Showing papers on "Mixture theory published in 2008"

Three-Dimensional Discrete Element Models for Asphalt Mixtures

Abstract: The main aim of this paper is to develop three-dimensional (3-D) microstructure-based discrete element models of asphalt mixtures to study the dynamic modulus from the stress-strain response under compressive loads. The 3-D microstructure of the asphalt mixture was obtained from a number of two-dimensional (2-D) images. In the 2-D discrete element model, the aggregate and mastic were simulated with the captured aggregate and mastic images. The 3-D models were reconstructed with a number of 2-D models. This stress-strain response of the 3-D model was computed under the loading cycles. The stress-strain response was used to predict the asphalt mixture's stiffness (modulus) by using the aggregate and mastic stiffness. The moduli of the 3-D models were compared with the experimental measurements. It was found that the 3-D discrete element models were able to predict the mixture moduli across a range of temperatures and loading frequencies. The 3-D model prediction was found to be better than that of the 2-D model. In addition, the effects of different air void percentages and aggregate moduli to the mixture moduli were investigated and discussed.

Variational formulation of pre-stressed solid-fluid mixture theory, with an application to wave phenomena

Abstract: Fluid saturated porous media are modelled by the theory of mixtures and the placement maps of the solid and of the fluid are considered. The momentum balance equations are derived in the framework of a variational approach: We take an action functional and two families of variations and assume that the sum of the virtual work of the external forces and the variation of such an action along each variation are zero. Constitutive equations for the two Cauchy stress tensors and for the interaction force are derived taking into account a general state of pre-stress for the solid and for the fluid species. Governing equations are therefore formulated, however, for the sake of simplicity, only the case of pure initial pressure is further investigated. The propagation of bulk (transversal and longitudinal) waves and the influence of pre-stress are studied: In particular, stability analyses are carried out starting from dispersion relations and the role of pre-stress is investigated. Finally, a numerical example is established for a given state of pre-stress, deriving the phase velocities and the attenuation coefficients of transversal and longitudinal waves.

Multiphase models of tumour growth

Abstract: The aim of this chapter is to give a pedagogical presentation of multiphase models in their application to the study of tumour growth Starting from the simplest concepts, we shall describe how to deduce multiphase models, paying attention to the general modelling framework and on how to model the different terms appearing in the equations A particular attention is also devoted to the definition of the interaction between cells and extracellular matrix In this way a general model is deduced which is then specialized in examples describing avascular phase and vascular phases of growth, and the formation of fibrosis

Blind Separation of Superimposed Shifted Images Using Parameterized Joint Diagonalization

Abstract: We consider the blind separation of source images from linear mixtures thereof, involving different relative spatial shifts of the sources in each mixture. Such mixtures can be caused, e.g., by the presence of a semi-reflective medium (such as a window glass) across a photographed scene, due to slight movements of the medium (or of the sources) between snapshots. Classical separation approaches assume either a static mixture model or a fully convolutive mixture model, which are, respectively, either under-or over-parameterized for this problem. In this paper, we develop a specially parameterized scheme for approximate joint diagonalization of estimated spectrum matrices, aimed at estimating the succinct set of mixture parameters: the static (gain) coefficients and the shift values. The estimated parameters are, in turn, used for convenient frequency-domain separation. As we demonstrate using both synthetic mixtures and real-life photographs, the advantage of the ability to incorporate spatial shifts is twofold: Not only does it enable separation when such shifts are present, but it also warrants deliberate introduction of such shifts as a simple source of added diversity whenever the static mixing coefficients form a singular matrix - thereby enabling separation in otherwise inseparable scenes.

TRUST-TECH-Based Expectation Maximization for Learning Finite Mixture Models

Abstract: The expectation maximization (EM) algorithm is widely used for learning finite mixture models despite its greedy nature. Most popular model-based clustering techniques might yield poor clusters if the parameters are not initialized properly. To reduce the sensitivity of initial points, a novel algorithm for learning mixture models from multivariate data is introduced in this paper. The proposed algorithm takes advantage of TRUST-TECH (TRansformation Under STability-reTaining Equilibria CHaracterization) to compute neighborhood local maxima on the likelihood surface using stability regions. Basically, our method coalesces the advantages of the traditional EM with that of the dynamic and geometric characteristics of the stability regions of the corresponding nonlinear dynamical system of the log-likelihood function. Two phases, namely, the EM phase and the stability region phase, are repeated alternatively in the parameter space to achieve local maxima with improved likelihood values. The EM phase obtains the local maximum of the likelihood function and the stability region phase helps to escape out of the local maximum by moving toward the neighboring stability regions. Though applied to Gaussian mixtures in this paper, our technique can be easily generalized to any other parametric finite mixture model. The algorithm has been tested on both synthetic and real data sets and the improvements in the performance compared to other approaches are demonstrated. The robustness with respect to initialization is also illustrated experimentally.

