Multiscale Coarsening for Linear Elasticity by Energy Minimization
In this work, we construct energy-minimizing coarse spaces for the finite element discretization of mixed boundary value problems for displacements in compressible linear elasticity. Motivated from the multiscale analysis of highly heterogeneous composite materials, basis functions on a triangular coarse mesh are constructed, obeying a minimal energy property subject to global pointwise constraints. These constraints allow that the coarse space exactly contains the rigid body translations, while rigid body rotations are preserved approximately. The application is twofold. Resolving the heterogeneities on the finest scale, we utilize the energy-minimizing coarse space for the construction of robust two-level overlapping domain decomposition preconditioners. Thereby, we do not assume that coefficient jumps are resolved by the coarse grid, nor do we impose assumptions on the alignment of material jumps and the coarse triangulation. Weonly assume that the size of the inclusions is small compared to the coarse mesh diameter. Ournumerical tests show uniform convergence rates independent of the contrast in the Young’s modulus within the heterogeneous material. Furthermore, we numerically observe the properties of the energy-minimizing coarse space in an upscaling framework. Therefore, we present numerical results showing the approximation errors of the energy-minimizing coarse space w.r.t. the fine-scale solution.
KeywordsLinear elasticity Domain decomposition Robust coarse spaces Energy-minimizing shape functions
The authors would like to thank Dr. Panayot Vassilevski and Prof. Ludmil Zikatanov for many fruitful discussions and their valuable comments on the subject of this paper.
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