A nonlinear manifold-based reduced order model for multiscale analysis of heterogeneous hyperelastic materials

Abstract

A new manifold-based reduced order model for nonlinear problems in multiscale modeling of heterogeneous hyperelastic materials is presented. The model relies on a global geometric framework for nonlinear dimensionality reduction (Isomap), and the macroscopic loading parameters are linked to the reduced space using a Neural Network. The proposed model provides both homogenization and localization of the multiscale solution in the context of computational homogenization. To construct the manifold, we perform a number of large three-dimensional simulations of a statistically representative unit cell using a parallel finite strain finite element solver. The manifold-based reduced order model is verified using common principles from the machine-learning community. Both homogenization and localization of the multiscale solution are demonstrated on a large three-dimensional example and the local microscopic fields as well as the homogenized macroscopic potential are obtained with acceptable engineering accuracy.

A physics-constrained data-driven approach based on locally convex reconstruction for noisy database

“…Recently, these approaches have been extended to the field of engineering mechanics, such as learning constitutive models in solid mechanics [12][13][14], surrogate models in fluid mechanics [15][16][17] and physical models or governing equations purely extracted from the collected data [18][19][20]. In conjunction with machine learning techniques such as manifold learning [21] or neural networks [22], the recent studies [23][24][25] offer a new paradigm for data-driven computing for various applications such as design of materials [26]. There is a vast body of literature devoted to these subjects, including the recent developments based on nonlinear dimensionality reduction [24], nonlinear regression, deep learning [27][28][29], among others.…”

“…In conjunction with machine learning techniques such as manifold learning [21] or neural networks [22], the recent studies [23][24][25] offer a new paradigm for data-driven computing for various applications such as design of materials [26]. There is a vast body of literature devoted to these subjects, including the recent developments based on nonlinear dimensionality reduction [24], nonlinear regression, deep learning [27][28][29], among others.…”

Hybrid multiscale integration for directionally scale separable problems

“…Substituting Equation 24 into the microscale weak form (Equation 23), results in the inelastic influence function problem for h :…”

“…These include computational homogenization (also known as FE 2 ), 9,10 multiscale finite element method, 11,12 and heterogeneous multiscale method, 13,14 among others. In view of the tremendous computational complexity associated with coupled evaluation of problems at multiple scales, these methods are often used in conjunction with reduced-order modeling such as Fast Fourier transform, 15 proper orthogonal decomposition, 16 transformation field analysis, [17][18][19][20] eigendeformation-based homogenization (EHM), [21][22][23] machine learning based reduced modeling, 24 hyper-reduced modeling, 25 among others, particularly in the presence of nonlinear behavior.…”

A second‐order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

“…Thus, the development of reduced order models for a heterogeneous continuum has been an active research area. Among noteworthy reduced order methods are the Voronoi cell method, 62,63 the spectral method, 64 the network approximation method, 65 the fast Fourier transforms, 66,67 the mesh-free reproducing kernel particle method, 68,69 the finite-volume direct averaging micromechanics, 70 the transformation field analysis, 71,72 the methods of cells 64 or its generalization, 73 methods based on control theory including balanced truncation, 74,75 the optimal Hankel norm approximation, 76 the proper orthogonal decomposition, 77,78 data-driven-based reduced order methods, [79][80][81][82] the reduced order homogenization methods for two scales 83,84 and more than two-scales, 85,86 and the nonuniform transformation field methods. [87][88][89] For a recent comprehensive review of various homogenization-like method, we refer to the works of Fish 90 and Geers et al 91 The primary objective of this manuscript is to develop an efficient computational framework for analyzing nonlinear periodic materials with large microstructure that combines nonlinear higher-order asymptotic homogenization methods that do not require higher-order continuity of the coarse-scale solution with an efficient model reduction scheme.…”