Simplifying Mixture Models Using the Unscented Transform

Abstract: Mixture of Gaussians (MoG) model is a useful tool in statistical learning. In many learning processes that are based on mixture models, computational requirements are very demanding due to the large number of components involved in the model. We propose a novel algorithm for learning a simplified representation of a Gaussian mixture, that is based on the Unscented Transform which was introduced for filtering nonlinear dynamical systems. The superiority of the proposed method is validated on both simulation experiments and categorization of a real image database. The proposed categorization methodology is based on modeling each image using a Gaussian mixture model. A category model is obtained by learning a simplified mixture model from all the images in the category.

Modeling cementitious materials as multiphase porous media: theoretical framework and applications

Abstract: In this paper a general model for the analysis of concrete as multiphase porous material, based on the so-called Hybrid Mixture Theory, is presented. The development of the model equations, taking into account both bulk phases and interfaces of the multiphase system is described, starting from the microscopic scale. An exploration of the second law of thermodynamics is also presented: it allows defining several quantities used in the model, like capillary pressure, disjoining pressure or effective stress, and to obtain some thermodynamic restrictions imposed on the evolution equations describing the material deterioration. Then, two specific forms of the general model adapted to the case of concrete at early ages and beyond and to the case of concrete structures under fire are shown. Some numerical simulations aimed to prove the validity of the approach adopted also are presented and discussed.

Mixture of Spherical Distributions for Single-View Relighting

Abstract: We present a method for simultaneously estimating the illumination of a scene and the reflectance property of an object from single view images-a single image or a small number of images taken from the same viewpoint. We assume that the illumination consists of multiple point light sources, and the shape of the object is known. First, we represent the illumination on the surface of a unit sphere as a finite mixture of von Mises-Fisher distributions based on a novel spherical specular reflection model that well approximates the Torrance-Sparrow reflection model. Next, we estimate the parameters of this mixture model including the number of its component distributions and the standard deviation of them, which correspond to the number of light sources and the surface roughness, respectively. Finally, using these results as the initial estimates, we iteratively refine the estimates based on the original Torrance-Sparrow reflection model. The final estimates can be used to relight single-view images such as altering the intensities and directions of the individual light sources. The proposed method provides a unified framework based on directional statistics for simultaneously estimating the intensities and directions of an unknown number of light sources, as well as the specular reflection parameter of the object in the scene.

A thermodynamic framework for a mixture of two liquids

Abstract: In this study, we extend a thermodynamic framework that has been used with some success for describing the response of a variety of single constituent continua. Using the thermodynamic framework, we obtain a model for the mixture of two compressible fluids that has a much simpler structure than the model obtained earlier within the context of mixture theory. We also investigate the response of a mixture of two fluids that is constrained to have a constant volume, using the same thermodynamic framework.

On the Theory of Viscoelastic Mixtures and Stability

Abstract: A theory is developed for binary mixtures of viscoelastic materials. The basic equations of a nonlinear theory of heat conducting viscoelastic mixtures are derived in Lagrangian description. The individual components of the mixture are modeled as Kelvin—Voigt viscoelastic materials. A nonlinear constitutive relation which generalizes Darcy's law is derived. The linearized version of the theory is established. In the present theory the diffusive force depends on relative displacement and relative velocity. Stability results are presented in the context of materials which are non-conductor of heat.

