Master-slave elimination scheme for arbitrary smooth nonlinear multi-point constraints

J. Boungard, J. Wackerfuß

Nonlinear multi-point constraints are essential in modeling various engineering problems, for example in the context of (a) linking individual degrees of freedom of multiple nodes to model nonlinear joints, (b) coupling different element types in finite element analysis, (c) enforcing various types of rigidity in parts of the mesh and (d) considering deformation-dependent Dirichlet boundary conditions. One method for addressing constraints is the master–slave elimination, which offers the benefit of reducing the problem dimension as opposed to Lagrange multipliers and the penalty method. However, the existing master–slave elimination method is limited to linear constraints. In this project, we introduce a new master–slave elimination method for handling arbitrary smooth nonlinear multi-point constraints in the system of equations of the discretized system. We present a rigorous mathematical derivation of the method. Within this method, new constraints can be easily considered as an item of a “constraint library”; i.e. no case-by-case-programming is required. In addition to the theoretical aspects, we also provide helpful remarks on the efficient implementation. Among others, we show that the new method results in a reduced computational complexity compared to the existing methods. The study also places emphasis on comparing the new approach with existing methods via numerical examples. We have developed innovative benchmarks which encompass all relevant computational properties, and provide analytical and reference solutions. Our findings demonstrate that our new method is as accurate, robust and flexible as the Lagrange multipliers, and more efficient due to the reduction of the total number of degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.

Boungard, J., & Wackerfuß, J. (2024). Master-slave elimination scheme for arbitrary smooth nonlinear multi-point constraints. Computational Mechanics, 14, published online. https://doi.org/10.1007/s00466-024-02463-7

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.

Boungard, J., & Wackerfuß, J. (2023). Consideration of nonlinear multipoint constraints in finite element analyses based on a master-slave elimination scheme operating at the global level. PAMM Proceedings in Applied Mathematics and Mechanics, 23, e202200311. https://doi.org/10.1002/pamm.202200311