Diffusion Matrix Research Articles

Exponential Time Decay of Solutions to Reaction-Cross-Diffusion Systems of Maxwell\u2013Stefan Type

The large-time asymptotics of weak solutions to Maxwell–Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite-dimensional inequality. The key elements of the proof are the existence of a unique detailed-balance equilibrium and the derivation of an inequality relating the entropy and the entropy production. The main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations. The idea is to enlarge the space of n partial concentrations by adding the total concentration, viewed as an independent variable, thus working with n+1 variables. Further results concern the existence of global bounded weak solutions to the parabolic system and an extension of the results to complex-balance systems.

Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

Abstract. The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

Atomic mobility evaluation and diffusion matrix for fcc_A1 Co–V–W alloys

On the basis of measured interdiffusivity matrices from our previous research and thermodynamic descriptions, atomic mobilities for fcc_A1 Co–V–W alloys were evaluated by diffusion controlled transformation software. For the purpose of verifying the reliability of the present atomic mobilities, comparisons between the predicted data containing interdiffusivity matrices, composition profiles, as well as diffusion paths and the experimental ones were conducted. The results demonstrate that presently obtained atomic mobilities can reproduce experimental data reasonably. Furthermore, three-dimensional surfaces for the diagonal interdiffusivities at 1373, 1423 and 1473 K were plotted and were compared with measured ones, from which the pre-exponential factor and activation energy for elements V and W were then derived by means of Arrhenius equation. It is worth mentioning that all of the composition profiles and diffusion paths at three temperatures can be reasonably predicted by the present atomic mobilities containing only two ternary interaction parameters. Such a high simulation accuracy is rarely reported in the literature. The present atomic mobilities contribute to computational design for novel Co-based superalloys and multi-component cemented carbides.

A priori Lipschitz estimates for solutions of local and nonlocal Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator

We establish a priori Lipschitz estimates for unbounded solutions of second-order Hamilton–Jacobi equations in \mathbb R^N in presence of an Ornstein–Uhlenbeck drift. We generalize the results obtained by Fujita, Ishii and Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a nonlocal integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear. These results are obtained for both degenerate and nondegenerate equations.

Adaptive discrete-smeared crack (A-DiSC) model for multi-scale progressive damage in composites

Apart from major cracks and delamination, extensive and diffused short matrix micro-cracks are generally observed in composite laminates. In principle, discrete crack models (DCM) offer greatest fidelity to the mechanics of crack growth, including interactions between different failure modes, but it is impractical to model numerous cracks explicitly beyond coupon-sized laminates. Smeared crack models (SCM), on the other hand, treat diffused damage by effective material degradation. However, homogenization of cracks does not model strong discontinuities and delamination-matrix crack interactions. An adaptive discrete-smeared crack model is proposed here to preserve fidelity to the fracturing process while maintaining efficiency and simplicity in modelling distributed damage. DCM is used to model individual matrix cracks at the initial stages of propagation. Subsequently, adaptive transition from DCM to coupled DCM/SCM is performed. Critical cracks are explicitly retained, while non-critical ones are modelled by dispersed damage. The method is demonstrated with off-axis and open-hole tension tests.

Dynamics of a stochastic SIR epidemic model with distributed delay and degenerate diffusion

In this paper, we study a stochastic SIR epidemic model with distributed delay and degenerate diffusion. Firstly, we transform the stochastic model into an equivalent system which contains three equations. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We verify the densities of the distributions of the solutions can converge in L1 to an invariant density. Then we establish sufficient conditions for extinction of the disease. Some examples and numerical simulations are introduced to illustrate our analytical results.

Hermite Expansion for Transition Densities of Irreducible Diffusions with An Application to Option Pricing

This paper shows that a small-time Hermite expansion is feasible for multivariate diffusions. By introducing an innovative quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, we derive explicit recursive formulas for the expansion coefficients of transition densities and European option prices for multivariate diffusions with jumps in return. These immediately available explicit formulas, particularly for the irreducbile, nonaffine, time-inhomogeneous model with different types of jump-size distribution, is new to the literature. The explicit formulas can lead to real-time derivatives pricing and hedging as well as model calibration. Extensive numerical experiments illustrate the accuracy and effectiveness of our approach.

