Solution Of Multibody Contact Problems Research Articles

On Conditioning of Constraints Arising from Variationally Consistent Discretization of Contact Problems and Duality Based Solvers

Abstract Variationally consistent approximation of the non-penetration conditions and friction laws was introduced by Wohlmuth [Acta Numer. 20 (2011) 569–734] as a powerful tool for the effective discretization of contact problems. This approach is especially useful when the potential contact interface is large and curved or when non-matching grids are applied. The point of this note is to give bounds on the singular values of the constraint matrices arising from the variationally consistent approximation and to indicate their application in the analysis of the scalable duality based domain decomposition algorithms for the solution of multi-body contact problems [Comput. Methods Appl. Mech. Eng. 247–248 (2012), 37–50.]. The theoretical results are illustrated by numerical experiments.

A theoretically supported scalable TFETI algorithm for the solution of multibody 3D contact problems with friction

A Total FETI (TFETI) based domain decomposition algorithm with preconditioning by a natural coarse grid defined by the rigid body motions of subdomains is adapted to the solution of multibody contact problems with friction in 3D and proved to be scalable for Tresca friction. The analysis admits “floating” bodies. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition and the discretization parameters is bounded. The analysis is based on the classical results by Farhat et al. on scalability of FETI for linear problems and on our development of optimal quadratic programming algorithms. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate that our algorithm is effective. The effectiveness of the method in the inner loop of the fixed point iterations for the solution of the contact problems with Coulomb’s friction is illustrated on the analysis of a mine support.

Scalable TFETI algorithm for the solution of multibody contact problems of elasticity

AbstractA Total FETI (TFETI)‐based domain decomposition algorithm with preconditioning by a natural coarse grid of the rigid body motions is adapted to the solution of multibody contact problems of elasticity in 2D and 3D and proved to be scalable. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded. The analysis is based on the classical results by Farhat, Mandel, and Roux on scalability of FETI with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate a high efficiency of our algorithm. In addition, its performance is illustrated on a real‐world problem of analysis of the ball bearing. Copyright © 2009 John Wiley & Sons, Ltd.