Solving Multi‐Scale problems with heterogeneous finite difference methods

Multi‐scale differential equations are problems in which the variables can have different length scales. The direct numerical solution of differential equations with multiple scales is often difficult due to the work for resolving the smallest scale. We present here a strategy which allows the use of finite difference methods for the numerical solution of parabolic multi‐scale problems, based on a coupling of macroscopic and microscopic models for the original equation.     

Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions

We provide a numerical study of the macroscopic model of Barré et al. (Multiscale Model Simul, 2017, to appear) derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodeling process is very fast, the macroscopic model takes the form of a single aggregation–diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the one-dimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the two-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modeling of dynamical networks at a large scale.     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Aspects of a direct homogenization procedure for electro-mechanically coupled problems

The aim of this work is to discuss a micro–macro homogenization procedure for electro–mechanically coupled problems. In this context a two–scale homogenization ansatz for ferroelectric ceramics based on an FE²-approach is presented. The microscopic discretization of the heterogeneous structure of the polycrystalline material allows for the incorporation of microscopic effects, which are necessary to determine the corresponding overall macroscopic material response. (© 2008 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)     

Upscaling of a parabolic system with a large nonlinear surface reaction term

Motivated by the study of the dynamics of calcium ions in biological cells, the authors derived in [33], via periodic homogenization, a macroscopic bidomain model, by considering in the corresponding microscopic two-component problem a properly scaled nonlinear exchange term. We study here, at the microscopic scale, a similar parabolic system, with a large nonlinear interfacial reaction term. At the macroscopic scale, the nonlinear effect of this reaction term is recovered in the homogenized diffusion matrix, which is not anymore constant. This nonstandard phenomenon shows the fine interplay between reaction and diffusion in such processes.     

Homogenization for contact problems with friction on rough interface

We consider a contact problem between a macroscopic solid with a smooth boundary and a technical textile, while the textile has a periodic microscopic structure and microscopically rough surface. Two–scale homogenization approach is applied to the problem. The microscopic solution is approximated in terms of macroscopic solution and some concentration factor, given as a solution of auxiliary boundary value or contact problems of elasticity on the periodicity cell. Local friction condition is represented as a continuous non–linear functional over the stress field. Two–scale convergence is used to prove the convergence of friction functional. The macroscopic initial frictional limit is found. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the asymmetry in traffic flow (b): Macroscopic approach

In [H. Xu, H. Liu, H. Gong, Modeling the asymmetry in traffic flow (a): microscopic approach, J. Appl. Math. Model. (submitted for publication)], the asymmetric characteristic of traffic flow has been studied from a microscopic approach through the modeling of car-following behavior. This paper further discusses the asymmetric traffic flow modeling at the macroscopic scale. The microscopic asymmetric full velocity difference model is extended to a continuum traffic flow model to study the anisotropic characteristic and diffusive influence under various traffic conditions. In order to accurately solve the mathematical problem, a weighted essentially no-oscillatory (WENO) approach is applied. The performance of the model is then demonstrated through thorough evaluation against select classic models and field data. The macroscopic model is the first of its kind that is directly developed from an asymmetric car-following approach. The results show that the model is able to present many complex traffic phenomena observed in the field such as shock waves, rarefaction waves, stop-and-go waves and local cluster effects at a better level of accuracy than most of the existing models.     

From vehicle–driver behaviors to first order traffic flow macroscopic models

This paper proposes a two scale modeling approach to vehicular traffic, where macroscopic conservation equations are closed by models at the microscopic scale obtained by a mathematical interpretation of driver behaviors to local flow conditions. The paper focuses on the closure of the mass conservation equations by phenomenological models derived by a detailed analysis at the scale of individual vehicles.     

Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium

Several choices of scaling are investigated for a coupled system of parabolic partial differential equations in a two‐phase medium at the microscopic scale. This system may be regarded as modelling a reaction–diffusion problem, the Stokes problem of single‐phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance), for example. It is shown that, starting with the same problem on the microscopic scale, different choices of scaling of the diffusion coefficients (resp. permeability or conductivity) and the interfacial‐exchange coefficient lead to different types of macroscopic systems of equations. The characterization of the limit problems in terms of the scaling parameters constitutes a modelling tool because it allows to determine the right type of limit problem. New macroscopic models, not previously dealt with, arise and, for some scalings, classical macroscopic models are recovered. Using the method of two‐scale convergence, a unified approach yielding rigorous proofs is given covering a very broad class of different scalings. Copyright © 2008 John Wiley & Sons, Ltd.     

