Mass Transport Problem Research Articles

Numerical Simulation of Stefan Problem Coupled with Mass Transport in a Binary System Through XFEM/Level Set Method

This paper deals with the application of the extended finite element method to simulate the phase change phenomenon in a binary system while considering the interaction with the mass transport of chemical species. To this end, the thermal conduction equation with the Stefan condition and the mass diffusion equation are solved to depict the temperature and solute concentration distributions, respectively. In dealing with the heat transfer problem with phase change, the temperature field is weakly enriched using the corrected abs-enrichment scheme to avoid the blending element problem. The melting temperature imposed by the penalty method at the solid–liquid interface is solute concentration dependent. On the other hand in dealing with the mass transport problem, due to the strong discontinuity in the concentration field at the interface, the sign-enrichment scheme is used. A constant separating out concentration is enforced on the solid side of the interface also by the penalty method, while a mass flux is naturally applied on the liquid side. The phase interface is captured implicitly by the level set method, which is applied on the same fixed finite element mesh. The robustness and the accuracy of the model are demonstrated through numerical case studies.

Lattice Boltzmann solution of advection-dominated mass transport problem: a comparison

This article addresses the abilities and limitations of the Lattice Boltzmann (LB) method in solving advection-dominated mass transport problems. Several schemes of the LB method, including D2Q4, D2Q5, and D2Q9, were assessed in the simulation of two-dimensional advection-dispersion equations. The concept of Single Relaxation Time (SRT) and Multiple Relaxation Time (MRT) in addition to linear and quadratic Equilibrium Distribution Functions (EDF) were taken into account. The results of LB models were compared to the well-known Finite Difference (FD) solutions, including Explicit Finite Difference (EFD) and Crank-Nicolson (CN) methods. All LB models are more accurate than the aforementioned FD schemes. The results also indicate the high potency of D2Q5 SRT and D2Q9 SRT in describing advection-controlled mass transfer problems. The numerical artificial oscillations are observed when the Grid Peclet Number (GPN) is greater than 10, 25, 20, 25, and 10 regarding D2Q4 SRT, D2Q5 SRT, D2Q5 MRT, D2Q9 SRT and D2Q9 MRT, respectively, while the corresponding GPN values obtained for the EFD and CN methods were 2 and 5, respectively. Finally, a coupled system of groundwater and solute transport equations were solved satisfactorily using several LB models. Considering computational time, all LB models are much faster than CN method.

Numerical and analytical approaches to an advection-diffusion problem at small Reynolds number and large Péclet number

Obtaining a detailed understanding of the physical interactions between a cell and its environment often requires information about the flow of fluid surrounding the cell. Cells must be able to effectively absorb and discard material in order to survive. Strategies for nutrient acquisition and toxin disposal, which have been evolutionarily selected for their efficacy, should reflect knowledge of the physics underlying this mass transport problem. Motivated by these considerations, in this paper we discuss the results from an undergraduate research project on the advection-diffusion equation at small Reynolds number and large Péclet number. In particular, we consider the problem of mass transport for a Stokesian spherical swimmer. We approach the problem numerically and analytically through a rescaling of the concentration boundary layer. A biophysically motivated first-passage problem for the absorption of material by the swimming cell demonstrates quantitative agreement between the numerical and analytical approaches. We conclude by discussing the connections between our results and the design of smart toxin disposal systems.

Solving Mass Transportation Problems Using Liouville Equations

Solving Mass Transportation Problems Using Liouville Equations

Modeling of non‐Fickian diffusion and dissolution from a thin polymeric coating: An application to drug‐eluting stents

SummaryIn this paper, we present a general model for non‐Fickian diffusion and drug dissolution from a controlled drug delivery device coated with a thin polymeric layer. First, we study the stability and deduce an analytic solution to the problem. Then, we consider this solution and provide suitable boundary conditions to replace the problem of mass transport in the coating of a coronary drug‐eluting stent. With this approach, we reduced the computational cost of performing numerical simulations in complex 3‐dimensional geometries. The model for mass transport by a coronary drug‐eluting stent is coupled with a non‐Newtonian blood model flow. In order to show the effectiveness of the method, numerical experiments and a model validation with experimental data are also included. In particular, we investigate the influence of the non‐Newtonian flow regime on the drug deposition in the arterial wall.

