Non intrusive method for parametric model order reduction using a bi-calibrated interpolation on the Grassmann manifold

Abstract

Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and leads generally to incorrect results. However, instead of using a straight forward interpolation on solutions, reduced order models can be interpolated. More particularly, Amsallem and Farhat proposed an efficient POD reduced order model interpolation technique based on differential geometry tools. This approach, named in this paper ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold), allows through the passage to the tangent space of the Grassmann manifold, to approximate accurately the reduced order basis associated to a new untrained parameter. This basis is used afterwards to build the interpolated ROM describing the temporal dynamics by performing the Galerkin projection on the high fidelity model. Such Galerkin ROMs require to access to the underlying high fidelity model, leading by that to intrusive ROMs. In this paper, and contrary to the ITSGM/Galerkin approach, we propose a non-intrusive reduced order modeling method which is independent of the governing equations. This method is named through this paper Bi-CITSGM (Bi-Calibrated ITSGM). It consists first to interpolate the spatial and temporal POD sampling bases considered as representatives of points on Grassmann manifolds, by the ITSGM method. Then, the resulting bases modes are reclassified by introducing two orthogonal matrices. These calibration matrices are determined as analytical solutions of two optimization problems. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces satisfyingly accurate solutions in a real computational time.