{text {FE}}^r method with surrogate localization model for hyperelastic composite materials

Ryo Hatano,Shuji Moriguchi,Julien Yvonnet,Kenjiro Terada,Seishiro Matsubara

doi:10.1186/s40323-020-00175-0

Abstract

This study presents a method for constructing a surrogate localization model for a periodic microstructure, or equivalently, a unit cell, to efficiently perform micro-macro coupled analyses of hyperelastic composite materials. The offline process in this approach is to make a response data matrix that stores the microscopic stress distributions in response to various patterns of macroscopic deformation gradients, which is followed by the proper orthogonal decomposition (POD) of the matrix to construct a reduced order model (ROM) of the microscopic analysis (localization) with properly extracted POD bases. Then, response surfaces of the POD coefficients are constructed so that the ROM can be continuous with respect to the input datum, namely, the macroscopic deformation gradient. The novel contributions of this study are the application of the L2 regularization to the interpolation approximations of the POD coefficients by use of radial basis functions (RBFs) to make the response surfaces continuous and the combined use of the cross-validation and the Bayesian optimization to search for the optimal set of parameters in both the RBFs and L2regularization formula. The resulting model can be an alternative to microscopic finite element (FE) analyses in the conventional {text {FE}}^2 method and realizes {text {FE}}^r with 1<r<<2 accordingly. Representative numerical examples of micro-macro coupled analysis with the {text {FE}}^r are presented to demonstrate the capability and promise of the surrogate localization model constructed with the proposed approach in comparison with the results with high-fidelity direct {text {FE}}^2.

Highlights

As a counterpart of theoretical mechanics for heterogenous media [1], which was developed for design support of composite materials since 1950s, mathematical theory of homogenization [2,3,4] was developed in applied mathematics around mid 70s as an area of variational methods and functional analysis
The unit cell is identified with a representative volume element (RVE) introduced in micromechanics, which is the domain for taking the volume averages of microscopic stress and strain to evaluate the macroscopic ones
This study proposes a surrogate localization model to perform micro-macro coupled analyses of hyperelastic composite materials with unit cells, for which microscopic analyses to be conducted at each macroscopic calculation point are replaced by a sort of reduced order model (ROM) constructed by the application of proper orthogonal decomposition (POD) to the response data set consisting of microscopic stresses

Summary

Introduction

As a counterpart of theoretical mechanics for heterogenous media [1], which was developed for design support of composite materials since 1950s, mathematical theory of homogenization [2,3,4] was developed in applied mathematics around mid 70s as an area of variational methods and functional analysis. Their more recent work has adopted neural network to more accurately construct NEXP, which carries the advantage over the use of PARAFAC in determining more than ten independent parameters in the three-dimensional (3D) potential function form [20] These can be regarded as data-driven approaches for two-scale analyses, as a database storing macroscopic responses is developed in advance, or equivalently, offline, by a number of high-fidelity computations to solve a microscopic BVP. This study proposes a surrogate localization model to perform micro-macro coupled analyses of hyperelastic composite materials with unit cells, for which microscopic analyses to be conducted at each macroscopic calculation point are replaced by a sort of ROM constructed by the application of POD to the response data set consisting of microscopic stresses. Since the surrogate localization model, which is a ROM of microscopic analysis, significantly reduces the computational cost of FE2, the method is referred to as FEr where 1 < r

Choose next parameter set by Bayesian optimization

Findings

Conclusion