Showing papers on "Discretization published in 1994"

REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems

Abstract: The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i.e., systems of linear equations whose coefficient matrix has the properties that its condition number is very large, and its singular values decay gradually to zero. Such problems typically arise in connection with discretization of Fredholm integral equations of the first kind, and similar ill-posed problems. Some form of regularization is always required in order to compute a stabilized solution to discrete ill-posed problems. The purpose of REGULARIZATION TOOLS is to provide the user with easy-to-use routines, based on numerical robust and efficient algorithms, for doing experiments with regularization of discrete ill-posed problems. By means of this package, the user can experiment with different regularization strategies, compare them, and draw conclusions from these experiments that would otherwise require a major programming effert. For discrete ill-posed problems, which are indeed difficult to treat numerically, such an approach is certainly superior to a single black-box routine. This paper describes the underlying theory gives an overview of the package; a complete manual is also available.

Efficient algorithms for globally optimal trajectories

Abstract: Presents serial and parallel algorithms for solving a system of equations that arises from the discretization of the Hamilton-Jacobi equation associated to a trajectory optimization problem of the following type. A vehicle starts at a prespecified point x/sub 0/ and follows a unit speed trajectory x(t) inside a region in /spl Rfr//sup m/, until an unspecified time T that the region is excited. A trajectory minimising a cost function of the form /spl int//sub 0//sup T/ r(x(t))dt+q(x(T)) is sought. The discretized Hamilton-Jacobi equation corresponding to this problem is usually served using iterative methods. Nevertheless, assuming that the function r is positive, one is able to exploit the problem structure and develop one-pass algorithms for the discretized problem. The first m resembles Dijkstra's shortest path algorithm and runs in time O(n log n), where n is the number of grid points. The second algorithm uses a somewhat different discretization and borrows some ideas from Dial's shortest path algorithm; it runs in time O(n), which is the best possible, under some fairly mild assumptions. Finally, the author shows that the latter algorithm can be efficiently parallelized: for two-dimensional problems and with p processors, its running time becomes O(n/p), provided that p=O(/spl radic/n/log n). >

Solution of a Class of Multistage Dynamic Optimization Problems. 2. Problems with Path Constraints

Abstract: This paper considers the treatment of general equality and inequality path constraints in the context of the control vector parametrization approach to the optimization of dynamic systems described by mixed sets of differential and algebraic equations (DAEs) of index not exceeding 1. Equality path constraints are handled by incorporating as many of them as possible within the DAE system itself without increasing its index. This allows a subset of the control variables to be determined from the solution of the augmented DAE system. The issues involved in establishing an appropriate partitioning of the control variable vector are examined. Inequality path constraints are handled through the combined application of the discretization of these constraints at a finite number of points, and forcing an integral measure of their violation to zero. Numerical experiments demonstrating the advantages of this hybrid technique over its individual components are presented.

Numerical methods for shallow-water flow

Abstract: Preface. 1. Shallow-water flows. 2. Equations. 3. Some properties. 4. Behaviour of solutions. 5. Boundary conditions. 6. Discretization in space. 7. Effect of space discretization on wave propagation. 8. Time integration methods. 9. Effects of time discretization on wave propagation. 10. Numerical treatment of boundary conditions. 11. Three-dimensional shallow-water flow. List of notations. References. Index.

Inexact and preconditioned Uzawa algorithms for saddle point problems

Abstract: Variants of the Uzawa algorithm for solving symmetric indefinite linear systems are developed and analyzed. Each step of this algorithm requires the solution of a symmetric positive- definite system of linear equations. It is shown that if this computation is replaced by an approximate solution produced by an arbitrary iterative method, then with relatively modest requirements on the accuracy of the approximate solution, the resulting inexact Uzawa algorithm is convergent, with a convergence rate close to that of the exact algorithm. In addition, it is shown that preconditioning can be used to improve performance. The analysis is illustrated and supplemented using several examples derived from mixed finite element discretization of the Stokes equations.

