A Petrov-Galerkin Method for Hyper-Reduced Order Modeling of Symmetric and Nonsymmetric Problems without Complementary Meshes
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Projection-based Reduced Order Model (PROM) methods aim to minimize the discrepancy between a high-fidelity full order model (FOM) and a lower-dimensional approximation while respecting the constraints imposed by the reduced space. For problems with symmetric operators, the Galerkin Projection method is a powerful tool to achieve an optimal minimization of the FOM residual, whereas for problems with nonsymmetric operators other techniques exists such as the Least-Squares Petrov-Galerkin (LSPG). Although LSPG ensures optimality of the solution when Galerkin cannot, its implementation requires a complementary mesh, which presents complications and challenges in hyper-reduction. To address this issue, the current work proposes a Petrov-Galerkin minimization technique as an alternative. By choosing a left basis, the method can perform a least-squares minimization procedure on a reduced problem, while guaranteeing that the discrete full order residual is minimized for both symmetric and non-symmetric problems. This method avoids the use of a complementary mesh and can simplify the implementation of finite element models by allowing the minimization problem to be assembled element by element. The resulting technique is amenable to hyper-reduction by the use of the Empirical Cubature Method (ECM), and can be applied in the context of nonlinear reduction procedures.
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