Finite Difference Method For Solution Research Articles

The FDM solution of unsteady inverse problem for two‐dimensional oscillating airfoils

PurposeThis paper seeks to develop an approach for the unsteady inverse problem of two‐dimensional oscillating airfoils based on the finite difference method (FDM) solution of the transient Euler equations.Design/methodology/approachThe solution strategies are determined according to the mathematical model for the inverse‐problem of oscillating airfoils. Then the unsteady nonreflecting far field boundary condition and the permeable wall boundary condition are employed to treat the boundary conditions. The applications are carried out for the modification of an oscillating airfoil according to the design targets of the unsteady pressure distribution in an oscillating period.FindingsThe results show that the pressure distributions over the new airfoils coincide with the design objects indicating that the mathematical model and solution strategy developed in this paper is rational and reliable.Research limitations/implicationsThis method is limited to frictionless flow.Originality/valueThe paper provides a new FDM solution of unsteady inverse problem for oscillating airfoils, which can be extended to treat the multipoint problem of airfoil design.

Solving coupled 3-D paraxial wave and thermal diffusion equations with mixed-mode parallel computations

The application of parallel computational methods to the modeling of laser light interactions with a thermally self-induced inhomogeneous medium is described. In these processes, laser light dynamically changes the local refractive index through heat absorption, which in-turn alters the beam profile. The mathematical model couples the 3-D heat diffusion and paraxial wave equations resulting in a system of non-linear equations. An explicit, finite-difference method (FDM) for solving the heat diffusion equation is coupled with an implicit FDM solution of the paraxial wave equation. The reasons for choosing these schemes are presented, along with a mixed-mode parallelization method in which OpenMP is used for the explicit solver, and MPI for the implicit solver. Finally, convergence, stability, accuracy and code performance is presented.

2D parallel and stable group explicit finite difference method for solution of diffusion equation

Recently various versions of alternating group explicit or alternating group explicit–implicit methods were proposed for solution of diffusion equation. The main benefits of these methods are: good stability, accuracy and parallelizing. But these methods were developed for 1D case and stability and accuracy were investigated for 1D case too. In the present study we extend the new group explicit method [R. Tavakoli, P. Davami, New stable group explicit finite difference method for solution of diffusion equation, Appl. Math. Comput. 181 (2006) 1379–1386] to 2D with operator splitting method. The implementation of method is discussed in details. Our numerical experiment shows that such 2D extension is unconditionally stable and it is more accurate that traditional unconditional stable explicit method.

Enhancement of Safety Analysis Capability for a CANDU-6 Reactor Using RELAP-CANDU/SCAN Coupled Code System

The purpose of this study is the development and verification of the coupled code system SCAN and RELAP-CANDU for transient analysis of a Canada deuterium uranium (CANDU) reactor. For this purpose, a spatial kinetics calculation module is developed and implemented in SCAN, a three-dimensional (3-D) CANDU–pressurized heavy water reactor neutronics design and analysis code. Then, a dynamic linked library of the SCAN code is generated for the integration with RELAP-CANDU.The RELAP-CANDU code has been developed for best-estimate transient simulation of CANDU reactor coolant systems based on the RELAP5 code. The SCAN code is a 3-D neutronic calculation code, which is composed of both unified nodal methods based on coarse-mesh finite difference method solutions to the time-dependent two-group diffusion equations.To verify the reliability of the coupled code system RELAP-CANDU/SCAN, the 40% reactor inlet header break accident, the 100% reactor outlet header break accident, and the pump suction pipe break are analyzed. The proposed coupled thermal-hydraulic and neutronic analyses methodology shows that there is an important margin in the traditional accident analysis.

New stable group explicit finite difference method for solution of diffusion equation

A new group explicit method for solution of diffusion equation is presented. This method is based on domain decomposition concept and using asymmetric Saul’yev schemes for internal nodes of each sub-domain and alternating group explicit method for sub-domain’s interfacial nodes. This new method has several advantages such as: good parallelism, unconditional stability, fully explicit nature and better accuracy than original Saul’yev schemes. The details of implementation and proving stability are briefly discussed. Numerical experiments on stability and accuracy are also presented.

Finite-difference methods for solution of nonlocal boundary value problems

The article is devoted to the construction and investigation of various types of difference schemes relevant to nonlocal boundary value problems stated for multidimensional elliptic equations. Suitable tools for performing the investigation are developed, including difference analogue of Green's formula and inequalities corresponding to nonlocal boundary conditions.

