A finite difference solution to a mixed boundary value problem for Laplace's equation
Finite difference solution to mixed boundary value problem for Laplace equation
Chu, S. C.↗
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An analysis of finite-difference techniques applied to equations governing convective transfer
Finite difference analysis of hyperbolic partial differential equations
Lomax, H.↗
Time-dependent viscous incompressible Navier-Stokes equations - The finite difference Galerkin formulation and streamfunction algorithms
Numerical techniques are developed to solve the Navier-Stokes equations for unsteady incompressible flow. The extension of the finite-difference Galerkin (FDG) method of Stephens et al. (1984) to the continuous-time case in two or three space dimensions is explained, and the numerical implementation of the method is discussed with particular attention to the staggered-MAC-grid primitive-variable discretization, the application of discrete mass balance to avoid problems inherent in FDG schemes, the direct interpretation of the FDG expansion variables as a discrete streamfunction, and a mass-balance approach to two-dimensional problems with throughflow or obstacles. Numerical results are presented graphically for the evolution of asymptotic steady flow in a driven cavity at Reynolds number 400, 1000, or 3200; good agreement with published experimental data is demonstrated, with accurate predictions of secondary-vortex formation from wall bubble recirculations at Reynolds number 1000.
Goodrich, John W.↗
A method for the numerical integration of coupled first-order differential equations with greatly different time constants.
Numerical integration of coupled first order ODE OF greatly differing time constant
DIFFERENTIAL EQUATION↗
Numerical techniques for the solution of symmetric positive linear differential equations
Finite difference method for solving symmetric positive linear differential equations
LINEAR EQUATION↗
An extension of MacCormack's method for flows with higher-order equations and in different configurations
The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives: (1) Korteweg-de Vries equation with a term of third-order derivative, (2) a system of nonlinear equations governing nonsteady one-dimensional plasma flow in cylindrical coordinate, (3) equations of solar wind. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed.
Ying, S. J.↗
Wing-Body Aeroelasticity Using Finite-Difference Fluid/Finite-Element Structural Equations on Parallel Computers
In recent years significant advances have been made for parallel computers in both hardware and software. Now parallel computers have become viable tools in computational mechanics. Many application codes developed on conventional computers have been modified to benefit from parallel computers. Significant speedups in some areas have been achieved by parallel computations. For single-discipline use of both fluid dynamics and structural dynamics, computations have been made on wing-body configurations using parallel computers. However, only a limited amount of work has been completed in combining these two disciplines for multidisciplinary applications. The prime reason is the increased level of complication associated with a multidisciplinary approach. In this work, procedures to compute aeroelasticity on parallel computers using direct coupling of fluid and structural equations will be investigated for wing-body configurations. The parallel computer selected for computations is an Intel iPSC/860 computer which is a distributed-memory, multiple-instruction, multiple data (MIMD) computer with 128 processors. In this study, the computational efficiency issues of parallel integration of both fluid and structural equations will be investigated in detail. The fluid and structural domains will be modeled using finite-difference and finite-element approaches, respectively. Results from the parallel computer will be compared with those from the conventional computers using a single processor. This study will provide an efficient computational tool for the aeroelastic analysis of wing-body structures on MIMD type parallel computers.
Byun, Chansup↗
A critical examination of the flap-lag dynamics of helicopter rotor blades in hover and in forward flight
A critical examination of flap-lag stability of a centrally hinged, spring-restrained rigid blade in both hover and forward flight is presented. Several differences in the equations of motion for blade flap-lag stability in the existing literature are identified. A rigorous and systematic development of these equations for a rigid articulated blade in forward flight shows the existence of some linear aerodynamic coupling terms associated with blade steady-state flapping and lagging in the perturbation equations. The differences identified are shown to be associated with the order in which the flap and lag transformations are taken in developing the equations of motion. The implications of these differences on stability are examined, and it is shown that the pitch-lag coupling terms associated with a flap-lag hinge transformation sequence have a marked influence on flap-lag stability. Some qualitative considerations on the role of the assumed transformation sequence in the development of the flap-lag equations for a hingeless elastic blade are also given. On the basis of these considerations, it is shown that aerodynamic coupling terms associated with blade steady-state flapping and lagging similar to those found for the rigid blade will also appear in the equations for the elastic blade.
Kaza, K. R. V.↗
Changing stepsize in the integration of differential equations using modified divided differences
Multistep methods for solving differential equations based on numerical integration formulas or numerical differentiation formulas (for stiff equations) require special provision for changing the stepsize. New algorithms are given which make the use of modified divided differences an attractive way to carry out the change in stepsize for such methods. Error estimation and some of the important factors in stepsize selection and the selection of integration order are also considered.
Krogh, F. T.↗
Newton's method applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations
Finite-difference approximations for steady-state compressible Navier-Stokes equations, whose two spatial dimensions are written in generalized curvilinear coordinates and strong conservation-law form, are presently solved by means of Newton's method in order to obtain a lifting-airfoil flow field under subsonic and transonnic conditions. In addition to ascertaining the computational requirements of an initial guess ensuring convergence and the degree of computational efficiency obtainable via the approximate Newton method's freezing of the Jacobian matrices, attention is given to the need for auxiliary methods assessing the temporal stability of steady-state solutions. It is demonstrated that nonunique solutions of the finite-difference equations are obtainable by Newton's method in conjunction with a continuation method.
