Strongly Implicit Research Articles

Comparison of Basic Iterative Methods Used to Solve of Heat and Fluid Flow Problems

In this research, a comparison of the convergence rate of different basic methods is made in order to solve the Poisson equation, which is similar to some of the resulting equations in computational fluid mechanics. The Jacobi method, the point Gauss-Seidel method, the successive over-relaxation method (SOR), the line Gauss method (TDMA), the ADI method and Strongly Implicit (SIP) method are among the iterative approaches investigated, which are then compared to find the most optimal method. The selection criteria included the number of iterations and the time needed to reach convergence. In both selection criteria, the SIP approach has been shown to be the most efficient.

Simulation of pollutant dispersion in urban street canyons using hybrid rans-les method with two-phase model

In this study, to simulate the pollution treatment as a separated phase a 3D incompressible turbulence model under isothermal conditions was considered. The hybrid RANS-LES method with a sub-grid scale(SGS) technique within the framework of two-phase flow was used to study pollution distribution in urban street canyons. An area source of pollutant with a constant concentration, located in the ground of the street canyon, was used to simulate vehicular emissions. The near-wall regions were simulated using the RANS(k-ε) turbulence model and the rest of the domain was simulated with the LES model. To measure the effect of building height on pollutant dispersion, the parameter AR(Aspect Ratio) was calculated. Also to determine the effect of street width, a new parameter SWV(Street Width Value) was introduced. Governing equations were discretized using the well-known finite volume method on a staggered grid. To increase the accuracy, the second-order Van Leer Method(VLM) was applied and the Volume of Fluid(VOF) method was employed for the description of the interfaces. The Iterative line-by-line TDMA solution procedure and the Strongly Implicit Procedure(SIP) was used for the Pressure Correction Equation. A fine comparison of the results revealed that by increasing SWV, the number of vertical vortices in the canyon streets generally decreased and the vortices grew larger. In addition, increasing AR caused a rise in the pollution volume fraction in the first canyon street. The current numerical results indicate that the AR and SWV influence the flow field and volume fraction of phases inside an isolated street canyon, significantly.

Current Profile Reconstruction Using Motional Stark Effect Polarimeter Data on HL-2A Tokamak

In order to reconstruct the plasma current density, the Current Profile Fitting (CPF) code has been successfully developed on the HL-2A tokamak. A seven-channel motional Stark effect (MSE) diagnostic based on dual photoelastic modulators is installed to measure the pitch angle of the magnetic field, which can be used as an internal magnetic field constraint for the CPF code. Recently, the MSE polarimeter was upgraded with a real-time wavelength matching system to improve the signal-to-noise ratio. The magnetic field angle (γpitch) with a temporal resolution of 10 ms can be provided. In the CPF code, the plasma current density is described as a polynomial, and the Least-Squares method is used to determine the coefficients of the polynomial. The Finite Difference method and the Strongly Implicit Procedure method are used to solve the Grad-Shafranov equation. The code operation is stable. With the improved-quality MSE data, the CPF calculation result of shot 30782 suggests that the safety factor q profile is monotonic. The minimum q value is less than 1 on-axis during sawtooth oscillations in shot 30782. And, the position of the q = 1 surface is consistent with the sawtooth inversion radius measured by electron cyclotron emission and soft X-ray diagnostics.

Numerical investigation of oscillating conjugate heat transfer in pulse tubes

This paper is aimed to investigate the influence of axial heat conduction on conjugate heat transfer for oscillating flow at the pulse tube section of Pulse Tube Cryocoolers (PTCs) via a numerical approach. Finite volume technique with collocated arrangement of grids has been employed to discretize momentum and energy equations. The well-established SIMPLE algorithm was followed for handling the pressure–velocity coupling. The set of individual algebraic equations was solved by the Strongly Implicit Procedure (SIP). Strouhal number (St), wall thickness and solid to fluid conductivity ratio (ksf) are three important parameters which affect the conjugate heat transfer in oscillating flow in pulse tubes. It is shown that the overall Nusselt number (Nu) increases from 5.97 to 28.45 by increasing St from 0.02 to 1. In addition, by increasing ksf from 0.5 to 15, Nu decreases almost 27%, however, for ksf>15, increasing the solid to fluid conductivity ratio has no influence on the Nusselt number. Furthermore, the cycle averaged friction coefficient at fully developed region increases almost 73% at St=1 in comparison to St=0.02.

