137 research outputs found

    Binary upscaling on complex heterogeneities: The role of geometry and connectivity

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    The equivalent conductivity (Keq) of a binary medium is known to vary with the proportion of the two phases, but it also depends on the geometry and topology of the inclusions. In this paper, we analyze the role of connectivity and shape of the connected components through a correlation study between Keq and two topological and geometrical indicators: the Euler number and the Solidity indicator. We show that a local measure such as the Euler number is weakly correlated to Keq and therefore it is not suitable to quantify the influence of connectivity on the global flux; on the contrary the Solidity indicator, related to the convex hull of the connected components, presents a direct correlation with Keq. This result suggests that, in order to estimate Keq properly, one may consider the convex hull of each connected component as the area of influence of its spatial distribution on flow and make a correction of the proportion of the hydrofacies according to that. As a direct application of these principles, we propose a new method for the estimation of Keq using simple image analysis operations. In particular, we introduce a direct measure of the connected fraction and a non-parametric correction of the hydrofacies proportion to compensate for the influence of the connected components shape on flow. This model, tested on a large ensemble of isotropic media, provides a good Keq approximation even on complex heterogeneities without the need for calibration

    Quantification of the influences of aggregate shape and sampling method on the overestimation of ITZ thickness in cementitious materials

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    The microstructure of the interfacial transition zone (ITZ) surrounding the aggregate in a cementitious composite is quite different from that of the bulk matrix, because of its distinct physical nature including relatively high porosity and low rigidity. The thickness and volume fraction of the ITZ play a major role in determining the transport and mechanical behavior of cementitious composites. However, the ITZ thickness may be overestimated when undertaking sectional plane analysis of these composites. Analysis of Platonic particles has previously shown that the sphericity of the particle is an important parameter in determining the overestimation of the ITZ thickness, but this raises the question of whether sphericity is sufficient to uniquely characterize the influence of aggregate shape. This paper investigates the influence of particle shape on overestimation of ITZ thickness for aggregate shapes which have the same sphericity values as Platonic particles; specifically, spheroids of differing geometries. A normal line sampling algorithm, which is designed to replicate the practical experimental process used in ITZ determination, is employed to obtain the apparent ITZ thickness. The influences of particle shape, sampling method and particle size distribution are investigated in terms of the overestimation of the ITZ volume fraction, and the effective diffusivity within three-phase composites, using the differential effective medium approximation

    A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

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    We present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Green’s functions. We describe fast solvers for the two‐dimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design.We split the infinite sum over the lattice of images into a directly summed “near” part plus a small number of auxiliary sources that represent the (smooth) remaining “far” contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank‐1 or rank‐3 correction and a Schur complement, leaves a well‐conditioned square system that can be solved iteratively using fast multipole acceleration plus a low‐rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimension‐ and PDE‐independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close‐to‐touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against high‐accuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 104 randomly located inclusions per unit cell. © 2018 Wiley Periodicals, Inc.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/146333/1/cpa21759.pd

    An indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity

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    Solute transport takes place in heterogeneous porous formations, with the log conductivity, Y = ln K, modeled as a stationary random space function of given univariate normal probability density function (pdf) with mean Y, variance σY2, and integral scale IY. For weak heterogeneity, the above mentioned quantities completely define the first-order approximation of the longitudinal macrodispersivity αL = σY2IY. However, in highly heterogeneous formations, nonlinear effects which depend on the multipoint joint pdf of Y, impact αL. Most of the past numerical simulations assumed a multivariate normal distribution (MVN) of Y values. The main aim of this study is to investigate the impact of deviations from the MVN structure upon αL. This is achieved by using the concept of spatial correlations of different Y classes, the latter being defined as the space domain where Y falls in the generic interval [Y,Y + ΔY]. The latter is characterized by a length scale λ(Y), reflecting the degree of connectivity of the domain (the concept is similar to the indicator variograms). We consider both “symmetrical” and “non-symmetrical” structures, for which λ(Yâ€Č) = λ(−Yâ€Č) (similar to the MVN), and λ(Yâ€Č) ≠ λ(−Yâ€Č), respectively, where Yâ€Č = Y − Y. For example, large Y zones may have high spatial correlation, while low Y zones are poorly correlated, or vice versa. The impact of λ(Y) on αL is investigated by adopting a structure model which has been used in the past in order to investigate flow and transport in highly heterogeneous media. It is found that the increased correlation in the low conductive zones with respect to the high ones generally leads to a significant increase in αL, for the same global IY. The finding is explained by the solute retention occurring in low Y zones, which has a larger effect on solute spreading than high Y zones. Conversely, αL decreases when the high conductivity zones are more correlated than the low Y ones. Dispersivity is less affected by the shape of λ(Y) for symmetrical distributions. It is found that the range of validity of the first-order dispersivity, i.e., αL = IYσY2, narrows down for non-symmetrical structures