Optimal partial mass transportation and obstacle Monge–Kantorovich equation

Optimal partial mass transport, which is a variant of the optimal transport problem, consists in transporting effectively a prescribed amount of mass from a source to a target. The problem was first studied by Caffarelli and McCann (2010) [6] and Figalli (2010) [12] with a particular attention to the quadratic cost. Our aim here is to study the optimal partial mass transport problem with Finsler distance costs including the Monge cost given by the Euclidian distance. Our approach is different and our results do not follow from previous works. Among our results, we introduce a PDE of Monge–Kantorovich type with a double obstacle to characterize active submeasures, Kantorovich potential and optimal flow for the optimal partial transport problem. This new PDE enables us to study the uniqueness and monotonicity results for the active submeasures. Another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper [14] (Igbida and Nguyen, 2018).

Model-supported characterization of a PEM water electrolysis cell for the effect of compression

This paper investigates the influence of the cell compression of a PEM water electrolysis cell. A small single cell is therefore electrochemically analyzed by means of polarization behavior and impedance spectroscopy throughout a range of currents (0.01 A cm−2 to 2.0 A cm−2) at two temperatures (60°C and 80°C) and eight compressions (0.77 MPa–3.45 MPa). Additionally, a computational model is utilized to support the analysis. The main findings are that cell compression has a positive effect on overall cell performance due to decreased contact resistances, but is subject to optimization. In this case, no signs of severe mass transport problems due to crushed transport layers are visible in either polarization curves or impedance plots, even at high currents. However, a Tafel plot analysis revealed more than one slope throughout the current range. The change in the Tafel slope is therefore discussed and connected to the electrochemical reaction or an ohmic contribution from a non-electrode component.

Variational interpretation of the eikonal calculation problem from the condition of generating a prescribed irradiance distribution

The problem of calculation of the light field eikonal defined on a certain surface from the condition of generating a prescribed irradiance distribution on another surface is formulated as a Monge-Kantorovich mass transportation problem. We show that the cost function in this mass transportation problem corresponds to the distance between a point on the initial surface (on which the eikonal function is defined) and a point on the target surface, where the prescribed irradiance distribution is to be generated. An analytical expression for the gradient of a “cost functional” describing the mass transportation problem is derived. It enables using gradient descent methods for the calculation of the eikonal function.

The connection between the phase problem in optics, focusing of radiation, and the Monge–Kantorovich problem

We discuss the use of variational principles for solving the phase problem in optics. In this paper, we consider the connection between four fundamental problems: the phase problem in optics, the inverse problem of focusing coherent radiation, the Monge – Kantorovich optimal mass transport problem, and the variational methods for solving the equation of a modified Monge – Ampere equation. It is shown that the solution of the phase problem in optics within the framework of the asymptotic approach is closely related to the solution of the problem of optimal mass transport with a nonquadratic cost function.

Optimal mass transportation problem and freeform optics design – the identity of optimization scheme and the numerical solution method

Optimal mass transportation problem and freeform optics design – the identity of optimization scheme and the numerical solution method

Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2016 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments including smooth manifolds, fractals and graphs, metric spaces and in random environments. The workshop brought together leading researchers from analysis, geometry and probability, and provided an excellent opportunity for interactions between scientists from these areas at different stages of their career. The unifying themes were heat kernel analysis, mass transportation problems and functional inequalities while the program straddled across a great variety of subjects and across the divide that exists between discrete and continuous mathematics. Other unifying concepts such as the notions of metric measure space, Otto Calculus and Lott-Sturm-Villani synthetic Ricci curvature bounds played an important part in the discussions. Novel directions including the study of Liouville quantum gravity were included. The workshop provided participants with an opportunity to discuss how these ideas and techniques can be used to approach problems regarding optimal transport, Riemannian and sub-Riemannian geometry, and analysis and stochastic processes in random media.