Laminar Confined Impinging Jet into a Porous Layer

This work aims at studying laminar impinging jets on surfaces covered with a layer of a porous material. This contribution may provide insight into the design and optimization of heat and mass transfer processes over surfaces. Numerical simulations are presented and the porous substrate is treated as a rigid, homogeneous, and isotropic medium. Macroscopic transport equations are written for a representative elementary volume (REV), yielding a set of equations that is valid for the entire computational domain, including both the porous layer attached to the surface and the fluid layer over the porous substrate. These equations are discretized using the control-volume method and the resulting system of algebraic equations is relaxed using the Strongly Implicit Procedure (SIP) methods. The SIMPLE algorithm is used to handle the pressure–velocity coupling. Results for flow, in both clear and porous flow domains, are given in terms of streamlines patterns, velocity profiles, pressure contours, and friction coefficient along the impinging wall. The influence of porosity on the flow pattern is shown to be very low in comparison to the effects caused by varying permeability, fluid-layer height, and porous-layer thickness. These finding could be used to advantage when designing engineering equipment, since the use of selected porous materials could reduce undesirable recirculation zones, promote quick flow redistribution, and adjust pressure to required levels.

A two-dimensional finite volume hydrodynamic model for coastal areas: Model development and validation

A two dimensional implicit finite volume scheme for solving the shallow-water equations is developed. The effects of the Coriolis force, surface wind stress, and waves are included. A non-uniform rectilinear forward staggered grid is used with Cartesian coordinates. The time integration is performed using the Euler implicit technique. The convective flux is treated using the deferred correction method. The viscous terms are discretized using a second order central difference approximation. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm is used for coupling the velocity components and the water elevation gradient for the water level correction. The system of equations is solved sequentially using the Strongly Implicit Procedure (SIP). To simulate wave driven current, a phase averaged wave model is used first to simulate wave transformation and calculate radiation stresses. The performance of the developed model is validated for different sources of external forces and different combinations of boundary conditions. The validation cases include tidal circulation in a harbor and wave induced currents behind a breakwater parallel to the coastline. The model is finally applied to simulate the flow pattern in a closed artificial lagoon and along the coastline near Damietta Port located along the Northern coast of Egypt. Results of the developed model agree well with the published results for the considered cases.

Parallel versions of Stone's strongly implicit algorithm

AbstractIn this paper, we describe various methods of deriving a parallel version of Stone's Strongly Implicit Procedure (SIP) for solving sparse linear equations arising from finite difference approximation to partial differential equations (PDEs). Sequential versions of this algorithm have been very successful in solving semi‐conductor, heat conduction and flow simulation problems and an efficient parallel version would enable much larger simulations to be run. An initial investigation of various parallelizing strategies was undertaken using a version of high performance Fortran (HPF) and the best methods were reprogrammed using the MPI message passing libraries for increased efficiency. Early attempts concentrated on developing a parallel version of the characteristic wavefront computation pattern of the existing sequential SIP code. However, a red‐black ordering of grid points, similar to that used in parallel versions of the Gauss–Seidel algorithm, is shown to be far more efficient. The results of both the wavefront and red‐black MPI based algorithms are reported for various size problems and number of processors on a sixteen node IBM SP2. Copyright © 2001 John Wiley & Sons, Ltd.

Speed-up of the strongly implicit procedure with application to subsonic/transonic potential flows

A modified version of the Strongly Implicit Procedure (SIP) for solving partial differential equations is presented. The capabilities of the modified scheme are demonstrated and analyzed for the case of two-dimensional subsonic or transonic potential flow over airfoils. The improved technique accelerates the convergence rate of the SIP considerably

Convergence of SIP

We prove that the Strongly Implicit Procedure (SIP), due to H. L. Stone, for solving selfadjoint elliptic difference equations is convergent.

A three-dimensional modified strongly implicit procedure for heat conduction

The application of discretizatio n techniques frequently leads to a system of algebraic equations having a welldefined coefficient structure. A modified strongly implicit procedure for solving the system resulting from the modeling of heat conduction in three dimensions is presented in this work. The method is derived for a 19 point scheme with the more common 7 point scheme emerging as a special case of the procedure. In this way, the asymmetric influence of the additional terms in the LU matrix product is weakened. As a consequence, the method is less sensitive to the iteration parameter and mesh aspect ratio and, in addition, provides considerably more rapid convergence than does the strongly implicit procedure. The increased convergence is exhibited by a significant reduction in the computational cost. The characteristics of the method are examined through application to several model problems and application is made to a more complex three-dimensional problem. Comparisons with the SIP (strongly implicit) and ADI (alternating direction implicit) methods are provided.

