Abstract
Multigrid methods for two-body contact problems are mostly based on special mortar discretizations, nonlinear Gauss-Seidel solvers, and solution-adapted coarse grid spaces. Their high computational efficiency comes at the cost of a complex implementation and a nonsymmetric master-slave discretization of the nonpenetration condition. Here we investigate an alternative symmetric and overconstrained segment-to-segment contact formulation that allows for a simple implementation based on standard multigrid and a symmetric treatment of contact boundaries, but leads to nonunique multipliers. For the solution of the arising quadratic programs, we propose augmented Lagrangian multigrid with overlapping block Gauss-Seidel smoothers. Approximation and convergence properties are studied numerically at standard test problems.
Highlights
Mortar methods are based on a nonsymmetric master-slave formulation and require the computation of a discrete L2 projection
Whereas the computation of the L2 projection for practical application is possible but technically demanding [13], the asymmetry of the approach necessitates a tedious assignment of master and slave sides, which is demanding for self-contact
Sample points xi with no potential contact partner xi found, i.e. g(xi) = ∞, can be omitted. This setting includes two-pass point-to-segment and segment-to-segment contact formulations depending on the distribution of sample points xi, but is different from integrated constraints used in mortar methods
Summary
The second class of contact formulation includes exterior penalty methods as well as barrier type approaches [22]. Their obvious advantage is the structural simplicity of the resulting unconstrained minimization problems. Stationary contact problems in linear elasticity, constraint formulations lead to quadratic programs to be solved. Both standard algorithms from optimization, such as projected gradient or active set/SQP methods, and algorithms adapted to the PDE structure at hand, such as multigrid and domain decomposition methods, are in use. The resulting solution method is investigated numerically using standard test examples
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