Sensitivity analysis with the modified Heaviside function for the optimal layout design of multi-component systems

Two kinds of design variables, i.e., pseudo-density variables associated with the framework structure and location design variables associated with connected components are involved in the layout design of multi-component systems. Although sensitivities with respect to the first ones can easily be carried out as in topology optimization, the semi-analytical method (SAM) is often used for sensitivity analysis with respect to the location design variables. Due to the geometric perturbation of the finite element mesh, the latter can then be regarded as a geometric perturbation model (GPM). In this paper, we propose a material perturbation model (MPM) using fixed finite element (FE) mesh for sensitivity analysis with respect to location design variables. The material discontinuity across the boundary between each component and the framework structure is smoothed approximately by means of a modified Heaviside function. When a location design variable of a certain component is perturbed, attached finite elements to the component boundary are assumed to undertake only a shift of material properties while the finite element mesh itself remains geometrically unchanged. As a result, analytical sensitivities with respect to location design variables are achieved as easily as for pseudo-density variables. The computing efficiency is thus improved because the velocity field for the mesh perturbation in the semi-analytical scheme is no longer needed. The MPM is illustrated by means of numerical tests, especially the design optimization of 3D multi-component systems.