Massively parallel finite difference elasticity using a block-structured adaptive mesh refinement with a geometric multigrid solver
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The finite element method (FEM) is, by far, the dominant method for performing elasticity calculations. The advantages are primarily (1) its ability to handle meshes of complex geometry using isoparametric elements, and (2) the weak formulation which eschews the need for computation of second derivatives. Despite its widespread use, FEM performance is sub-optimal when working with adaptively refined meshes, due to the excess overhead involved in reconstructing stiffness matrices. Furthermore, FEM is no longer advantageous when working with representative volume elements (RVEs) that use regular grids. Blockstructured AMR (BSAMR) is a method for adaptive mesh refinement that exhibits good scaling and is well-suited for many problems in materials science. Here, it is shown that the equations of elasticity can be efficiently solved using BSAMR using the finite difference method. The boundary operator method is used to treat different types of boundary conditions, and the "reflux-free" method is introduced to efficiently and easily treat the coarse-fine boundaries that arise in BSAMR. Examples are presented that demonstrate the use of this method in a variety of cases relevant to materials science, including Eshelby inclusions, material discontinuities, and phase field fracture. It is shown that the implementation scales very well to tens of millions of grid points and exhibits good AMR efficiency
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