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

Giacoma, Anthony; Dureisseix, David; Gravouil, Anthony

doi:10.1186/s40323-016-0067-7

Cited by 11 publications

(7 citation statements)

References 30 publications

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“…The computation of a full SVD, in case of n > n t , requires O(nn 2 t ) floating point operations (flops) while seeking a truncated SVD requires O(nn t k) flops. Due to the high computational cost of applying an SVD at each enrichment step in a PGD context, a quasi-optimal iterative orthonormalisation scheme was proposed in [2,16]. However, another appealing straightforward approach to provide a direct compression of the PGD modes into a minimal set is utilised here.…”

Section: Compute a Thin/truncated Svd Of The Solutionũmentioning

confidence: 99%

“…In the context of reusing an ROB from a previous computation, a learning strategy has been proposed in [14,15] to extract an optimal basis from the reduced order model (ROM) through a Karhunen-Loève expansion. In a PGD framework, recompression based on SVD has been evaluated in [16]. However, the SVD step turns out to be numerically expensive prohibiting its implementation at each iteration.…”

Section: Introductionmentioning

confidence: 99%

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Toward Optimality of Proper Generalised Decomposition Bases

Alameddin

Fau

Néron

et al. 2019

MCA

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The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method.

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Section: Compute a Thin/truncated Svd Of The Solutionũmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Toward Optimality of Proper Generalised Decomposition Bases

Alameddin

Fau

Néron

et al. 2019

MCA

View full text Add to dashboard Cite

show abstract

“…The computation of a full SVD, in case of n > n t , requires O(nn 2 t ) floating point operations (flops) while seeking a truncated SVD requires O(nn t k) flops. Due to the high computational cost of applying an SVD at each enrichment step in a PGD context, a quasi-optimal iterative orthonormalisation scheme was proposed in [2,15]. However, another appealing straightforward approach to provide a direct compression of the PGD modes into a minimal set is utilised here.…”

Section: Datamentioning

confidence: 99%

“…In the context of reusing a ROB from a previous computation, a learning strategy has been proposed in [14] to extract an optimal basis from the ROM through a Karhunen-Loève expansion. In a PGD framework, recompression based on SVD has been evaluated in [15]. However, the SVD step turns out to be numerically expensive prohibiting its implementation at each iteration.…”

Section: Introductionmentioning

confidence: 99%

Toward Optimality of Proper Generalised Decomposition Bases

Alameddin¹,

Fau²,

Néron³

et al. 2019

Preprint

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The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the reduced basis, i.e. the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of the PGD basis using a randomised SVD algorithm. Comparing to conventional approaches such as Gram-Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the reduced basis. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method.

show abstract

“…So far, all the existing results using the RB method deal with linear constraints. Yet, we mention that another class of model order reduction methods, namely the Proper Generalized Decomposition (PGD), is used in Reference to address nonlinear contact problems.…”

Section: Introductionmentioning

confidence: 99%

A reduced basis method for parametrized variational inequalities applied to contact mechanics

Benaceur

Ern

Ehrlacher

2019

Numerical Meth Engineering

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Summary We investigate new developments of the reduced‐basis method for parametrized optimization problems with nonlinear constraints. We propose a reduced‐basis scheme in a saddle‐point form combined with the Empirical Interpolation Method to deal with the nonlinear constraint. In this setting, a primal reduced‐basis is needed for the primal solution and a dual one is needed for the Lagrange multipliers. We suggest to construct the latter using a cone‐projected greedy algorithm that conserves the non‐negativity of the dual basis vectors. The reduction strategy is applied to elastic frictionless contact problems including the possibility of using nonmatching meshes. The numerical examples confirm the efficiency of the reduction strategy.

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

Cited by 11 publications

References 30 publications

Toward Optimality of Proper Generalised Decomposition Bases

Toward Optimality of Proper Generalised Decomposition Bases

Toward Optimality of Proper Generalised Decomposition Bases

A reduced basis method for parametrized variational inequalities applied to contact mechanics

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