Showing papers on "Finite difference published in 1993"

Direct numerical simulation of turbulent flow in a square duct

Abstract: A direct numerical simulation of a fully developed, low-Reynolds-number turbulent flow in a square duct is presented. The numerical scheme employs a time-splitting method to integrate the three-dimensional, incompressible Navier-Stokes equations using spectral/high-order finite-difference discretization on a staggered mesh ; the nonlinear terms are represented by fifth-order upwind-biased finite differences. The unsteady flow field was simulated at a Reynolds number of 600 based on the mean friction velocity and the duct width, using 96 x 101 x 101 grid points. Turbulence statistics from the fully developed turbulent field are compared with existing experimental and numerical square duct data, providing good qualitative agreement. Results from the present study furnish the details of the corner effects and near-wall effects in this complex turbulent flow field; also included is a detailed description of the terms in the Reynolds-averaged streamwise momentum and vorticity equations. Mechanisms responsible for the generation of the stress-driven secondary flow are studied by quadrant analysis and by analysing the instantaneous turbulence structures. It is demonstrated that the mean secondary flow pattern, the distorted isotachs and the anisotropic Reynolds stress distribution can be explained by the preferred location of an ejection structure near the corner and the interaction between bursts from the two intersecting walls. Corner effects are also manifested in the behaviour of the pressure-strain and velocity-pressure gradient correlations.

The global dynamics of discrete semilinear parabolic equations

Abstract: A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as $t \to \infty $ The dynamical properties of various finite difference and finite element schemes for the equations are analysed The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied

A numerical method for solving the neutrino Boltzmann equation coupled to spherically symmetric stellar core collapse

Abstract: We present a numerical method to solve the neutrino Boltzmann equation coupled to stellar core collapse and lepton conservation, with no approximations made to the neutrino scattering kernels. Spherical symmetry is assumed. Our finite differencing of the Boltzmann equation is similar to its finite difference representation in the discrete ordinates method, but our auxiliary equations relating the zone-center and zone-edge distribution functions are different. We also differ from the discrete ordinates method in the way we solve the finite differenced Boltzmann equation

Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics

Abstract: Most of the recently proposed computational methods for solving partial differential equations on multiprocessor architectures stem from the 'divide and conquer' paradigm and involve some form of domain decomposition. For those methods which also require grids of points or patches of elements, it is often necessary to explicitly partition the underlying mesh, especially when working with local memory parallel processors. In this paper, a family of cost-effective algorithms for the automatic partitioning of arbitrary two- and three-dimensional finite element and finite difference meshes is presented and discussed in view of a domain decomposed solution procedure and parallel processing. The influence of the algorithmic aspects of a solution method (implicit/explicit computations), and the architectural specifics of a multiprocessor (SIMD/MIMD, startup/transmission time), on the design of a mesh partitioning algorithm are discussed. The impact of the partitioning strategy on load balancing, operation count, operator conditioning, rate of convergence and processor mapping is also addressed. Finally, the proposed mesh decomposition algorithms are demonstrated with realistic examples of finite element, finite volume, and finite difference meshes associated with the parallel solution of solid and fluid mechanics problems on the iPSC/2 and iPSC/860 multiprocessors.

Calculation of Stiffness and Damping Coefficients for Elastically Supported Gas Foil Bearings

Abstract: The stiffness and damping coefficients of an elastically supported gas foil bearing are calculated. A perfect gas is used as the lubricant, and its behavior is described by the Reynolds equation. The structural model consists only of an elastic foundation. The fluid equations and the structural equations are coupled. A perturbation method is used to obtain the linearized dynamic coefficient equations. A finite difference formulation has been developed to solve for the four stiffness and the four damping coefficients. The effect of the bearing compliance on the dynamic coefficients is discussed in this paper.

