Parameter–identification of macroscopic material models based on virtual testing of given material mesostructures

For many heterogeneous materials such as composites and polycrystals, the material modeling for the constituents on a representative mesoscale can be considered as known, including concrete values of their inherent material parameters. Typical examples are isotropic elastic–plastic models for the constituents of composites or anisotropic crystal–plasticity models for the grains of polycrystals. This knowledge can be exploited with regard to the modeling of the homogenized macroscopic response. In particular, parameters in macroscopic models may be identified by virtual experiments provided by a computational deformation–driving of representative mesostructures. This paper outlines the general concept for the parameter–identification of macroscopic materialmodels based on the virtual testing of given material mesostructures. The virtual test data are obtained in the form of multi–dimensional stress–strain paths by applying different deformation gradients to a given mesostructure. After specifying a corresponding macroscopic material model covering the observed effects on the macroscale, the material parameters are identified by a least–square–type optimization procedure that optimizes the macroscopic material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical homogenization of foam-like structures based on the FE2-approach

The computation of foam–like structures is still a topic of research. There are two basic approaches: the microscopic model where the foam–like structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE²-approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point. In this contribution, we present a two–dimensional geometrically nonlinear FE²-framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p-version. The p-version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuum–based approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beam–like structures. In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Macroscopic current induced boundary conditions for Schrödinger-type operators

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.     

Reversible dynamics in discrete dissipative systems

The random walk model of Brownian motion is an example of a stochastic system which exhibits intrinsically irreversible behaviour. In spite of this, a simple discrete version of the model has been shown to harbour dynamics which are reversible and are described by a discrete form of Schrödinger's equation. The reversible dynamics appear as second order effects in this diffusive model, and the usual relationship between macroscopic irreversibility and microscopic reversibility is itself reversed. This will be discussed in the context of the `Brussels' school' on irreversibility.     

FE-Analysis of concentration dependent healing processes in a polymer matrix

In this contribution, a macroscopic four-phase model, based on the Theory of Porous Media, is presented to simulate healing processes in a polymer matrix which depend on the amount (concentration) of catalysts. Therefore, the healing process is described by the phase transition from liquid like healing agents to solid like healed material. This phase transition is a function depending on the concentration. To show the applicability of the developed model, a numerical example will be presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An explicitly solvable kinetic model for vehicular traffic and associated macroscopic equations

In the present paper, a kinetic model for vehicular traffic is presented and investigated in detail. For this model, the stationary distributions can be determined explicitly. A derivation of associated macroscopic traffic flow equations from the kinetic equation is given. The coefficients appearing in these equations are identified from the solutions of the underlying stationary kinetic equation and are given explicitly. Moreover, numerical experiments and comparisons between different macroscopic models are presented.     

Two-scale computational homogenization of magneto-electric composites

This contribution presents a two-scale computational homogenization framework for the micro-macro simulation of magneto-electro-mechanically coupled materials. Energetically consistent micro-macro transition conditions will be derived from a generalized form of the classical Hill-Mandel condition. A focus of the work is on the computation of effective magneto-electric moduli which are derived on the basis of an algorithmically consistent linearization of the macroscopic field equations. The method will be applied to the homogenization of magneto-electric composites which are composed of piezomagnetic and piezoelectric phases. The effective magneto-electric moduli of two-phase composites will be computed. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of fibre composites using X-FEM

A successful material design process for novel textile reinforced composites requires an integrated simulation of the material behaviour and the estimation of the effective properties used in a macroscopic structural analysis. In this context the Extended Finite Element Method (X-FEM) is used to model the behavior of materials that show a complex structure on the mesoscale efficiently. A homogenization technique is applied to compute effective macroscopic stiffness parameters. This contribution gives an outline of the implementation of the X-FEM for complex multi-material structures. A modelling procedure is presented that allows for the automated generation of an extended finite element model for a specific representative volume element. Furthermore, the problem of branching material interfaces arising from complex textile reinforcement architectures in combination with high fibre volume fractions will be addressed and an appropriate solution is proposed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong Convergence in Stabilised Degenerate Convex Problems

Solutions to non–convex variational problems typically exhibit enforced finer and finer oscillations called microstructures such that the infimal energy is not attained. Those oscillations are physically meaningful, but finite element approximations typically experience dramatic difficulty in their reproduction. The relaxation of the non–convex minimisation problem by (semi–)convexification leads to a macroscopic model for the effective energy. The resulting discrete macroscopic problem is degenerate in the sense that it is convex but not strictly convex. This paper discusses a modified discretisation by adding a stabilisation term to the discrete energy. It will be announced that, for a wide class of problems, this stabilisation technique leads to strong H¹–convergence of the macroscipic variables even on unstructured triangulations. This is in contrast to the work [2] for quasi–uniform triangulations and enables the use of adaptive algorithms for the stabilised formulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dérivation de modèles couplés dérive-diffusion/cinétique par une méthode de décomposition en vitesseDerivation of a kinetic/drift-diffusion model describing fast and slow particles

