Showing papers on "Finite difference method published in 1977"
Book•
01 May 1977
TL;DR: A comprehensive and illustrated account of the use of finite difference computational methods for heat transfer calculations is presented in this paper, which is oriented towards the practical man who needs a complete work to enable him to understand and apply the methods to solve his problems.
Abstract: A comprehensive and illustrated account of the use of finite difference computational methods for heat transfer calculations is presented. The methods are basically simple but offer a powerful tool to the engineering designer or researcher faced with heat transfer problems of a difficult, or more often impossible, analytical nature. The text is oriented towards the practical man who needs a complete work to enable him to understand and apply the methods to solve his problems. Information is provided about all the many facets of analyzing and solving conductive heat transfer problems, including the computer programming aspects. The general problem considered is that of calculating the distribution of temperature or temperature history in a physical system in which heat transfer is taking place. A special attribute offered is the strong practical emphasis, and recent ideas on numerical solution techniques, and their implementation via Fortran computer programs. The various numerical solution schemes are illustrated through a series of worked examples, tabular computations, Fortran programs and case studies.
186 citations
TL;DR: In this paper, the use of the artificial compression method for the computation of discontinuous solutions of a single conservation law by finite difference methods is discussed, and the numerical implementation of artificial compression is described.
Abstract: The paper discusses the use of the artificial compression method for the computation of discontinuous solutions of a single conservation law by finite difference methods. The single conservation law has either a shock or a contact discontinuity. Any monotone finite difference scheme applied to the original equation smears the discontinuity, while the same scheme applied to the equation modified by an artificial compression flux produces steady progressing profiles. If L is any finite difference scheme in conservation form and C is an artificial compressor, the split flux artificial compression method CL is a corrective scheme: L smears the discontinuity while propagating it; C compresses the smeared transition toward a sharp discontinuity. Numerical implementation of artificial compression is described.
132 citations
TL;DR: In this article, a simple numerical solution for the coupled-power equation in optical fibers was obtained by using the finite-difference method of numerical analysis, which yields all the quantities of interest in the interior of the fiber: power distribution, attenuation, and far-field radiation pattern as functions of length.
Abstract: By use of the finite-difference method of numerical analysis, a simple numerical solution is obtained for the coupled-power equation in optical fibers. For a specified arbitrary coupling coefficient and launching condition, the solution yields all the quantities of interest in the interior of the fiber: power distribution, attenuation, and far-field radiation pattern as functions of length. Results for buffered and cabled Corning fibers are reported. Attention is mainly focused on the influence of the microbends on the optical losses.
99 citations
01 Jun 1977
TL;DR: In this paper, a numerical method is presented for analyzing the transonic potential flow past a lifting, swept wing, which is solved in a coordinate system which is nearly conformally mapped from the physical space in planes parallel to the symmetry plane, and reduces the wing surface to a portion of one boundary of the computational grid.
Abstract: A numerical method is presented for analyzing the transonic potential flow past a lifting, swept wing. A finite-difference approximation to the full potential equation is solved in a coordinate system which is nearly conformally mapped from the physical space in planes parallel to the symmetry plane, and reduces the wing surface to a portion of one boundary of the computational grid. A coordinate invariant, rotated difference scheme is used, and the difference equations are solved by relaxation. The method is capable of treating wings of arbitrary planform and dihedral, although approximations in treating the tips and vortex sheet make its accuracy suspect for wings of small aspect ratio. Comparisons of calculated results with experimental data are shown for examples of both conventional and supercritical transport wings. Agreement is quite good for both types, but it was found necessary to account for the displacement effect of the boundary layer for the supercritical wing, presumably because of its greater sensitivity to changes in effective geometry.
87 citations
TL;DR: In this article, the fast Karhunen-Loeve transform is extended to images with nonseparable or nearly isotropic covariance functions, or both, for image restoration, data compression, edge detection, image synthesis, etc.
Abstract: Stochastic representation of discrete images by partial differential equation operators is considered. It is shown that these representations can fit random images, with nonseparable, isotropic covariance functions, better than other common covariance models. Application of these models in image restoration, data compression, edge detection, image synthesis, etc., is possible. Different representations based on classification of partial differential equations are considered. Examples on different images show the advantages of using these representations. The previously introduced notion of fast Karhunen-Loeve transform is extended to images with nonseparable or nearly isotropic covariance functions, or both.