Theory of eddy current inversion

Abstract: The inverse eddy current problem can be described as the task of reconstructing an unknown distribution of electrical conductivity from eddy‐current probe impedance measurements recorded as a function of probe position, excitation frequency, or both. In eddy current nondestructive evaluation, this is widely recognized as a central theoretical problem whose solution is likely to have a significant impact on the characterization of flaws in conducting materials. Because the inverse problem is nonlinear, we propose using an iterative least‐squares algorithm for recovering the conductivity. In this algorithm, the conductivity distribution sought minimizes the mean‐square difference between the predicted and measured impedance values. The gradient of the impedance plays a fundamental role since it tells us how to update the conductivity in such a way as to guarantee a reduction in the mean‐square difference. The impedance gradient is obtained in analytic form using function‐space methods. The resulting expression is independent of the type of discretization ultimately chosen to approximate the flaw, and thus has greater generality than an approach in which discretization is performed first. The gradient is derived from the solution to two forward problems: an ordinary and an ‘‘adjoint’’ problem. In contrast, a finite difference computation of the gradient requires the solution of multiple forward problems, one for each unknown parameter used in modeling the flaw. Two general types of inverse problems are considered: the reconstruction of a conductivity distribution, and the reconstruction of the shape of an inclusion or crack whose conductivity is known or assumed to be zero. A layered conductor with unknown layer conductivities is treated as an example of the first type of inversion problem. An ellipsoidal crack is presented as an example of the second type of inversion problem.

Numerical solution of an evolution equation with a positive-type memory term

Abstract: We study the numerical solution of an initial-boundary value problem for a Volterra type integro-differential equation, in which the integral operator is a convolution product of a positive-definite kernel and an elliptic partial-differential operator. The equation is discretised in space by the Galerkin finite-element method and in time by finite differences in combination with various quadrature rules which preserve the positive character of the memory term. Special attention is paid to the case of a weakly singular kernel. Error estimates are derived and numerical experiments reported.

A brief description of a new numerical framework for solving conservation laws — The method of space-time conservation element and solution element

Abstract: A new numerical method for solving conservation laws is being developed. It differs substantially from the well established methods, i.e., finite difference, finite volume, finite element, and spectral methods, in both concept and methodology. It is much simpler than a typical high resolution method. No flux limiter or any technique related to characteristics is involved. No artificial viscosity or smoothing is introduced, and no moving mesh is used. Yet this method is capable of generating highly accurate shock tube solutions. The slight numerical overshoot and/or oscillations generated can be removed if a simple averaging formula initially used is replaced by a weighted formula. This modification has little effect on other parts of the solution. Because of its simplicity, generalization of this new method for multi-dimensional problems is straightforward.

The node-centred finite volume approach: bridge between finite differences and finite elements

Abstract: The paper describes a node-centred finite volume approach for the solution of the Euler equations. This approach leads to discretizations which result from both classical finite volume (finite difference for Cartesian grids) and finite element theories; this depends only on the method used to build the control volume surrounding a node of the grid. In particular, the method proposed here is able to deal with structured, unstructured and hybrid grids. Solutions for transonic flow past an airfoil are presented which illustrate the versatility of the method.

A Method of “Exact” Numerical Differentiation for Error Elimination in Finite-Element-Based Semi-Analytical Shape Sensitivity Analyses*

Abstract: The traditional, simple numerical differentiation of finite-element stiffness matrices by a forward difference scheme is the source of severe error problems that have been reported recently for certain problems of finite-element-based, semi-analytical shape design sensitivity analysis. In order to develop a method for elimination of such errors, without a sacrifice of the simple numerical differentiation and other main advantages of the semi-analytical method, the common mathematical structure of a broad range of finite-element stiffness matrices is studied in this paper. This study leads to the result that element stiffness matrices can generally be expressed in terms of a class of special scalar functions and a class of matrix functions of shape design variables that are defined such that the members of the classes admit “exact” numerical differentiation (exact up to round-off error) by means of very simple correction factors to upgrade standard computationally inexpensive first-order finite di...

Numerical Simulations of Large Water Waves due to Landslides

Abstract: A mathematical model based on the hydrodynamic shallow water equations is developed for numerical simulation of slide generated waves in fjords. The equations are solved numerically by a finite difference technique. To examine the performance of the numerical model we have simulated the slide catastrophe in Tafjord, western Norway, 1934. The predicted runup heights are in good agreement with measured runup heights. The effects of wave amplification are estimated in runup zones with gentle beach slopes. The model results reveal wave energy trapping due to the fjord geometry. This causes standing wave oscillations in accordance with the observations.

Testing four elastic finite-difference schemes for behavior at discontinuities

Abstract: Three second-order and one fourth-order finite-difference schemes are theoretically and numerically investigated for their behavior at elastic discontinuities. One of them is extended with new formulas for a flat free surface. Two of the schemes are consistent with the stress-continuity condition for P-SV waves at discontinuities coinciding with horizontal (or vertical) grid lines; none of them is consistent at diagonal discontinuities. Despite these significant theoretical differences, the numerical results from all four schemes are very similar. Moreover, the results compare well with semianalytic solutions for three different models. A practical conclusion is that the recent finite-difference schemes are by no means free from the accuracy problems at elastic discontinuities. Nevertheless, the schemes provide synthetic seismograms whose differences are well below the level normally introduced by structural and focal uncertainties.

