Showing papers on "Mixture theory published in 2009"

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching

Abstract: We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.

On the effective stress in unsaturated porous continua with double porosity

Abstract: Using mixture theory we formulate the balance laws for unsaturated porous media composed of a double-porosity solid matrix infiltrated by liquid and gas. In this context, the term ‘double porosity’ pertains to the microstructural characteristic that allows the pore spaces in a continuum to be classified into two pore subspaces. We use the first law of thermodynamics to identify energy-conjugate variables and derive an expression for the ‘effective’, or constitutive, stress that is energy-conjugate to the rate of deformation of the solid matrix. The effective stress has the form σ ¯ = σ + B p ¯ 1 , where σ is the total Cauchy stress tensor, B is the Biot coefficient, and p ¯ is the mean fluid pressure weighted according to the local degrees of saturation and pore fractions. We identify other emerging energy-conjugate pairs relevant for constitutive modeling of double-porosity unsaturated continua, including the local suction versus degree of saturation pair and the pore volume fraction versus weighted pore pressure difference pair. Finally, we use the second law of thermodynamics to determine conditions for maximum plastic dissipation in the regime of inelastic deformation for the unsaturated two-porosity mixture.

Wrapped Gaussian Mixture Models for Modeling and High-Rate Quantization of Phase Data of Speech

Abstract: The harmonic representation of speech signals has found many applications in speech processing. This paper presents a novel statistical approach to model the behavior of harmonic phases. Phase information is decomposed into three parts: a minimum phase part, a translation term, and a residual term referred to as dispersion phase. Dispersion phases are modeled by wrapped Gaussian mixture models (WGMMs) using an expectation-maximization algorithm suitable for circular vector data. A multivariate WGMM-based phase quantizer is then proposed and constructed using novel scalar quantizers for circular random variables. The proposed phase modeling and quantization scheme is evaluated in the context of a narrowband harmonic representation of speech. Results indicate that it is possible to construct a variable-rate harmonic codec that is equivalent to iLBC at approximately 13 kbps.

Learning Gaussian Mixture Models With Entropy-Based Criteria

Abstract: In this paper, we address the problem of estimating the parameters of Gaussian mixture models. Although the expectation-maximization (EM) algorithm yields the maximum-likelihood (ML) solution, its sensitivity to the selection of the starting parameters is well-known and it may converge to the boundary of the parameter space. Furthermore, the resulting mixture depends on the number of selected components, but the optimal number of kernels may be unknown beforehand. We introduce the use of the entropy of the probability density function (pdf) associated to each kernel to measure the quality of a given mixture model with a fixed number of kernels. We propose two methods to approximate the entropy of each kernel and a modification of the classical EM algorithm in order to find the optimum number of components of the mixture. Moreover, we use two stopping criteria: a novel global mixture entropy-based criterion called Gaussianity deficiency (GD) and a minimum description length (MDL) principle-based one. Our algorithm, called entropy-based EM (EBEM), starts with a unique kernel and performs only splitting by selecting the worst kernel attending to GD. We have successfully tested it in probability density estimation, pattern classification, and color image segmentation. Experimental results improve the ones of other state-of-the-art model order selection methods.

Direct Importance Estimation with Gaussian Mixture Models

Abstract: The ratio of two probability densities is called the importance and its estimation has gathered a great deal of attention these days since the importance can be used for various data processing purposes. In this paper, we propose a new importance estimation method using Gaussian mixture models (GMMs). Our method is an extention of the Kullback-Leibler importance estimation procedure (KLIEP), an importance estimation method using linear or kernel models. An advantage of GMMs is that covariance matrices can also be learned through an expectation-maximization procedure, so the proposed method—which we call the Gaussian mixture KLIEP (GM—KLIEP)—is expected to work well when the true importance function has high correlation. Through experiments, we show the validity of the proposed approach.

The micromechanics of fluid-solid interactions during growth in porous soft biological tissue.

Abstract: In this paper, we address some modelling issues related to biological growth. Our treatment is based on a formulation for growth that was proposed within the context of mixture theory (J Mech Phys Solids 52:1595-1625, 2004). We aim to make this treatment more appropriate for the physics of porous soft tissues, paying particular attention to the nature of fluid transport, and mechanics of fluid and solid phases. The interactions between transport and mechanics have significant implications for growth and swelling. We also reformulate the governing differential equations for reaction-transport of solutes to represent the incompressibility constraint on the fluid phase of the tissue. This revision enables a straightforward implementation of numerical stabilisation for the advection-dominated limit of these equations. A finite element implementation with operator splitting is used to solve the coupled, non-linear partial differential equations that arise from the theory. We carry out a numerical and analytic study of the convergence of the operator splitting scheme subject to strain- and stress-homogenisation of the mechanics of fluid-solid interactions. A few computations are presented to demonstrate aspects of the physical mechanisms, and the numerical performance of the formulation.

