Showing papers on "Discretization published in 1997"

A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers

Abstract: The numerical implementation of an ocean model based on the incompressible Navier Stokes equations which is designed for studies of the ocean circulation on horizontal scales less than the depth of the ocean right up to global scale is described. A "pressure correction" method is used which is solved as a Poisson equation for the pressure field with Neumann boundary conditions in a geometry as complicated as that of the ocean basins. A major objective of the study is to make this inversion, and hence nonhydrostatic ocean modeling, efficient on parallel computers. The pressure field is separated into surface, hydrostatic, and nonhydrostatic components. First, as in hydrostatic models, a two-dimensional problem is inverted for the surface pressure which is then made use of in the three-dimensional inversion for the nonhydrostatic pressure. Preconditioned conjugate-gradient iteration is used to invert symmetric elliptic operators in both two and three dimensions. Physically motivated preconditioners are designed which are efficient at reducing computation and minimizing communication between processors. Our method exploits the fact that as the horizontal scale of the motion becomes very much larger than the vertical scale, the motion becomes more and more hydrostatic and the three- dimensional Poisson operator becomes increasingly anisotropic and dominated by the vertical axis. Accordingly, a preconditioner is used which, in the hydrostatic limit, is an exact integral of the Poisson operator and so leads to a single algorithm that seamlessly moves from nonhydrostatic to hydrostatic limits. Thus in the hydrostatic limit the model is "fast," competitive with the fastest ocean climate models in use today based on the hydrostatic primitive equations. But as the resolution is increased, the model dynamics asymptote smoothly to the Navier Stokes equations and so can be used to address small- scale processes. A "finite-volume" approach is employed to discretize the model in space in which property fluxes are defined normal to faces that delineate the volumes. The method makes possible a novel treatment of the boundary in which cells abutting the bottom or coast may take on irregular shapes and be "shaved" to fit the boundary. The algorithm can conveniently exploit massively parallel computers and suggests a domain decomposition which allocates vertical columns of ocean to each processing unit. The resulting model, which can handle arbitrarily complex geometry, is efficient and scalable and has been mapped on to massively parallel multiprocessors such as the Connection Machine (CM5) using data-parallel FORTRAN and the Massachusetts Institute of Technology data-flow machine MONSOON using the implicitly parallel language Id. Details of the numerical implementation of a model which has been designed for the study of dynamical processes in the ocean from the convective, through the geostrophic eddy, up to global scale are set out. The "kernel" algorithm solves the incompressible Navier Stokes equations on the sphere, in a geometry as complicated as that of the ocean basins with ir- regular coastlines and islands. (Here we use the term "Navier Stokes" to signify that the full nonhydrostatic equations are being employed; it does not imply a particular constitutive relation. The relevant equations for modeling the full complex- ity of the ocean include, as here, active tracers such as tem- perature and salt.) It builds on ideas developed in the compu- tational fluid community. The numerical challenge is to ensure that the evolving velocity field remains nondivergent. Most

An Elastic–Viscous–Plastic Model for Sea Ice Dynamics

Abstract: The standard model for sea ice dynamics treats the ice pack as a visco‐plastic material that flows plastically under typical stress conditions but behaves as a linear viscous fluid where strain rates are small and the ice becomes nearly rigid. Because of large viscosities in these regions, implicit numerical methods are necessary for time steps larger than a few seconds. Current solution methods for these equations use iterative relaxation methods, which are time consuming, scale poorly with mesh resolution, and are not well adapted to parallel computation. To remedy this, the authors developed and tested two separate methods. First, by demonstrating that the viscous‐plastic rheology can be represented by a symmetric, negative definite matrix operator, the much faster and better behaved preconditioned conjugate gradient method was implemented. Second, realizing that only the response of the ice on timescales associated with wind forcing need be accurately resolved, the model was modified so that it reduces to the viscous‐plastic model at these timescales, whereas at shorter timescales the adjustment process takes place by a numerically more efficient elastic wave mechanism. This modification leads to a fully explicit numerical scheme that further improves the model’s computational efficiency and is a great advantage for implementations on parallel machines. Furthermore, it is observed that the standard viscous‐plastic model has poor dynamic response to forcing on a daily timescale, given the standard time step (1 day) used by the ice modeling community. In contrast, the explicit discretization of the elastic wave mechanism allows the elastic‐viscous‐plastic model to capture the ice response to variations in the imposed stress more accurately. Thus, the elastic‐viscous‐plastic model provides more accurate results for shorter timescales associated with physical forcing, reproduces viscous‐plastic model behavior on longer timescales, and is computationally more efficient overall.

