially for studies involving contact. On the other hand, in the case of studies where the boundary conditions are fixed, this approach allows gains in robustness, and in performance in the case of a large number of boundary conditions. First, we recall the general technique for eliminating boundary conditions, and then we present the approach used to eliminate boundary conditions in practice, as well as the various associated algorithms. Finally, the elimination technique is presented in the case of an eigenvalue problem.