A two‐scale FE‐FFT approach to nonlinear magneto‐elasticity

Abstract

Fourier-based approaches are a well-established class of methods for the theoretical and computational characterization of microheterogeneous materials.Driven by the advent of computational homogenization techniques, Fourier schemes gained additional momentum over the past decade. In recent contributions, the interpretation of Green operators central to Fourier solvers as projections opened up a new perspective. Based on such a viewpoint, the present work addresses a multiscale framework for magneto-mechanically coupled materials at finite strains. The key ingredient for the solution of magneto-mechanic boundary value problems at the microscale is the construction of suitable operators in Fourier space that project vector fields onto either curl-free or divergence-free subspaces. The resulting linear system of equations is solved by a conjugate gradient method. In addition to that, we describe the computation of the consistent macroscopic tangent operator based on the same linear operators as the microscopic equilibrium with appropriately defined right-hand sides. We employ the framework for the simulation of representative two-scale boundary value problems and compare the results with pure finite element schemes.

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

“…Therefore, a regular grid in the components of F might lead to a prohibitively large amount of samples, and even to a violation of (39). For instance, such a grid with a rather moderate resolution of just 10 samples of each component would require 1 billion solutions of the FOM.…”

“…The assembly of the FE residual and of the FE stiffness depend on N qp . Future research should aim at an application of the introduced Reduced Basis method within realistic two-scale simulations, in analog to [12,[38][39][40]. Hyperreduction methods, cf.…”

Magneto-Mechanical Coupling in Magneto-Active Elastomers

“…In the following, the procedure according to [ 42 , 68 ], where a macro-continuum model is calibrated from micro-continuum simulations for the simplified two-dimensional case, is briefly presented. The framework represents a decoupled multiscale scheme which—in contrast to, e.g., FE -approaches [ 39 , 45 ]—can only account for microstructural evolution within the limits of the model calibration. Since isotropic MAEs are considered, it is sufficient to describe the macroscopic behavior by the averaged principal stretches and the three additional magneto-mechanical invariants …”

“…Since this yields a system of fully coupled, nonlinear partial differential equations which require computationally demanding numerical methods—such as a finite element (FE) analysis—for their solution, only comparably small MAE samples as well as representative microstructures in the surrounding of a material point can be considered. To this end, the approach allows for identifying mechanisms leading to magnetically induced deformations and enhanced mechanical moduli on the microstructural level but requires an appropriate computational homogenization procedure to predict the effective material behavior [ 36 , 37 , 38 , 39 ]. In macro-continuum approaches, the MAE is modeled as a homogeneous continuum in which microstructural information are captured via suitable coupling terms; see Figure 1 c. Here, a phenomenological material model is typically fitted to data obtained from experiments [ 40 , 41 ] or more resolved, microscopic analyses [ 42 , 43 , 44 ].…”

A surrogate model for computational homogenization of elastostatics at finite strain using high‐dimensional model representation‐based neural network

“…More importantly, the method did not require a set of reference mediums. It has been lately applied to computational homogenization for electroelastically and magnetoelastically coupled materials in Göküzüm et al 30 and Rambausek et al, 31 respectively. These contributions employed a technique of computing the constitutive tangent introduced by Göküzüm and Keip 32 based on the fluctuation sensitivitiesf.…”