Our purpose is to derive a model describing the evolution of charged particles in a plasma, at various scales following their kinetic energy. Fast particles will be described through a collisional kinetic equation of Boltzmann type. This equation will be coupled with a drift-diffusion model that describes the evolution of slower particles. The main interest of this approach is to reduce the cost of numerical simulations. This gain is due to the use of a macroscopic model for slow particles instead of a kinetic model for all the particles, which would involve a larger number of variables. To cite this article: N. Crouseilles, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 827–832.     

Solutions for the Hyperbolic Dirac Equation on R1, m

In this article the hyperbolic unit ball in R ^m will be identified with the manifold of rays in the future null cone in R ^m+1. By means of the induced Clifford algebra structure there, one can introduce a definition of Dirac operators on sections of homogeneous line bundles. An infinite class of solutions for the resulting hyperbolic Dirac-equation will be constructed, in case of an odd dimension. In order to obtain these solutions a geometrical picture will be used, because each ray in the future cone will be identified with a point on a surface Σ in the future cone. The original Dirac-equation can then be rewritten in terms of the coordinates on this surface, and the resulting equation will be solved by means of Frobenius' method.     

活体大变形的能量原理

活体是具有自组织和自调控能力的生命系统.讨论活体的能量原理包含有力学和热力学原理两大部分.经典的小变形力学和可逆平衡态热力学理论巳不足于描述活体的运动.本文从大变形非对称应力理论力学描述活体宏观运动的力学能量原理.有关不可逆热力学问题将另文讨论.     

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement. To cite this article: A. Abdulle, A. Nonnenmacher, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A computational two scaled model for the simulation of micro-heterogeneous low-alloyed TRIP steels

In transformation induced plasticity (TRIP) steel a diffusionless austenitic-martensitic phase transformation induced by plastic deformation can be observed, resulting in excellent macroscopic properties. In particular low-alloyed TRIP steels, which can be obtained at lower production costs than high-alloyed TRIP steel, combine this mechanism with a heterogeneous arrangement of different phases at the microscale, namely ferrite, bainite, and retained austenite. The macroscopic behavior is governed by a complex interaction of the phases at the micro-level and the inelastic phase transformation from retained austenite to martensite. A reliable model for low-alloyed TRIP steel should therefore account for these microstructural processes to achieve an accurate macroscopic prediction. To enable this, we focus on a multiscale method often referred to as FE² approach, see [6]. In order to obtain a reasonable representative volume element, a three-dimensional statistically similar representative volume element (SSRVE) [1] can be used. Thereby, also computational costs associated with FE² calculations can be significantly reduced at a comparable prediction quality. The material model used here to capture the above mentioned microstructural phase transformation is based on [3] which was proposed for high alloyed TRIP steels, see also e.g. [8]. Computations based on the proposed two-scale approach are presented here for a three dimensional boundary value problem to show the evolution of phase transformation at the microscale and its effects on the macroscopic properties. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational micro-macro transitions at large strains for curvilinear physical directions

Biological tissues such as those involved in the eye, heart, veins or arteries are heterogeneous on one or another spatial scale and can undergo very large elastic strains. Frequently, these tissues are characterized by shell-like structures at the macroscopic scale and the physical material directions follow curvilinear paths. We consider a homogenized macro-continuum formulated in curvilinear convective coordinates with locally attached representative micro-structures. Micro-structures attached to different macroscopic points are assumed to be rotated counterparts according to the curvilinear path of the physical material directions at the macro-scale. The solution of such multi-scale problems according to the computational homogenization scheme [1, 2, 3] would need a different RVE at each macroscopic point. The goal of this paper is to use the same initial RVE at each macroscopic point by generalizing the computational homogenization scheme to a formulation considering different physical spaces at the micro- and macro-scale. The deformation and the reference frame of the micro-structure are assumed to be coupled with the local deformation and the local reference frame at the corresponding point of the macrocontinuum. For a consistent formulation of micro-macro transitions physical reference directions are defined on both scales, where the macroscopic one follows a curvilinear path. To formulate the generalized micro-macro transitions in absolute tensor notation the operations scale-up and scale-down are introduced. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)