2013
DOI: 10.1016/j.cma.2013.08.009
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A PGD-based homogenization technique for the resolution of nonlinear multiscale problems

Abstract: 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 drast… Show more

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Cited by 40 publications
(23 citation statements)
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“…In the specific context of two-scale homogenization, it has been recently explored by Boyaval [10], Yvonnet et al [62], and Monteiro et al [51]. Traces of this idea can also be found in articles dealing with more general hierarchical multiscale techniques -that do not presuppose either scale separation or periodicity/statistical homogeneity, or both-, namely, in the multiscale finite element method [53,26,27], in the heterogeneous multiscale method [2,1], and in multiscale approaches based on the Proper Generalized Decomposition (PGD) [21]. However, it should be noted that none of the above cited papers confronts the previously described, crucial question of how to efficiently integrate the resulting reduced-order equations, simply because, in most of them [10,53,26,27,2,1], integration is not an issue -the fine-scale BVPs addressed in these works bear an affine relation with the corresponding coarse-scale, input parameter, as in linear elasticity, and, consequently, all integrals can be pre-computed, i.e., evaluated offline, with no impact in the online computational cost.…”
Section: Originality Of This Workmentioning
confidence: 99%
See 1 more Smart Citation
“…In the specific context of two-scale homogenization, it has been recently explored by Boyaval [10], Yvonnet et al [62], and Monteiro et al [51]. Traces of this idea can also be found in articles dealing with more general hierarchical multiscale techniques -that do not presuppose either scale separation or periodicity/statistical homogeneity, or both-, namely, in the multiscale finite element method [53,26,27], in the heterogeneous multiscale method [2,1], and in multiscale approaches based on the Proper Generalized Decomposition (PGD) [21]. However, it should be noted that none of the above cited papers confronts the previously described, crucial question of how to efficiently integrate the resulting reduced-order equations, simply because, in most of them [10,53,26,27,2,1], integration is not an issue -the fine-scale BVPs addressed in these works bear an affine relation with the corresponding coarse-scale, input parameter, as in linear elasticity, and, consequently, all integrals can be pre-computed, i.e., evaluated offline, with no impact in the online computational cost.…”
Section: Originality Of This Workmentioning
confidence: 99%
“…Thus, for the reduced-order problem to be well-posed, the approximation space V apr σ cannot be only formed by statically admissible stresses, but it must also include statically inadmissible fields -i.e. stress functions that do not satisfy the reduced-order equilibrium equation (21).…”
Section: Proposed Remedy: the Expanded Space Approachmentioning
confidence: 99%
“…the so-called proper generalized decomposition (PGD) originally proposed by Ladevèze with a different terminology and recently applied e.g. in [7][8][9][10].…”
Section: Introductionmentioning
confidence: 99%
“…To resolve the high computational costs of evaluating the nonlinear integrand, a number of methods have been proposed, such as Proper Generalized Decomposition [30], which reduces the costs through separation of variables. Transformation Field Analysis is another semi-analytical approach, introduced by Dvorak [31], in which the computational costs of the FE 2 scheme are reduced by assuming constant plastic strain fields.…”
Section: Introductionmentioning
confidence: 99%