A Sparse Basis POD for Model Reduction of Multiphase Compressible Flow

A Sparse Basis POD for Model Reduction of Multiphase Compressible Flow

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Abstract

We develop a sparse basis model-order reduction technique for approximation of flux/pressure fields based on local proper orthogonal decompositions (PODs) glued together using the Multiscale Mixed FEM (MsMFEM) framework on a coarse grid. Based on snapshots from one or more simulation run, we perform singular value decompositions (SVDs) for the flux distribution over coarse grid interfaces and use the singular vectors corresponding the largest singular values as boundary conditions for the multiscale flux basis functions. The span of these basis functions matches (to prescribed accuracy) the span of the snapshots over coarse grid faces. Accordingly, the complementary span (what’s left) can be approximated by local PODs on each coarse block giving a second set of local/sparse basis functions. The reduced system unknowns corresponding to the second set of basis functions can be eliminated to keep the system size low.
Content

We develop a sparse basis model-order reduction technique for approximation of flux/pressure fields based on local proper orthogonal decompositions (PODs) glued together using the Multiscale Mixed FEM (MsMFEM) framework on a coarse grid. Based on snapshots from one or more simulation run, we perform singular value decompositions (SVDs) for the flux distribution over coarse grid interfaces and use the singular vectors corresponding the largest singular values as boundary conditions for the multiscale flux basis functions. The span of these basis functions matches (to prescribed accuracy) the span of the snapshots over coarse grid faces. Accordingly, the complementary span (what’s left) can be approximated by local PODs on each coarse block giving a second set of local/sparse basis functions. The reduced system unknowns corresponding to the second set of basis functions can be eliminated to keep the system size low.

  To assess the accuracy, we apply the methodology to two test problems (including compressibility and gravity) and compare to results obtained from full order simulations. The methodology produces accurate results for a large variation of coarse grids, but we do observe that a large number of basis vectors are needed where the flow is strongly dominated by gravity.

 Compared to standard POD, the suggested sparse version results in a larger number of basis functions, but requires overall less storage. Also the sparse POD appears to be more process independent. An additional benefit is that SVDs are performed on multiple small matrices rather than on one big.

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