Many‐scale finite strain computational homogenization via Concentric Interpolation

Abstract

A method for efficient computational homogenization of hyperelastic materials under finite strains is proposed. Multiple spatial scales are homogenized in a recursive procedure: starting on the smallest scale, few high fidelity FE computations are performed. The resulting fields of deformation gradient fluctuations are processed by a snapshot POD resulting in a reduced basis (RB) model. By means of the computationally efficient RB model, a large set of samples of the homogenized material response is created. This data set serves as the support for the Concentric Interpolation (CI) scheme, interpolating the effective stress and stiffness. Then, the same procedure is invoked on the next larger scale with this CI surrogating the homogenized material law. A three-scale homogenization process is completed within few hours on a standard workstation. The resulting model is evaluated within minutes on a laptop computer in order to generate fourth-scale results. Open source code is provided.

Material modeling for parametric, anisotropic finite strain hyperelasticity based on machine learning with application in optimization of metamaterials

“…Naturally, as visible in Table 1, smaller networks achieve comparable quality. The present work simply As a first evaluation of the chosen model  with 𝜖 = 0.0367 for D C and 𝜖 0.0466 for D V , we consider the cumulative distribution function (cdf) of 𝜖 [t,c] , compare (35). It is evaluation for D C and D V is illustrated in Figure 11 in blue and red, respectively, and the vertical lines correspond to 𝜖 accordingly.…”

“…Besides potential and stress values, also the wMSE of the stress tangent tensor C(F) = d 2 W∕dF 2 could be added to the objective function 𝜖 from (35), which might further improve accuracy or regularity of the calibrated models, compare Reference 37. However, this approach is not pursued here, because (i) the homogenization of C from the beam model is rather elaborate, 19 (ii) for other applications, C might not be available from the microstructural simulations and is generally not available for experimental characterization data, and (iii) it would increase the computational effort for evaluating 𝜖 tremendously, since d 2 W ML ∕dF 2 would have to be evaluated for each parameter [t, c, s] in each iteration of the optimization, including the symmetrization resulting from (11).…”

“…Starting from a small strain highly nonlinear model, the radial numerically explicit potentials were suggested in Reference 32 as an extension of the NEXP proposed first in Reference 33 which employs a tensor decomposition into amplitude and direction followed by kernel interpolation. The method was later refined, compare References 34,35, and combined with ROM for the creation of efficient surrogate models for finite strain hyperelasticity in two‐ and three‐scale simulations. In Reference 36, microstructural simulations have been conducted to generate data for the calibration of a FFNN identifying the constitutive behavior of nonlinear elastic structures at finite strains, but without consideration of material symmetries.…”