Application of the finite-element method to the diffusion and reaction of chemical species in multilayered polymeric bodies

Application of the finite-element method (FEM) to chemical species diffusion and reaction in polymers by Fickian mass transport is described. The method is developed by analogy to heat conduction and is extended to include multiple, reactive chemical species dissolved in multilayered polymeric materials. Because of the analogy to conductive heat transfer, existing FEM thermal codes can be readily adapted to solve chemical diffusion problems. The method described is limited to Fickian diffusion at a constant material temperature.     

Finite-element method simulation of rotating disk flow: effect of the transport of a chemical species

Electrochemical cells containing an iron rotating disk electrode which is dissolved in the electrolyte, an 1 M H₂SO₄ solution present a current instability in the plateau region, where the current is controlled by the mass transport. Dissolution of the electrode gives rise to a thin concentration boundary layer, due to a Schmidt number Sc = 2000. This boundary layer, together with the potential applied to the electrode, leads to an increase in the fluid viscosity and in a decrease in the diffusion coefficient, coupling the concentration and the chemical species field. Since the current is proportional to the concentration gradient at the interface, an instability of the coupled fields at Reynolds numbers attained in experimental conditions could be responsible for the current instability. Mangiavacchi [1] performed a linear stability analysis of the problem and showed that this is indeed the case. In this paper we review the main results of the stability analysis and present the main features of the FEM code recently developed in our group, to proceed with the investigation of the current instability observed in electrochemical cells. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pointwise superconvergence of the streamline diffusion finite-element method

In this article, we analyze the local superconvergence property of the streamline-diffusion finite-element method (SDFEM) for scalar convection-diffusion problems with dominant convection. By orienting the mesh in the streamline direction and imposing a uniformity condition on the mesh, the theoretical order of pointwise convergence is increased from O(h^11/8|log h|) to O(h²|log h|). Numerical tests show that this result cannot be extended to arbitrary quasi-uniform meshes. © 1996 John Wiley & Sons, Inc.     

Advanced continuum modelling of gas-particle flows beyond the hydrodynamic limit

The accurate prediction of dilute gas-particle flows using Euler–Euler models is challenging because particle–particle collisions are usually not dominant in such flows. In other words, in dilute flows the particle Knudsen number is not small enough to justify a Chapman–Enskog expansion about the collision-dominated near-equilibrium limit. Moreover, due to the fluid drag and inelastic collisions, the granular temperature in gas-particle flows is often small compared to the mean particle kinetic energy, implying that the particle-phase Mach number can be very large. In analogy to rarefied gas flows, it is thus not surprising that two-fluid models fail for gas-particle flows with moderate Knudsen and Mach numbers. In this work, a third-order quadrature-based moment method, valid for arbitrary Knudsen number, coupled with a fluid solver has been applied to simulate dilute gas-particle flow in a vertical channel with particle-phase volume fractions between 0.0001 and 0.01. In order to isolate the instabilities that arise due to fluid-particle coupling, a fluid mass flow rate that ensures that turbulence would not develop in a single phase flow (Re = 1380) is employed. Results are compared with the predictions of a two-fluid model with standard kinetic theory based closures for the particle phase. The effect of the particle-phase volume fraction on flow instabilities leading to particle segregation is investigated, and differences with respect to the two-fluid model predictions are examined. The influence of the discretization on the solution of both models is investigated using three different grid resolutions. Radial profiles of phase velocities and particle concentration are shown for the case with an average particle volume fraction of 0.01, showing the flow is in the core-annular regime.     

Application of the Courant-Isaacson-Rees method to solve the shallow-water hydrodynamic equations

A finite-difference scheme is presented to solve the shallow-water hydrodynamic equations which describe the behavior of natural water bodies under the influence of wind stress, atmospheric pressure gradients, and tides. The numerical technique, which is an extension of the two-dimensional method of Courant, Isaacson, and Rees (1952), proved to be accurate and computationally stable for two-dimensional oceanographic investigations. Numerical simulations are given for wind-driven circulation in the Ligurian Sea.     

