# Fictitious Domain Methods and Topology Optimization

**Martin Berggren
**

Department of Computing Science

UMIT Research Lab

Umeå Universitet

### Abstract:

An important reason for the success of the finite element method is its flexibility with respect geometry. The method supports unstructured "body-fitted" meshes that can efficiently be generated to fit a given computational domain, even if its shape is complicated and irregular. However, if the geometry of the domain is changing with time, or if the domain itself is unknown and a part of the solution process, the standard “body-fitted” meshing procedure can be cumbersome and impractical. In such cases, in can be an advantage to instead rely on so-called fictitious domain (also called domain embedding) methods. Instead of fitting the mesh exactly to the computational domain, fictitious domain methods rely on a fixed, often regular mesh that cover the region in which any possible configuration is allowed to be located. The physical boundary will occur somewhere inside the fixed computational mesh, and the appropriate boundary condition needs to be applied at these “fictitious” boundaries. The different methods within the class of fictitious domain methods differ according to how these boundary conditions are applied.

A simple fictitious domain approach relies on a varying coefficient in the governing equation, where the extreme values of the coefficient correspond to, for instance, presence or absence of material. The geometry will then be described in an image-like way by pixel values of the coefficient. This type of geometry description is exploited in the so-called material distribution approach to topology optimization. In this approach, the layout of material is optimized in order to maximize performance of a component. The classical problem of this kind is to find the distribution of a fixed amount of material that maximizes the elastic stiffness of a machine part for a given load pattern. Similar approaches can be used for other optimization problem, for instance, for the purpose of optimizing the radiation properties of loudspeaks.