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Certi fied Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation

TitleCerti fied Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation
Publication TypeBook Chapter
Year of Publication2017
AuthorsDevaud, D, Rozza, G
Book TitleSpectral and High Order Methods for Partial Differential Equations
Volume 119
EditionBittencourt, Dumont, Hesthaven. (Eds).
Series VolumeLNCSE
ISBN Number978-3-319-65869-8
Other NumbersDOI 10.1007/978-3-319-65870-4

In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based onNURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrizationof domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis (IGA) is a growing research theme in scientic computing and computational mechanics, as well as reducedbasis methods for parametric PDEs. Their combination enhances the solution of some class of problems,especially the ones characterized by parametrized geometries we introduced in this work. For a generaloverview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstratethat it is also possible for some class of problems to deal with ane geometrical parametrization combinedwith a NURBS IGA formulation. This is what this work brings as original ingredients with respect to otherworks dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2] sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this workwe show a certication of accuracy and a complete integration between IGA formulation and parametriccertified greedy RB formulation. Section 2 recalls the abstract setting for parametrized PDEs, Section 3recalls IGA setting, Section 4 deals with RB formulation, and Section 5 illustrates two numerical examples in heat transfer with different parametrization.

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