Application of the finite-element method to improve the quasi-exact reinforcement theory of fibrous polymeric composites

In the paper, the WL quasi-exact reinforcement theory of fibrous polymeric composites is improved. An optimum compatibility condition related to the transverse shear problem for a unit cell, which brings solutions closest to reality, is derived. This condition is formulated in the form of a linear combination of maximum radial and circumferential displacements. Optimum coefficients of this combination are determined by comparing analytical and numerical solutions for a test specimen in the form of a rectangular thin plate, which is in a plane strain state and is subject to selected loading schemes. The analytic solutions are obtained for a homogenized material by using the WL reinforcement theory. The numerical solutions are found for an actual heterogeneous composite material by using the finite-element method, and they verify the WL reinforcement theory, in particular, the admissibility of Hills assumption. An analysis performed for two composite materials shows that the improved WL reinforcement theory gives adequate displacement fields.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 79–92, January–Febrauary, 2005.     

Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method

In this work, accurate solutions to linear and nonlinear diffusion equations were introduced. A combination of a sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time have been used for treatment of these equations. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. Here, the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation. The present method is seen to be a very good alternative method to some existing techniques for realistic problems.     

Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method

In this paper, by introducing the fractional derivative in the sense of Caputo, the generalized two-dimensional differential transform method (DTM) is directly applied to solve the coupled Burgers equations with space- and time-fractional derivatives. The presented method is a numerical method based on the generalized Taylor series formula which constructs an analytical solution in the form of a polynomial. Several illustrative examples are given to demonstrate the effectiveness of the generalized two-dimensional DTM for the equations.     

Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

A numerical method is developed for simulation of stochastic chemical reactions. The system is modeled by the Fokker–Planck equation for the probability density of the molecular state. The dimension of the domain of the equation is reduced by assuming that most of the molecular species have a normal distribution with a small variance. The numerical approximation preserves properties of the analytical solution such as non-negativity and constant total probability. The method is applied to a nine dimensional problem modelling an oscillating molecular clock. The oscillations stop at a fixed point with a macroscopic model but they continue with our two dimensional, mixed macroscopic and mesoscopic model. Dedicated to the memory of Germund Dahlquist (1925–2005). AMS subject classification (2000)  65M20, 65M60     

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.     

Nonconformal schemes of the finite-element method for nonlinear hyperbolic conservation laws

We suggest a method for constructing grid schemes for initial-boundary value problems for many-dimensional nonlinear systems of first-order equations of hyperbolic type on the basis of the Galerkin-Petrov limit approximation to the mixed statement of an original problem. Our grid schemes are versions of the nonconformal finite-element method in which the approximate solution is constructed in the space of piecewise polynomial functions that admit discontinuities on the boundary of triangulation elements of the design domain.     

Numerical analysis of heat and moisture transport with a finite difference method

In this article, we study a system of nonlinear parabolic partial differential equations arising from the heat and moisture transport through textile materials with phase change. A splitting finite difference method with semi‐implicit Euler scheme in time direction is proposed for solving the system of equations. We prove the existence and uniqueness of a classical positive solution to the parabolic system as well as the existence and uniqueness of a positive solution to the splitting finite difference system. We provide optimal error estimates for the splitting finite difference system under the condition that the mesh size and time step size are smaller than a positive constant which solely depends upon the physical parameters involved. Numerical results are presented to confirm our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Numerical modelling of rarefied gas flow through a slit at arbitrary pressure ratio based on the kinetic equation

A rarefied gas flow through a thin slit at an arbitrary gas pressure ratio is calculated on the basis of the kinetic model equations (BGK and S-model) applying the discrete velocity method. The calculations are carried out for the whole range of the gas rarefaction from the free-molecular regime to the hydrodynamic one. Numerical data on the flow rate and distributions of density, bulk velocity and temperature are reported. Comparisons of the present results with those based on the direct simulation Monte Carlo method and on the linearized BGK kinetic equation are performed. The conditions of applicability of the linearized theory are discussed.