Global renormalized solutions to reaction-cross-diffusion systems with self-diffusion

The global-in-time existence of renormalized solutions to reaction-cross-diffusion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions is proved. The cross-diffusion part describes the segregation of population species and is a generalization of the Shigesada-Kawasaki-Teramoto model. The diffusion matrix is not diagonal and generally neither symmetric nor positive semi-definite, but the diffusion operator possesses a formal gradient-flow or entropy structure. The reaction part includes reversible reactions of mass-action kinetics and does not obey any growth condition. The existence result generalizes both the condition on the reaction part required in the boundedness-by-entropy method and the proof of J. Fischer for reaction-diffusion systems with diagonal diffusion matrices. Moreover, we do not assume local Lipschitz continuity of the reaction terms but only continuity.

Homogenization results for a coupled system of reaction–diffusion equations

The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction–diffusion processes in periodic porous media is analyzed. Our mathematical model can be used for studying several metabolic processes taking place in living cells, in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the organellar membranes. The coupling of the concentrations of the biochemical species is realized via various properly scaled nonlinear reaction terms. These nonlinearities, which model, at the microscopic scale, various volume or surface reaction processes, give rise in the macroscopic model to different effects, such as the appearance of additional source or sink terms or of a non-standard diffusion matrix.

Homogenization of the Eigenvalues of the Neumann–Poincaré Operator

In this article, we investigate the spectrum of the Neumann–Poincare operator $${{\mathcal {K}}}_\varepsilon ^*$$ (or equivalently, that of the associated Poincare variational operator $$T_\varepsilon $$ ) associated to a periodic distribution of small inclusions with size $$\varepsilon $$ , and its asymptotic behavior as the parameter $$\varepsilon $$ vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the ‘trivial’ eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to $$\varepsilon $$ . This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light on the issue of the homogenization of the voltage potential $$u_\varepsilon $$ caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possible negative) conductivity a, surrounded by a dielectric medium, with unit conductivity. In particular, we prove that the limit behavior of $$u_\varepsilon $$ is strongly related to the (possibly ill-defined) homogenized diffusion matrix predicted by the homogenization theory in the standard elliptic case. Additionally, we prove that the homogenization of $$u_\varepsilon $$ is always possible when a is either positive, or negative with a ‘small’ or ‘large’ modulus.

Homogenization of degenerate cross-diffusion systems

Two-scale homogenization limits of parabolic cross-diffusion systems in a heterogeneous medium with no-flux boundary conditions are proved. The heterogeneity of the medium is reflected in the diffusion coefficients or by the perforated domain. The diffusion matrix is of degenerate type and may be neither symmetric nor positive semi-definite, but the diffusion system is assumed to satisfy an entropy structure. Uniform estimates are derived from the entropy production inequality. New estimates on the equicontinuity with respect to the time variable ensure the strong convergence of a sequence of solutions to the microscopic problems defined in perforated domains.

Global existence of solutions for the Poisson–Nernst–Planck system with steric effects

In this work, we study the global existence of a modified Poisson–Nernst–Planck system with steric effects. This model involves cross-diffusion and non-local terms. The main idea of the proof is to approximate the system by truncating the diffusion matrix and apply the Schauder’s fixed-point theorem. Moreover, we obtain L2 uniform in time estimates from the energy inequality (3.5). The global existence then follows.

Species diffusion in clinopyroxene solid solution in the diopside\u2013anorthite system

