Topology Optimization of 3D Elastic Structures Using Boundary Elements

Abstract

A numerical approach for the topological optimization of 3D linear elastic problems using boundary elements and the topological derivative is presented in this work.

Scalable topology optimization with the kernel-independent fast multipole method

“…Checkerboards appear because of high stiffness of the checkerboard pattern in finite-element discretization, in comparison with a continuous density distribution with the same total mass. Consistently with previous work [6,7] we do not observe checkerboard patterns in our formulation. In all our simulations, in spite the low-order approximation and absence of explicit regularization, we did not observe anything similar to typical FEM checkerboards.…”

“…These were, however, applied to solve inverse scattering problems in elastodynamics [38], which are related to topology/shape optimization. First applications of BEM to topological optimization of elastic structures were presented in [6,7]. These early works demonstrated conceptual applicability of BEM in combination with a hard-kill algorithm of material removal to the problems of topology optimization.…”

Fast topological-shape optimization with boundary elements in two dimensions

“…In case if admissible designs include only homogeneous regions with piecewise-constant properties (microstructured composite designs are prohibited), the problem of optimization reduces to finding optimal configuration of the domain boundaries. Few recent papers [1,2] suggested that boundary integral approaches, and in particular boundary element method (BEM), can be a convenient tool to address this class of problems. The implementation of optimization algorithms within BEM utilizes the concept of topological derivative (TD) [11][12][13] -a cost of making an infinitesimal circular (spherical) cavity with a center in a given point of the domain.…”

“…Several mathematical approaches were developed to address the problem of finding optimal design of an engineered structure. Recent works [1,2] have demonstrated the feasibility of boundary element method as a tool for topological-shape optimization. However, it was noted that the approach has certain drawbacks, and in particular high computational cost of the iterative optimization process.…”

Engineering optimization with the fast boundary element method

“…A number of works illustrated that boundary variation optimization approaches are naturally treated with boundary element method (BEM) [5]. Few recent works [6,7] have also demonstrated that BEM can be a convenient tool for topological-shape optimization. Significant work has been done in the adjacent field of shape sensitivity analysis in elastodynamics using fast multipole BEM [8].…”