scholarly journals On the space separated representation when addressing the solution of PDE in complex domains

2016 ◽  
Vol 9 (2) ◽  
pp. 475-500 ◽  
Author(s):  
Alain Cimetiere ◽  
Francisco Chinesta ◽  
Michel Visonneau ◽  
Adrien Leygue ◽  
Guangtao Xu ◽  
...  
Keyword(s):  
Separated Representation

scholarly journals A Regression Based Non-Intrusive Method Using Separated Representation for Uncertainty Quantification

2012 ◽  
Author(s):  
Prashant Rai ◽  
Mathilde Chevreuil ◽  
Anthony Nouy ◽  
Jayant Sen Gupta
Keyword(s):  
Tensor Product ◽  
Uncertainty Propagation ◽  
Sparse Regularization ◽  
Underlying Assumption ◽  
Tensor Product Space ◽  
Regularization Techniques ◽  
Separated Representation ◽  
Regression Techniques ◽  
Regression Problems ◽  
Well Posed

This paper aims at handling high dimensional uncertainty propagation problems by proposing a tensor product approximation method based on regression techniques. The underlying assumption is that the model output functional can be well represented in a separated form, as a sum of elementary tensors in the stochastic tensor product space. The proposed method consists in constructing a tensor basis with a greedy algorithm and then in computing an approximation in the generated approximation space using regression with sparse regularization. Using appropriate regularization techniques, the regression problems are well posed for only few sample evaluations and they provide accurate approximations of model outputs.

scholarly journals A separated representation involving multiple time scales within the Proper Generalized Decomposition framework

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Angelo Pasquale ◽  
Amine Ammar ◽  
Antonio Falcó ◽  
Simona Perotto ◽  
Elías Cueto ◽  
...  
Keyword(s):  
Time Scales ◽  
Time Discretization ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Proper Generalized Decomposition ◽  
Partial Differential ◽  
Time Coordinate ◽  
Separated Representation ◽  
The Time Domain ◽  
Fine Time

AbstractSolutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.

scholarly journals A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains

2018 ◽  
Vol 210 ◽  
pp. 135-144 ◽  
Author(s):  
G. Quaranta ◽  
B. Bognet ◽  
R. Ibañez ◽  
A. Tramecon ◽  
E. Haug ◽  
...  
Keyword(s):  
Dynamic Problems ◽  
Out Of Plane ◽  
Separated Representation

Parametric Shape and Material Simulations for Optimized Parts Design

2012 ◽  
Author(s):  
B. Bognet ◽  
A. Leygue ◽  
F. Chinesta ◽  
F. Bordeu
Keyword(s):  
Parametric Model ◽  
Curse Of Dimensionality ◽  
Elasticity Problem ◽  
Single Solution ◽  
Separated Representation

In this paper we consider a parametric model related to the elasticity problem in which many parameter describing he material and/or the geometry will considered as extra-coordinates, allowing for the construction, from a single solution, of an abaqus containing the solution for each choice of the parameters considered as extra-coordinates. In order to circumvent the curse of dimensionality related to the high number of dimensions considered in the model, a separated representation will be considered.

Numerical Simulation of Friction Stir Welding by an Efficient 3D Updated Lagrangian Technique

2014 ◽  
Author(s):  
D. Canales ◽  
Ch. Ghnatios ◽  
A. Leygue ◽  
F. Chinesta ◽  
I. Alfaro ◽  
...  
Keyword(s):  
Friction Stir Welding ◽  
Solution Strategy ◽  
Proper Generalized Decomposition ◽  
Technological Parameters ◽  
Friction Stir ◽  
Natural Element Method ◽  
Separated Representation ◽  
Natural Element ◽  
Updated Lagrangian ◽  
Welding Technique

Friction Stir Welding (FSW) is a welding technique which since its invention in 1991 is of great interest to the industry for its many advantages. Despite being widely used, its physical foundations and its relation to the technological parameters of the process are not known in detail. Numerical simulations are a powerful tool to achieve a greater understanding of the physics of the problem. Although several approaches can be found in the literature for FSW, all of them present different limitations that restrict their applicability to the industry. This paper presents a new solution strategy that combines a meshless method, the Natural Element Method (NEM), with a solution separated representation making use of the Proper Generalized Decomposition (PGD), for creating a new powerful updated-Lagrangian method for addressing the 3D model while maintaining a 2D computational complexity.

