Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system. (English) Zbl 1227.49046

Antil, H.; Heinkenschloss, M.; Hoppe, R. H. W.

Optim. Methods Softw. 26, No. 4-5, 643-669 (2011).

Summary: 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.

MSC:

49Q10	Optimization of shapes other than minimal surfaces
65M55	Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
76D07	Stokes and related (Oseen, etc.) flows

Keywords:

shape optimization; time-dependent Stokes system; domain decomposition; balanced truncation model reduction; microfluidic biochips

Software:

INTLAB; KELLEY

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