    Computational Prediction of Conductivities of Disk-Shaped Particulate Composites

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    The effective conductivities are determined for randomly oriented disk-shaped particles using an efficient computational algorithm based on the finite element method. The pairwise intersection criteria of disks are developed using a set of vector operations. An element partition scheme has been implemented to connect the elements on different disks across the lines of intersection. The computed conductivity is expressed as a function of the density and the size of the circular disks or elliptical plates. It is further expressed in a power-law form with the key parameters determined from curve fittings. The particle number and the trial number of simulations vary with the disk size to minimize the computational effort in search of the percolation paths. The estimated percolation threshold agrees well with the result reported in the literature. It has been confirmed that the statistical invariant for percolation is a cubic function of the characteristic size, and that the definition of percolation threshold is consistent with that of the equivalent system containing spherical particles. The effect of aspect ratio to the percolation threshold has been studied in this article. High aspect ratio will decrease the percolation threshold. Binary dispersions of disks of different radii have also been investigated to study the effect of the size distribution. The approximate solutions in the power-law function have potential applications in advanced composites with embedded graphene nanoplatelets

    Computational Modeling of Percolation Conduction and Diffusion of Heterogeneous Materials

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    Heterogeneous materials provide a unique combination of desirable mechanical, thermal or electrical properties. This dissertation presents several micro-structure modeling approaches to predict the effective properties of heterogeneous materials and demonstrates its certain application toward two highly heterogeneous, unconventional structural composite materials (carbon fiber reinforced composite materials and graphene nanoplatelets composite). By using the efficient computational algorithm based on the FEA, a randomly oriented disk-shaped particles system are generated. A new element partition scheme based on the vector operations and geometry of inclusion has been implemented to mesh the intersected disks. The computed equivalent conductivity is expressed as a power-law function form with the key parameters determined from curve fitting. Also, we proposed a novel random walk method to study the 2-D circular or elliptical and 3-D spherical or ellipsoidal non-overlapping system diffusion process. A Monte-Carlo scheme is applied to generate the particulate system for simulation. The effective diffusion coefficient has been predicted and compared to the finite element method and effective medium theory. The aspect ratio effect also investigated and compared to other numerical studies

    Liquid metal flows in continuous casting molds: A numerical study of electromagnetic flow control, turbulence and multiphase phenomena

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    Der Effekt eines externen Magnetfeldes auf die mehrphasige und turbulente Strömung in Stranggußkokillen und deren Wechelspiel fĂŒhrt in den wissenschaftlichen Arbeiten zu widersprĂŒchlichen Aussagen. Die verschiedenen Prozessparameter können innerhalb eines kleinen Varianzbereichs entscheidenden Einfluss auf die Aussage haben, ob ein Magnetfeld begĂŒnstigend oder schĂ€digend auf die QualitĂ€t des Produkts wirkt. Um wichtige Einflussfaktoren zu identifizieren, werden daher numerische Strömungssimulationen des Prozesses durchgefĂŒhrt. Dazu wird zunĂ€chst ein mehrphasiger und inkompressibler Mehrregionen-CFD-Löser fĂŒr magnetohydrodynamische Strömungen entwickelt und validiert, um die komplexe Strömung in einer Stranggußkokille mit hoher Genauigkeit simulieren zu können. Darauf aufbauend wird das numerische Setup anhand einer Modellkokille mit aktuellen Messdaten validiert. Durch die neuartige Kombination Lagrange'scher Lösungsmethoden mit angepassten Termen fĂŒr die Magnetohydrodynamik sowie der skalenaufgelösten magnetohydrodynamischen Turbulenz, können erstmals Aussagen zur optimalen Magnetfeldverteilung im Hinblick auf StrömungsstabilitĂ€t, Turbulenzmodulation und Blasenverteilung getroffen werden. Mit Hilfe dieses Wissens können neuartige Konzepte elektromagnetischer Bremssysteme fĂŒr den Stranggußprozess entwickelt werden

    Computational Study of the Effect of Interparticle Contact in Conductive Properties of Random Particulate Systems

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    The effective conductivity of random particulate material system is computationally investigated using the Monte Carlo scheme and finite element method. A cubic system consisting of randomly-dispersed, equal-sized, deformable ellipsoids are modeled in this study. The steady-state conduction analysis along with a finite element analysis are performed to evaluate the electrical or thermal conductivity for the mechanical contact system. To represent more realistic material system, interfacial friction and gap conductance (or contact resistance) are included among the contacting particles. The Monte Carlo simulations are implemented to give a quantitative relationship between the effective conductivity and the inclusion volume fraction. Several parametric studies are performed to quantify the relationships, for example, (1) the particle number, (2) the particle shape, (3) the random distribution of particle, (4) the interfacial friction, (5) gap conductance. The study reveals the nonlinear relationship of the gap conductance with respect to the overall conductivity. Therefore, the mechanical properties of particulate system are strongly dependent on the interactions among inclusions. The study of microstructure of material is merited in advanced composite manufacture
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