Engineering optimization with the fast boundary element method

Abstract

A wide variety of engineering design tasks can be formulated as optimization problems where the shape and topology of an elastic domain are optimized to reduce costs, e.g. global compliance, while satisfying certain constraints, such as volume constraint. We propose an application of a fast 3D boundary element code to the problems of shape and topology optimization. Our algorithm is based on the formalism of topological derivatives. Adaptive tree strategy of sampling of topological derivatives inside the domain, high performance algebraic solver and the analysis of optimization problem in reduced dimensions promise state of the art performance in the problems of engineering optimization. The approach can be applied to various optimization problems, such as minimization of compliance of an elastic structure or minimization of the distance from a current homogenized elasticity tensor of a periodic structure to the desired one. The efficiency of the approach is illustrated with a numerical example.

Parallel optimization with boundary elements and kernel independent fast multipole method

“…The level of details in the final solution depends on the threshold of the topological derivative and the number of iterations. However, both obtained solutions are in qualitative agreement with 2D and 3D solutions of similar problems obtained earlier [15,16,23]. The quality of the surface of the optimal configuration is improved with Laplassian smoothing [24] post-processing step.…”

“…In our recent works [15,16] we have demonstrated the application of 2D and 3D boundary element codes equipped with fast algebraic solvers to the problem of topological-shape optimization of elastic structures. However, the tools we used did not allow us to reach true scalability, and evaluate the capabilities of our approach on large problems.…”

Scalable topology optimization with the kernel-independent fast multipole method

“…The level of details in the final solution depends on the threshold of the topological derivative and the number of iterations. However, both obtained solutions are in agreement with 2D and 3D solutions of similar problems obtained earlier [23, 24,4]. The quality of the surface of the optimal configuration is improved with Laplacian smoothing [25] post-processing step (Fig.…”

“…In our earlier works [23,24] we have demonstrated that if the change in boundary configuration at every iteration is relatively small (Fig. ( 5) (B)), one can use fast update techniques for the volume and surface solutions, which would be faster than full re-computation of the BVP.…”

“…These early works demonstrated conceptual applicability of BEM in combination with a hard-kill algorithm of material removal to the problems of topology optimization. The first applications of algebraically accelerated BEM to two-and tree-dimensional problems of elasticity were presented in our papers [23,24]. In these works we used Shur complements [36] for fast updates of the BVP solution [23], and H 2 -matrices for fast solutions of the BVP [24].…”