Dictionary-based probability density function estimation for high-resolution SAR data

Abstract: In the context of remotely sensed data analysis, a crucial problem is represented by the need to develop accurate models for the statistics of pixel intensities. In this work, we develop a parametric finite mixture model for the statistics of pixel intensities in high resolution synthetic aperture radar (SAR) images. This method is an extension of previously existing method for lower resolution images. The method integrates the stochastic expectation maximization (SEM) scheme and the method of log-cumulants (MoLC) with an automatic technique to select, for each mixture component, an optimal parametric model taken from a predefined dictionary of parametric probability density functions (pdf). The proposed dictionary consists of eight state-of-the-art SAR-specific pdfs: Nakagami, log-normal, generalized Gaussian Rayleigh, Heavy-tailed Rayleigh, Weibull, K-root, Fisher and generalized Gamma. The designed scheme is endowed with the novel initialization procedure and the algorithm to automatically estimate the optimal number of mixture components. The experimental results with a set of several high resolution COSMO-SkyMed images demonstrate the high accuracy of the designed algorithm, both from the viewpoint of a visual comparison of the histograms, and from the viewpoint of quantitive accuracy measures such as correlation coefficient (above 99,5%). The method proves to be effective on all the considered images, remaining accurate for multimodal and highly heterogeneous scenes.

Numerical Implementation of Polymer Viscoplastic Equations for High Strain-Rate Composite Models

Abstract: For polymer matrix composites subjected to large strain rates, it is important to correctly characterize the nonlinear and strain-rate dependent response of polymers. Viscoplastic constitutive material models have been developed to account for the effects of hydrostatic effects and inelastic strains in polymer materials. The effective implementation of such viscoplastic models is important for development of composite models geared toward practical applications. Goldberg’s polymer model numerical implementation into a commercial finite-element code constitutes the main objective of this paper. Special attention is given to the use of effective algorithms for solving the model nonlinear rate dependent viscoplastic equations. Existent experimental data are used to verify the accuracy and robustness of the computational polymer model. A phenomenological fiber model and a simplified iso-strain mixture theory used to obtain the resultant stresses in the composite by averaging the stresses of the individual con...

Hybrid‐Trefftz stress elements for incompressible biphasic media

Abstract: SUMMARY A wavelet-based, high-order time integration method is applied to replace the parabolic problem governing the response of incompressible biphasic media by a set of uncoupled Helmholtz problems. Their formal solutions are used to formulate the stress model of the hybrid-Trefftz finite element formulation. The stress, pressure and displacement fields are directly approximated and designed to satisfy locally the equilibrium condition in each phase of the mixture. This basis is used to enforce on average the compatibility conditions and the constitutive relations of the mixture. The displacements in the solid and the normal displacement in the fluid are approximated independently on the boundary of the element and the basis is used to enforce in weak form the boundary equilibrium conditions. The resulting solving system is sparse, well suited to adaptive refinement and parallel processing. The energy statements associated with the formulation are recovered and sufficient conditions for the uniqueness of the finite element solutions are stated. Testing problems reported in the literature are used to illustrate the quality of the pressure, stress, displacement and velocity estimates obtained with the hybrid-Trefftz stress element. Copyright q 2009 John Wiley & Sons, Ltd.

Evolutionary Algorithm Using Mutual Information for Independent Component Analysis

Abstract: Independent Component Analysis (ICA) is a method for finding underlying factors from multidimensional statistical data. ICA differs from other similar methods in that it looks for components that are both statistically independent and nongaussian. Blind Source Separation (BSS) consists in recovering unobserved signals from a known set of mixtures. Thus, ICA and BSS are equivalent when the mixture is assumed to be linear up to possible permutations and invertible scalings. However, when the mixing model is nonlinear, additional constraints are needed to assure that independent components correspond to the original signals. In this paper, we propose a simple though effective method based on estimating the probability densities of the outputs for solving the BSS problem in linear and nonlinear mixtures making use of genetic algorithms. A post-nonlinear mixture model is assumed so that the solution space in the nonlinear case is restricted to signals equivalent to the original ones.

