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 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

High-performance model reduction techniques in computational multiscale homogenization

“…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.…”

“…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).…”

Coupled domain decomposition–proper orthogonal decomposition methods for the simulation of quasi-brittle fracture processes

“…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].…”

Integration efficiency for model reduction in micro-mechanical analyses

“…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.…”