A priori derivation of the lattice Boltzmann equation

Abstract: The lattice Boltzmann equation (LBE) is directly derived from the Boltzmann equation by discretization in both time and phase space. A procedure to systematically derive discrete velocity models is presented. A LBE algorithm with arbitrary mesh grids is proposed and a numerical simulation of the backward-facing step is conducted. The numerical result agrees well with experimental and previous numerical results. Various improvements on the LBE models are discussed, and an explanation of the instability of the existing LBE thermal models is also provided.

Wavelet and multiscale methods for operator equations

Abstract: More than anything else, the increase of computing power seems to stimulate the greed for tackling ever larger problems involving large-scale numerical simulation. As a consequence, the need for understanding something like the intrinsic complexity of a problem occupies a more and more pivotal position. Moreover, computability often only becomes feasible if an algorithm can be found that is asymptotically optimal. This means that storage and the number of floating point operations needed to resolve the problem with desired accuracy remain proportional to the problem size when the resolution of the discretization is refined. A significant reduction of complexity is indeed often possible, when the underlying problem admits a continuous model in terms of differential or integral equations. The physical phenomena behind such a model usually exhibit characteristic features over a wide range of scales. Accordingly, the most successful numerical schemes exploit in one way or another the interaction of different scales of discretization. A very prominent representative is the multigrid methodology; see, for instance, Hackbusch (1985) and Bramble (1993). In a way it has caused a breakthrough in numerical analysis since, in an important range of cases, it does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and a unified treatment of several variants, such as multiplicative or additive subspace correction methods, see Bramble, Pasciak and Xu (1990), Oswald (1994), Xu (1992), and Yserentant (1993). Although there remain many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparatively clear profile. They are particularly powerful for elliptic and parabolic problems.

Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems

Abstract: In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.

Modelling landscape evolution on geological time scales: a new method based on irregular spatial discretization

Abstract: We present simulations of large-scale landscape evolution on tectonic time scales obtained from a new numerical model which allows for arbitrary spatial discretization. The new method makes use of efficient algorithms from the field of computational geometry to compute the set of natural neighbours of any irregular distribution of points in a plane. The natural neighbours are used to solve geomorphic equations that include erosion/deposition by channelled flow and diffusion. The algorithm has great geometrical flexibility, which makes it possible to solve problems involving complex boundaries, radially symmetrical uplift functions and horizontal tectonic transport across strike-slip faults. The algorithm is also ideally suited for problems which require large variations in spatial discretization and/or self-adaptive meshing. We present a number of examples to illustrate the power of the new approach and its advantages over more ‘classical’ models based on regular (rectangular) discretization. We also demonstrate that the synthetic river networks and landscapes generated by the model obey the laws of network composition and have scaling properties similar to those of natural landscapes. Finally we explain how orographically controlled precipitation and flexural isostasy may be easily incorporated in the model without sacrificing efficiency.

Feature selection via discretization

Abstract: Discretization can turn numeric attributes into discrete ones. Feature selection can eliminate some irrelevant and/or redundant attributes. Chi2 is a simple and general algorithm that uses the /spl chi//sup 2/ statistic to discretize numeric attributes repeatedly until some inconsistencies are found in the data. It achieves feature selection via discretization. It can handle mixed attributes, work with multiclass data, and remove irrelevant and redundant attributes.

Quantum Integrable Models and Discrete Classical Hirota Equations

Abstract: The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived.

A Detailed Analysis of Film–Cooling Physics: Part I — Streamwise Injection With Cylindrical Holes

Abstract: A previously documented systematic computational methodology is implemented and applied to a jet-in-crossflow problem in order to document all of the pertinent flow physics associated with a film-cooling flowfield Numerical results are compared to experimental data for the case of a row of three-dimensional, inclined jets with length-to-diameter ratios similar to a realistic film-cooling application A novel vorticity-based approach is included in the analysis of the flow physics Particular attention has been paid to the downstream coolant structures and to the source and influence of counterrotating vortices in the crossflow region It is shown that the vorticity in the boundary layers within the film hole is primarily responsible for this secondary motion Important aspects of the study include: (1) a systematic treatment of the key numerical issues, including accurate computational modeling of the physical problem, exact geometry and high-quality grid generation techniques, higher-order numerical discretization, and accurate evaluation of turbulence model performance; (2) vorticity-based analysis and documentation of the physical mechanisms of jet-crossflow interaction and their influence on film-cooling performance; (3) a comparison of computational results to experimental data; and (4) comparison of results using a two-layer model near-wall treatment versus generalized wall functions Solution of the steady, time-averaged Navier-Stokes equations were obtained for all cases using an unstructured/adaptive grid, fully explicit, time-marching code with multigrid, local time stepping, and residual smoothing acceleration techniques For the case using the two-layer model, the solution was obtained with an implicit, pressure-correction solver with multigrid The three-dimensional test case was examined for two different film-hole length-to-diameter ratios of 175 and 35, and three different blowing ratios, from 05 to 20 All of the simulations had a density ratio of 20, and an injection angle of 35 deg An improved understanding of the flow physics has provided insight into future advances to film-cooling configuration design In addition, the advantages and disadvantages of the two-layer turbulence model are highlighted for this class of problems

