A parallel multigrid method for constrained minimization problems and its application to friction, contact, and obstacle problems. (English) Zbl 1360.65175
Summary: The parallel solution of constrained minimization problems requires special care to be taken with respect to the information transfer between the different subproblems. Here, we present a nonlinear decomposition approach which employs an additional nonlinear correction step along the processor interfaces. Our approach is generic in the sense that it can be applied to a wide class of minimization problems with strongly local nonlinearities, including even nonsmooth minimization problems. We also describe the implementation of our nonlinear decomposition method in the object oriented library ObsLib\(++\). The flexibility of our approach and its implementation is presented along different problem classes as obstacle problems, frictional contact problems and biomechanical applications. For the same examples, number of iterations, computation time, and parallelization speedup are measured, and the results demonstrate that the implementation scales reasonably well up to 4096 processors.
MSC:
| 65K10 | Numerical optimization and variational techniques |
| 65Y05 | Parallel numerical computation |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15A23 | Factorization of matrices |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74P10 | Optimization of other properties in solid mechanics |
| 74M10 | Friction in solid mechanics |
| 74M15 | Contact in solid mechanics |
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