# Domain Decomposition Techniques in Topology Optimization

Michal Kocvara^{*}, Daniel Loghin and James Turner

We investigate several approaches to the solution of topology optimization problems using decomposition of the computational domain. The first part of this talk will discuss the application of domain decomposition to a typical topology optimization problem via an interior point approach. This method has the potential to be carried out in parallel, and therefore can exploit recent developments in the area. The second part of the talk will focus on a nonlinear reaction diffusion system solved using Newton’s method. Current work considers applying domain decomposition to such a system using a Newton Krylov Schur (NKS) type approach. However, strong local nonlinearities can have a drastic effect on the rate of convergence. Our aim is to instead consider a three step procedure that applies Newton’s method locally on subdomains in order to address this issue. In the second part we will use reformulation of the original problem as a semidefinite optimization problem. This formulation is particularly suitable for problems with vibration or global buckling constraints. Standard semidefinite optimization solvers exercise high computational complexity when applied to problems with many variables and a large semidefinite constraint. To avoid this unfavourable situation, we use results from the graph theory that allow us to equivalently replace the original large-scale matrix constraint by several smaller constraints associated with the subdomains. This leads to a significant improvement in efficiency, as will be demonstrated by numerical examples.

Mathematics Subject Classification: 74P05 65N55

Keywords: Topology optimization, domain decomposition

Minisymposion: Material and Topology Optimization: Theory, Methods and Applications