02450nas a2200169 4500008004100000020002200041024003400063245010200097210006900199250004300268260002500311490000900336520177800345100001802123700002102141856011802162 2017 eng d a978-3-319-65869-8 aDOI 10.1007/978-3-319-65870-400aCerti fied Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation0 aCerti fied Reduced Basis Method for Affinely Parametric Isogeome aBittencourt, Dumont, Hesthaven. (Eds). aHeildebergbSpringer0 v 1193 a

In this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on

NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization

of 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 reduced

basis 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 general

overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate

that it is also possible for some class of problems to deal with ane geometrical parametrization combined

with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other

works 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 work

we show a certication of accuracy and a complete integration between IGA formulation and parametric

certified greedy RB formulation. Section 2 recalls the abstract setting for parametrized PDEs, Section 3

recalls IGA setting, Section 4 deals with RB formulation, and Section 5 illustrates two numerical examples in heat transfer with different parametrization.

1 aDevaud, Denis1 aRozza, Gianluigi uhttps://math.sissa.it/publication/certi-fied-reduced-basis-method-affinely-parametric-isogeometric-analysis-nurbs01890nas a2200145 4500008004100000245011800041210006900159260001300228520137300241653003501614100001801649700002001667700002101687856003601708 2013 en d00aA combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices0 acombination between the reduced basis method and the ANOVA expan bElsevier3 a

We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.

10aPartial differential equations1 aDevaud, Denis1 aManzoni, Andrea1 aRozza, Gianluigi uhttp://hdl.handle.net/1963/738900527nas a2200121 4500008004100000245011700041210006900158300001100227490000700238100001800245700002100263856012100284 2013 eng d00aReduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)0 aReduced Basis Approximation for the StructuralAcoustic Design ba a98-1150 v481 aDevaud, Denis1 aRozza, Gianluigi uhttps://math.sissa.it/publication/reduced-basis-approximation-structural-acoustic-design-based-energy-finite-element