Abstract: A comprehensive presentation is given of virtually all numerical methods that are suitable for the analysis of the various heat transverse and fluid flow problems that occur in research, practice, and university instruction. After reviewing basic methodologies, the following topics are covered: finite difference and finite element methods for parabolic, elliptic, and hyperbolic systems; a comparative appraisal of finite difference versus finite element methods; integral and integrodifferential systems; perturbation methods; Monte Carlo methods; finite analytic methods; moving boundary problems; inverse problems; graphical display methods; grid generation methods; and programing methods for supercomputers.

Abstract: The frequency-dependent characteristics of the microstrip discontinuities have previously been analyzed using full-wave approaches. The time-domain finite-difference (TD-FD) method presented here is an independent approach and is relatively new in its application for obtaining the frequency-domain results for microwave components. The validity of the TD-FD method in modeling circuit components for MMIC CAD applications is established. >

Abstract: We consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains. We demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids. Extensions to nonselfadjoint and parabolic problems are discussed.

Abstract: Two approximate methods, which have not previously been used for structural dynamics problems, are applied to the free vibration analysis of various structural components. The first method is a new version of the complementary energy method. It is shown to be considerably more accurate than the conventional Rayleigh and Rayleigh-Schmidt methods when applied to spatially one-dimensional free vibration problems: prismatic and tapered bars, prismatic beams, and axisymmetric motion of circular membranes. The second method is the differential quadrature method introduced by Bellman and his associates. It is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates.

Abstract: A simple accurate method, which automatically takes full account of the discontinuities in the normal electric field components across any arbitrary distribution of internal dielectric interfaces, is presented for the determination of polarised solutions of the Helmholtz wave equation. The application of the shifted inverse power iteration method to the resulting matrices, enables any required propagation eigenvalue (not necessarily the fundamanetal mode) to be determined, together with its corresponding electric field profile. It is found that the results, which are presented for various semiconductor rib waveguide structures, compare favourably with published vector finite element and scalar results.

Abstract: A direct time-domain finite-difference method is used to recharacterize the microstrip. Maxwell's equations are discretized both in time and space and a Gaussian pulse is used to excite the microstrip. The frequency-domain data are obtained from the Fourier transform of the calculated time-domain field values. Since this method is completely independent of all the above-mentioned investigations, the results can be considered as an impartial verification of the published results. The comparison of the time-domain results and those from the frequency-domain methods has shown the integrity of the time-domain computations. This method is very general and can be applied to model many other microwave components. >

Abstract: The problem of concentrated brine transport arises in the study of transport of pollutants released from a repository in a rock salt formation. An important characteristic of brine, as compared to other solutions normally encountered in groundwater problems, is that it contains a high concentration of solutes. This factor requires special attention in the development of mathematical models for brine transport problems. In this work we discuss certain important physical and mathematical differences between low- and high-concentration situations. In particular, we consider three primary aspects of a model: basic equations, boundary conditions, and numerical techniques. Recognizing the fact that in high-concentration situations, the fluid motion is not independent of the solutes movement, a new formulation of Darcy's and Fick's law are proposed. The basic equations comprise a set of two nonlinear coupled partial differential equations to be solved for the pressure p and the solute mass fraction ω. These equations have to be solved by means of iterative methods. Various possibilities involving finite difference methods have been studied. In one case, after discretizing the equations in a fully implicit way, the Newton-Raphson method has been employed to solve the system of nonlinear difference equations simultaneously. In another case, after removing part of the nonlinearity by a transformation of the dependent variable ω, a procedure of sequential solution of the two equations by successive substitution is employed. It turns out that the latter method is considerably faster than the former one as a result of the quasi-linearization. Finally, considering boundary conditions, it is shown that often they are also nonlinear and coupled. Appropriate conditions for a rock salt boundary and an outflow boundary are developed and their significance in high-concentration situations are discussed. In particular, a nonlinear time-dependent boundary condition at a rock salt boundary is developed which takes into account the process of salt dissolution and cap rock formation.

Abstract: The heat transfer arising due to the movement of a continuous heated plate in processes such as hot rolling and hot extrusion has been studied. Of particular interest were the resulting temperature distribution in the solid and the proper imposition of the boundary conditions at the location where the material emerges from a furnace or die. These considerations are important in the simulation and design of practical systems. A numerical study of the thermal transport process has been carried out, assuming a two-dimensional steady circumstance. The boundary layer equations, as well as full governing equations including buoyancey effects, are solved employing finite difference techniques. The effect of various physical parameters, which determine the temperature and flow fields, is studied in detail. The significance of these results in actual manufacturing processes is discussed.

Abstract: THE MATHEMATICAL MODELS FOR FLUID FLOW SIMULATIONS AT VARIOUS LEVELS OF APPROXIMATION. The Basic Equations of Fluid Dynamics. The Dynamic Levels of Approximation. The Mathematical Nature of the Flow Equations and Their Boundary Conditions. BASIC DISCRETIZATION TECHNIQUES. The Finite Difference Method. The Finite Element Method. Finite Volume Method and Conservative Discretizations. THE ANALYSIS OF NUMERICAL SCHEMES. The Concepts of Consistency, Stability, and Convergence. The Von Neumann Method for Stability Analysis. The Method of the Equivalent Differential Equation for the Analysis of Stability. The Matrix Method for Stability Analysis. THE RESOLUTION OF DISCRETIZED EQUATIONS. Integration Methods for Systems of Ordinary Differential Equations. Iterative Methods for the Resolution of Algebraic Systems. Appendix. Index.

