Enhanced parametric shape descriptions in PGD-based space separated representations

Mohammad Javad Kazemzadeh-Parsi,Francisco Chinesta,Jean Louis Duval,Amine Ammar

doi:10.1186/s40323-021-00208-2

Abstract

Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more sophisticated geometrical transformations are needed to map the complex physical domain into a regular one where computations are performed. This paper aims at proposing a very efficient route for accomplishing such space separation. A NURBS-based geometry representation, usual in computer aided design—CAD—, is retained and combined with a fully separated representation for allying efficiency (ensured by the fully separated representations) and generality (by addressing complex geometries). Some numerical examples are considered to prove the potential of the proposed methodology.

Highlights

A generic problem in physics consists of a differential operator acting on the so-called unknown field
In what follows for the sake of clarity we will assume a scalar unknown field depending on space x and time t, as well as on a series of parameters, grouped in vector μ, related to the considered physics, the loading or the domain in which the problem is defined
The present paper aims at exploring the use of NURBS-based geometrical descriptions combined with the PGD-based fully 3D space separation

Summary

Introduction

A generic problem in physics consists of a differential operator acting on the so-called unknown field. In what follows for the sake of clarity we will assume a scalar unknown field depending on space x and time t, as well as on a series of parameters, grouped in vector μ, related to the considered physics, the loading or the domain in which the problem is defined. Where x ∈ x(μ) ⊂ R3, t ∈ t ⊂ R and the parameteres μ ∈ μ ⊂ RP. For approximating the unknown function, and in absence of a priori knowledge, any multi-purpose polynomial approximation basis can be used. To approximate very general functions the approximation basis must be rich enough, with the computational efficiency impact, in particular when addressing nonlinear tran-

Objectives

Conclusion