Dynamic Domain Decomposition and Error Correction for Reduced Order Models

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Abstract

The use of reduced order models based on Proper Orthogonal Decomposition (POD) has been demonstrated to be of practical use for modeling low-speed aerodynamics. However, the use of such techniques for transonic flows with the presence of shocks has been more limited. This is because the reduced order model will only be able to represent a shock in a given location if one of the snapshots used to build the model has a discontinuity at that same location. Lucia et. al. proposed domain decomposition as a means of overcoming this limitation. This technique uses a reduced order model over the majority of the computational domain while solving the full governing equations in a small region which would otherwise require an unacceptably high number of snapshots to achieve sucient accuracy of the approximate solution. In this work, we extend the concept of domain decomposition as a dynamic, a posteriori verification and if necessary, correction of the approximate solution. Given an approximate solution with unknown accuracy generated with a set of POD basis functions the error is estimated by augmenting the POD basis with top hat basis functions and computing the first order change in the solution due to the additional basis functions. By comparing the results from a solution of known accuracy, such as one of the snapshots used to generate the POD basis, with the results for a solution of unknown accuracy the need for domain decomposition and its spatial extent can be determined. Once the boundaries of the domain decomposition have been determined, a POD solver coupled with a full order solver can be used to generate solutions of a coupled system. Results for 2-D transonic flow show that this method can generate good approximate solutions for flows with significant shock movement. Application to a sample drag minimization problem indicate that this type of variable fidelity model would be a good candidate for use in a multilevel optimization framework.

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