80 citations
Book•
01 Jan 1977
TL;DR: In this paper, the origins of the first scheme were discussed and the second and third schemes were executed on the intermediate layer and the final layer, respectively, and the first and second schemes on the third and fourth layers, respectively.
Abstract: 1. General Introduction.- 1.1 Introduction.- 1.2 Boundary Value Problems and Initial Problems.- 1.3 One-Dimensional Unsteady Flow Characteristics.- 1.4 Steady Supersonic Plane or Axi-Symmetric Flow. Equations of Motion in Characteristic Form.- 1.5 Basic Concepts Used in Finite Difference Methods.- References.- 2. The Godunov Schemes.- 2.1 The Origins of Godunov's First Scheme.- 2.2 Godunov's First Scheme. One-Dimensional Eulerian Equations.- 2.3 Godunov's First Scheme in Two and More Dimensions.- 2.4 Godunov's Second Scheme.- 2.5 The Double Sweep Method.- 2.6 Execution of the Second Scheme on the Intermediate Layer.- 2.7 Boundary Conditions on the Intermediate Layer.- 2.8 Procedure on the Final Layer.- 2.9 Applications of the Second Godunov Scheme.- 2.10 Glimm's Method.- 2.11 Outline of Solution for Gas Dynamic Equations.- 2.12 The Glimm Scheme for Simple Acoustic Waves.- 2.13 Random Choice for the Gas Dynamic Equations.- 2.14 Solution of the Riemann Problem.- 2.15 Extension to Unsteady Flow with Cylindrical or Spherical Symmetry.- 2.16 Remarks on Multi-Dimensional Problems.- References.- 3. The BVLR Method.- 3.1 Description of Method for Supersonic Flow.- 3.2 Extensions to Mixed Subsonic-Supersonic Flow. The Blunt Body Problem.- 3.3 The Double Sweep Method for Unsteady Three-Dimensional Flow.- 3.4 Worked Problem. Application to Circular Arc Airfoil.- 3.5 Results and Discussion.- Appendix-Shock Expansion Theory.- References.- 4. The Method of Characteristics for Three-Dimensional Problems in Gas Dynamics.- 4.1 Introduction.- 4.2 Bicharacteristics Method (Butler).- 4.3 Optimal Characteristics Methods (Bruhn and Haack, Schaetz).- 4.4 Near Characteristics Method (Sauer).- References.- 5. The Method of Integral Relations.- 5.1 Introduction.- 5.2 General Formulation. Model Problem.- 5.3 Flow Past Ellipses.- 5.4 The Supersonic Blunt Body Problem.- 5.5 Transonic Flow.- 5.6 Incompressible Laminar Boundary Layer Equations. Basic Formulation.- 5.7 The Method in the Compressible Case.- 5.8 Laminar Boundary Layers with Suction or Injection.- 5.9 Extension to Separated Flows.- 5.10 Application to Supersonic Wakes and Base Flows.- 5.11 Application to Three-Dimensional Laminar Boundary Layers.- 5.12 A Modified Form of the Method of Integral Relations.- 5.13 Application to Viscous Supersonic Conical Flows.- 5.14 Extension to Unsteady Laminar Boundary Layers.- 5.15 Application to Internal Flow Problems.- Model Problem (Chu and Gong).- References.- 6. Telenin's Method and the Method of Lines.- 6.1 Introduction.- 6.2 Solution of Laplace's Equation by Telenin's Method.- 6.3 Solution of a Model Mixed Type Equation by Telenin's Method.- 6.4 Application of Telenin's Method to the Symmetrical Blunt Body Problem.- 6.5 Extension to Unsymmetrical Blunt Body Flows.- 6.6 Application of Telenin's Method to the Supersonic Yawed Cone Problem.- 6.7 The Method of Lines. General Description.- 6.8 Applications of the Method of Lines.- 6.9 Powell's Method Applied to Two Point Boundary Value Problems.- Telenin's Method. Model Problems (Klopfer).- References.
67 citations
TL;DR: Peck, Hamilton, and Shaw as mentioned in this paper developed a method for numerical solution of problems in transient heat flow and applied it to cooling models of Hawaiian lava lakes, which is an adaptation of a computer method that is called the method of explicit cell balances.
