On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics

Abstract

This paper addresses an efficient and robust automatic adaptive local multilevel mesh refinement strategy for unilateral frictional contact problems in elastostatics. The proposed strategy couples the Local Defect Correction multigrid method LDC (Hackbusch, 1984) with the ZZ (Zienkiewicz and Zhu, 1987) a posteriori error estimator. An extension of LDC method to frictional contact problems is introduced. An interesting feature of this extended LDC algorithm is that it still only lies on interpolations of displacement fields. Neither forces conservation nor exchange of contact status is required between the refinement levels. The ZZ a posteriori error estimator is exploited to automatically build the sub-grids of the LDC method. A criterion linked to the relative error in stress is used. The efficiency of the proposed strategy is analyzed on examples derived from nuclear engineering. Practical numerical choices are proposed and justified. The refinement process automatically varies and stops with respect to a given tolerance. Post-treatments show that the sub-grids focus around the contact areas and that the converged LDC solution always respects the prescribed tolerance.

Original geometrical stopping criteria associated to multilevel adaptive mesh refinement for problems with local singularities

“…One can remark that on the initial grid (k = 0), only volume forces This method has been initially introduced and applied for computational fluids problems and has been recently successfully extended to solids me-chanics: elastostatics problems in [4], elastostatics problems with frictional contact in [30] and Norton-type material behaviour in [5].…”

“…Various stopping criteria are often combined in order to optimize the refinement strategy, see for example [12,30].…”

“…This kind of methodology is cheap because only additional little size problems are solved. Error estimators can be easily coupled to such methods in order to automatically detect the region of interest [5,30]. As the multilevel process is inherently iterative, no density function is required: a fixed refinement ratio can be applied on the detected zones.…”

“…The restriction operator slightly differs from the one proposed by Hackbusch [24]. Indeed the subset A l (and consequentlyÅ l ) has been enlarged in order to take into account the contribution of the natural BCs on the next finer grid, see [30] for more details.…”

Superconvergent patch recovery with constraints for three‐dimensional contact problems within the Cartesian grid Finite Element Method

“…The ZZ estimator has been adapted to contact problems with different approaches. Some works in this direction are the use of the global version of the ZZ estimator for Coulomb's frictional contact and its extension to multigrid methods in Reference .…”