Mathematical foundation of the master-slave elimination for arbitrary nonlinear multi-point constraints

Nonlinear multi-point constraints are essential in modeling various engineering problems, for example, joints undergoing large rotations or coupling of different element types in finite element analysis. They can be handled using Lagrange multipliers, the penalty method, or master-slave elimination. The latter satisfies the constraints exactly and reduces the dimension of the resulting system of equations, which is particularly advantageous when a large number of constraints have to be considered. However, the existing schemes in literature are limited to linear constraints. Therefore, the authors introduced an extension of the method to arbitrary nonlinear constraints. A mathematically rigorous derivation of this new method is presented. The starting point is the optimization problem with constraints. It is transformed into a modified optimization problem without constraints using the implicit function theorem. For this, an appropriate selection of slave degrees of freedom (dofs) is crucial, ensuring that the Jacobian constraints meets specific conditions. This implies the consideration of three aspects: The handling of redundant constraints, the automatic selection of slave dofs, and the change of slave dofs in the context of large deformations. Additionally, it allows for the combination of the new method with existing constraint methods.