Shape optimization of embedded solids using implicit Vertex-Morphing

One of the biggest challenges in optimizing the shape of complex solids is the requirement to maintain a reasonable mesh quality not only at the boundary but also for the bulk discretization of the interior. Thus, additional regularization and, in many cases, re-meshing of the structure during the iterative process is unavoidable with a Lagrangian description. By tracking the shape update using an Eulerian representation, embedded boundary methods are a promising technique for eliminating mesh distortion problems. This work consistently combines the unique features of implicit Vertex-Morphing and embedded boundary methods, facilitating the node-based shape optimization of solids with industrial complexity. One of the crucial elements for solving the primal problem on a fixed background grid is an efficient and robust quadrature scheme. To this end, we incorporate the open-source C++ framework QuESo (https://github.com/manuelmessmer/QuESo) developed for the numerical integration of arbitrarily complex embedded solids defined by oriented boundary meshes, e.g., in STereoLithography (STL) format. Meanwhile, applying the Helmholtz/Sobolev-based (implicit) filter to the vertices of the embedded boundary mesh not only exploits the extensive design space of node-based optimization but also ensures robust control over the feature size. To realize the above methodology, this work introduces a novel sensitivity analysis that yields mesh-independent shape gradients with respect to the immersed boundary. Through a specific sensitivity weighting, we recover a continuous gradient field from discrete values calculated at the nodes of the background grid. In addition, the presented workflow ensures robust enforcement of challenging geometrical constraints, including minimum wall thicknesses and design space limitations. We critically assess our approach with benchmarks and structures of industrial relevance. In all examples, trivariate B-Spline bases span the background grid, providing highly accurate finite element solutions and shape sensitivities at every iteration. Moreover, the elimination of mesh distortion problems enables a successful termination at the local optimum, even for large shape modifications.