|Authors||J. S. Dokken, S. W. Funke, A. Johansson and S. Schmidt|
|Title||Shape Optimization with Multiple Meshes|
|Project(s)||OptCutCell: Simulation-based optimisation with dynamic domains, Center for Biomedical Computing (SFF)|
|Publication Type||Talks, contributed|
|Year of Publication||2017|
|Location of Talk||FEniCS 2017 conference, Luxembourg|
For shape optimization problems, the computational domain is the design variable. Changing the shape of an airfoil in a channel to minimize drag is such a problem. The evolving domains complicate the numerical solution of shape optimization problems, and typically require large mesh deformations with quality checks and a re-meshing software as a fallback. We propose an approach for solving shape optimization problems on multiple overlapping meshes. In this approach, each mesh can be moved freely and hence the multi-mesh approach allows larger deformation of the domain than standard single-mesh approaches. The approach has been implemented in FEniCS and dolfin-adjoint, by employing the already tested environment for multi-mesh. We give examples of derivation of the shape-optimization problem for a Stokes flow and present implementation of this in FEniCS.
Consider a general PDE constrained shape optimization problem we want to minimize a goal functional J, which is subject to a state equation F, with solution u over the domain Omega. We choose to divide the domain Omega into two non-overlapping domains by creating an artificial interface Gamma, such that the union of Omega0 and Omega1 is the original domain Omega. This is depicted in Figure 1. Extension to an arbitrary number of overlapping domains is possible. The weak formulation of the state equations are then formulated and the continuity over the artificial boundary is enforced by using Nitsches method.
For minimization, we choose a gradient based scheme, and find the gradient by using the adjoint method. By employing the Hadamard formulas for Volume and Surface objective functions one can achieve the functional sensitivities as a function of the moving boundary and not the domain.
A concrete example of this approach is the shape optimization of an obstacle in Stokes-flow in the domain specified in Figure 2.
For deformation of the mesh, we have used two different deformation equations, a Laplacian smoothing and a set of Eikonal convection equations. For the multi-mesh problem, deformation is only done one the front mesh, while the background mesh is stationary. Figure 3 shows that with the Laplacian deformation the mesh degenerates in both the single-mesh and multi-mesh-case. Figure 4 shows that the Eikonal convection equations preserves the mesh-quality in the multi-mesh-case, but not in the single-mesh case, where the mesh degenerates at the boundary. We conclude that with a multi-mesh-approach, the meshes are preserved better than with a single-mesh approach.