Prevascularization of natural nanofibrous extracellular matrix for engineering completely biological three-dimensional prevascularized tissues for diverse applications.

Self-sustainability after implantation is one of the critical obstacles facing large engineered tissues. A preformed functional vascular network provides an effective solution for solving the mass transportation problem. With the support of mural cells, endothelial cells (ECs) can form microvessels within engineered tissues. As an important mural cell, human mesenchymal stem cells (hMSCs) not only stabilize the engineered microvessel network, but also preserve their multi-potency when grown under optimal culture conditions. A prevascularized hMSC/extracellular matrix (ECM) sheet fabricated by the combination of hMSCs, ECs and a naturally derived nanofibrous ECM scaffold offers great opportunity for engineering mechanically strong and completely biological three-dimensional prevascularized tissues. The objective of this study was to create a prevascularized hMSC/ECM sheet by co-culturing ECs and hMSCs on a nanofibrous ECM scaffold. Physiologically low oxygen (2% O2 ) was introduced during the 7day hMSC culture to preserve the stemness of hMSCs and thereby their capability to secrete angiogenic factors. The ECs were then included to form microvessels under normal oxygen (20% O2 ) for up to 7days. The results showed that a branched and mature vascular network was formed in the co-culture condition. Angiogenic factors vascular endothelial growth factor (VEGF), basic fibroblast growth factor (bFGF) and angiopoietin-1 (Ang-1) were significantly increased by low-oxygen culture of hMSCs, which further stabilized and supported the maturation of microvessels. A differentiation assay of the prevascularized ECM scaffold demonstrated a retained hMSC multi-potency in the hypoxia cultured samples. The prevascularized hMSC/ECM sheet holds great promise for engineering three-dimensional prevascularized tissues for diverse applications.

Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport

In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semi-definiteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise, and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes.

Kantorovich potentials and continuity of total cost for relativistic cost functions

In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is ct(x,y)=h(y−xt), h being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every t larger than some critical time T>0, existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time t has been only proved to be a decreasing right-continuous function of t. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined “chain lemma” and the result that, for t>T, the points moving at maximal distance are negligible for the optimal plan.

Variational approach to calculation of light field eikonal function for illuminating a prescribed region.

The problem of calculation of the light field eikonal function providing focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is obtained that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the source region (in which the eikonal function is defined) and a point of the target region. This result demonstrates that the sought-for eikonal function corresponds to a mapping, for which the total distance between the points of the original plane and the target region is minimized. The formalism proposed in the present work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. Besides, the calculation of the "ray mapping" corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed approach is illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangle and a beam of square section into a ring.

Portrait Lighting Transfer Using a Mass Transport Approach

Lighting is a critical element of portrait photography. However, good lighting design typically requires complex equipment and significant time and expertise. Our work simplifies this task using a relighting technique that transfers the desired illumination of one portrait onto another. The novelty in our approach to this challenging problem is our formulation of relighting as a mass transport problem. We start from standard color histogram matching that only captures the overall tone of the illumination, and we show how to use the mass-transport formulation to make it dependent on facial geometry. We fit a three-dimensional (3D) morphable face model to the portrait, and for each pixel, we combine the color value with the corresponding 3D position and normal. We then solve a mass-transport problem in this augmented space to generate a color remapping that achieves localized, geometry-aware relighting. Our technique is robust to variations in facial appearance and small errors in face reconstruction. As we demonstrate, this allows our technique to handle a variety of portraits and illumination conditions, including scenarios that are challenging for previous methods.

