A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations

Nouy, Anthony

doi:10.1016/j.cma.2010.01.009

Cited by 226 publications

(281 citation statements)

References 35 publications

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“…While some progress has been made in this area, in particular with greedy sampling approaches, further work is needed to develop methods that exploit system structure to avoid the curse of dimensionality. Promising recent efforts towards this goal use tensor techniques [182,186]. The combination of tensor calculus [125] and parametric model reduction techniques for timedependent problems is still in its infancy, but offers a promising research direction.…”

Section: Equation-free Model Reductionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Equation-free Model Reductionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…Relation (29) means that once the algorithm has converged L W E can be viewed as the local macrobehavior near the calculated solution, defined over I ¼ ½0; T and in each subdomain X E . This behavior establishes a relation between a small perturbation in terms of macrodisplacements and a small perturbation in terms of macroforces.…”

Section: Discussion Of the Computation And Storage Costsmentioning

confidence: 99%

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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“…A priori model reduction techniques [42] do not require any knowledge of the solution and operate in an iterative way, where a set of simple problems, that can be seen as pseudo eigenvalue problems, need to be solved. They are based on the proper generalized decomposition (PGD) [8,43] where the deterministic vectors and stochastic functions in (6) are initially unknown and computed at the same time.…”

Section: A Priori Techniquesmentioning

confidence: 99%

Uncertainties in structural dynamics: overview and comparative analysis of methods

Daouk

Louf

Dorival

et al. 2015

Mechanics & Industry

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-When studying mechanical systems, engineers usually consider that mathematical models and structural parameters are deterministic. However, experimental results show that these elements are uncertain in most cases, due to natural variability or lack of knowledge. Therefore, engineers are becoming more and more interested in uncertainty quantification. In order to improve the predictability and robustness of numerical models, a variety of methods and techniques have been developed. In this work we propose to review the main probabilistic approaches used to model and propagate uncertainties in structural mechanics. Then we present the Lack-Of-Knowledge theory that was recently developed to take into account all sources of uncertainties. Finally, a comparative analysis of different parametric probabilistic methods and the Lack-Of-Knowledge theory in terms of accuracy and computation time provides useful information on modeling and propagating uncertainties in structural dynamics.

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A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations

Cited by 226 publications

References 35 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

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

Uncertainties in structural dynamics: overview and comparative analysis of methods

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