Mass Transport Problem Research Articles

Robust and versatile ionic liquid microarrays achieved by microcontact printing.

Lab-on-a-chip and miniaturized systems have gained significant popularity motivated by marked differences in material performance at the micro-to-nano-scale realm. However, to fully exploit micro-to-nano-scale chemistry, solvent volatility and lack of reproducibility need to be overcome. Here, we combine the non-volatile and versatile nature of ionic liquids with microcontact printing in an attempt to establish a facile protocol for high throughput fabrication of open microreactors and microfluidics. The micropatterned ionic liquid droplets have been demonstrated as electrochemical cells and reactors for microfabrication of metals and charge transfer complexes, substrates for immobilization of proteins and as membrane-free high-performance amperometric gas sensor arrays. The results suggest that miniaturized ionic liquid systems can be used to solve the problems of solvent volatility and slow mass transport in viscous ionic liquids in lab-on-a-chip devices, thus providing a versatile platform for a diverse number of applications.

Unconditionally stable numerical scheme for natural convection problems

Purpose – Both the importance of the natural convection in science and engineering and the shortage of publications in the field of numerical features of time-stepping schemes for the simulation of coupled heat and fluid flow problems motivate the present work. The paper aims to discuss these issues. Design/methodology/approach – The paper presents the unconditionally stable time-stepping scheme for simulation of coupled problems of mass and heat transport. The paper is divided into two parts. The first part concerns the mathematical formulation of the scheme and discusses its implementation. The second part focuses on the numerical simulation and its results. A detailed investigation of the temporal order of the scheme with respect to the L2-norms of the errors of the pressure, velocity, temperature and divergence of velocity fields has also been given. Findings – The work shows that it is possible to formulate a numerical scheme which is unconditionally stable with respect to the time step size. Moreover, application of the spectral element method for the spatial discretization results in a high order of approximation in space and very good overall accuracy. Furthermore, the investigation of the numerical features of the scheme showed that the formal temporal order of the scheme (formally second order) has been deferred very slightly and the order of 1.8-1.9 is achieved for all unknown fields. Originality/value – The paper presents a new unconditionally stable scheme for simulation of unsteady flows with bidirectional coupling of heat transfer and the fluid flow. It also carefully investigates the numerical behaviour of the method.

Mass transport problems obtained as limits of p-Laplacian type problems with spatial dependence

Abstract. We consider the following problem: given a bounded convex domain Ω ⊂ ℝ N ${\Omega \subset \mathbb {R}^N}$ we consider the limit as p → ∞ of solutions to - div ( b p - p | D u | p - 2 D u ) = f + - f - ${- \operatorname{div} (b_{p}^{-p} |Du|^{p-2} Du)=f_+ - f_-}$ in Ω and b p - p | D u | p - 2 ∂ u ∂ η = 0 ${ b_{p}^{-p} |Du|^{p-2} \frac{\partial u}{\partial \eta }=0}$ on ∂ Ω ${\partial \Omega }$ . Under appropriate assumptions on the coefficients bp that in particular verify that lim p → ∞ b p = b ${ \lim _{p\rightarrow \infty } b_p = b }$ uniformly in Ω ¯ ${\overline{\Omega }}$ , we prove that there is a uniform limit of u p j ${u_{p_j}}$ (along a sequence p j → ∞ ${p_j \rightarrow \infty }$ ) and that this limit is a Kantorovich potential for the optimal mass transport problem of f + ${f_+}$ to f - ${f_-}$ with cost c ( x , y ) ${c(x,y)}$ given by the formula c ( x , y ) = inf σ ( 0 ) = x , σ ( 1 ) = y ∫ σ b d s ${c(x,y) = \inf _{\sigma (0)=x,\,\sigma (1)=y} \int _\sigma b\, ds}$ .

An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary

In this paper we analyze a mass transportation problem in a bounded domain in which there is the possibility of import/export mass across the boundary paying a tax in addition to the transport cost that is assumed to be given by the Euclidean distance. We show a general duality argument and for the dual problem we find a Kantorovich potential as the limit as p\to \infty of solutions to p -Laplacian type problems with nonlinear boundary conditions. In addition, we show that this limit encodes all the relevant information for our problem. It provides the masses that are exported and imported from the boundary and also allows the construction of an optimal transport plan. Finally we show that the arguments can be adapted to deal with the case in which the mass that can be exported/imported is bounded by prescribed functions.

Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2013 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments: finite and infinite-dimensional manifolds, metric spaces, fractals and graphs, including random environments. The workshop brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas. Unifying themes were heat kernel analysis, mass transport problems and related functional inequalities such as Poincar´e, Sobolev, logarithmic Sobolev, Bakry-Emery, Otto-Villani and Talagrand inequalities. These concepts were at the heart of Perelman’s proof of Poincar´e’s conjecture, as well as of the development of the Otto calculus, and the synthetic Ricci bounds of Lott-Sturm-Villani. The workshop provided participants with an opportunity to discuss how these techniques can be used to approach problems in optimal transport for non-local operators, subelliptic operators in finite and infinite dimensions, analysis on singular spaces, as well as random walks in random media.

Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles

In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of −g1(x) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at y. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p→∞ to a double obstacle problem (with obstacles g1, g2) for the p-Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved. We also show that this problem can be interpreted as an optimal mass transport problem in which one can make the transport directly (paying a cost given by the Euclidean distance) or may hire a courier that cost g2(y)−g1(x) to pick up a unit of mass at y and deliver it to x. For this different interpretation we provide examples and a decomposition of the optimal transport plan that shows when we have to use the courier.

An optimal transportation problem related to the limits of solutions of local and nonlocal $$p$$ p -Laplace-type problems

We investigate the asymptotic behavior of minimizers of problems related to the \(p\)-Laplace equation \(-\Delta _p u=f\). As \(p\rightarrow \infty \), the minimizers converge (up to a subsequence) to a function, which maximizes the functional \(I(u)= \int fu\) with appropriate constraints. The main result of this paper is that the problem of maximizing \(I\) with Dirichlet boundary condition can be identified as a dual problem of a certain mass transportation problem. Our approach applies to the limits of both \(p\)-Laplace type problems in classical Sobolev spaces and analogous nonlocal problems in fractional Sobolev spaces.

Online health monitoring of a fuel cell using total harmonic distortion analysis

The present work aims to show the nonlinear behavior of a PEM fuel cell under different mass transport conditions. Understanding this behavior helps in online state-of-health monitoring and control of a fuel cell stack. To analyze the health of an operating stack, a total harmonic distortion analysis (THDA) system requires only the sum of voltages or currents of the stack to be monitored. A low-frequency current or voltage signal is impressed on the fuel cell stack, and the resulting voltage or current signal is measured. To determine any change in harmonics, the measured signal is processed with a harmonic analyzer. The operational states of individual cells of the fuel cell stack may be inferred from at least one change in the harmonic content of the impressed signal. The mass transport problem related to the cathode and anode is distinguished using mixed-frequency signals. The present study found experimentally that hydrogen starvation is dominantly observed in the harmonic analysis only below a frequency of 15 Hz, whereas air starvation showed harmonic changes at frequencies below 100 Hz. Total harmonic distortions were observed to rise to 2–2.5% under both the starvation conditions but with different frequency signals.

Electrocatalytic activity of ZnS nanoparticles in direct ethanol fuel cells

Low temperature fuel cells consuming ethanol without reformation would be a major step toward the use of renewable energy sources from biomass. However, the necessary electrodes and electrocatalysts still are far from being perfect and suffer from various poisoning and deactivation processes. This work describes investigations on systems using carbon/ZnS-based electrocatalysts for ethanol oxidation in complete membrane electrode assemblies (MEAs). MEAs were built on Nafion membranes with active masses prepared from ZnS nanoparticles and Vulcan carbon support. Under operation, acetic acid and acetaldehyde were identified and quantified as soluble oxidation products, whereas the amount of CO2 generated could not be quantified directly. Overall conversion efficiencies of up to 25% were estimated from cells operated over prolonged time. From polarization curves, interrupt experiments and analysis of reaction products, mass transport problems (concentration polarization) and breakthrough losses were found to be the main deficiencies of the ethanol oxidation electrodes fabricated so far.

Study on the fabrication of gold electrode by laser assembling

We proposed the fabrication of gold micro-electrode and grating electrode through laser assembling of gold nanoparticles and realized the electrical interconnection of the single carbon nanotube and gold nanolines, which can decrease the damage of the functional unit to a great extent. This method can also solve the problem of inadequate mass transport in the fabrication of ions. The microstructure could keep unoxidized in the atomasphere with excellent continuity, integrity, and electrical properties, which made this technique have wide application prospects.

Area-Preservation Mapping using Optimal Mass Transport

We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n2) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newton's method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining conformal mapping and optimal mass transport mapping, serves as a powerful tool for a broad range of applications in visualization and graphics, especially for medical imaging. We provide a variety of experimental results to demonstrate the efficiency, robustness and efficacy of our novel framework.

As-Rigid-AsPossible Distance Field Metamorphosis

Widely used for morphing between objects with arbitrary topology, distance field interpolation (DFI) handles topological transition naturally without the need for correspondence or remeshing, unlike surface-based interpolation approaches. However, lack of correspondence in DFI also leads to ineffective control over the morphing process. In particular, unless the user specifies a dense set of landmarks, it is not even possible to measure the distortion of intermediate shapes during interpolation, let alone control it. To remedy such issues, we introduce an approach for establishing correspondence between the interior of two arbitrary objects, formulated as an optimal mass transport problem with a sparse set of landmarks. This correspondence enables us to compute non-rigid warping functions that better align the source and target objects as well as to incorporate local rigidity constraints to perform as-rigid-aspossible DFI. We demonstrate how our approach helps achieve flexible morphing results with a small number of landmarks.

