Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system

Abstract

The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multi-scale, multi-physics problem. Therefore, reduced-order modelling techniques such as balanced truncation model reduction (BTMR), proper orthogonal decomposition, or reduced basis methods are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. In this paper, we are interested in such shape optimization problems where the design issue is restricted to a relatively small portion of the computational domain. In this case, it appears to be natural to rely on a full-order model only in that specific part of the domain and to use a reduced-order model elsewhere. A convenient methodology to realize this idea consists of a suitable combination of domain decomposition techniques and BTMR. We will consider such an approach for shape optimization problems associated with the time-dependent Stokes system and derive explicit error bounds for the modelling error. As an application in life sciences, we will be concerned with the optimal design of capillary barriers as part of a network of microchannels and reservoirs on microfluidic biochips that are used in clinical diagnostics, pharmacology, and forensics for high-throughput screening and hybridization in genomics and protein profiling in proteomics.