The coupled multicomponent diffusion of the species Ca2Si2O6, CaAl2SiO6 and Mg2Si2O6 was determined in diopside crystals in the diopside/anorthite (Di/An) system at temperatures (T) of 1110–1260 °C and oxygen fugacities (fO2) between 1.0 log unit below and above the fayalite–magnetite–quartz equilibrium (FMQ ± 1). Diffusion couples were prepared by the seed overgrowth technique. Element concentration profiles were measured perpendicular to the rim/core interface by step-scanning profiling with a field emission gun scanning electron microscope (FEG-SEM). The multicomponent diffusion matrix was solved by fitting its eigenvalues (λ) and eigenvectors (v) to the measured concentration profiles. The full diffusion matrix D can be recovered by using the formula D = PvarLambda P^{ - 1} resulting in the following equation: DDi/An=1.00-0.67-0.381.00λ1T00λ2T1.00-0.67-0.381.00-1.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$D_{\\text{Di/An}} = \\left[ {\\begin{array}{*{20}c} {1.00} & { - 0.67} \\\\ { - 0.38} & {1.00} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}c} {\\lambda 1\\left( T \\right)} & 0 \\\\ 0 & {\\lambda 2\\left( T \\right)} \\\\ \\end{array} } \\right]\\left[ {\\begin{array}{*{20}c} {1.00} & { - 0.67} \\\\ { - 0.38} & {1.00} \\\\ \\end{array} } \\right]^{ - 1} .$$\\end{document}The eigenvalues (λ1 and λ2) represent upper limit values and are described by the following Arrhenius-type equations: λ1Di/An=10-15.98±1.17×exp-114.4±32.8kJ/molRT,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\lambda 1 \\left( {\\text{Di/An}} \\right) = 10^{ - 15.98 \\pm 1.17} \\times \\exp \\left[ {\\frac{{ - 114.4 \\pm 32.8 {\\text{kJ/mol}}}}{RT}} \\right],$$\\end{document}λ2Di/An=10-16.23±1.17×exp-114.4±32.8kJ/molRT,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\lambda 2 \\left( {\\text{Di/An}} \\right) = 10^{ - 16.23 \\pm 1.17} \\times { \\exp }\\left[ {\\frac{{ - 114.4 \\pm 32.8 {\\text{kJ/mol}}}}{RT}} \\right],$$\\end{document} where λ1 and λ2 are the first and second eigenvalue of the diffusion matrix in m2 s−1, R is the gas constant and T is the temperature in K. The dominant eigenvalue (λ1) is one quarter order of magnitude larger than the second eigenvalue (λ2). The eigenvectors are constant for all experiments inferring that the entire D matrix can be described with the eigenvalues as the only T-dependent parameter. Additionally, the derived diffusion data and modeling approach were applied to constrain the duration of magmatic processes recorded in zoned clinopyroxene (cpx) phenocrysts from a basaltic, post-plutonic dyke of the Tertiary Adamello batholiths (N-Italy). The results reveal residence times of the overgrown cpx prior to final emplacement in the range of 0.25–1.7 years (lower limit values) testifying that the data and method can be applied to model cpx diffusion profiles in complex natural cpx.

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

AbstractEntropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.

Sparse-TFM Imaging of Lamb Waves for the Near-Distance Defects in Plate-Like Structures

The ultrasonic phased array total focusing method (TFM) has the advantages of high imaging resolution and high sensitivity to small defects. However, it has a long imaging time and cannot realize near-distance defect imaging, which limits its application for industrial detection. A sparse-TFM algorithm is adopted in this work to solve the problem regarding rapid imaging of near- distance defects in thin plates. Green’s function is reconstructed through the cross-correlation of the diffuse full matrix captured by the ultrasonic phased array. The reconstructed full matrix recovers near-distance scattering information submerged by noise. A sparse array is applied to TFM for rapid imaging. In order to improve the imaging resolution, the location of active array elements in the sparse array can be optimized using the genetic algorithm (GA). Experiments are conducted on three aluminium plates with near-distance defects. The experimental results confirm that the sparse-TFM algorithm of Lamb waves can be used for near-distance defects imaging, which increases the computational efficiency by keeping the imaging accuracy. This paper provides a theoretical guidance for Lamb wave non-destructive testing of the near-distance defects in plate-like structures.

A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data

In this article, we investigate the problem of identifying the source term f in the elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data (j, g) with noise level In this context, the diffusion matrix Q is given. A variational method of Tikhonov-type regularization with specific misfit term of Kohn–Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem. The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov and Lavrentiev regularization. Using the variational discretization concept, where the PDE is discretized with piecewise linear and continuous finite elements, we show the convergence of finite element approximations to a sought source function. Moreover, we derive an error bound and corresponding convergence rates provided a suitable range-type source condition is satisfied. For the numerical solution, we propose a conjugate gradient method. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.

Conditioning of the finite volume element method for diffusion problems with general simplicial meshes

The conditioning of the linear finite volume element discretization for general diffusion equations is studied on arbitrary simplicial meshes. The condition number is defined as the ratio of the maximal singular value of the stiffness matrix to the minimal eigenvalue of its symmetric part. This definition is motivated by the fact that the convergence rate of the generalized minimal residual method for the corresponding linear systems is determined by the ratio. An upper bound for the ratio is established by developing an upper bound for the maximal singular value and a lower bound for the minimal eigenvalue of the symmetric part. It is shown that the bound depends on three factors, the number of the elements in the mesh, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity measured in the metric specified by the inverse diffusion matrix. It is also shown that the diagonal scaling can effectively eliminates the effects from the mesh nonuniformity measured in the Euclidean metric. Numerical results for a selection of examples in one, two, and three dimensions are presented.