Equation of State for Energetic Structural Materials

Abstract: Current investigations of energetic structural materials involve shock-induced and shock-assisted chemical reactions in which complex physical processes are currently only elucidated by computational models of gas-gun experiments. The equation of state is one of the most important parts of the constitutive models incorporated into models that describe the processes in shock-induced and shock-assisted chemical reactions. Implementation of current methods typically requires simplifying assumptions in the mixture rules. In this paper, two new equation-of-state methods are proposed that 1) are used to physically interpret both homobaric and uniform-strain assumptions, 2) do not require mixture-averaged equation-of-state model parameters, and 3) do not have any restrictions on the form of constituent equation-of-state or pore-collapse model. The proposed methods are compared with other mixture equation-of-state methods, and cases in which the mixture is porous are demonstrated. Gas-gun experiments are simulated and compared with experimental data for a material with 2Al, Fe 2 O 3 , 20 wt % Epon 828, and voids, in which the reaction initiation threshold was not reached. The simulation integrates conservation equations and momentum balance using a second-order finite volume scheme.

A micropolar mixture theory of multi-component porous media

Abstract: A mixture theory is developed for multi-component micropolar porous media with a combination of the hybrid mixture theory and the micropolar continuum theory. The system is modeled as multi-component micropolar elastic solids saturated with multi-component micropolar viscous fluids. Balance equations are given through the mixture theory. Constitutive equations are developed based on the second law of thermodynamics and constitutive assumptions. Taking account of compressibility of solid phases, the volume fraction of fluid as an independent state variable is introduced in the free energy function, and the dynamic compatibility condition is obtained to restrict the change of pressure difference on the solid-fluid interface. The constructed constitutive equations are used to close the field equations. The linear field equations are obtained using a linearization procedure, and the micropolar thermo-hydro-mechanical component transport model is established. This model can be applied to practical problems, such as contaminant, drug, and pesticide transport. When the proposed model is supposed to be porous media, and both fluid and solid are single-component, it will almost agree with Eringen’s model.

A numerical study of fluids with pressure dependent viscosity flowing through a rigid porous medium

Abstract: In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications like enhanced oil recovery and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton-Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is also to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution.

On the flame structure in multi-temperature mixture theory

Abstract: We study the simple case of stationary one-dimensional flame propagation, modeled by the stoichiometric formula A → B To this aim we consider a binary mixture where the constituent A prevails before the flame while B outnumbers A behind the flame For the mathematical description of the problem, we refer to Mixture Theory and compare the predictions obtained by multi-temperature mixture balance laws with those (known in the literature) obtained by single-temperature equations

Binary Mixtures of Elastic Solids with Microstructure

Abstract: In this paper we establish a nonlinear theory of binary mixtures of elastic solids with microstructure. The independent constitutive variables are the displacement fields, displacement gradients, microdeformation tensors and their gradients. The basic equations are derived in Lagrangian description. The theory is linearized and a uniqueness theorem with no definiteness assumption on the constitutive coefficients is presented. The theory is used to study a special kind of microstructure in which the microdeformation tensor is isotropic. The problem of a concentrated body moment is investigated.

Introduction to Mixture Theory

Abstract: After a general description of mixtures and multi-phase systems and their difference, reasons are given why their distinctions are premature prior to a complete thermodynamic exploitation of postulated constitutive relations by the Second Law of Thermodynamics. Consequently, both systems are here denoted as mixtures. Kinematics is treated first. Then, the general balance laws and their specializations for constituent mass, momenta, energy and entropy are discussed in global and local forms as well as jump conditions across singular surfaces. Based on Truesdell’s metaphysical principles the sum relations define the corresponding mixture quantities which obey the physical balance laws for the mixture as a whole.

Estimation method of CCD and CMOS response functions based on a single image

Abstract: Response function is one of the important characteristics of CCD and CMOS. Traditional measure method of response function need to fulfill strict measurement conditions, and has the shortcoming of low measurement efficiency and high measurement cost. This paper use PCA method to compute 5 eigenvectors corresponding to 5 maximum eigenvalues of the response function space base on 201 existing response functions. Express the existing response functions by the linear combination of the 5 eigenvectors. Combine all the linear coefficient in a probability distribution in the form of Gaussian Mixture Model (GMM). Estimate the parameter of this Gaussian Mixture Model. Base on this GMM, process the color edge pixels clusters in the image, estimate the RGB response functions. This method has high precision and wide applying prospect.