Optimum shapes of tire‐treads for avoiding lateral slippage between tires and roads

AbstractOptimum design of tire‐tread sections is an important practical issue. However, useful study of the problem that can suggest a reliable guideline for determining the optimum tread sections had hardly been made in the past. The present paper describes a new analysis of the state of stresses in tire‐tread sections in contact with the road surface, taking special care of the boundary conditions. Based on the analysis, a method is proposed to determine the optimum tread shapes for avoiding lateral slippage between tires and roads. The displacement potential function formulation, an ideal mathematical model for the practical stress problems, has been used in conjunction with finite‐difference method of solution. For the present analysis, lateral slipping in absence of frictional resistance as well as the no‐slip conditions of the tire‐tread contact surface have been considered along with a large number of tread aspect ratios. The present computational approach proves to be a powerful tool for determining the optimum tread shapes for avoiding the lateral slippage of tire‐treads. Copyright © 2005 John Wiley & Sons, Ltd.

Finite Difference Solution Methods for a System of the Nonlinear Schrödinger Equations

This paper investigates finite difference schemes for solving a system of the nonlinear Schrödinger (NLS) equations. Several types of schemes, including explicit, implicit, Hopscotch-type and Crank-Nicholson-type are defined. Cubic spline interpolation is used for solving time-shifting part of equations. The numerical results of the different solution methods are compared using two analytical invariant properties.

Neutral particle collisional relaxation of suprathermal distributions in aeronomy

There are numerous situations in laboratory experiments and in atmospheric science that are characterized by distributions of energetic neutral species with extended high-energy tails. This paper provides a detailed analysis of the relaxation of such isotropic nonequilibrium distributions of neutral species. We consider a minor constituent, referred to as the test-particle dilutely dispersed in a second component that acts as a heat bath at equilibrium. A hard sphere cross section is assumed for the collisions of the test particles and the heat bath particles, and collisions between test particles are not included in the analysis. We study the approach to equilibrium with a finite difference method of solution of the Boltzmann equation. The solution of the Boltzmann equation is also presented as an expansion in the eigenfunctions of the Boltzmann collision operator. The main objective of the paper is the calculation of the energy-dependent relaxation times for the distribution function. It is anticipated that these relaxation times will not exhibit a strong energy dependence owing to the energy-independent hard sphere cross section. This relaxation behaviour is important in the characterization of nonthermal populations of energetic atoms in aeronomy. The results are compared with a similar analysis for Coulomb collisions described in the preceeding paper.

Coulomb collisional processes in space plasmas; relaxation of suprathermal particle distributions

Nonequilibrium distributions of space plasmas are often characterized by extended high-energy tails. This paper provides a detailed analysis of the relaxation of such isotropic nonequilibrium plasmas. We consider an energetic charged species dilutely dispersed in a fully ionized plasma, which acts as a heat bath at equilibrium. The minor constituent is referred to as a “test particle” and collisions between test particles are not included. We study the approach to equilibrium with a finite difference method of solution of the Fokker–Planck equation appropriate for collisions between charged particles. The solution of the Fokker–Planck equation is also presented formally as an expansion in the eigenfunctions of the Fokker–Planck operator. The main objective of the paper is the calculation of the energy-dependent relaxation times of the distribution function. A strong energy dependence for these relaxation times is anticipated since, for Coulomb collisions, the Rutherford cross-section varies with relative speed g as g −4. Analogous results for neutral species are presented for comparison in the following paper.

Examination of the Phenomena Concerning Optical Particle Measurements Using the Direct Finite-Difference Solution of Maxwell's Equation(Influence of Multiple Scattering and Three-Dimensionality of Scatterer on Diffraction)

At first, the accuracy of the scattering patterns predicted for typical examples by a direct finite-difference solution technique was examined by comparing them with the analytical solutions. Comparisons with regard to the scattering patters formed by a single cylinder, a pair of cylinders or a single true sphere showed good agreement. Based on this confirmation of accuracy of the direct finite-difference solution method, the influence of multiple-scattering and three-dimensionality of scatterer on the diffraction was discussed. Even if the volume fraction of particles is same, larger number density of scatterers leads to more remarkable distortion of scattering pattern and subsequent erroneous measurement of particle size distribution. When a cylindrical scatterer is long in the direction of optical axis, its scattering pattern is remarkably different from that formed by a true sphere of the same diameter. That is, the diffraction pattern is affected by the three-dimensionality of scatterer.

