On a multiscale computational strategy with time and space homogenization for structural mechanics

Ladevèze, Pierre; Nouy, Anthony

doi:10.1016/s0045-7825(03)00341-4

Cited by 136 publications

(131 citation statements)

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“…The resolution of this system is carried out in a four-step procedure whose complete details can be found in [16]. Only the key points are recalled here.…”

Section: Relation With Newton-schur Domain Decomposition Methodsmentioning

confidence: 99%

“…It has been shown in previous works [19,37,29] that under small-displacement assumptions this approach reduces the computational cost drastically. All the details can be found in [16].…”

Section: Radial Loading Approximationmentioning

confidence: 99%

“…Here, we follow a recently introduced multiscale computational strategy for nonlinear evolution problems. This strategy involves an automatic homogenization technique in space as well as in time [15,16] which is an extension of previous works limited to space alone [17,18]. This strategy, developed in a general framework, makes no a priori assumption regarding the form of the solution and, therefore, does not suffer from the limitations of standard homogenization techniques; moreover, it relies on an iterative algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…The third characteristic, and not the least, is the need to solve a set of similar and, generally, numerous microproblems over the time-space domain. A time-radial approximation, leading to the construction of a reduced basis of the space which is updated at each iteration, was introduced in [19,16,20]. Here, in 2 order to favor a parallel computational strategy, we propose to group the substructures into several families in such a way that at each iteration the same reduced basis can be used for all members of a given family.…”

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confidence: 99%

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On a mixed and multiscale domain decomposition method

Ladevèze

Néron

Gosselet

2007

Computer Methods in Applied Mechanics and Engineering

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This paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems, focusing on its suitability for parallel computing. Spatially, this strategy can be viewed as a mixed, multilevel domain decomposition method (or, more accurately, as a "structure decomposition" method). Regarding time, a "parallel" property is also described. We also draw bridges between this and other current approaches.Key words: domain decomposition, multiscale, computational mechanics IntroductionIn the structural mechanics field, one can observe a surge of interest in the multiscale analysis of structures with complex microstructural geometry and/or complex behavior. When accurate solutions are required, calculations must be performed on a finely discretized model of the structure (defined on what is called the "micro" level) consistent with short lengths of variation, both in space and in time. A major application is the analysis of composite structures described on the microscale or on the mesoscale [1]; there are also other applications, such as in [2,3]. These types of situations lead to problems with very large numbers of degrees of freedom and computation costs which are prohibitive if one uses classical finite element codes. One of the main objectives * Corresponding author. E-mail: pierre.ladeveze@lmt.ens-cachan.fr Preprint of the last few decades has been to develop efficient and robust computational strategies suitable for these types of problems. One of these strategies uses the theory of the homogenization of periodic media [4][5][6]. Other developments and the associated computational approaches can be found in [7][8][9][10][11][12][13][14]. Besides periodicity, these strategies rely on the fundamental assumption that the ratio between the two scales is small. Thus, the boundary zones require specific treatment because in these zones the microstructure cannot be homogenized. Here, we follow a recently introduced multiscale computational strategy for nonlinear evolution problems. This strategy involves an automatic homogenization technique in space as well as in time [15,16] which is an extension of previous works limited to space alone [17,18]. This strategy, developed in a general framework, makes no a priori assumption regarding the form of the solution and, therefore, does not suffer from the limitations of standard homogenization techniques; moreover, it relies on an iterative algorithm. Until now, this strategy has been developed in the framework of small displacements of (visco)plastic structures under possible contact with or without friction. This paper is a reconsideration of this computational strategy in order to assess its adaptability to parallel computing. Therefore, we examine its three main characteristics. The first characteristic is that it can be viewed as a mixed, multilevel domain decomposition method or, more precisely as will explained further on, as a "structure decomposition" method. Ind...

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“…The resolution of this system is carried out in a four-step procedure whose complete details can be found in [16]. Only the key points are recalled here.…”

Section: Relation With Newton-schur Domain Decomposition Methodsmentioning

confidence: 99%

“…It has been shown in previous works [19,37,29] that under small-displacement assumptions this approach reduces the computational cost drastically. All the details can be found in [16].…”

Section: Radial Loading Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On a mixed and multiscale domain decomposition method

Ladevèze

Néron

Gosselet

2007

Computer Methods in Applied Mechanics and Engineering

Self Cite

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“…PGD was originally introduced into the LATIN method as part of the solver under the name ''radial approximation'' [28]. It has been shown in previous works that PGD leads to a significant reduction in computation time [36,37,44].…”

Section: Proper Generalized Decompositionmentioning

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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Multiscale Methods

Néron¹

2010

Encyclopedia of Aerospace Engineering

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On a multiscale computational strategy with time and space homogenization for structural mechanics

Cited by 136 publications

References 20 publications

On a mixed and multiscale domain decomposition method

On a mixed and multiscale domain decomposition method

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

Multiscale Methods

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