Topology optimization of unsteady flows using the spectral element method

Abstract

We investigate the applicability of a high-order Spectral Element Method (SEM) to density based topology optimization of unsteady flows in two dimensions. Direct Numerical Simulations (DNS) are conducted relying on Brinkman penalization to describe the presence of solid within the domain. The optimization procedure uses the adjoint-variable method to compute gradients and a checkpointing strategy to reduce storage requirements. A nonlinear filtering strategy is used to both enforce a minimum length scale and to provide smoothing across the fluid–solid interface, preventing Gibbs oscillations. This method has been successfully applied to the design of a channel bend and an oscillating pump, and demonstrates good agreement with body fitted meshes. The precise design of the pump is shown to depend on the initial material distribution. However, the underlying topology and pumping mechanism is the same. The effect of a minimum length scale has been studied, revealing it to be a necessary regularization constraint for the oscillating pump to produce meaningful designs. The combination of SEM and density based optimization offer some unique challenges which are addressed and discussed, namely a lack of explicit boundary tracking exacerbated by the interface smoothing. Nevertheless, SEM can achieve equivalent levels of precision to traditional finite element methods, while requiring fewer degrees of freedom. Hence, the use of SEM addresses the two major bottlenecks associated with optimizing unsteady flows: computation cost and data storage.