A New Iterative Method For Solving Reservoir Simulation Equations

Abstract Recently Poussin and Watts have presented some interesting iterative methods of solving matrix equations resulting from the finite-difference approximation of partial differential equations. These methods are however, not competitive with the strongly implicit (SIP) method of Stone for highly anisotropic and heterogeneous reservoirs. Our method is an extension of the method of ‘Watts and it is competitive with SIP from a computational point of view; furthermore our method is easier to program. In this paper, we first formulate the matrix problem using the notation of Stone. Next we present Watts’ method in the notation of our paper. Following this, the new method is developed. Computational algorithms for the implementation of both of these methods are also presented. Finally, we present numerical results and a comparison of several numerical methods. Introduction In RESERVOIR SIMULATION and in many other problems involving partial differential equations, we are usually faced with the problem of solving equations of the parabolic type, (Equation Available In Full Paper) where n is the direction normal to the bounding surface. In spite of the fact that computers are getting bigger and faster and the cost per computation is being reduced the cost of reservoir simulation in many important cases is still too high and cannot be justified, Efforts are being made to reduce the cost of reservoir simulation. Along with improvements in computers, it is also necessary that more efficient methods be developed for the solution of reservoir simulation equations. This is indeed the case and many new methods are continually appearing in the literature. Many useful techniques are often buried in mathematical details which the engineer usually tries to avoid. In another paper(1), we have given mathematical details of a new method for solving the matrix equations resulting from the finite-difference approximations of Equations (1) and (2), In addition to the consideration of the boundary condition given by Equation (3), we have also discussed other boundary conditions in that paper. Our objective here is to present important practical results without giving any mathematical details which are not necessary as far as applications of the method are concerned. In order to keep our discussion brief and simple, here we will deal in detail with the solution of the parabolic Equation n) and then show how the same method can be applied to Equation (2). Many important theoretical results not presented here are contained in the companion paper(1), and some of the practical considerations given here are not discussed elsewhere. Derivation of Matrix Equation The method to be presented is quite general and can be applied to irregularly shaped domains (like natural petroleum reservoirs); however, to illustrate the method we will choose a;simple example of a rectangular domain as shown in Figure L As usual, we divide the domain of interest into grid blocks and locate a grid point inside each grid block.

Comparative Evaluation of the Strongly Implicit Procedure (SIP) For Gas Reservoir Simulation

Abstract The present work was principally concerned with an evaluation of the new SIP algorithm for solving the matrix problem arising from the application of the various implicit difference techniques to the nonlinear partial differential equation for the flow of a real gas in two dimensions, expressed ill two different forms, utilizing the original and the transformed variables, respective. A number of con,- emotional difference methods were also tested for comparison purposes. The equations resulting from the application of the completely implicit scheme were solved by both SIP and SOR (successive over-relaxation), in order to evaluate the speed of convergence using the two methods. In addition, a series of numerical experiments was carried out using the SIP algorithm in order to study the effect of the number of iteration parameters on the number of iterations required for convergence, the rate of convergence and the material balance error induced. It was found that the explicit schemes tested suffered from time step size limitations and increasing material balance errors with time. The implicit schemes, in particular the modified implicit scheme, is therefore recommended. Computational results show that the SIP algorithm is three times as fast as SOR. The optimized SIP algorithm, although involving more computational work per iteration requires fewer iterations to converge than SOR, for a given accuracy. The number of SIP iteration parameters should be optimized where feasible. INTRODUCTION NUMEROUS AUTHORS have discussed gas reservoir simulation schemes, involving both explicit and implicit differencing methods. The present study gives a comparison of some of the schemes, and was principally concerned with an evaluation of the Strongly Implicit Procedure (SIP) recently proposed by Weinstein, Stone and Kwan(1) for solving the equations resulting from the conventional as well as from the completely implicit scheme due to Blair and Weinaug(2). Mathematical Model The equation describing the flow of a gas in a two dimensional porous medium can be derived by a consideration of the continuity and Darcy's equations, and is as follows: (Equation Available In Full Paper) where -q represents the production term; the remaining symbols are explained in the Nomenclature. Equation (1) can be rendered quasilinear by use of the gas potential, P*, proposed by AI-Hussainy, Ramey and Crawford(3) and by Russell, Goodrich, Perry and Bruskotter(4): (Equation Available In Full Paper) By use of Eq. (2), Eq. (1) can be transformed to the following form (later referred to as Model 1) : (Equation Available In Full Paper) and may be expressed as a function of P*. In the same manner, Eq. (4) can be used to transform Eq. (1) as follows (later referred to as Model 2): (Equation Available In Full Paper) Eqs. (3) and (5) were solved in the present study by use of a number of explicit and implicit differencing schemes. The hypothetical reservoir employed for numerical computations was a 320-acre square (side = 3,734 ft) with a formation thickness of 20 ft, porosity of 25 per cent and permeability of 100 md (in the homogeneous case;