A second-order sharp numerical method for solving the linear elasticity equations on irregular domains and adaptive grids – Application to shape optimization

Abstract

We present a numerical method for solving the equations of linear elasticity on irregular domains in two and three spatial dimensions. We combine a finite volume and a finite difference approaches to derive discretizations that produce second-order accurate solutions in the L^∞-norm. Our discretization is 'sharp' in the sense that the physical boundary conditions (mixed Dirichlet/Neumann-type) are imposed at the interface and the solution is computed inside the irregular domain only, without the need of smearing the solution across the interface. The irregular domain is represented implicitly using a level-set function so that this approach is applicable to free moving boundary problems; we provide a simple example of shape optimization to illustrate this capability. In addition, we provide an extension of our method to the case of adaptive meshes in both two and three spatial dimensions: we use non-graded quadtree (2D) and octree (3D) data structures to represent the grid that is automatically refined near the irregular domain's boundary. This extension to quadtree/octree grids produces second-order accurate solutions albeit non-symmetric linear systems, due to the node-based sampling nature of the approach. However, the linear system can be solved with simple linear solvers; in this work we use the BICGSTAB algorithm.

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