Two-scale simulation of damage in unidirectionally fibre reinforced composite materials

The finite element analysis of engineering structures usually assumes a homogeneous as well as a continuous medium. The heterogeneity of matter, which is always found on a sufficiently small length scale is neglected by replacing the inhomogeneous medium through a model of a mathematically homogenized material. The macroscopic constitutive behaviour is derived from volume averaging procedures that smear the microscopic heterogeneities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topological sensitivity analysis of a multi‐scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi‐scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non‐penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth‐order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi‐scale framework changes when a small circular inclusion is introduced at the micro‐scale level. Copyright © 2009 John Wiley & Sons, Ltd.     

Cases,clusters, densities: Modeling the nonlinear dynamics of complex health trajectories

In the health informatics era, modeling longitudinal data remains problematic. The issue is method: health data are highly nonlinear and dynamic, multilevel and multidimensional, comprised of multiple major/minor trends, and causally complex—making curve fitting, modeling, and prediction difficult. The current study is fourth in a series exploring a case‐based density (CBD) approach for modeling complex trajectories, which has the following advantages: it can (1) convert databases into sets of cases (k dimensional row vectors; i.e., rows containing k elements); (2) compute the trajectory (velocity vector) for each case based on (3) a set of bio‐social variables called traces; (4) construct a theoretical map to explain these traces; (5) use vector quantization (i.e., k‐means, topographical neural nets) to longitudinally cluster case trajectories into major/minor trends; (6) employ genetic algorithms and ordinary differential equations to create a microscopic (vector field) model (the inverse problem) of these trajectories; (7) look for complex steady‐state behaviors (e.g., spiraling sources, etc) in the microscopic model; (8) draw from thermodynamics, synergetics and transport theory to translate the vector field (microscopic model) into the linear movement of macroscopic densities; (9) use the macroscopic model to simulate known and novel case‐based scenarios (the forward problem); and (10) construct multiple accounts of the data by linking the theoretical map and k dimensional profile with the macroscopic, microscopic and cluster models. Given the utility of this approach, our purpose here is to organize our method (as applied to recent research) so it can be employed by others. © 2015 Wiley Periodicals, Inc. Complexity 21: 160–180, 2016     

Measurements of partially lubricated contacts on different scales

This paper focuses on experimental investigations for lubricated contacts with partially filled gaps. The measurements are performed on two different scales. A pin-on-disc test bench represents the microscopic scale whereas a novel test bench with asperities in the range of centimeters corresponds to a rather macroscopic scale. The results are compared and conclusions towards the tribological properties are drawn. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structural macroscopic theory of stiff and soft composites. Invariant description

A structural macroscopic theory of stiff and soft composites, which generalizes the theory in [1] constructed with application of a model of one-dimensional stressed state of reinforcing fibers in the current configuration of a composite is presented. The theory combines the micro- and macromechanics of composite materials. The two trends in the mechanics of composites are based on the idea of a field of macroscopic displacements and the concept of macroscopic stresses of the composite material when changes in the metrics of the matrix and reinforcing fibers in the current state of a composite medium are taken into consideration. The fibers of the reinforcing systems and matrix are analyzed on the basis of a general 3D model of deformation. No limits on the stiffness of the materials of the structural components are imposed. The analysis of the composite medium, on the macromechanical level, includes a definition of macrodisplacement and macrodeformation fields, as well as parametric structural fields in the current configuration. On the micromechanical level, the fields of macroscopic stresses in the medium, together with the fields of microscopic strains and stresses in the structural components, are defined on the basis of information obtained from the analysis of the field of the macroscopic displacements. With the corresponding interpretation of the field of macroscopic displacements, the structural macroscopic theory is applied to composite media with fibrous, laminated, and matrix structures.     

Theoretical study and numerical simulation of textiles

We propose a new approach for developing continuum models fit to describe the mechanical behavior of textiles. We develop a physically motivated model, based on the properties of the yarns, which can predict and simulate the textile behavior. The approach relies on the selection of a suitable topological model for the patch of the textile, coupled with constitutive models for the yarn behavior. The textile structural configuration is related to the deformation through an energy functional, which depends on both the macroscopic deformation and the distribution of internal nodes. We determine the equilibrium positions of these latter, constrained to an assigned macroscopic deformation. As a result, we derive a macroscopic strain energy function, which reflects the possibly nonlinear character of the yarns as well as the anisotropy induced by the microscopic topological pattern. By means of both analytical estimates and numerical experiments, we show that our model is well suited for both academic test cases and real industrial textiles, with particular emphasis on the tricot textile.