Optimal shape design with domain decomposition

TR Number



Journal Title

Journal ISSN

Volume Title


Virginia Tech


In this work, we considered an “inverse-design” problem, where we specified the flow distribution in the computational domain or on its subset and sought the geometrical configuration that produced this flow. Based on this idea, we formulated a control problem, in which the optimization procedure minimizes the error between the target flow and the actual flow through successive adjustment of the design parameters. We were interested in exploring the computational efficiency of the numerical solution of this problem, particularly the implementation of workstation cluster environment to the solution of the control problem by employing numerical algorithms, which would allow coarse-grained parallelization. These aspects were studied with an example of one-dimensional heat transfer in surfaces of non-uniform cross-sectional area and the optimal design of a two-dimensional nozzle. We compared the computational cost and convergence properties of the optimization procedure for two approaches: the “Black-Box” method and domain decomposition method. When employing the “Black-Box” technique, the unconstrained control problem was solved in terms of the design variables and in the case of domain decomposition implementation, the constrained control problem was solved in terms of design variables and boundary data on interfaces. Also, we implemented the grid-embedding chimera technique to the solution of a one-dimensional heat transfer problem. The formulation of this scheme in terms of domain decomposition leads to an overlapping domain decomposition method.

It was concluded from one-dimensional results that domain decomposition methods can be successfully incorporated into the optimization-based design framework. The original analysis problem can be split into problems on subdomains, which can be solved in parallel with data transfer at each step limited to exchanging boundary information between the neighboring subdomains and between the “master” processor. In the case of a two-dimensional flow, the optimization was applied to supersonic flow and the discontinuous flow. It was concluded, that if one wishes to implement this algorithm in a parallel environment, the computations should be spread between the processors in such a way, that the number of processors is proportional to the number of control variables.