On predicting incompressible flows by using a stabilized finite difference method with penalty

This paper presents a finite difference solution method for the incompressible Navier–Stokes equations, in primitive variables form. To provide the necessary coupling between continuity and momentum penalty method is introduced into continuity equation. Differential terms arising in Navier–Stokes equations are discretized by stabilized finite differences of the SU type (streamline upwind) to calculate the convective terms. The use of a penalty function method in a finite difference method of staggered grids, allows to eliminate the pressure and the result is a system of equations of the velocities arises. The aim of this paper is to describe the techniques used in the approximations to calculate each term of the discretized model. In particular, time integration techniques, solution of non-linearities associated to convective terms and constitutive laws, stabilization procedures and discretization of Navier–Stokes equation via finite differences are explained. Solving three standard fluid mechanics problems validates the method: the flow of a Newtonian fluid in a sudden contraction, the developing non-Newtonian fluid in a circular tube and a transient laminar axisymmetric flow trough a tube with a moderate local area reduction. Comparison with available experimental results is used to examine the accuracy of the method, reported a close agreement between the numerical prediction and the experimental data.

Second-order stochastic fluid models with fluid-dependent flow rates

In this paper, the analysis of second-order stochastic fluid models, where the fluid rate is dependent on the fluid level, is addressed. The boundary conditions are presented for the fluid models under consideration, which have extended previous work with only reflecting barrier assumptions. To obtain the transient solution of the fluid dynamics, a finite difference solution method is presented, which confirms to the boundary conditions and satisfies the normalization condition at the same time. With our approach, the modeling power of second-order fluid models is directly extended to include the case with fluid-dependent rates. As an application example, a statistical multiplexing problem is analyzed with our proposed method.

Finite difference solution of the 2-dimensional Poisson–Boltzmann equation for spheres in confined geometries

Abstract Published experimental data has shown a long-range attractive interaction between identical colloidal particles close to a charged surface. However, previous numerical calculations for a capillary geometry, which indicated that such attractions could arise from solution of the non-linear Poisson–Boltzmann equation (PBE) have been shown to be both qualitatively and quantitatively in error. The present paper uses a finite difference method for solving the PBE in 2-dimensions. Test case calculations are performed on two spheres interacting within a charged capillary. These calculations verify that, as long as the confining walls are parallel to the line joining the centres of the particles, the interaction between identical particles is always repulsive irrespective of whether or not the particles are isolated or confined in a capillary. The finite difference solution method also enables the calculation of the interaction between two spheres when there is a very small slope or protrusion on the confining walls. An apparent attractive interaction is predicted under such conditions. These calculations indicate that surface roughness (protrusion) is unlikely to be the origin of the experimentally determined attraction, but that great care is needed in the alignment of surfaces in such experiments.

A finite cell method for simulating the mass transport process in porous media

This paper presents a new numerical method, the finite cell method (FCM), for simulating complex biological and chemical transport phenomena in porous media. It has the same advantages as the random walk method (RW): there is no advection‐dispersion equation to be solved, it is unnecessary to calculate the concentration distribution in each time step, and there is no numerical dispersion for the case of high Peclet number. The concept of FCM, however, is different from the RWM in several aspects. The minimum unit in the FCM is a cell that has a certain volume and carries variable mass, while in the RWM the minimum unit is a particle that has no volume but carries a certain mass. In the FCM we use multiple sets of cells to represent different phases in a porous medium. A mobile or an immobile cell may carry multiple biological and/or chemical components. Mass exchanges may occur not only between cells but also within cells. These mass exchanges can be described by different rules: equilibrium or nonequilibrium, linear or nonlinear. In this paper, a one‐dimensional FCM code is developed and used to simulate the solute transport with nonequilibrium adsorption, colloid transport with kinetic deposition and release, and colloid facilitated transport with kinetic mass exchanges. The FCM solutions have been compared with available analytical solutions and the solutions obtained by the finite difference method (FDM). Results show that both FCM and FDM solutions match analytical solutions quite well when the concentration front is flat. When the concentration front is sharp, however, the FDM solutions become inaccurate because of numerical dispersion, but the FCM can still produce very accurate peak and tailing concentrations. From the example of colloid‐facilitated transport, we can see how a complex chemical and biological transport procedure in porous media can be simulated directly by the FCM without solving any advection‐dispersion equation.