Abstract: Four methods for the calculation of derivatives of vibration mode shapes (eigenvectors) with respect to design parameters are described. These are finite-difference method, modal method, a modified modal method and Nelson's method. The methods are implemented in a general-purpose commercial finite-element program and applied to the following test problems: a cantilever beam and a stiffened cylinder with a cutout. Design variables are a beam tip mass, a beam root height, and specific dimensions of the cylinder model. The methods are compared on the basis of central processor (CP) seconds required to obtain the derivatives, and two of the methods are also evaluated for the rapidity of convergence. Data is presented showing the amount of CP time used to compute the first four eigenvector derivatives for each example problem. A scalar measure of the error in the mode shape derivative is defined, and numerical results illustrating the rapidity of convergence of the approximate derivative to the exact derivative are presented. Results indicate an advantage in using Nelson's method because this method is exact and requires less CP time, especially when derivatives with respect to several design variables are computed.

Abstract: A version of the finite-difference time-domain method adapted to the needs of S-matrix calculations of microwave two-dimensional circuits is presented. The analysis is conducted by simulating the wave propagation in the circuit terminated by matched loads and excited by a matched pulse source. Various aspects of the method's accuracy are investigated. Practical computer implementation of the method is discussed, and an example of its application to an arbitrarily shaped microstrip circuit is presented. It is shown that the method in the proposed form is an effective tool of circuit analysis in engineering applications. The method is compared to two other methods used for a similar purpose, namely the contour integral method and the transmission-line matrix method. >

Abstract: A model describing the propagation of a binary mixture at finite concentration in nonlinear liquid chromatography is discussed. This model consists of two mass balance equations, one for each solute. A finite difference method is used to derive numerical solutions of this set of nonlinear partial differential equation with boundary conditions corresponding to the elution of large concentration bands. These solutions describe the shape of the elution profiles of partially resolved compounds. Although the model used corresponds to ideal chromatography (constant equilibrium between the mobile and stationary phase, i.e., infinite column efficiency), it is possible to simulate the smoothing effect of a finite column efficiency by properly selecting the differential space element in the numerical integration. The numerical solutions appear to converge satisfactorily toward a stable solution of the system of equations provided the Courant-Friedrichs-Lewy (CFL) criterion is met in the choice of the integration parameters. The profiles obtained are very realistic and fare quite well with experimental results retrieved from the literature. Some of the results obtained are discussed in detail.

Abstract: Natural convection in laminar boundary layers along slender vertical cylinders is analyzed for the situation in which the wall temperature T{sub w}(x) varies arbitrarily with the axial coordinate x. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form by a nonsimilar transformation and the resulting system of equations is then solved by a finite difference method in conjunction with the cubic spline interpolation technique. As an example, numerical results were obtained for the case of T{sub w}(x) = T{infinity} + ax{sup n}, a power-law wall temperature variation. They cover Prandtl numbers of 0.1, 0.7, 7, and 100 over a wide range of values of the surface curvature parameter. Representative local Nusselt number as well as velocity and temperature profiles are presented. Correlation equations for the local and average Nusselt numbers are also given.

Abstract: Understanding seismic wave propagation in realistic seafloor environments is essential for many problems in marine seismology. Finite difference methods are becoming increasingly popular in solving propagation problems as the limitations of other techniques which apply only at high frequency or for flat-lying media become fully appreciated. The seafloor problem, a high contrast in Poisson's ratio at a rough sharp interface, is particularly challenging, and many published formulations fail to solve it accurately. The purpose of this paper is to summarize the published finite difference formulations for the elastic wave equation and to outline their applicability to seafloor problems.

Abstract: On presente une methode de calcul des ecoulements stationnaires laminaires a grand nombre de Reynolds. Discretisation des equations de Navier-Stokes par differences finies sur une grille en quinconce et utilisation d'une methode de direction alternee implicite

Abstract: The variable grid central finite difference method was used in the solution of simultaneous heat and moisture transfer (SHMT) equations with variable transport parameters, providing predictions of local moisture content, temperature, and dimensional changes in an infinite slab at any time during processing. The model was applied to the air drying of ocean perch (Sebastes marinus). Good agreement between experimental results and model predications indicated that the method could be successfully applied to SHMT operations in foods involving dimensional changes.

Abstract: Numerical investigations are conducted into steady laminar combined convection flows in vertical parallel plate ducts with asymmetric constant wall temperature boundary conditions. The streamwise diffusion terms in the governing equations are neglected and the resulting parabolic equations are expressed in an implicit finite difference scheme and solved using a marching technique. In certain situations the combination of the size of the ratio {vert bar}Gr/Re{vert bar} and the difference in temperature between the walls of the duct is such that the fully developed flow profile, as they streamwise coordinate tends toward infinity, includes reverse flow in the vicinity of the cold wall. Techniques first used in a previous study are employed in finding, a solution through the whole domain of the fluid for those situations involving reverse flow in the fully developed region. Comparisons with existing data near the duet entrance and far along the duct show very good agreement.

Abstract: A method for the spatial analysis of EEG and EP data, based on the spherical harmonic Fourier expansion (SHE) of scalp potential measurements, is described. This model provides efficient and accurate formulas for: (1) the computation of the surface Laplacian and (2) the interpolation of electrical potentials, current source densities, test statistics and other derived variables. Physiologically based simulation experiments show that the SHE method gives better estimates of the surface Laplacian than the commonly used finite difference method. Cross-validation studies for the objective comparison of different interpolation methods demonstrate the superiority of the SHE over the commonly used methods based on the weighted (inverse distance) average of the nearest three and four neighbor values.

Abstract: Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.

Abstract: In Evans and Sahimi (1988), the Alternating Group Explicit (AGE) method was applied successfully to solve 2 space dimensional problems involving parabolic partial differential equations. Here the operator splitting technique used there is further extended to solve 3 dimensional parabolic problems.