A novel two-grid method for semilinear elliptic equations

Abstract: A new finite element discretization technique based on two (coarse and fine) subspaces is presented for a semilinear elliptic boundary value problem The solution of a nonlinear system on the fine space is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse space and a linear system on the fine space It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation As a result, the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

Implicit free‐surface method for the Bryan‐Cox‐Semtner ocean model

Abstract: The surface pressure reformulation of the Bryan-Cox-Semtner global ocean model, which we developed recently, has been further improved by eliminating the rigid-lid approximation in favor of an implicit free-surface method for the barotropic mode. This retains the advantages of the surface pressure formulation and in addition (1) substantially improves the computational efficiency of the barotropic mode; (2) computes surface height directly, allowing comparison with and assimilation of altimetry data (with a rigid-lid case the surface height is accurately computed only in the limit of a steady state); (3) improves accuracy in the computation of long-wavelength Rossby waves and includes the effects of long-wavelength surface gravity waves; (4) provides robustness by selectively damping computational modes; (5) simplifies satisfying global energetic balances required for energetic consistency; and (6) alleviates “checkerboarding” in the surface height or pressure fields. A variable thickness top-layer model is developed to account for the free surface, and a generalized form of implicit time discretization of the barotropic equations is introduced. This discretization is analyzed in detail from the point of view of accuracy, stability, and damping properties, and certain cases are selected for their advantageous properties. The performance of the method is demonstrated on the Connection Machine, CM-200, and the results of numerical simulations are compared using the stream function, rigid-lid surface pressure, and free-surface formulations.

Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow☆

Abstract: The stability analysis, the accuracy and the efficiency of a semi-implicit finite difference scheme for the numerical solution of a three-dimensional shallow water model are presented and discussed. The governing equations are the three-dimensional Reynolds equations in which pressure is assumed to be hydrostatic. The pressure gradient in the momentum equations and the velocities in the vertically integrated continuity equation are discretized with the θ-method, with θ being an implicitness parameter. It is shown that the method is stable for 1 2 ≤ θ ≤ 1, unstable for θ 1 2 and highest accuracy and efficiency is achieved when θ = 1 2 . The resulting algorithm is mass conservative and naturally allows for the simulation of flooding and drying of tidal flats.

Stress-penalty method for unilateral contact problems: mathematical formulation and computational aspects

Abstract: The unilaterality of non penetration constraints, the velocity jumps which occur in case of collisions, the irregularity of the law of dry friction are 'nonsmooth features of the dynamical systems in view. Numerical methods are presented, which face nonsmoothness without resorting to mollifying approximation procedures. A careful formulation of contact laws generates algorihms which, at every step of the time discretization, are ready to face possible collisions on the same footing as permanent contacts

Three-dimensional electromagnetic modeling using finite difference equations: The magnetotelluric example

Abstract: We have developed a robust and efficient finite difference algorithm for computing the magnetotelluric response of general three-dimensional (3-D) models using the minimum residual relaxation method. The difference equations that we solve are second order in H and are derived from the integral forms of Maxwell's equations on a staggered grid. The boundary H field values are obtained from two-dimensional transverse magnetic mode calculations for the vertical planes in the 3-D model. An incomplete Cholesky decomposition of the diagonal subblocks of the coefficient matrix is used as a preconditioner, and corrections are made to the H fields every few iterations to ensure there are no H divergences in the solution. For a plane wave source field, this algorithm reduces the errors in the H field for simple 3-D models to around the 0.01% level compared to their fully converged values in a modest number of iterations, taking only a few minutes of computation time on our desktop workstation. The E fields can then be determined from discretized versions of the curl of H equations.

On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations

Abstract: Convergence estimates in terms of the data are shown for multistep methods applied to non-homogeneous linear initial-boundary value problems. Similar error bounds are derived for a new class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems, e.g., for the single-layer potential equation of the wave equation. In both cases, the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions. These estimates, which are also proved here, depend on bounds of the Laplace transform of the (distributional) convolution kernel outside the stability region scaled by the time stepsize, and on the smoothness of the data.

On Nonlinear Modes of Continuous Systems

Abstract: We use several methods to study the nonlinear modes of one-dimensional continuous systems with cubic inertia and geometric nonlinearities. Invariant manifold and perturbation methods applied to the discretized system and the method of multiple scales applied to the partial-differential equation and boundary conditions are discussed and their equivalence is demonstrated. The method of multiple scales is then applied directly to the partial-differential equation and boundary conditions governing several nonlinear beam problems.

