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.
Summary The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.
Artificial neural networks (ANNs) have aroused research's and industry's interest due to their excellent approximation properties and are broadly used nowadays in the field of machine learning. In the present contribution, ANNs are used for finding solutions of periodic homogenization problems. The construction of ANN‐based trial functions that satisfy the given boundary conditions on the microscale allows for the unconstrained optimization of a global energy potential. Goal of the present approach is a memory efficient solution scheme as ANNs are known to fit complicated functions with a relatively small number of internal parameters. The method is tested for a three‐dimensional example using a global trial function and is qualitatively compared to a fast Fourier transform (FFT) based simulation.
Electroactive polymers (EAPs) are a group of materials that is able to respond with large strains to applied electric fields, making them candidates for applications such as artificial muscles or smart fins. The present contribution addresses the computational homogenization of electroactive materials based on fast-Fourier-transforms. The focus lies on the formulation of a coupled Lippmann-Schwinger equation with respect to the deformation gradient as well as the electric displacement, where the reference medium introduced in the Lippmann-Schwinger equation is also fully coupled. As the reference medium is acting as a preconditioner on the system, this has an impact on the convergence rate and stability of the iterative solver. We provide an algorithmically consistent coupled tangent operator as alternative to finite-difference based approaches. Coupled balance equationsIn the context of homogenization, the effective material response is usually computed through a representative volume element (RVE) deformed at microscale, whose microstructure is statistically representative for the local continuum properties. In the geometrically nonlinear setting, we distinguish between two configurations when describing material behavior, namely the reference configuration B 0 and the current configuration B of the microscopic body. The deformation of the body from reference coordinates X to current coordinates x is described through the deformation map ϕ(X), where the associated tangent map F = ∇ϕ is the deformation gradient. In this work, we want to predict the behavior of an electroactive body under static conditions. Maxwell's equations can therefore be simplified to Faraday's and Gauss' law of electrostatics. On the mechanical side, the kinematic compatibility as well as the balance of linear momentum must be fulfilled. The static balance equations then appear aswhere E and D are the electric field and electric displacement in the reference configuration and T is the first Piola-Kirchhoff stress. The electric displacement D = D + D and deformation gradient F = F + F in Equations (1) are split into constant macroscopic contributions D, F and fluctuative contributions D, F in dependence of X. In line with the concepts of firstorder homogenization, we impose a zero-average condition on the fluctuative quantities within the RVE. The relation between microscopic and macroscopic quantities in the absence of body forces appears asThe Hill-Mandel condition is fulfilled by applying periodic boundary conditions on the RVE's surface. Finally, the system is closed by introducing the constitutive relation for the electric field and stress E = ∂ψ ∂D and T = ∂ψ ∂F in terms of the energy density function ψ(F , D). For more details, the reader is referred to MIEHE ET AL. [1]. Fourier-transform based solution strategyThe solution of the coupled differential Equation (1) under periodic boundary conditions can be formulated in terms of Green operators Γ (•) , where the solution then takes the form of the well-known Lippmann-Schwinger equatio...
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