On entropy-constrained vector quantization using gaussian mixture models

Abstract: A flexible and low-complexity entropy-constrained vector quantizer (ECVQ) scheme based on Gaussian mixture models (GMMs), lattice quantization, and arithmetic coding is presented. The source is assumed to have a probability density function of a GMM. An input vector is first classified to one of the mixture components, and the Karhunen-Loeve transform of the selected mixture component is applied to the vector, followed by quantization using a lattice structured codebook. Finally, the scalar elements of the quantized vector are entropy coded sequentially using a specially designed arithmetic coder. The computational complexity of the proposed scheme is low, and independent of the coding rate in both the encoder and the decoder. Therefore, the proposed scheme serves as a lower complexity alternative to the GMM based ECVQ proposed by Gardner, Subramaniam and Rao. The performance of the proposed scheme is analyzed under a high-rate assumption, and quantified for a given GMM. The practical performance of the scheme was evaluated through simulations on both synthetic and speech line spectral frequency (LSF) vectors. For LSF quantization, the proposed scheme has a comparable performance to at rates relevant for speech coding (20-28 bits per vector) with lower computational complexity.

An Anisotropic Constitutive Equation for the Stress Tensor of Blood Based on Mixture Theory

Abstract: Based on ideas proposed by Massoudi and Rajagopal (M-R), we develop a model for blood using the theory of interacting continua, that is, the mixture theory. We first provide a brief review of mixture theory, and then discuss certain issues in constitutive modeling of a two-component mixture. In the present formulation, we ignore the biochemistry of blood and assume that blood is composed of red blood cells (RBCs) suspended in plasma, where the plasma behaves as a linearly viscous fluid and the RBCs are modeled as an anisotropic nonlinear density-gradient-type fluid. We obtain a constitutive relation for blood, based on the simplified constitutive relations derived for plasma and RBCs. A simple shear flow is discussed, and an exact solution is obtained for a very special case; for more general cases, it is necessary to solve the nonlinear coupled equations numerically.

An ICA Mixture Hidden Markov Model for Video Content Analysis

Abstract: In this paper, a new theoretical framework based on hidden Markov model (HMM) and independent component analysis (ICA) mixture model is presented for content analysis of video, namely ICAMHMM. Unlike the Gaussian mixture observation model commonly used in conventional HMM applications, the observations in the new ICAMHMM are modeled as a mixture of non-Gaussian components. Each non-Gaussian component is formulated by an ICA mixture, reflecting the independence of different components across video frames. In addition, to construct a compact feature space to represent a video frame, ICA is applied on video frames and the ICA coefficients are used to form a compact 2-D feature subspace that makes the subsequent modeling computationally efficient. The model parameters can be identified using supervised learning by the training sequences. The new re-estimation learning formulae of iterative ICAMHMM parameter estimation are derived based on a maximum likelihood function. Employing the identified model, maximum likelihood algorithms are developed to detect and recognize video events. As a case study, golf video sequences are used to test the effectiveness of the proposed algorithm. Experimental results show that the presented method can effectively detect and recognize the recurrent event patterns in video data. The presented new ICAMHMM is generic and can be applied to sequential data analysis in other applications.

A Mixture Theory for Microstretch Thermoviscoelastic Solids

Abstract: A nonlinear theory is developed for a heat-conducting viscoelastic composite which is modelled as a mixture consisting of a microstretch Kelvin–Voigt material and a microstretch elastic solid. The strain measures, the basic laws and the constitutive equations are established and presented in Lagrangian description. The initial boundary value problem associated to such model is also formulated. Then the linearized theory is considered and the constitutive equations are given for both anisotropic and isotropic bodies. Finally, a uniqueness result is established within the framework of the linear theory.

Volume-weighted mixture theory for granular materials

Abstract: In the present work we treat granular materials as mixtures composed of a solid and a surrounding void continuum, proposing then a continuum thermodynamic theory for it. In contrast to the common mass-weighted balance equations of mass, momentum, energy and entropy for mixtures, the volume-weighted balance equations and the associated jump conditions of the corresponding physical quantities are derived in terms of volume-weighted field quantities here. The evolution equations of volume fractions, volume-weighted velocity, energy, and entropy are presented and explained in detail. By virtue of the second law of thermodynamics, three dissipative mechanisms are considered which are specialized for a simple set of linear constitutive equations. The derived theory is applied to the analysis of reversible and irreversible compaction of cohesionless granular particles when a vertical oscillation is exerted on the system. In this analysis, a hypothesis for the existence of a characteristic depth within the granular material in its closely compacted state is proposed to model the reversible compaction.

