DCDS-S
On the space separated representation when addressing the solution of PDE in complex domains
Chady Ghnatios Guangtao Xu Adrien Leygue Michel Visonneau Francisco Chinesta Alain Cimetiere
Separated representations allow impressive computational CPU time savings when applied in different fields of computational mechanics. They have been extensively used for solving models defined in multidimensional spaces coming from (i) its proper physics, (ii) model parameters that were introduced as extra-coordinates and (iii) 3D models when the solution can be separated as a finite sum of functional products involving lower dimensional spaces. The last route is especially suitable when models are defined in hexahedral domains. When it is not the case, different possibilities exist and were considered in our former works. In the present work, we are analyzing two alternative routes. The first one consists of immersing the real non-separable domain into a fully separable hexahedral domain. The second procedure consists in applying a geometrical transformation able to transform the real domain into a hexahedra in which the model is solved by using a fully separated representation of the unknown field.
keywords: model order reduction. separated representation space separation Proper generalized decomposition
DCDS-S
Parametric nonlinear PDEs with multiple solutions: A PGD approach
Marianne Beringhier Adrien Leygue Francisco Chinesta
This paper presents some insights into the determination, using the Proper Generalized Decomposition, of multiple solutions of nonlinear parametric partial differential equations. Although the Proper Generalized Decomposition (PGD) is well suited for computing the solution of, possibly nonlinear, parametric problems that vary smoothly with a physical parameter, no work has been achieved for the case of problems that exhibit multiple solutions for some values of a parameter. For two representative cases, we show how an appropriate parametrization, combined to a nonlinear solution procedure can be devised to describe and compute the multiple solutions of a PDE.
keywords: non-linear parametric PDE model reduction. PGD

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