PGD-BEM Applied to the Nonlinear Heat Equation

2012 ◽  
Author(s):  
Pierre Joyot ◽  
Nicolas Verdon ◽  
Gaël Bonithon ◽  
Francisco Chinesta ◽  
Pierre Villon
Keyword(s):  
Heat Equation ◽  
Physical Space ◽  
Weak Form ◽  
Space Time ◽  
Kernel Functions ◽  
Nonlinear Heat Equation ◽  
Integration Strategy ◽  
Time Problem ◽  
Separated Representation ◽  
Steady State Problem

The Boundary Element Method (BEM) allows efficient solution of partial differential equations whose kernel functions are known. The heat equation is one of these candidates when the thermal parameters are assumed constant (linear model). When the model involves large physical domains and time simulation intervals the amount of information that must be stored increases significantly. This drawback can be circumvented by using advanced strategies, as for example the multi-poles technique. We propose radically different approach that leads to a separated solution of the space and time problems within a non-incremental integration strategy. The technique is based on the use of a space-time separated representation of the unknown field that, introduced in the residual weighting formulation, allows to define a separated solution of the resulting weak form. The spatial step can be then treated by invoking the standard BEM for solving the resulting steady state problem defined in the physical space. Then, the time problem that results in an ordinary first order differential equation is solved using any standard appropriate integration technique (e.g. backward finite differences). When considering the nonlinear heat equation, the BEM cannot be easily applied because its Green’s kernel is generally not known but the use of the PGD presents the advantage of rewriting the problem in such a way that the kernel is now clearly known. Indeed, the system obtained by the PGD is composed of a Poisson equation in space coupled with an ODE in time so that the use of the BEM for solving the spatial part of the problem is efficient. During the solving, we must however separate the nonlinear term into a space-time representation that can limit the method in terms of CPU time and storage, that is why we introduce in the second part of the paper a new approach combining the PGD and the Asymptotic Numerical Method (ANM) in order to efficiently treat the nonlinearity.

First Steps in the Space Separated Representation of Models Defined in Complex Domains

2012 ◽  
Author(s):  
Chady Ghnatios ◽  
Amine Ammar ◽  
Alain Cimetiere ◽  
Aziz Hamdouni ◽  
Adrien Leygue ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Penalty Method ◽  
3D Models ◽  
Geometrical Parameters ◽  
Regular Domain ◽  
Inverse Technique ◽  
One Dimensional ◽  
Separated Representations ◽  
Separated Representation ◽  
Alternative Routes

Separated representations allow substituting the resolution of 3D models by a sequence of three one-dimensional problems [1]. This route is especially suitable when models are defined in hexahedral domains. When it is not the case, different possibilities exist. In a former work [2], we explored the route of immersing the domain into a hexahedral domain, and then use a kind of penalty method to solve the model whilst enforcing the boundary conditions. In the present work, we are analyzing two alternative routes. The first one consists of using an inverse technique in order to compute the boundary conditions on the border of the hexahedra in which the complex domain is immersed. The second one consists in solving the model in a regular domain for a number of geometrical parameters considered as extra-coordinates from which we could have access to the solution in any geometry resulting from a choice of those parameters.

scholarly journals On the fully 3D simulations of thermoelastic models defined in plate and shell geometries

2012 ◽  
Author(s):  
Brice Bognet ◽  
Adrien Leygue ◽  
Francisco Chinesta
Keyword(s):  
Computational Complexity ◽  
Dimensionality Reduction ◽  
3D Model ◽  
Three Dimensional ◽  
3D Models ◽  
Polymer Processing ◽  
Two Dimensional ◽  
Composites Manufacturing ◽  
Plate And Shell ◽  
Separated Representation

Many models in polymer processing and composites manufacturing are defined in degenerated three-dimensional domains (3D), involving plate or shell geometries. The reduction of models from 3D to two-dimensional (2D) is not obvious when complex physics or particular geometries are involved. The hypotheses to be introduced for reaching this dimensionality reduction are unclear, and most of the possible proposals will have a narrow interval of validity. The only gateway is to explore new discretisation strategies able to circumvent or at least alleviate the drawbacks related to mesh-based discretisations of fully 3D models defined in plate or shell domains. Appropriate separated representation of the involved fields within the context of the proper generalised decomposition allows solving the fully 3D model by keeping a 2D characteristic computational complexity.

scholarly journals Enhanced parametric shape descriptions in PGD-based space separated representations

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Mohammad Javad Kazemzadeh-Parsi ◽  
Amine Ammar ◽  
Jean Louis Duval ◽  
Francisco Chinesta
Keyword(s):  
Computer Aided Design ◽  
Characteristic Functions ◽  
Proper Generalized Decomposition ◽  
Separated Representations ◽  
Complex Shapes ◽  
Separated Representation ◽  
Efficient Route ◽  
Shape Descriptions ◽  
Aided Design ◽  
Space Separation

AbstractSpace 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.

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