A mixture theory based method for three-dimensional modeling of reinforced concrete members with embedded crack finite elements

Abstract: The paper presents a methodology to model three-dimensional reinforced concrete members by means of embedded discontinuity elements based on the Continuum Strong Discontinuous Approach (CSDA). Mixture theory concepts are used to model reinforced concrete as a 3D composite material constituted of concrete with long fibers (rebars) bundles oriented in different directions embedded in it. The effects of the rebars are modeled by phenomenological constitutive models devised to reproduce the axial non-linear behavior, as well as the bond-slip and dowel action. The paper presents the constitutive models assumed for the components and the compatibility conditions chosen to constitute the composite. Numerical analyses of existing experimental reinforced concrete members are presented, illustrating the applicability of the proposed methodology.

On structural mixture theory applied to elastic isotropic materials with internal three-component nanoscale structure

Abstract: Two types of isotropic materials are considered: two- and three-component elastic particulate composite materials. Two elastic mixture models describing the propagation of plane waves in such materials are briefly characterized. The basic features of plane wave motion are analyzed for both models. The descriptions of the wave picture are compared to reveal some general cases of similarity and difference. To this end, the material is modeled by a three-component elastic mixture and, additionally, plane waves are analyzed using this model

Component Reduction for Gaussian Mixture Models

Abstract: The mixture modeling framework is widely used in many applications. In this paper, we propose a component reduction technique, that collapses a Gaussian mixture model into a Gaussian mixture with fewer components. The EM (Expectation-Maximization) algorithm is usually used to fit a mixture model to data. Our algorithm is derived by extending mixture model learning using the EM-algorithm. In this extension, a difficulty arises from the fact that some crucial quantities cannot be evaluated analytically. We overcome this difficulty by introducing an effective approximation. The effectiveness of our algorithm is demonstrated by applying it to a simple synthetic component reduction task and a phoneme clustering problem.

A Composite Material Model for High Strain Rates

Abstract: For polymer matrix composites subjected to large strain rates, it is important to correctly characterize the nonlinear and strain rate depen dent response of polymers. For this purpose, viscoplastic constitutive equations ori ginally implemented for metals have been modified to account for the effects of hydrostatic e ffects and inelastic strains in polymers. The resultant stress in the composite can then be obtained by using a mixture theory that averages the individual stresses of the polymer and fiber (assumed to be elastic) constituents. The implementation of such analytical models using finite el ement methods constitutes one of the first objectives of the current effort. A c ontinuum model that combines the individual constituents is defined using mixture t heory. Experimental tests will be used to validate the preliminary analytical model and verify its efficiency and applicability for engineering applications

Analysis of macroscopic momentum equation in a columnar mushy zone

Abstract: Purpose – This paper aims to tackle the problem of some ambiguity of the momentum equation formulation in the commonly used macroscopic models of two‐phase solid/liquid region, developing during alloy solidification. These different appearances of the momentum equation are compared and the issue is addressed of how the choice of the particular form affects velocity and temperature fields.Design/methodology/approach – Attention is focused on the ensemble averaging method, which, owing to its stochastic nature, is a new promising tool for setting up the macroscopic transport equations in highly inhomogeneous multiphase micro‐ and macro‐structures, with morphology continuously changing in time when the solidification proceeds. The basic assumptions of the two other continuum models, i.e. based on the classical mixture theory and on the volume‐averaging technique, are also unveiled. These three different forms of the momentum equation are then compared analytically and their impact on calculated velocity and ...

Effective stress principle in modelling shrinkage and creep of concrete at early ages and beyond

Abstract: In this work a novel numerical model of hygro-thermal and hydration phenomena in concrete at early ages and beyond is presented. This is a solidification-type model where all changes of material properties are expressed as functions of the hydration degree, and not the maturity nor equivalent hydration period as in the maturity-type models. A mechanistic approach, typical in the mechanics of porous media, was used to obtain the governing equations, by means of a hybrid mixture theory. The final equations, mass (water species and dry air), energy and momentum balances are written in terms of the chosen primary variables: gas pressure, capillary pressure, temperature and displacements. The model [1-3] takes into account full coupling between hygral, thermal and chemical phenomena, as well as changes of concrete properties caused by hydration process, i.e. porosity, density, permeability. Phase changes and chemical phenomena, as well as the related heat and mass sources are considered. For a more detailed description of the mathematical model, the governing equations and the constitutive relationships, see [1-3]. Kinetics of cement hydration is described by means of an evolution equation which relates the internal variable, hydration degree, with the hydration rate through the chemical affinity. The latter one is considered as the driving force of the chemical reaction, [1-3].