A model reduction technique based on the PGD for elastic-viscoplastic computational analysis

Relun, Nicolas; Néron, David; Boucard, Pierre-Alain

doi:10.1007/s00466-012-0706-x

Cited by 39 publications

(44 citation statements)

References 23 publications

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“…This innovative non-incremental approach is well known for its ability to solve efficiently various non-linear time dependent problems such as frictional contact problems [27,60,61], large displacement [62], non-linear materials [46,63,64], transient dynamics [65][66][67]. …”

Section: Formulation Of the Latin-pgd For Frictional Contact Problemsmentioning

confidence: 99%

Toward an optimal a priori reduced basis strategy for frictional contact problems with LATIN solver

Giacoma

Dureisseix

Gravouil

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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In this paper, an efficient a priori model reduction strategy for frictional contact problems is presented. We propose to solve this problem by using the finite element method and the non-linear LATIN solver. Basically, this non-linear solver assumes a space-time separated representation presaging nowadays PGD strategies. We extend this family of solvers to frictional engineering applications with reduced subspaces and no prior knowledge about the solution (contrary to a posteriori model reduction techniques). Hereinafter, a hybrid a priori/a posteriori LATIN-PGD formulation for frictional contact problems is proposed. Indeed, the suggested algorithm may or may not start with an initial guess of the reduced basis and is able to enrich the basis in order to reach a given level of accuracy. Moreover, it provides progressively the solution of the considered problem into a quasi-optimal space-time separated form compared to the singular value decomposition (SVD). Some examples are provided in order to illustrate the efficiency and quasi-optimality of the proposed a priori reduced basis LATIN solver.

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Section: Formulation Of the Latin-pgd For Frictional Contact Problemsmentioning

confidence: 99%

Toward an optimal a priori reduced basis strategy for frictional contact problems with LATIN solver

Giacoma

Dureisseix

Gravouil

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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show abstract

“…Conversely, in the case of nonlinear behavior, parameters H and h are recalculated throughout the iterations and the homogenized operator must be updated [48]. In practice, the search directions are recalculated only for the first iterations, then remain fixed when their variations become negligible.…”

Section: Discussion Of the Computation And Storage Costsmentioning

confidence: 99%

“…Other types of constitutive laws, such as elastic-viscoplastic behavior, which requires the introduction of internal variables, can be found in [47,48]. Dealing with such laws does not require any change in the strategy which will be developed in this paper.…”

Section: Description Of the Reference Problemmentioning

confidence: 99%

“…If the quality of the solution is insufficient, the ROB is enriched by adding new PGD functions using a greedy power algorithm based on the minimization of a functional built from the verification of the search direction (A.8) and (A.10). Further details on the introduction of PGD into the microproblems and examples showing the ability of the method to reduce the computation cost of the microproblems can be found in [37,44,48]. The novelty of the present work lies in the reduction of the computation and storage cost of the homogenized operator by extending the use of the PGD technique to its construction.…”

Section: Proper Generalized Decompositionmentioning

confidence: 99%

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A PGD-based homogenization technique for the resolution of nonlinear multiscale problems

Cremonesi

Néron

Guidault

et al. 2013

Computer Methods in Applied Mechanics and Engineering

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This paper deals with the offline resolution of nonlinear multiscale problems leading to a PGD-reduced model from which one can derive microinformation which is suitable for design. Here, we focus on an improvement to the Proper Generalized Decomposition (PGD) technique which is used for the calculation of the homogenized operator and which plays a central role in our multiscale computational strategy. This homogenized behavior is calculated offline using the LATIN multiscale strategy, and PGD leads to a drastic reduction in CPU cost. Indeed, the multiscale aspect of the LATIN method is based on splitting the unknowns between a macroscale and a microcomplement. This macroscale is used to capture the homogenized behavior of the structure and then to accelerate the convergence of the iterative algorithm. The novelty of this work lies in the reduction of the computation cost of building the homogenized operator. In order to do that, the problems defined within each subdomain are solved using a new PGD representation of the unknowns in which the separation of the unknowns is performed not only in time and in space, but also in terms of the interface macrodisplacements. The numerical example presented shows that the convergence rate of the approach is not affected by this new representation and that a significant reduction in computation cost can be achieved, particularly in the case of nonlinear behavior. The technique used herein is then particularly important to enhance the performances of the LATIN strategy but is not limited to this method. The more general path which is followed in this paper is to solve the microproblems arising in the homogenization for all the configurations that can be encountered in the calculation. An additional and novel benefit of this approach is that it includes the calculation of the homogenized tangent operator which gives, over the whole time interval and for any cell, the relation between the perturbation in terms of macroforces and the perturbation in terms of macrodisplacements. In linear viscoelasticity, this operator represents the homogenized behavior of the structure exactly

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“…This method, close to augmented Lagrangian methods, is well-known for its ability to solve difficult non-linear and time-dependent large problems with a global time-space approach (non-linear material [19], contact problems [4,15,20], large displacement [21], transient dynamics [22,23], fracture mechanics [24,25]...). The non-incremental LATIN method was proposed as a commitment of three principles, which are, for the elastic frictional contact problems: …”

Section: The Large Time Increment Methodsmentioning

confidence: 99%

An efficient quasi-optimal space-time PGD application to frictional contact mechanics

Giacoma

Dureisseix

Gravouil

2016

Adv. Model. and Simul. in Eng. Sci.

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The proper generalized decomposition (PGD) aims at finding the solution of a generic problems into a low rank approximation. On the contrary to the singular value decomposition (SVD), such a low rank approximation is generally not the optimal one leading to memory issues and loss of computational efficiency. Nonetheless, the computational cost of the SVD is generally prohibitive to be performed. In this paper, authors suggest an algorithm to address this issue. First, the algorithm is described and studied in details. It consists in a cheap iterative method compressing a low rank expansion. It will be shown that given a low rank approximation, the SVD of a provided low rank approximation can be reached at convergence. Behavior of the method is exhibited on a numerical application. Second, the algorithm is embedded into a general space-time PGD solver to compress the iterated separated form for the solution. An application to a quasi-static frictional contact problem is illustrated. Then, efficiency of such a compressing method will be demonstrated.

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A model reduction technique based on the PGD for elastic-viscoplastic computational analysis

Cited by 39 publications

References 23 publications

Toward an optimal a priori reduced basis strategy for frictional contact problems with LATIN solver

Toward an optimal a priori reduced basis strategy for frictional contact problems with LATIN solver

A PGD-based homogenization technique for the resolution of nonlinear multiscale problems

An efficient quasi-optimal space-time PGD application to frictional contact mechanics

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