Model-order reduction with optimal morphings for poorly reducible problems with geometric variability

We propose a new model-order reduction framework for poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, standard projection-based model-order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, the reducibility of the corresponding family is improved. We formulate a global optimization problem on the morphings so as to maximize the energy captured by the first few modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process Regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions. The overall methodology, called O-MMGP (optimal mesh morphing Gaussian process regression) provides a general framework that can be applied to many-query applications with either parameterized or non-parameterized geometries. The methodology is illustrating on challenging CFD datasets related to airfoil and turbine blade simulations.