Showing papers on "Finite difference published in 1971"

Calculation of plane steady transonic flows

Abstract: Transonic small disturbance theory is used to solve for the flow past thin airfoils including cases with imbedded shock waves. The small disturbance equations and similarity rules are presented, and a boundary value problem is formulated for the case of a subsonic freestream Mach number. The governing transonic potential equation is a mixed (elliptic-hyperbolic) differential equation which is solved numerically using a newly developed mixed finite difference system. Separate difference formulas are used in the elliptic and hyperbolic regions to account properly for the local domain of dependence of the differential equation. An analytical solution derived for the far field is used as a boundary condition for the numerical solution. The difference equations are solved with a line relaxation algorithm. Shock waves, if any, and supersonic zones appear naturally during the iterative process. Results are presented for nonlifting circular arc airfoils and a shock free Nieuwland airfoil. Agreement with experiment for the circular arc airfoils, and exact theory for the Nieuwland airfoil is excellent.

Quantitative Evaluation of Numerical Diffusion (Truncation Error)

Abstract: Truncation error limits the use of numerical finite difference approximations to solve partial differential equations. In the solution of convection-diffusion equations such as occur in miscible displacement and thermal transport, truncation error results in an artificial dispersion term often denoted as numerical diffusion. The differential equations describing 2-phase fluid flow can also be rearranged into a convection-diffusion form. Miscible and immiscible differential equations have been shown to be completely analogous. In this form, it is easy to infer that numerical diffusion will result in an additional term resembling flow due to capillarity. Many users of numerical programs and probably all numerical analysts recognize that the magnitude of the numerical diffusivity for convection-diffusion equations can depend on both block size and time step. Most expressions developed in the literature have been used primarily to determine the order of the error rather than to quantify it. The primary purpose of this study is to give the user more than just a qualitative feel for the importance of truncation error. Insofar as possible, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity. (14 refs.)

Alternating-direction galerkin methods on rectangles

Abstract: Publisher Summary This chapter presents an overview of alternating-direction Galerkin methods on rectangles. Alternating-direction methods in several forms have proved to be very valuable in the approximate solution of partial differential equations problems involving several space variables by finite differences. The methods have been applied to transient problems directly and to stationary problems as iterative procedures. The chapter presents highly efficient procedures for the numerical solution of second-order parabolic and hyperbolic problems in two or more space variables and for the iterative solution of the algebraic equations arising from the Galerkin treatment of elliptic problems. The results presented are limited to rectangular domains. The chapter presents heat equation on a rectangle and extensions to variable coefficients and nonlinear parabolic equations and systems. It describes an iterative procedure for elliptic equations.

Transient Currents Induced on a Metallic Body of Revolution by an Electromagnetic Pulse

Abstract: A numerical technique is presented that may be used to predict the current induced on a thin metallic body of revolution excited by an electromagnetic pulse. Examples are given. Introduced here is the use of the radiation condition in a finite difference solution. This development alleviates the requirement that finite difference techniques be applied to a bounded region of space.

Solving the Biharmonic Equation as Coupled Finite Difference Equations

Abstract: A technique is proposed for solving the finite difference biharmonic equation as a coupled pair of harmonic difference equations. Essentially, the method is a general block SOR method with convergence rate $O(h^{{1 / 2}} )$ on a square, where h is mesh size.

Comparison of two independent methods for the solution of wave-scattering problems: Response of a sedimentary basin to vertically incident SH waves

Abstract: The transient solution to several problems that was obtained by numerical integration of equations of motion using a finite difference (FD) technique is compared with the complex-frequency solutions obtained by the approximate wave theoretical (AL) method of K. Aki and K. L. Lamer. The excellent agreement between the two solutions not only provides a comparative check on the accuracies of the two techniques, but also demonstrates that the interpretation of the AL solution is comparable to the Fourier transform of the transient solution premultiplied by an exponential window. Most of the paper is devoted to a discussion of two models that are relevant to the engineering-seismological study of earthquake motions in soft layers of varying thicknesses. The FD and AL solutions show that lateral reverberations of waves produced by the nonplanar structure form complex interference patterns that are not predicted by the usual flat-layer approximations. In one example, constructive interference enhances the peak amplitude of the transient motion over the center of the basin by a factor of 3 relative to the flat-layer solutions. The results indicate that a realistic appraisal of earthquake hazards in areas underlain by soft surficial layers should include the effect of nonuniformity in the structure.

The finite element method and approximation theory

Abstract: Publisher Summary This chapter focuses on the finite element method and approximation theory. By a suitable choice of the trial functions , the Galerkin equations for the coefficients turn out to be difference equations. It is to this combination of two fundamental techniques—to the fact that it is a variational method and at the same time takes the form of a difference equation—that the finite element method owes much of its success. The variational aspect is crucial to the formulation of the approximating scheme, allowing flexibility in the geometry and a physically intuitive derivation of accurate discrete analogues; in these respects, it is superior to more conventional difference equations. In the solution of these discrete analogs, it is the finite difference aspect that dominates. This chapter describes a systematic approach to the choice of such trial functions .