A finite-volume Eulerian-Lagrangian Localized Adjoint Method for solution of the advection-dispersion equation

Abstract: A new mass-conservative method for solution of the one-dimensional advection-dispersion equation is derived and discussed. Test results demonstrate that the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) outperforms standard finite-difference methods, in terms of accuracy and efficiency, for solute transport problems that are dominated by advection. For dispersion-dominated problems, the performance of the method is similar to that of standard methods. Like previous ELLAM formulations, FVELLAM systematically conserves mass globally with all types of boundary conditions. FVELLAM differs from other ELLAM approaches in that integrated finite differences, instead of finite elements, are used to approximate the governing equation. This approach, in conjunction with a forward tracking scheme, greatly facilitates mass conservation. The mass storage integral is numerically evaluated at the current time level, and quadrature points are then tracked forward in time to the next level. Forward tracking permits straightforward treatment of inflow boundaries, thus avoiding the inherent problem in backtracking, as used by most characteristic methods, of characteristic lines intersecting inflow boundaries. FVELLAM extends previous ELLAM results by obtaining mass conservation locally on Lagrangian space-time elements. Details of the integration, tracking, and boundary algorithms are presented. Test results are given for problems in Cartesian and radial coordinates.

Preconditioning and the Limit to the Incompressible Flow Equations

Abstract: The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations are considered. The relation between them for both the continuous problem and the finite difference approximation is also considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Hence, the steady state of the preconditioned system is the same as the steady state of the original system. For finite difference methods the preconditioning can change and improve the steady state solutions. An application to flow around an airfoil is presented.

9 – Numerical Modeling of Discontinua

Abstract: Publisher Summary This chapter focuses on numerical modeling of discontinua. It discusses the important aspects in the modeling of systems of discrete bodies—both physical and numerical aspects— and explains the diversity of applications. A discontinuous medium is distinguished from a continuous one by the existence of contacts or interfaces between the discrete bodies that comprise the system. An important component of any discrete element method is the formulation for representing contacts. In the direct method of introducing deformability, the body is divided into internal elements or boundary elements to increase the number of degrees of freedom. The possible complexity of deformation depends on the number of elements into which the body is divided. A complex deformation pattern can also be achieved in a body by the superposition of several mode shapes for the whole body. In the naive approach, each body is checked against every other body to determine if contact can occur. Many finite element, boundary element and Lagrangian finite difference programs have interface elements or slide lines that enable them to model a discontinuous material to some extent.

A generalized two-dimensional analytical solute transport model in bounded media for flux-type finite multiple sources

Abstract: This paper presents a generalized two-dimensional analytical solute transport model with unidirectional flow field from time- and space-dependent sources in a bounded homogeneous medium using the flux-type (or third-type or Cauchy) boundary condition at the inlet location of the medium. The solute transport equation incorporates terms accounting for advection, dispersion, linear equilibrium sorption, and first-order decay. General solutions were determined for arbitrary input boundary conditions with the help of Fourier analysis and Laplace transform techniques. After presenting a generally applicable solution, special solutions are given for a single and two finite sources having different inlet solute flux distributions. The expressions for convective-dispersive flux components are also derived. Expressions for concentration distribution and convective-dispersive flux components for steady state transport case are also presented. Comparison of the results of this analytical model and a finite difference numerical model showed very good agreement. The solutions may be used for predicting solute concentrations in a unidirectional porous media flow field as well as in rivers and canals. The solutions may also be used for verification of more comprehensive numerical models, and laboratory or field determination of solute transport parameters.