Abstract: A computer method for numerical solution of problems in transient heat flow was developed and applied to cooling models of Hawaiian lava lakes. This report and Part II by Peck, Hamilton, and Shaw apply the method to Alae lava lake as the first detailed test against a systematic program of temperature measurements in lava lakes carried out by the Hawaii Volcano Observatory. The method is neither a finite element nor finite difference method, as the terminology of numerical analysis is currently employed, though there are similarities to both. It is an adaptation of a computer method that is called the method of explicit cell balances. Input-output balances are computed between cells in a grid system simulating the lava lake geometry. Computation explicitly performs the cell balances sequentially in the same way the ''cooling wave'' moves into the lava body without recourse to numerical solutions of sets of simultaneous equations. The procedure clearly identifies the physical processes governing thermal effects: the effects of the latent heat of crystallization, effects of rainfall permeating the lava surface, and effects of large variations in density and thermal conductivity. The accuracy of numerical approximations was tested against exact solutions for an extrusive sheet without sources.more » Deviations were less than 1/sup 0/C in all cases. Agreement with data for Alae lava lake given in Part II indicates that the method can be used with good confidence for refinement of thermal properties and identification of heat transfer mechanisms in other lava lakes as well as for predictive tests of cooling histories for hypothetical systems involving complicated heat transfer mechanisms.« less
57 citations
TL;DR: In this paper, a higher order beam finite element is developed for dynamic response of beams subjected to impact of elastic spheres, and the Hertzian law is used to evaluate the contact force.
Abstract: : A higher order beam finite element is developed for dynamic response of beams subjected to impact of elastic spheres. Hertzian law is used to evaluate the contact force. A step by step finite difference method is employed to integrate the time variable. The finite elements are first evaluated for homogeneous isotropic beams and excellent results are found. Impact of glass- epoxy laminates are then considered. The total energy imparted from the projectile to the laminate is computed and compared with experimental data. Good agreement is found. The present finite element procedure also allows one to separate the vibrational energy from the damage energy which is to be related to the residual strength of the composite after impact.
57 citations
49 citations
TL;DR: Optimized iteration methods for the solution of large-scale fast reactor finite difference diffusion theory calculations are presented, and the performance of a computer code employing these methods is compared with that of several existing production diffusion theory codes for a range of typical problems.
Abstract: Optimized iteration methods for the solution of large-scale fast reactor finite difference diffusion theory calculations are presented, along with their theoretical basis. The computational and dat...
48 citations
TL;DR: In this article, experimental freezing studies on model and actual biological systems have been simulated by finite difference methods with a three time level scheme and boundary conditions of the third kind, for one dimensional heat transfer.
TL;DR: In this paper, the authors presented a new sixth order finite difference method for the second order differential equation subject to the boundary conditionsy(a)=A,y(b)=B. An interesting feature of their method is that each discretization of the differential equation at an interior grid point is based on five evaluations off; the classical second order method is based in one and the well-known fourth order method of Noumerov is also based on three evaluations off.
Abstract: We present a new sixth order finite difference method for the second order differential equationy″=f(x,y) subject to the boundary conditionsy(a)=A,y(b)=B. An interesting feature of our method is that each discretization of the differential equation at an interior grid point is based onfive evaluations off; the classical second order method is based on one and the well-known fourth order method of Noumerov is based on three evaluations off. In case of linear differential equations, our finite difference scheme leads to tridiagonal linear systems. We establish, under appropriate conditions, the sixth order convergence of the finite difference method. Numerical examples are considered to demonstrate computationally the sixth order of the method.
TL;DR: In this article, a general finite difference formulation is presented for deriving an accurate and stable finite-difference scheme, which introduces a new concept of "decay function" which is determined analytically at each grid point.
TL;DR: In this paper, an iterative algorithm for solving nonlinear inverse problems in remote sensing of density profiles of a simple ocean model by using acoustic pulses is developed, where the adiabatic sound velocity is assumed to be proportional to the inverse square root of the density.
TL;DR: In this paper, orthogonal curvilinear mesh networks are generated numerically between the wavy walls of two-dimensional peristaltic channels using an implicit finite-difference technique to obtain transient solutions of the Navier-Stokes equations.