Numerical analysis of gas-bubble flow in water-filled natural fractures

Abstract A novel numerical procedure has been developed, and is being investigated to model gas-bubble flow in natural rock fractures. Experimental observations and numerical computations are combined in the process of developing a finite-difference method of solution for the problem. Two-phase fluid mechanics are used for the formulation of the equations that govern the problem, and the velocity and pressure distribution of the two phases is pursued inside the fracture. A simplified analytical solution has also been developed for the purpose of comparing it to the numerical scheme.

A more accurate finite difference approach to the pricing of contingent claims

Numerical approximations of contingent claim partial differential equations (PDEs) are quickly becoming one of the most accepted techniques used in derivative security valuation. The most common methodology is the finite difference method (FDM). This procedure can be used as long as a well-posed PDE can be derived and therefore lends itself to contingent claims. The FDM requires prescribed conditions at the boundary. These boundary conditions are not readily available (at all boundaries) for most contingent claim PDEs. This study presents an accurate and computationally inexpensive method for providing these boundary conditions. These absorbing and adjusting boundary conditions when applied to the contingent claim PDEs presented in this study increased the accuracy of the FDM solution at relatively little cost.

FILMWISE CONDENSATION ON A VERTICAL POROUS ELLIPSOID WITH UNIFORM SUCTION VELOCITY

A model is developed for study of laminar, filmwise condensation from quiescent vapors onto a vertical ellipsoid with uniform suction at its porous wall. The local film thickness and heat transfer coefficient around the ellipsoidal periphery have been analyzed numerically for various values of ellipticity, suction parameters Sw and the inverse Bond number 1/Bo.The dimension-less mean heat transfer coefficient Nu¯(Ja/Rarpar;¼1 increases with increasing ellipticity and Sw. But, the effect of surface tension on Nu¯(Ja/Rarpar;¼1 is insignificant especially for small ellipticity near zero. For the case of no wall suction, Sw= 0, the present solution reduces to an analytical result, which agrees well with finite difference method's solution by Karimi.

Measuring Donnan-related phenomena using a solid-state ion sensor and a concentration-step method

Measurements are performed with a device consisting of an ISFET pH-sensor in the middle of a Ag AgCl electrode, on top of which a microporous composite membrane is deposited. A sudden change of the salt concentration in the bathing electrolyte causes a transient change in the electrical potential of these sensors when measured vs. a reference electrode in the bathing electrolyte. The potential transient is modulated by adsorption of protein to the membrane. To explain the measured transients, a model is presented for the measuring device describing the ion transport by the Nernst-Planck and Poisson equations, incorporating the different proton-dissociation reactions occurring in the system, and the sensor responses to their potential determining ions (the proton or the Cl − ion). A finite-difference solution method is presented to solve the resulting differential equations. Measurements are performed before and after the adsorption of the model protein lysozyme to the membrane. Analysis of the measurement results indicates that the measured potential transient is caused by a change of the Donnan potential of the membrane, followed by a compensating change in the concentration of the potential determining ion. It is proven that no diffusion potential is generated. In addition, it is shown that an interlayer of electrolyte between membrane and measuring electrode will not influence the measured response. The potential transients measured by the ISFET have a larger amplitude and a longer duration than the Ag AgCl -measured transients. An analysis shows that this is caused by the buffering action of the proton-dissociating membrane groups. The longer duration results from the release of a large amount of protons from binding to fixed groups, while chloride ions are not bound. The larger amplitude can be explained by refining the Donnan model to account for the inhomogeneous charge distribution in the membrane. The proton-dissociating groups reside at the surface of the polystyrene beads, at which place the potential change on an ion step is larger than the average in the membrane pore solution. This surface-potential change can be measured by the pH-sensitive ISFET because a proton release occurs from the surface-bound groups into the membrane pores, changing the pore pH.

Numerical Analysis of Structure of Free-Surface Shock Wave About a Wedge Model

Detailed characteristics of the free-surface shock wave (FSSW) are numerically studied for the case of waves generated about a wedge model. The MAC-type finite-difference solution method called TUMMAC-VIII is employed. The density function method is used for implementation of the nonlinear free-surface condition in the framework of a rectangular coordinate system. The computed results show that the wave front angle varies systematically and that the shock condition is fulfilled in the thin layer near the free surface, which is one of the most typical characteristics of the FSSW. Some important mechanisms of wave breaking accompanied by the FSSW, such as overturning of wave front, vortex generation and energy dissipation, are also approximately simulated.