Tensor-based methods for numerical homogenization from high-resolution images

Abstract

International audienceWe present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images. We first introduce specific numerical treatments for the translation of image-based homogenization problems into a tensor framework. It includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images. We then introduce some variants of proper generalized decomposition (PGD) methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems. A new definition of PGD is introduced which allows the progressive construction of a Tucker decomposition of the solution. This tensor format is well adapted to the present application and improves convergence properties of tensor decompositions. Finally, we use a dual-based error estimator on quantities of interest which was recently introduced in the context of PGD. We exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material. We also provide a complete goal-oriented adaptive strategy for the progressive construction of tensor decompositions (of primal and dual solutions) yielding to predictions of homogenized quantities with a prescribed accuracy

Automatic image‐based stress analysis by the scaled boundary finite element method

“…Hence, this information can be translated into an appropriate mesh for various computer analyses. This technique is often referred to as image-based analysis, which has been researched in many fields including material characterization [1][2][3][4], fracture analysis [5][6][7] and biomedical applications [8][9][10][11][12] among others.…”

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

“…In that case the random vector Z ∼ N(0, Σ) follows the same distribution as X l where X ∼ N(0, 1) is a standard normal random variable (scalar). The random dual problem (23)…”

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“…In contrast, the framework we present in this paper often allows using a dual PGD approximation with significantly less terms than the primal approximation. An adaptive strategy for the PGD approximation of the dual problem for the error estimator proposed in [2] was suggested in [23] in the context of computational homogenization. In [1] the approach proposed in [2] is extended to problems in nonlinear solids by considering a suitable linearization of the problem.An error estimator for the error between the exact solution of a parabolic PDE and the PGD approximation in a suitable norm is derived in [34] based on the constitutive relation error and the Prager-Synge theorem.…”

Prediction of apparent properties with uncertain material parameters using high‐order fictitious domain methods and PGD model reduction

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