TL;DR: An interactive computer program which has some capability for solving systems of finite difference equations is described, and a knowledgeable user can, with the help of the program, solve a wider class of equations.
Abstract: An interactive computer program which has some capability for solving systems of finite difference equations is described. Although this capability is limited to linear systems, a knowledgeable user can, with the help of the program, solve a wider class of equations. Over 100 examples, covering a variety of cases, have been solved by using the program. Some representative examples are presented. Additional features which would improve the versatility of the program are also discussed.
TL;DR: In this article, the transient responses to a step change in an inlet temperature are analyzed as an important example of dynamic characteristics of cross-flow heat exchangers with both fluids unmixed.
Abstract: The transient responses to a step change in an inlet temperature are analyzed as an important example of dynamic characteristics of cross-flow heat exchangers with both fluids unmixed. The numerical calculations are performed by two methods, namely, a method expanding a finite difference method used by Myers et al. to calculate accurately in the case where temperature changes discontinuously, and a central finite difference method to calculate more accurately with a small number of nodes in large values of a heat capacity of a solid wall. Consequently, the effects of an initial condition and the various parameters concerning heat transfer performance on the transient responses in the temperature efficiency are shown.
TL;DR: In this article, the authors considered the problem of finding an orbitally asymptotically stable (unstable) solution of the differential equation, where the eigenvalues of the matrix A are 1 =b hi.
Abstract: The eigenvalues of the matrix A are 1 =b hi. Hence all the iterates 2/1,2/2, ■ • • , of (2) spiral away from the origin. On the other hand, every non-trivial solution of (1) is periodic with period 2ir. The essential difficulty with the differential equation (1) is that the equilibrium solution x = x = 0 is a center. Equivalently, the periodic solutions of (I) are not orbitally asymptotically stable, and thus they may be destroyed under an arbitrary small perturbation. The situation is very different, however, when x =
TL;DR: In this article, the coupled system of equations governing the thermomechanical deformations of a viscoelastic sheet while it is being cold rolled is solved numerically by finite difference method and the mechanical problem is solved by the finite element method using uniform first order rectangular elements.
Abstract: The coupled system of equations governing the thermomechanical deformations of a viscoelastic sheet while it is being cold rolled is solved numerically. The pertinent energy equation is solved by the finite difference method and the mechanical problem is solved by the finite element method using uniform first order rectangular elements. The developed computer program enables one to compute the complete deformation and temperature fields in the sheet. Results presented graphically include the temperature distribution, the stress distribution at the middle surface, the contact pressure distribution and the asymmetric surface deformation of the sheet.
TL;DR: In this paper, an extended integral equation method for transonic flows is developed, where velocities in the flow field are calculated in addition to values on the aerofoil surface.
Abstract: An extended integral equation method for transonic flows is developed. In the extended integral equation method velocities in the flow field are calculated in addition to values on the aerofoil surface, in contrast with the less accurate 'standard' integral equation method in which only surface velocities are calculated. The results obtained for aerofoils in subcritical flow and in supercritical flow when shock waves are present compare satisfactorily with the results of recent finite difference methods.
TL;DR: In this article, it is shown that a linear approximation is adequate for the likely levels of flow encountered in the fast breeder reactor and consequently two linearised solutions are used to solve the non-linear problem by using a time marching method.
TL;DR: In this paper, a set of finite difference equations for the numerical solution of a thermally expandable fluid transient is presented, and the authors apply these ideas to a problem involving flow in the preheater section of a steam generator.
Abstract: In this paper we discuss equations that define a thermally expandable fluid transient. We then present a set of finite difference equations for the numerical solution of such transients, discuss their solvability, and investigate certain aspects involving the convergence of the numerical solution. Finally, we apply these ideas to a problem involving flow in the preheater section of a steam generator.
TL;DR: A review of the application of neutron diffusion theory to reactor design and analysis is given in this article, with a focus on finite difference methods, synthesis methods, nodal calculations, finite element methods, and perturbation theory.
Abstract: Current work and trends in the application of neutron diffusion theory to reactor design and analysis are reviewed. Specific topics covered include finite difference methods, synthesis methods, nodal calculations, finite element methods, and perturbation theory.
