# Identification, elimination and handling of redundant nonlinear multi-point constraints

J. Boungard, J. Wackerfuß

An important challenge that arises frequently, when dealing with constraints, is redundancy. A set of constraints is redundant if there are more constraints than necessary to realize the modeling that the constraints are intended to achieve (e.g. distance constraints to model a rigid inclusion in an elastic matrix). With the exception of a few special cases, redundancy is an undesirable property. For the Lagrange multipliers, the augmented Lagrange multipliers and the existing master-slave elimination schemes for linear constraints, redundancy leads to a singular effective stiffness matrix resulting in an undesired termination of the simulation. Therefore, it is inevitable to employ a robust identification and elimination method of redundant constraints.

Redundancy has various causes that can be categorized as follows: First, there are error-related causes. This covers incorrect modeling and incorrect input in a finite element software by the user. The simplest example of this is the repeated input of an existing constraint. Errors in modeling or input can even lead to contradictory constraints. Second, the complexity of the model can forbid the input of a non-redundant set of constraints by the user. In this case, the model and input are actually correct and a non-redundant set of constraints exists in theory. However, it is impossible to find this set by hand. This is due to (i) a large number of coupled constraints, (ii) the dependency of the topology and coupling of constraints on the finite element mesh, (iii) the concentration of coupled constraints at a certain location, (iv) three-dimensionality or (v) a combination of these issues. In addition, often, there exist no algorithm for an automated input that generates a non-redundant set of constraints. Third, strong nonlinearity of constraints can lead to redundancy. A well-known example for this is contact. Fourth, related to the last category, there are nonlinear constraints that are redundant in equilibrium configuration but not in non-equilibrium configuration during the Newton-Raphson iteration.

A critical problem related to redundancy is the contradiction of constraints. A set of constraints is contradictory if there exists no solution fulfilling all constraints. In contrast to redundancy, which cannot always be prevented and which does not necessarily lead to difficulties in finding a solution, contradiction is always an undesirable property, impairs the solvability of the problem and must therefore be avoided.

Similarly to redundancy, contradiction has various causes that can be categorized as follows: First, there are error-related causes. These are the same causes as for redundancy. Second, there are ill-posed problems that are inherently contradictory or become contradictory over the course of the deformation.

## Publikationen

Boungard, J., Wackerfuß, J. Identification, elimination and handling of redundant nonlinear multi-point constraints (in preparation)

Wackerfuß, J., & Boungard, J. (2024). On the embedding of nonlinear multipoint constraints in the finite element method.  In ECCOMAS (Hrsg.), The 9th European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS Congress 2024, 3–7 June 2024, Lisboa, Portugal.

Wackerfuß, J., & Boungard, J. (2024). A computationally efficient method for considering a large number of nonlinear multi-point constraints within the finite element method. In A. Korobenko, M. Laforest, S. Prudhomme, & R. Vaziri (Hrsg.), 16th World Congress in Computational Mechanics (WCCM) 21-26 July 2024, Vancouver, Canada.