The implicit bulk-surface filtering method for node-based shape optimization and a comparison of explicit and implicit filtering techniques

Abstract This work studies shape filtering techniques, namely the convolution-based (explicit) and the PDE-based (implicit), and introduces the implicit bulk-surface filtering method to control the boundary smoothness and preserve the internal mesh quality simultaneously in the course of bulk (solid) shape optimization. To that end, the volumetric mesh is governed by the pseudo-solid smoothing equations, which are stiffened by the mesh-Jacobian and endowed with the Robin boundary condition, which involves the Laplace-Beltrami operator on the mesh boundaries. Its superior performance from the non-simultaneous (sequential) treatment of boundary and internal meshes is demonstrated for the shape optimization of complex solid structures. Well-established explicit filters, namely Gaussian and linear, and the Helmholtz/Sobolev-based (implicit) filter are critically examined for shell optimization in terms of consistency (rigid-body-movement production), geometric characteristics, and computational cost. It is demonstrated that implicit filtering is more numerically efficient and robustly enforces fixed boundaries compared to explicit filtering. Supported by numerical experiments, a regularized Green’s function is introduced as an equivalent explicit form of the Helmholtz/Sobolev filter. Furthermore, we give special attention to deriving mesh-independent filtered sensitivities for node-based shape optimization with non-uniform meshes. It is shown that mesh-independent filtering can be achieved by scaling discrete sensitivities with the inverse of the mesh mass matrix.