Routes for Efficient Computational Homogenization of Nonlinear Materials Using the Proper Generalized Decompositions

By Hajer Lamari, Amine Ammar, P. Cartraud, Grégory Legrain, Francisco Chinesta, F. Jacquemin. 2010

68 citations recorded by the source corpus.

Computational homogenization is nowadays one of the most active research topics in computational mechanics. Different strategies have been proposed, the main challenge being the computing cost induced by complex microstructures exhibiting nonlinear behaviors. Two quite tricky scenarios lie in (i) the necessity of applying the homogenization procedure for many microstructures (e.g. material microstructure evolving at the macroscopic level or stochastic microstructure); the second situation concerns the homogenization of nonlinear behaviors implying the necessity of solving microscopic problems for each macroscopic state (history independent nonlinear models) or for each macroscopic history (history dependant nonlinear models). In this paper we present some preliminary results concerning the application of Proper Generalized Decompositions—PGD—for addressing the efficient solution of homogenization problems. This numerical technique could allow to compute the homogenized properties for any microstructure or for any macroscopic loading history by solving a single but highly multidimensional model. The PGD allows circumventing the so called curse of dimensionality that mesh based representations suffer. Even if this work only describes the first steps in a very ambitious objective, many original ideas are launched that could be at the origin of impressive progresses.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Mechanics of Materials

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “• computational homogenization [43, 97, 177];”
MULTISCALE CONTINUOUS AND DISCONTINUOUS MODELING OF HETEROGENEOUS MATERIALS: A REVIEW ON RECENT DEVELOPMENTS (2011, methodology): “Recently, Lamari et al.74 used proper generalized decomposition (PGD) to reduce the cost of computational homogenization models.”
Multiscale structural topology optimization with an approximate constitutive model for local material microstructure (2015, unknown): “Most of the existing ROMs constructed using reduction strategies such as Proper Orthogonal Decomposition (POD) [24,25], Proper Generalized Decomposition (PGD) [26–28], material map model [29,30], eigendeformation-based reduction [31,32], Nonuniform Transformation Field Analysis (NTFA) [33–36], and N”
A PGD-based homogenization technique for the resolution of nonlinear multiscale problems (2013, methodology): “A similar procedure has been described in [45], where”
Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method (2016, methodology): “adaptive strategies to solve the micro-scale problem only the minimum number of times necessary [93, 124], adaptive sub-incremental strategies to ensure the convergence of the multiscale solution in the presence of several sources of non-linearity [100, 110], model-order reduction techniques [41, 64”
Tensor-based methods for numerical homogenization from high-resolution images (2013, methodology): “In [30, 31], the authors already introduced a similar computational method for homogenization in the EBC case and for simple geometries.”