Groundwater flow in a sandy tidal beach: 2. Two‐dimensional finite element analysis

Abstract: A 31-day time series of observations of beach water table and tidal fluctuations was obtained from 13 wells along a profile perpendicular to the shoreline at Virginia Beach, Virginia. Finite element techniques were applied to solve the one-dimensional, unsteady state, nonlinear equation for groundwater movement. For the finite element analysis, the semi-infinite mass (unconfined aquifer) had to be replaced by a finite mass. The boundary conditions were found from the field data by directly solving the flow equation with a finite difference technique. The finite element method, using the variational principle, provided a reasonable solution and afforded economy in computer time. Field data were compared with the corresponding finite element solution. The results indicate the general accuracy of the methodology.

A note on finite-difference schemes for the surface and planetary boundary layers

Abstract: The solution of the planetary boundary-layer equations by finite-difference methods has recently become very popular. Among recent papers using such methods, several use somewhat arbitrary finite-difference meshes and some do not make use of a constant flux or wall layer near the ground. It is shown that the use of finite differences right down to the ground can be a very inaccurate procedure when used in conjunction with an eddy viscosity or mixing length proportional to (z +z0) orz near the ground. Such an approach can lead to results that are highly dependent on the finite-difference scheme used and virtually independent of the roughness length,z0. A scheme using an expanding grid, based on the form chosen for mixing length or eddy viscosity, is proposed which gives good results with or without a surface layer in the case of a neutrally stratified atmosphere.

Analysis of expression operations

Abstract: The internal mechanisms of expression operations are analysed in view of the variations of flow rate through filter-cake and of the modified consolidation coefficient through consolidated cake, and basic equations for consolidation processes are presented in this paper. Expression operations are classified into three types according to the relations of pressure and flow rate to time. The equations for flow variations in filter-cakes are numerically solved on the basis of the so-called modern filtration theory. The generalized basic equations for consolidation are solved by using the calculus of finite difference with variable coefficients. It has been demonstrated that the operational variables for expression operations can be more reasonably determined from calculations based upon these numerical solutions rather than predicted by the approximate methods formerly reported. Favourable coincidence between theory and experiments are assured under constant pressure, constant rate and variable pressure-variable rate conditions.

Dynamic behavior of liquids in elastic tanks

Abstract: The finite element procedure is used in conjunction with a simple source distribution on all surfaces of a liquid, which is contained in a tank of virtually arbitrary geometry, to obtain expressions for the mass and stiffness (because of the gravity) matrices of the liquid. The accuracy of the formulation is evaluated by the numerical solution of eigenvalue problems appropriate to simple rigid tanks for which exact analytical solutions are available. Good agreement is obtained. Once the mass and stiffness matrices of the liquid are obtained, they can be combined with those of an elastic tank in order to study the dynamic behavior of liquids in elastic tanks. The matrices for the elastic tank can be generated by any of a variety of currently available finite element or finite difference programs. The dynamic behavior of a liquid in an elastic circular cylindrical tank is treated by making use of a standard finite element program. Very good agreement with the known exact analytical results is obtained. In addition, a tank geometry which cannot be readily analyzed by any of the previously available methods is considered.

A General Nonlinear Relaxation Iteration Technique for Solving Nonlinear Problems in Mechanics

Abstract: : The nonlinear relaxation method, an iterative approach used in conjunction with finite difference approximations, is illustrated via the solution to a very simple problem. Subsequently, the method is used to solve three geometrically nonlinear problems in mechanics: finite bending of a circular thin walled tube, the large deflection membrane response of a spherical cap, and finite deformations of a uniformly loaded circular membrane. Formulations for the three problems are quite different but this difference does not inhibit the use of the nonlinear relaxation technique. Solutions were obtained in approximately one man day per problem including the total time devoted to examining, planning, programming, debugging, etc. Solutions compare very favorably with results found elsewhere in the literature. The essential and important advantages of the nonlinear relaxation technique are: (a) versatility and ease of application, (b) efficiency with respect to people and computer time utilized, (c) insensitivity to starting values as far as convergence is concerned and (d) simplicity of logic that makes it a trivial task to learn how to employ it. (Author)

Numeric calculation of turbulent diffusion

Abstract: A simple, flexible method involving a random number generator is given for simulating time dependent dispersion. The diffusion is simulated by letting a series of particles move with the local mean wind plus random fluctuations of this wind. This simulation method is not greatly complicated by introduction of horizontal and vertical shear, buoyancy, or anisotropic turbulence, and generally requires less computer time and storage than needed for finite difference computations of comparable accuracy over a network of fixed grid points. Solutions of particular cases compare well with known solutions.