FEM-FDM coupled liquefaction analysis of a porous soil using an elasto-plastic model

Abstract: The phenomenon of liquefaction is one of the most important subjects in Earthquake Engineering and Coastal Engineering. In the present study, the governing equations of such coupling problems as soil skeleton and pore water are obtained through application of the two-phase mixture theory. Using au-p (displacement of the solid phase-pore water pressure) formulation, a simple and practical numerical method for the liquefaction analysis is formulated. The finite difference method (FDM) is used for the spatial discretization of the continuity equation to define the pore water pressure at the center of the element, while the finite element method (FEM) is used for the spatial discretization of the equilibrium equation. FEM-FDM coupled analysis succeeds in reducing the degrees of freedom in the descretized equations. The accuracy of the proposed numerical method is addressed through a comparison of the numerical results and the analytical solutions for the transient response of saturated porous solids. An elasto-plastic constitutive model based on the non-linear kinematic hardening rule is formulated to describe the stress-strain behavior of granular materials under cyclic loading. Finally, the applicability of the proposed numerical method is examined. The following two numerical examples are analyzed in this study: (1) the behavior of seabed deposits under wave action, and (2) a numerical simulation of shaking table test of coal fly ash deposit.

Acoustical finite-difference time-domain simulation in a quasi-cartesian grid

Abstract: The finite‐difference time‐domain (FDTD) approximation can be used to solve acoustical field problems numerically. Mainly because it is a time‐domain method, it has some specific advantages. The basic formulation of the FDTD method uses an analytical grid for the discretization of an unknown field. This is a major disadvantage. In this paper, FDTD equations that allow us to use a nonuniform grid are derived. With this grid, tilted and curved boundaries can be described more easily. This gives a better accuracy to CPU–resource ratio in a number of circumstances. The paper focuses on the new formulation and its accuracy. The problem of automatically generating the mesh in a general situation is not addressed. Simulations using quasi‐Cartesian grids are compared to Cartesian grid results.

Normalized variable and space formulation methodology for high-resolution schemes

Abstract: The normalized variable formulation (NVF) methodology of Leonard [1] provides the proper framework for the development and analysis of high-resolution convection-diffusion schemes, which combine the accuracy of higher-order schemes with the stability and boundedness of the first-order upwind scheme. However, in its current form the NVF methodology helps in deriving connective schemes for uniformly or nearly uniformly discretized spaces. To remove this shortcoming, a new, normalized variable and space formulation (NVSF) methodology is developed. In the newly developed technique, spatial parameters are introduced so as to extend the applicability of the NVF methodology to nonuniformly discretized domains. Furthermore, the required conditions for accuracy and boundedness of connective schemes on nonuniform grids are also derived. Several schemes formulated using NVF are generalized to nonuniform grids using the suggested method. Both formulations are tested on nonuniform grids by solving two problem...

Global spherical harmonic analysis by least‐squares and numerical quadrature methods in historical perspective

Abstract: SUMMARY Methods of global spherical harmonic analysis of discrete data on a sphere are placed in a historical context. The paper concentrates on the loss of orthogonality in the direction of latitude, due to the transition from continuous to discretized functions. Special attention is paid to Neumann's (1838) solution to this problem. By recasting the formulae of spherical harmonic analysis into matrix-vector notation, both least-squares solutions and quadrature methods are represented in a general framework of weighted least squares. It is also shown that the two-step formulation of global spherical harmonic computation was applied already by Neumann (1838) and Gauss (1839). Computational modifications to Neumann's method are reviewed as well.

Modelling tides in the western North Atlantic using unstructured graded grids

Abstract: This paper describes grid convergence studies for a finite-element-based tidal model of the western North Atlantic, Gulf of Mexico and Caribbean. The very large computational domain used for this tidal model encompasses both the coastal and the deep ocean and facilitates the specification of boundary conditions. Due to the large variability in depths as well as scale content of the tides within the model domain, an optimal unstructured graded grid with highly variable finite element areas is developed which significantly reduces the size of the discrete problem while improving the accuracy of the computations. The convergence studies include computations for a sequence of regularly discretized grids ranging from a very coarse 1.6° × 1.6° mesh to a very fine 6′ × 6′ to 12′ × 12′ mesh as well as unstructured graded grids with resolutions varying between 1.6° and 5′ within each mesh. Resolution requirements are related to depth, gradients in topography as well as the resolution of the coastal boundary. The final optimal graded grid has a tidal response which is comparable to that of the finest regular grid in most regions. The optimal graded grid is then forced with Schwiderski's (1979, 1980, 1981a–g) global model on the open ocean boundary and tidal potential forcing functions within the interior domain. The structure of the tides is examined, computed co-tidal charts are presented and comparisons are made between the computed results and field data at 77 stations within the model domain. DOI: 10.1034/j.1600-0870.1994.00007.x

Monotone multigrid methods for elliptic variational inequalities I

Abstract: We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify well-known relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse-grid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case.