On the application of SUPG/θ-method in 2D advection–diffusion-reaction simulation

The SUPG/θ-method for applications in 2D advection–diffusion transport simulation is reviewed and it is presented in details. Spatial discretization is done with finite elements method with stabilized streamline upwind Petrov–Galerkin method, and time discretization was done with θ-stable finite difference operator. Numerical tests showed that this algorithm is general and robust and it can be applied to simulate simple problems of mass transport; for example, pure advection phenomenon in structured mesh, as well as reactive advection–diffusion transport with an unstructured mesh, and not well-behaved velocity field. The method was applied in a simulation of the transport of total phosphorus in the Lake Agua Preta, and the results were validated by comparison with experimental data. The computational cost is directly associated with the value of the safety factor, introduced to assure the stability of the method.

A mathematical model of the single freeform surface design for collimated beam shaping

Incoherent collimated beam has a wide application, and reshaping the collimated beam with freeform optics has become a popular and challenging topic of noniamging design. In this paper, we address this issue, embedded in three-dimensional space without any symmetry, with a freeform surface from a new perspective. A mathematical model is established for achieving the one-freeform surface design based on the problem of optimal mass transport. A numerical technique for solving this design model is disclosed for the first time, and boundary conditions for balancing light are presented. Besides, some key issues in achieving complex illuminations are addressed, and the influence of caustic surface on this design model is also discussed. Design examples are given to verify these theories. The results show elegance of the design model in tackling complex illumination tasks. The conclusions obtained in this paper can be generalized to achieve LED illumination and tackle multiple freeform surfaces illumination design.

Re-examination of steady-state concentration profile near a uniformly accessible rotating disk electrode

In this article we re-examine the steady-state concentration profile of electroactive-dissolved species near a uniformly accessible rotating disk electrode (RDE), using both symbolic and numerical computation procedures. First, the one-dimensional diffusion–convection equation pertaining to RDE is solved numerically in order to get highly-accurate benchmark data. Next, the analytical solution of mass-transport problem is derived under the assumption of infinite Schmidt number (Sc). Finally, series expansion of concentration profile with respect to Sc−1/3 is performed at finite values of Schmidt number. Both analytical and series solutions are compared to the numerical data. Series expansion of the limiting current density pertaining to RDE follows directly from the above derivations.

Improvement of nonlinear response for the power conversion efficiency with light intensities in cobalt complex electrolyte system

A vertically aligned TiO2 photoelectrode with sufficient space between TiO2 nanorods was synthesized by sacrificial templating of ZnO nanorods grown on a dense TiO2 layer/FTO film. The more expanded pore space in the TiO2 nanorod film than conventional mesoscopic TiO2 film greatly reduces the mass transport problem of cobalt electrolyte in the vicinity of the Sb2S3 sensitizer in Sb2S3-sensitized photoelectrochemical solar cells. Therefore, the nonlinearity of the power conversion efficiency (PCE) under illumination intensity was greatly relieved to 5.0% (0.1sun), 4.7% (0.5sun), and 3.7% (1sun) in a TiO2-nanorod-based Sb2S3-sensitized solar cell, whereas the efficiency of a mesoscopic-TiO2-based solar cell was significantly degraded to 5.3% (0.1 sun), 3.8% (0.5sun), and 2.3% (1sun).

Comparing dynamical systems by a graph matching method

In this paper, we consider comparing dynamical systems by using a method of graph matching, either between the graphs representing the underlying symbolic dynamics, or between the graphs approximating the action of the systems on a fine but otherwise non-generating partition. For conjugate systems, the graphs are isomorphic and we show that the permutation matrices that relate the adjacency matrices coincide with the solution of Monge’s mass transport problem. We use the underlying earth mover’s distance (EMD) to generate the “approximate” matching matrix to illustrate the association of graphs which are derived from equal-distance partitioning of the phase spaces of systems. In addition, for one system which embeds into the other, we show that the comparison of these two systems by our method is an issue of subgraph matching.

An Explicit Martingale Version of Brenier's Theorem

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in cite{BeiglbockHenry-LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

Maximal Coupling of Euclidean Brownian Motions

We prove that the mirror coupling is the unique maximal Markovian coupling of two Euclidean Brownian motions starting from single points and discuss the connection between the uniqueness of maximal Markovian coupling of Brownian motions and certain mass transportation problems.

Freeform illumination design: a nonlinear boundary problem for the elliptic Monge–Ampére equation

We propose an approach to deal with the problem of freeform surface illumination design without assuming any symmetry based on the concept that this problem is similar to the problem of optimal mass transport. With this approach, the freeform design is converted into a nonlinear boundary problem for the elliptic Monge-Ampére equation. The theory and numerical method are given for solving this boundary problem. Experimental results show the feasibility of this approach in tackling this freeform design problem.