Velocity field level‐set method for topological shape optimization using freely distributed design variables

Abstract

Summary The velocity field level‐set topological shape optimization method combines the implicit representation in the standard level‐set method and the capabilities of general mathematical programming algorithms in handling multiple constraints and additional design variables. The key concept is to construct the normal velocity field using basis functions and the velocity design variables at specified points (referred to as velocity knots) in the entire design domain. In this study, the velocity design variables are decoupled from the level‐set grid points. Making use of this property, we can adaptively change the arrangement of the velocity knots as the structural boundary evolves. This provides more design freedom in the optimization and allows for a significant reduction in the number of design variables. Several numerical examples in two‐ and three‐dimensional design domains are presented to demonstrate the robustness and efficiency of the proposed method. We also show that changing the number of velocity knots may implicitly exert certain control on topological complexity and length scale.

Merging automatic differentiation and the adjoint method for photonic inverse design

“…At iteration i, the variation deforms the shape Ω i+1 = (1 + δΩ)(Ω i ). n denotes the normal vector on the boundary, and V Ω (s) denotes the so-called sensitivity field (also known as shape gradient field/density [19,39] or velocity field [41]). Since our goal is to minimize the loss functional, we see that this can be achieved by setting the geometry deformation to δΩ = −n V Ω (s).…”

“…Then, the loss functional is guaranteed to decrease to first order in every iteration. The sensitivity field acts on the shape as a vector field that drags the boundary along the direction of the vector field [41,52]. For a second-order elliptical PDE, such as the Maxwell equations, the sensitivity field V Ω (s) is given by [19,25].…”

“…This approach simplifies the simulation and optimization of physical problems, as the task moves from formulating comprehensive complex problems to building the components of the successive steps as shown in figure 1, a fundamental concept within differentiable programming. This enables users to couple black-box photonic optimization easily to other components which are important for the problem such as the geometry treatment for topology [39], shape optimization [40,41], multiphysics [42] or neural networks [43][44][45].…”