FE design for the numerical modelling of failure induced by differential straining in meso-scale concrete: Algorithmic implementation based on operator split method

The present paper focuses on the design of FE for the numerical modelling of failure induced by differential straining in concrete at the meso-scale. The differential straining is here considered as a deformation field added to the mechanical strain field and coming from the solution of a mass transport problem for instance. This differential straining can be induced by changes in degree of saturation, drying shrinkage or fluid overpressure for example. The meso-structure is based on a two-phase 3D representation of concrete, where stiff aggregates are embedded into a mortar matrix. In order to explicitly take into account these aggregates without any mesh adaptation, a weak discontinuity is introduced into the strain field. In addition, a strong discontinuity is also added to take into account cracking. The FE implementation is performed within the Enhanced Finite Element Method (E-FEM). An operator split method equipped with a return mapping algorithm accounting for differential straining is implemented for the computation of the discontinuities values, treated as local variables. In regards to the operator split method proposed in this paper, a broad range of problems can be solved such as basic mass transport problems or failure problem induced both by mechanical loading and differential straining. This feature is an advantage for the design of FE for durability issues. Finally 1D simulations are presented to assess the FE implementation and the performance of the numerical method. We also demonstrate the capability of the method for solving and analyzing 3D mesoscale problems.

Portrait Lighting Transfer Using a Mass Transport Approach

Lighting is a critical element of portrait photography. However, good lighting design typically requires complex equipment and significant time and expertise. Our work simplifies this task using a relighting technique that transfers the desired illumination of one portrait onto another. The novelty in our approach to this challenging problem is our formulation of relighting as a mass transport problem. We start from standard color histogram matching that only captures the overall tone of the illumination, and we show how to use the mass-transport formulation to make it dependent on facial geometry. We fit a three-dimensional (3D) morphable face model to the portrait, and for each pixel, we combine the color value with the corresponding 3D position and normal. We then solve a mass-transport problem in this augmented space to generate a color remapping that achieves localized, geometry-aware relighting. Our technique is robust to variations in facial appearance and small errors in face reconstruction. As we demonstrate, this allows our technique to handle a variety of portraits and illumination conditions, including scenarios that are challenging for previous methods.

Volume Averaged Equations for Mass Transport and Reaction for In-Situ Combustion

Abstract In-situ combustion (ISC) is an oil recovery technique where many phenomena can take place simultaneously such as: chemical reactions, phase change, heat transfer, mass transport, thermodynamic equilibrium, and so on. Each one of these phenomena may have important contributions over the ISC behavior at any scale of interest as lab-scale, inter-wells or reservoir-scale. In this work, a mass transport study is presented. Firstly, the appropriate phase and interface governing equations at pore-scale are set up. Later, the volume averaged equations valid at macroscale are rigorously derived using the volume averaging method (VAM). The theoretical analysis is general and applies for typical oil-water-gas-rock systems found in petroleum reservoirs, and for any number of chemical species distributed in the phases. The model also allows the existence of several heterogeneous and homogeneous chemical reactions. From this general point of view, the volume averaged equations governing species and phase mass transport at macroscale, along its closure scheme to predict the effective transport parameters, are presented. We have clearly identified the length scale constraints and assumptions that support our derivations. In future works, we shall expand the range of applicability of the model by relaxing some of these assumptions. To demonstrate the applicability of the average models, we numerically predicted the longitudinal mass dispersion of oxygen for passive and reactive mass transport problems at lab-scale. The general trends of theoretical results are in concordance with previous works.

Portrait lighting transfer using a mass transport approach

Lighting is a critical element of portrait photography. However, good lighting design typically requires complex equipment and significant time and expertise. Our work simplifies this task using a relighting technique that transfers the desired illumination of one portrait onto another. The novelty in our approach to this challenging problem is our formulation of relighting as a mass transport problem. We start from standard color histogram matching that only captures the overall tone of the illumination, and we show how to use the mass-transport formulation to make it dependent on facial geometry. We fit a three-dimensional (3D) morphable face model to the portrait, and for each pixel, we combine the color value with the corresponding 3D position and normal. We then solve a mass-transport problem in this augmented space to generate a color remapping that achieves localized, geometry-aware relighting. Our technique is robust to variations in facial appearance and small errors in face reconstruction. As we demonstrate, this allows our technique to handle a variety of portraits and illumination conditions, including scenarios that are challenging for previous methods.