A least‐squares finite element method for time‐dependent incompressible flows with thermal convection

Abstract: SUMMARY The time-dependent Navier-Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity-pressure-vorticity-temperature-heat-flux (u-P-w- T-q) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the 12-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to lo6, lid-driven cavity flow at Reynolds numbers up to lo4 and flow over a square obstacle at Reynolds number 200, are presented to validate the method. The past decade has witnessed a great deal of progress in the area of computational fluid dynamics. Numerous flow problems have been successfully solved by finite difference, finite volume and finite element methods. Most finite element methods are based on the Galerkin method, 9 ' the Taylor-Galerkin method and the Petrov-Galerkin method. 3-5 Mixed-order interpolation and penalty approach are commonly used in these methods. It is well known that these methods often lead to large, sparse, unsymmetric linear systems which are difficult to solve numerically. This explains why finite element analysis for three-dimensional fluid flow problems is not a common practice. To overcome this difficulty, we propose and develop a Least-Squares Finite Element Method (LSFEM) for time-dependent incompressible flow problems. The linear systems resulting from the discretization of LSFEM are always symmetrical and positive-definite. Therefore, they can be solved more easily and efficiently. This is the main reason for investigating the least-squares finite element approach. Least-squares finite element methods have already been applied with some success to com- pressible Euler and hyperbolic equations. Jiang and Carey6- ' and Jiang and Povinelli' used an implicit method for compressible flows. To further the capabilities of the method, Lefebvre et al.' applied unstructured triangular meshes to compressible flow problems. For transient advection problems, Donea and Quartapelle lo classified four different LSFEM approaches: characteristic LSFEM proposed by Li," LSFEM by Carey and Jiang," Taylor LSFEM by Park and Liggett l3 and space-time LSFEM by Nguyen and Reynen.I4

A numerical treatment of geodynamic : viscous flow problems involving the advection of material interfaces

Abstract: Effective numerical treatment of multicomponent viscous flow problems involving the advection of sharp interfaces between materials of differing physical properties requires correction techniques to prevent spurious diffusion and dispersion. We develop a particular algorithm, based on modern shock-capture techniques, employing a two-step nonlinear method. The first step involves the global application of a high-order upwind scheme to a hyperbolic advection equation used to model the distribution of distinct material components in a flow field. The second step is corrective and involves the application of a global filter designed to remove dispersion errors that result from the advection of discontinuities (e.g., material interfaces) by high-order, minimally dissipative schemes. The filter introduces no additional diffusion error. Nonuniform viscosity across a material interface is allowed for by the implementation of a compositionally weighted-inverse interface viscosity scheme. The combined method approaches the optimal accuracy of modern shock-capture techniques with a minimal increase in computational time and memory. A key advantage of this method is its simplicity to incorporate into preexisting codes be they finite difference, element, or volume of two or three dimensions.

Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory

Abstract: A general framework is proposed for proving convergence of high-order accurate difference schemes for the approximation of conservation laws with several space variables. The standard approach deduces compactness from a BV (bounded variation) stability estimate and Helly's theorem. In this paper, it is proved that an a priori estimate weaker than a BV estimate is sufficient. The method of proof is based on the result of uniqueness given by Di Perna in the class of measure-valued solutions. Several general theorems of convergence are given in the spirit of the Lax-Wendroff theorem. This general method is then applied to the high-order schemes constructed with the modified-flux approach.

Mathematical Modelling of Heat Transfer Phenomena in Continuous Casting of Steel

Abstract: A steady-state three-dimensional heat flow model based on the concept of artificial effective thermal conductivity has been developed. On the basis of available literature information, boundary conditions to the governing heat flow equation have been applied, and the equation was solved via the control-volume based finite difference procedure. The model is sufficiently general and can be applied to various geometrical shapes of relevance to continuous casting of steel. Sensitivity of the predicted results to various numerical approximation including grid configurations, as well as to other modelling parameters such as axial conduction, mushy zone modelling procedure, choice of value of Keff have been extensively studied. It has been shown that assumptions and numerical procedures influence the computed results significantly. Finally, numerical predictions have been compared with three sets of experimental measurements reported in literature on shell thickness in industrial casters. In contrast to some earlier claims, these indicated only poor to moderate agreement between model predictions and experimental results.

WAVELET BASED GREEN'S FUNCTION APPROACH TO 2D PDEs

Abstract: We describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multigrid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. To demonstrate the wavelet technique, we consider Poisson's equation in two dimensions. By comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund and others. The convergence of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method.

Rosenbrock methods for partial differential equations and fractional orders of convergence

Abstract: The authors apply Rosenbrock methods for the solution of initial value problems for linear partial differential equations $u_t (x,t) = \mathcal {L}(x,\partial )u(x,t) + f(x,t)$. Under weak assumptions on the operator $\mathcal {L}$ and the source function f, a sharp lower bound for the order of convergence of these numerical methods is given. It is further shown that the order of convergence is, in general, fractional. It depends on the $L^r$-norm used to estimate the global error. This analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques. Numerical examples illustrate the results.