TL;DR: On etudie des methodes de discretisation a 2 pas pour le probleme aux valeurs initiales suivant: y'=f(x,y), y(x 0 )=y 0, ou la fonction f(x,y) est suffisamment lisse.
Abstract: On etudie des methodes de discretisation a 2 pas pour le probleme aux valeurs initiales suivant: y'=f(x,y), y(x 0 )=y 0 , ou la fonction f(x,y) est suffisamment lisse
01 Dec 1977
TL;DR: In this paper, a cyclic procedure has been developed for representation of adjacent blade-to-blade passages which asymptotically achieves the correct phase between all passages of a stage.
Abstract: A numerical method of solution of the inviscid, compressible, two-dimensional unsteady flow on a blade-to-blade stream surface through a stage (rotor and stator) or a single blade row of an axial flow compressor or fan is described. A cyclic procedure has been developed for representation of adjacent blade-to-blade passages which asymptotically achieves the correct phase between all passages of a stage. A shock-capturing finite difference method is employed in the interior of the passage, and a method of characteristics technique is used at the boundaries. The blade slipstreams form two of the passage boundaries and are treated as moving contact surfaces capable of supporting jumps in entropy and tangential velocity. The Kutta condition is imposed by requiring the slipstreams to originate at the trailing edges, which are assumed to be sharp. Results are presented for several transonic fan rotors and compared with available experimental data, consisting of holographic observations of shock structure and pressure contour maps. A subcritical stator solution is also compared with results from a relaxation method. Finally, a periodic solution for a stage consisting of 44 rotor blades and 46 stator blades is discussed.
TL;DR: In this paper, four different formulations were used to approximate the surface radiation boundary condition while retaining an implicit formulation for the interior temperature nodes, and the results of these methods in predicting surface temperature on the space shuttle orbiter thermal protection system model under a variety of heating rates.
Abstract: For the problem of predicting one-dimensional heat transfer between conducting and radiating mediums by an implicit finite difference method, four different formulations were used to approximate the surface radiation boundary condition while retaining an implicit formulation for the interior temperature nodes. These formulations are an explicit boundary condition, a linearized boundary condition, an iterative boundary condition, and a semi-iterative boundary method. The results of these methods in predicting surface temperature on the space shuttle orbiter thermal protection system model under a variety of heating rates were compared. The iterative technique caused the surface temperature to be bounded at each step. While the linearized and explicit methods were generally more efficient, the iterative and semi-iterative techniques provided a realistic surface temperature response without requiring step size control techniques.
TL;DR: In this paper, a method of solution of a monoenergetic neutron transport equation in PL approximation using the finite Fourier transformation was presented, where the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone.
Abstract: A method of solution of a monoenergetic neutron transport equation in PL approximation is presented for x-y and x-y-z geometries using the finite Fourier transformation. A reactor system is assumed to consist of multiregions in each of which the nuclear cross sections are spatially constant. Since the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone, the three- and two-dimensional equations are reduced to two- and one-dimensional equations, respectively. The present approach therefore gives fewer unknowns than in the usual series expansion method or in the finite difference method. Some numerical examples are shown for the criticality problem.
01 Jan 1977
TL;DR: In this article, three surface temperature algorithms were evaluated: linear Fourier series, finite difference, and Laplace transform, and they were compared in terms of uncertainty in thermal inertia and compared with the thermal inertia values of geologic materials.
Abstract: The errors incurred in producing a thermal inertia map are of three general types: measurement, analysis, and model simplification. To emphasize the geophysical relevance of these errors, they were expressed in terms of uncertainty in thermal inertia and compared with the thermal inertia values of geologic materials. Thus the applications and practical limitations of the technique were illustrated. All errors were calculated using the parameter values appropriate to a site at the Raft River, Id. Although these error values serve to illustrate the magnitudes that can be expected from the three general types of errors, extrapolation to other sites should be done using parameter values particular to the area. Three surface temperature algorithms were evaluated: linear Fourier series, finite difference, and Laplace transform. In terms of resulting errors in thermal inertia, the Laplace transform method is the most accurate (260 TIU), the forward finite difference method is intermediate (300 TIU), and the linear Fourier series method the least accurate (460 TIU).