Bailey, Harry E.↗
Methods of Attenuation Correction for Dual-Wavelength and Dual-Polarization Weather Radar Data
In writing the integral equations for the median mass diameter and number concentration, or comparable parameters of the raindrop size distribution, it is apparent that the forms of the equations for dual-polarization and dual-wavelength radar data are identical when attenuation effects are included. The differential backscattering and extinction coefficients appear in both sets of equations: for the dual-polarization equations, the differences are taken with respect to polarization at a fixed frequency while for the dual-wavelength equations, the differences are taken with respect to frequency at a fixed polarization. An alternative to the integral equation formulation is that based on the k-Z (attenuation coefficient-radar reflectivity factor) parameterization. This-technique was originally developed for attenuating single-wavelength radars, a variation of which has been applied to the TRMM Precipitation Radar data (PR). Extensions of this method have also been applied to dual-polarization data. In fact, it is not difficult to show that nearly identical equations are applicable as well to dualwavelength radar data. In this case, the equations for median mass diameter and number concentration take the form of coupled, but non-integral equations. Differences between this and the integral equation formulation are a consequence of the different ways in which attenuation correction is performed under the two formulations. For both techniques, the equations can be solved either forward from the radar outward or backward from the final range gate toward the radar. Although the forward-going solutions tend to be unstable as the attenuation out to the range of interest becomes large in some sense, an independent estimate of path attenuation is not required. This is analogous to the case of an attenuating single-wavelength radar where the forward solution to the Hitschfeld-Bordan equation becomes unstable as the attenuation increases. To circumvent this problem, the equations can be expressed in the form of a final-value problem so that the recursion begins at the far range gate and proceeds inward towards the radar. Solving the problem in this way traditionally requires estimates of path attenuation to the final gate: in the case of orthogonal linear polarizations, the attenuations at horizontal and vertical polarizations (same frequency) are required while in the dual-wavelength case, attenuations at the two frequencies (same polarization) are required.
Meneghini, R.↗
Examination of the flap-lag stability of rigid articulated rotor blades
A critical examination of flap-lag stability of a centrally hinged, spring-restrained rigid blade in both hover and forward flight is presented. Several differences in the equations of motion for blade flap-lag stability in the existing literature are identified. A rigorous and systematic development of these equations for a rigid articulated blade in forward flight shows the existence of some linear aerodynamic coupling terms associated with blade steady-state flapping and lagging in the perturbation equations. The differences identified are shown to be associated with whether or not the lag hinge flaps with the blade. The implications of these differences on stability are examined, and it is shown that the pitch-lag coupling terms associated with a hinge arrangement in which the lag hinge flaps with the blade have a marked influence on flap-lag stability, depending on the system parameters.
Kaza, K. R. V.↗
Transient wave propagation in composite plates due to impact
Stress-wave propagation in a composite plate due to an impact has been examined using anisotropic elasticity theory. The plate is modelled as a number of identical anisotropic layers whose elastic moduli are given. Mindlin's (1959) approximate theory of plates is then applied to each layer to obtain a set of difference-differential equations of motion by using the interlaminar stresses and displacements as explicit variables. Dispersion relationships for harmonic waves are found when traction-free boundary conditions are applied to both surfaces of the plate. The difference-differential equations are reduced to difference equations via integral transforms. With given impact boundary conditions these equations are solved for an arbitrary number of layers in the plate, and the transient propagation of a stress wave is calculated by means of a fast Fourier transform algorithm.
Sung Kim, B.↗
Construction of stable explicit finite-difference schemes for Schroedinger type differential equations
A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented.
Mickens, Ronald E.↗
A conservative implicit finite difference algorithm for the unsteady transonic full potential equation
An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form. Computational efficiency is maintained by use of approximate factorization techniques. The numerical algorithm is first order in time and second order in space. A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm.
Steger, J. L.↗
An Exponential Finite Difference Technique for Solving Partial Differential Equations
An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.
Handschuh, Robert F.↗
Method and models for R-curve instability calculations
This paper presents a simple method for performing elastic R-curve instability calculations. For a single material-structure combination, the calculations can be done on some pocket calculators. On microcomputers and larger, it permits the development of a comprehensive program having libraries of driving force equations for different configurations and R-curve model equations for different materials. The paper also presents several model equations for fitting to experimental R-curve data, both linear elastic and elastoplastic. The models are fit to data from the literature to demonstrate their viability.
Orange, Thomas W.↗
Method and models for R-curve instability calculations
This paper presents a simple method for performing elastic R-curve instability calculations. For a single material-structure combination, the calculations can be done on some pocket calculators. On microcomputers and larger, it permits the development of a comprehensive program having libraries of driving force equations for different configurations and R-curve model equations for different materials. The paper also presents several model equations for fitting to experimental R-curve data, both linear elastic and elastoplastic. The models are fit to data from the literature to demonstrate their viability.
Orange, Thomas W.↗