Reduced order modelling of nonaffine problems on parameterized NURBS multipatch geometries

TL;DR

This contribution considers the Empirical Interpolation Method (EIM) to recover an affine parametric dependence and combine domain decomposition to reduce the dimensionality of reduced basis methods and IsoGeometric Analysis (IGA) in the context of parameterized partial differential equations.

Abstract

This contribution explores the combined capabilities of reduced basis methods and IsoGeometric Analysis (IGA) in the context of parameterized partial differential equations. The introduction of IGA enables a unified simulation framework based on a single geometry representation for both design and analysis. The coupling of reduced basis methods with IGA has been motivated in particular by their combined capabilities for geometric design and solution of parameterized geometries. In most IGA applications, the geometry is modelled by multiple patches with different physical or geometrical parameters. In particular, we are interested in nonaffine problems characterized by a high-dimensional parameter space. We consider the Empirical Interpolation Method (EIM) to recover an affine parametric dependence and combine domain decomposition to reduce the dimensionality. We couple spline patches in a parameterized setting, where multiple evaluations are performed for a given set of geometrical parameters, and employ the Static Condensation Reduced Basis Element (SCRBE) method. At the common interface between adjacent patches a static condensation procedure is employed, whereas in the interior a reduced basis approximation enables an efficient offline/online decomposition. The full order model over which we setup the RB formulation is based on NURBS approximation, whereas the reduced basis construction relies on techniques such as the Greedy algorithm or proper orthogonal decomposition (POD). We demonstrate the developed procedure using an illustrative model problem on a three-dimensional geometry featuring a multi-dimensional geometrical parameterization.

Figures (7)

• Figure 1: Coarse geometries (a) and (b) for different values of a geometrical parameter $\mu$ representing the radius of the circular hole. The refined mesh (c) is employed for the analysis.
• Figure 2: Geometry of horseshoe obtained by B-spline mapping
• Figure 3: Geometry and parameterization.
• Figure 4: Singular value decay vs. number of port modes for each interface.
• Figure 5: Error decay in $L^\infty$ norm of the EIM approximations.