20 January – ‪Isabelle Ramiere‬ : Automatic multigrid adaptive mesh refinement with controlled accuracy for quasi-static nonlinear solid mechanics

Isabelle Ramiere‬ Thursday 20th January at 11:00



Many real industrial problems involve localized effects (nonlinearity, contact, heterogenity,…). Adaptive Mesh Refinement (AMR) approaches are well-suited numerical techniques to take into account mesoscale phenomena in simulation processes. For implicit solvers (such as for quasi-static mechanics problems), classical h and/or p-adaptive refinement strategies consisting in generating a unique global mesh locally refined (in mesh step and/or in degree of basis function) are limited by the resulting size of problems to be solved (cf. number of DoFs). Hence, we were interested in local multigrid methods, consisting in adding local refined nested meshes in zones of interest without modifying the initial computation mesh. An iterative process (similar to standard multigrid solvers) enables to correct to various levels solutions. We have extended the multigrid Local Defect Correction (LDC) method (Hackbusch, 1984), initially introduced in Computational Fluid Dynamics, to elastostaticity (Barbié et al., 2014) with a multilevel generalization of the algorithm. In order to automatically detect the zone of interest and hence to avoid the pollution error, the LDC method has been coupled with an a posteriori error estimate of Zienckiewicz-Zhu type (Barbié et al., 2014; Barbié et al., 2015; Liu et al., 2017). We also proposed an original stopping criterion in case of local singularity (Ramière et al., 2019). We have compared in (Koliesnikova et al.,2021) within a unified AMR framework the efficiency of the LDC method with respect to conforming and nonconforming h-adaptive strategies. We have also extended the LDC method to structural mechanics nonlinearities. In (Liu et al., 2017), an efficient algorithm has been developed in order to deal with frictional contact via the LDC method. For nonlinear material behaviours, a one time step algorithm has been first introduced in (Barbié et al., 2015) while a fully automatic algorithm in time with error control and fields transfer have been recently proposed (Koliesnikova et al.,2022). Finally, we have shown in (Koliesnikova et al.,2020) that the two-level LDC method can be seen as a mesoscale homogenization method and introduced a unified algorithm for nonintrusive multiscale numerical coupling methods (numerical homogenization methods (FE2, HMM,…), glocal/local coupling,…).

(Hackbusch, 1984) Hackbusch W. . Local Defect Correction Method and Domain Decomposition Techniques. Computing Suppl. Springer-Verlag, 5:89–113, 1984.

(Barbié et al., 2014) Barbié L., Ramière I. et Lebon F. . Strategies around the Local Defect Correction multi-level refinement method for three-dimensional linear elastic problems.
Computers and Structures, 130:73–90, 2014.

(Barbié et al., 2015) Barbié L., Ramière I. et Lebon F. . An automatic multilevel refinement technique based on nested local meshes for nonlinear mechanics. Computers and Structures,
147:14–25, 2015.

(Liu et al., 2017) Liu H., Ramière I. et Lebon F. . On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics.
Computer Methods in Applied Mechanics and Engineering, 323:1–26, 2017.

(Ramière et al., 2019) Ramière I., Liu H. et Lebon F. . Original geometrical stopping criteria associated to multilevel adaptive mesh refinement for problems with local singularities.
Computational Mechanics, 64(3):645–661, 2019.

(Koliesnikova et al.,2020) Koliesnikova D., Ramière I. et Lebon F. . Analytical comparison of two multiscale coupling methods for nonlinear solid mechanics. Journal of Applied Mechanics,
87(9):094501, 2020.

(Koliesnikova et al.,2021) Koliesnikova D., Ramière I. et Lebon F. . A unified framework for the computational comparison of adaptive mesh refinement strategies for all-quadrilateral and all-hexahedral meshes : locally adaptive multigrid methods versus h-adaptive
methods. Journal of Computational Physics, 437:110310, 2021.

(Koliesnikova et al.,2022) Koliesnikova D., Ramière I. et Lebon F. . Fully automatic multigrid adaptive mesh refinement strategy with controlled accuracy for nonlinear quasi-static problems, preprint, 2022.

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