Multilevel Adaptive Mesh Refinement with controlled accuracy for nonlinear quasi-static mechanics
Résumé
We propose an adaptive mesh refinement (AMR) algorithm dedicated to the simulation of nonlinear
quasi-static solid mechanics problems with phenomena localized at the structural scale [1]. The proposed
method allows us to follow in time the evolution of studied phenomena in a fully-automatic (thanks to an
a posteriori error estimator [2]), precise (respecting user-prescribed accuracies) and efficient (in terms of
memory space and computational time) way.
We adopt the multilevel Local Defect Correction (LDC) refinement method [3] for its great potentialities
for solving elliptic problems [4]. We propose an algorithmic extension of the LDC method to nonlinear
quasi-static problems and provide key aspects associated to its practical implementation. We highlight
its efficiency in the nonlinear context and show its natural ability to generate a hierarchy of meshes of
limited sizes that dynamically follow the evolution over time of studied phenomena.
We address generic AMR-related questions associated to dynamic mesh adaptation, such as fields trans-
fer between time steps as well as the discretization error control over time. We propose a straightforward
and efficient equilibration strategy lying on the introduction of the initial non equilibrated residual as a
source term of the problem. We also develop a reliable remeshing algorithm aiming to limit the number
of mesh regeneration over time while guaranteeing the fulfillment of prescribed errors.
Several numerical experiments, in 2D and 3D, with different types of material behavior as well as variable
loadings are proposed to validate the efficiency of the developed algorithm.
REFERENCES
[1] Koliesnikova, D., Ramière, I. and Lebon, F. Fully automatic multigrid adaptive mesh refinement
strategy with controlled accuracy for nonlinear quasi-static problems. in progress.
[2] Verfürth, R. A review of a posteriori error estimation and adaptive mesh-refinement techniques.
Wiley, Chichester. (1996).
[3] Hackbusch, W. Local Defect Correction method and Domain Decomposition techniques, In: Defect
Correction Methods. Springer, Vienna (1984) 89-113.
[4] Koliesnikova, D., Ramière, I. and 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. J. of Computational Physics (2021) 110310.