Fast topological-shape optimization with boundary elements in two dimensions

Abstract

Wide variety of engineering design tasks can be formulated as constrained optimization problems where the shape and topology of the domain are optimized to reduce costs while satisfying certain constraints. 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. In this short note we suggest ways to address critical limitations of boundary element method as a tool for topological-shape optimization. We validate our approaches by supplementing the existing complex variables boundary element code for elastostatic problems with robust tools for fast topological-shape optimization. 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.…”

“…Since the surface solution is found, the solution in stresses at internal points is found via PVFMM summation of the kernels 5, 6. It is worth noting here, that the sampling of the points inside the domain should not necessarily be uniform -one could successfully use adaptive strategies of sampling points inside the domain, which reduces the computational complexity of the domain computation to O(N s ), where N s is the number of surface elements [15].…”

“…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

“…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].…”