Unconfined Transient Seepage in Sloping Banks

Abstract: A finite difference scheme is used for the problem of unsteady flow with arbitrary variations in water levels in nonhomogeneous and anisotropic porous soils. The solutions are obtained by using the finite difference scheme for an approximate nonlinear and the linearized version of the free surface equation of flow. The phreatic surface at any time level is located by satisfying the special boundary condition at that surface. A special iterative scheme is employed to locate the point of exit of the phreatic surface at the entrance face. Numerical solutions are compared with experimental results with a large parallel-plate viscous flow model. Good agreement is obtained between the two results. A flow net is constructed for typical numerical results to indicate their application for field problems. On the basis of good comparisons, it is concluded that the proposed numerical method can be employed for the location of the free surface in riverbanks and embankments subjected to transient flow conditions.

Calculation by a finite difference method of supersonic turbulent boundary layers with tangential slot injection

Abstract: Finite difference calculation based on eddy diffusivity and mixing length flow theory to characterize supersonic turbulent boundary layer with tangential slot injection

Computation of nonequilibrium merged stagnation shock layers by successive accelerated replacement

Abstract: Results are presented for fully merged shock layer computations in the stagnation region of a hypersonic body. Numerical solutions are obtained with a new technique which avoids several troublesome aspects of previous methods. The effects of nonequilibrium chemistry are studied for a multicomponent air mixture and the resulting electron concentrations are calculated. The analysis assumes a continuum approach and employs the well known "locally similar" flow model. The governing system of equations constitutes a two-point boundary value problem which is solved using a simple finite difference method called successive accelerated replacement, or SAR. Special attention is given to a singularity that appears in the continuity equation, and the importance of an acceleration factor on the convergence of the solution is discussed. Solutions are obtained for a Reynolds number range of 814 to 4000 and for flight speeds from 15,000 to 26,000/fps. Predicted results for the electron concentration are compared with experimental data and good agreement is obtained.

On the numerical solution of elliptic boundary value problems by least squares approximation of the data

Abstract: Publisher Summary This chapter discusses the numerical solution of elliptic boundary value problems by least squares approximation of the data. In the approximate solution of boundary value, problems arising in the theory of elliptic partial differential equations, several rather general approaches have been taken. The method perhaps most extensively used has been the method of finite differences. The chapter presents certain classes of finite dimensional subspaces of Sobolev spaces and highlights a theorem concerning their approximation theoretic properties relative to general boundary problems. It presents an approximation scheme of least squares type for 2mth order elliptic equations and general boundary conditions. The chapter also discusses a second-order equation with coefficients discontinuous across an interface. It presents an approximation scheme in addition to corresponding error estimates. The chapter also presents a few examples of specific problems and specific choices of subspaces.

The linear two-point boundary-value problem on an infinite interval

Abstract: A numerical method, using finite-difference approximations to the second-order differential equation, is given which tests the suitability of the finite point chosen to represent infinity before computing the numerical solution. The theory is illustrated with examples and suggestions for further applications of the method are presented. 1. Introduction. Consider the linear two-point boundary-value problem on an infinite interval (1) Ly = -y" + p(x)y' + q(x)y= f(x) with y(a) a, y(Acm) = 0. More general boundary conditions will be considered later. A typical method is to apply the second boundary condition at a finite point or at several finite points and observe the variation of the solutions (Fox (2)). The basic objective is to obtain an accurate numerical solution in a relatively small finite interval starting at a. The proposed method, involving finite-difference approxima- tions, finds a solution over such an interval and also produces values b and ,3 such that the solution of (1) on (a, b) is given by the solution of

Em Response of a Rectangular Thin Plate

Abstract: By formulating the fundamental electromagnetic (EM) equations in terms of the stream potential of the surface current density, we can express the EM response of a rectangular thin plate as the solution of a single equation subject to simple boundary conditions. A finite difference approximation of this equation reduces the problem to that of solving a large set of linear algebraic equations. The solution of these equations by a modified Gauss‐Seidel iterative method yields the stream potential and thus permits visualization of the eddy currents circulating inside the plate conductors. The secondary field calculated from the stream potential compares well with that given by scale model measurements provided that the grid spacings used in the finite differences are small enough. Using a further approximation, we can also simulate inductively thick conductors if the conductors are not geometrically thick.

De Saint-Venant Equations Experimentally Verified

Abstract: The unsteady spatially varied flow equations (De Saint-Venant equations) are being solved by implicit finite differences with explicit description at the boundaries. Imposition of improper boundary conditions which violate the physics of the problem resulted into either violation of continuity or numerical instability problems or meaningless results. The magnitude of the spatial increment used in this implicit solution scheme was critical on steep slopes (2.0%). The temporal distribution of flow and time to equilibrium was altered considerably depending on the specified value of Δ x . Hydrographs on milder slopes (0.5% and 1.0%) were affected to a progressively lesser extent as the channel slope was decreased. The time to equilibrium flow starting from a dry bed was nonlinear with respect to change of channel length, channel slope, and rate of lateral inflow. For a given channel length and slope, time to equilibrium approached a constant for high rates of later inflow.