A Mathematical Framework for Modelling the Metastatic Spread of Cancer

Cancer is a complex disease that starts with mutations of key genes in one cell or a small group of cells at a primary site in the body. If these cancer cells continue to grow successfully and, at some later stage, invade the surrounding tissue and acquire a vascular network, they can spread to distant secondary sites in the body. This process, known as metastatic spread, is responsible for around 90% of deaths from cancer and is one of the so-called hallmarks of cancer. To shed light on the metastatic process, we present a mathematical modelling framework that captures for the first time the interconnected processes of invasion and metastatic spread of individual cancer cells in a spatially explicit manner—a multigrid, hybrid, individual-based approach. This framework accounts for the spatiotemporal evolution of mesenchymal- and epithelial-like cancer cells, membrane-type-1 matrix metalloproteinase (MT1-MMP) and the diffusible matrix metalloproteinase-2 (MMP-2), and for their interactions with the extracellular matrix. Using computational simulations, we demonstrate that our model captures all the key steps of the invasion-metastasis cascade, i.e. invasion by both heterogeneous cancer cell clusters and by single mesenchymal-like cancer cells; intravasation of these clusters and single cells both via active mechanisms mediated by matrix-degrading enzymes (MDEs) and via passive shedding; circulation of cancer cell clusters and single cancer cells in the vasculature with the associated risk of cell death and disaggregation of clusters; extravasation of clusters and single cells; and metastatic growth at distant secondary sites in the body. By faithfully reproducing experimental results, our simulations support the evidence-based hypothesis that the membrane-bound MT1-MMP is the main driver of invasive spread rather than diffusible MDEs such as MMP-2.

Two-dimensional simulations and experiments of homogenization and Kirkendall effect in single-phase diffusion triples

A fully discrete local discontinuous Galerkin (LDG) method has been successfully adopted to simulate two-dimensional (2D) homogenization in single-phase diffusion triples controlled by diffusion process. Two Co-Ni and Co-Fe-Ni diffusion triples were prepared and analyzed to verify the simulation results at 1473 K. In areas far away from the triple conjunction of a diffusion triple, there exists composition gradient in only one direction, which can be efficiently predicted by one-dimensional (1D) sharp-interface diffusion simulations (for example, DICTRA: DIffusion-Controlled TRAnsformation simulation software). It has been extensively adopted in the study of diffusional phase transformations readily supported by multi-component multi-phase thermodynamics and full diffusion matrix establishing an integrated thermodynamics-kinetics simulation platform for engineering materials and industrial applications. However, there are composition gradients in two independent directions in the vicinity of triple conjunction (i.e. triple diffusion zone), which cannot be handled on this platform. In the present work, the LDG method combining with alloy thermodynamics and diffusion kinetics was demonstrated to tackle this 2D homogenization problem very well and the effectiveness of this method has been proved from the aspect of numerical analysis by realizing it with a standalone C++ code. In addition to the composition and flux evolutions during 2D homogenization processes, one can reasonably simulate the shift of the Kirkendall plane in the 2D single-phase diffusion zone. This standalone code serves as a solid basis to further consider 2D diffusional multi-phase transformations assuming sharp interfaces in between.

Diffusion of colloidal rods in corrugated channels.

In many natural and artificial devices diffusive transport takes place in confined geometries with corrugated boundaries. Such boundaries cause both entropic and hydrodynamic effects, which have been studied only for the case of spherical particles. Here we experimentally investigate the diffusion of particles of elongated shape confined in a corrugated quasi-two-dimensional channel. The elongated shape causes complex excluded-volume interactions between particles and channel walls which reduce the accessible configuration space and lead to novel entropic free-energy effects. The extra rotational degree of freedom also gives rise to a complex diffusivity matrix that depends on both the particle location and its orientation. We further show how to extend the standard Fick-Jacobs theory to incorporate combined hydrodynamic and entropic effects, so as, for instance, to accurately predict experimentally measured mean first passage times along the channel. Our approach can be used as a generic method to describe translational diffusion of anisotropic particles in corrugated channels.