The scattering of sound waves by a vortex: numerical simulations and analytical solutions

Abstract: The scattering of plane sound waves by a vortex is investigated by solving the compressible Navier-Stokes equations numerically, and analytically with asymptotic expansions. Numerical errors associated with discretization and boundary conditions are made small by using high-order-accurate spatial differentiation and time marching schemes along with accurate non-reflecting boundary conditions. The accuracy of computations of flow fields with acoustic waves of amplitude five orders of magnitude smaller than the hydrodynamic fluctuations is directly verified. The properties of the scattered field are examined in detail. The results reveal inadequacies in previous vortex scattering theories when the circulation of the vortex is non-zero and refraction by the slowly decaying vortex flow field is important. Approximate analytical solutions that account for the refraction effect are developed and found to be in good agreement with the computations and experiments. The prediction of the sound produced by turbulent flow requires a detailed knowledge of acoustic source terms. Direct computation of both the acoustic sources and far-field sound using the unsteady Navier-Stokes equations allows direct validation of aeroacoustic theories. In a recent review by Crighton (1988), the difficulties involved in direct computations of aeroacoustic fields are discussed. These include: the large extent of the acoustic field compared with the flow field; the small energy of the acoustic field compared to the flow field; and the possibility that numerical discretization may introduce a significant sound source due to the acoustic inefficiency of low-Mach-number flows. In order to address these difficulties, Crighton proposed that direct computations be performed on elementary model aeroacoustic problems whose physics are well understood. For this reason, and to validate our numerical scheme for direct computation of aeroacoustic problems, we investigate the scattering of sound waves by a compressible viscous vortex. This problem has received significant attention, and thus provides a large database of theory, numerics and experiment with which detailed comparisons may be made. Yet there is significant disagreement amongst the various theories, which has not yet been fully rectified. Therefore, the purpose of the current work is twofold: to validate our numerical scheme for direct computation of aeroacoustic problems using the unsteady Navier-Stokes equations, and to investigate the scattering of sound waves by a compressible viscous vortex.

The Multiple reciprocity boundary element method

Abstract: The boundary element method (BEM) is a numerical technique which is now emerging as a viable alternative to finite difference and finite element methods for solving a wide range of engineering problems. The main advantage of the BEM is its unique ability to confine the dependence of the problem solution to the boundary values only. However, the main drawback of BEM occurs in problems such as those with body forces, time-dependent effects or non-linearities. In these cases, the domain integrals, that appear in the integral equation, can be evaluated by using cell integration. Although this technique is effective in general, it affects the overall efficiency of the BEM and detracts from its elegance owing to the additional internal discretization. In an effort to avoid the internal discretization, many different approaches have been developed. One of the more successful, is the multiple reciprocity method (MRM). This method employs a sequence of higher-order fundamental solutions which permit the application of the reciprocity theorem recurrently. This book presents recent developments in MRM as it applies to BEM, with contributions from researchers worldwide.

A field theoretical derivation of TLM

Abstract: A field theoretical foundation of the TLM method is presented in this paper. In the derivation of the condensed symmetric TLM node, the Method of Moments is applied to Maxwell's equations to obtain discretized field equations. It is shown that the traditional mapping between wave amplitudes and electric and magnetic field components incorporates serious problems. Therefore, a new mapping between the wave amplitudes and the electric and magnetic field components is introduced. Applying the new mapping to the discretized field equations, the fundamental equations of the three-dimensional TLM method with condensed symmetric node are derived from Maxwell's equations. >

Adaptive multilevel methods for obstacle problems

Abstract: The authors consider the discretization of obstacle problems for second-order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned conjugate gradient iterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement semilocal and local a posteriors error estimates are derived, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.