Discretized LMI set in the stability problem of linear uncertain time-delay systems

Abstract: The stability problem of linear uncertain time-delay systems is considered using a quadratic Lyapunov functional. The resulting stability criterion is a constrained linear matrix inequality set. The condition is necessary and sufficient if it is applied to uncertainty-free systems. A discretization scheme is proposed to reduce the constrained LMI set to a regular LMI problem. Conservatism due to discretization can be made small through finer discretization. Comparison with a previous example shows significant improvements even under very coarse discretization.

Enhanced simulated annealing for globally minimizing functions of many-continuous variables

Abstract: A new global optimization algorithm for functions of many continuous variables is presented, derived from the basic Simulated annealing method. Our main contribution lies in dealing with high-dimensionality minimization problems, which are often difficult to solve by all known minimization methods with or without gradient. In this article we take a special interest in the variables discretization issue. We also develop and implement several complementary stopping criteria. The original Metropolis iterative random search, which takes place in a Euclidean space Rn, is replaced by another similar exploration, performed within a succession of Euclidean spaces Rp, with p

A space-time discretization criterion for a stable time-marching solution of the electric field integral equation

Abstract: Numerical techniques based on a time-domain recursive solution of the electric field integral equation (EFIE) may exhibit instability phenomena induced by the joint space-time discretization. The above problem is addressed with specific reference to the evaluation of electromagnetic scattering from perfectly conducting bodies of arbitrary shape. We analyze a particular formulation of the method of moments which relies on a triangular-patch geometrical model of the exterior surface of the scattering body and operates according to a "marching-on-in-time" scheme, whereby the surface current distribution at a given time step is recursively evaluated as a function of the current distribution at previous steps. A heuristic stability condition is devised which allows us to define a proper time step, as well as a geometrical discretization criterion, ensuring convergence of the numerical procedure and, therefore, eliminating insurgence of late-time oscillations. The stability condition is discussed and validated by means of a few working examples.

Development of a Third Generation Shallow-Water Wave Model with Unstructured Spatial Meshing

Abstract: A numerical third-generation wave model dedicated both to deep water and nearshore applications is presented and applied to several test-cases to highlight its capabilities. Among its main features, this model uses a finite-elements technique for the discretization of the modelled area, which makes it suitable to represent complex bottom topographies and irregular shorelines. Furthermore, the piece-wise ray method used for wave propagation allows to use rather large time-steps, which in turn allows to keep the computational time at a very moderate level. The implementation of shallow-water physics in the model is also described, in particular with respect to depth-induced breaking. Several applications of the model are presented and compared to field or laboratory data for their validation. Finally, the main research and development items are mentioned and discussed.

Preconditioning in H (div) and applications

Abstract: We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I-grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

Large deformation analysis of rubber based on a reproducing kernel particle method

Abstract: A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal shape functions derived on␣the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation analysis. Since the current approach employs global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing with large material distortions.

Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations

Abstract: Introduction - Preliminareis - Stationary Quasi-Compressibility Methods: The Penalty Method and the Pressure Stabilization Method - Nonstationary Quasi-Compressibility Methods - Mixed Quasi-Compressibility Methods - The Projection Scheme of Chorin - The Projection Scheme of Van Kan - Two Modified Chorin Schemes - Multi-Component Schemes - Time Discretization on Time-Grids with Structure from Euler to Revised Projection Schemes

A finite‐volume integration method for computing pressure gradient force in general vertical coordinates

Abstract: A finite-volume integration method is proposed for computing the pressure gradient force in general vertical coordinates. It is based on fundamental physical principles in the discrete physical space, rather than on the common approach of transforming analytically the pressure gradient terms in differential form from the vertical physical (i.e., height or pressure) coordinate to one following the bottom topography. The finite-volume discretization is compact, involving only the four vertices of the finite volume. The accuracy of the method is evaluated statically in a two-dimensional environment and dynamically in three-dimensional dynamical cores for general circulation models. The errors generated by the proposed method are demonstrated to be very low in these tests.

A survey of Hirota's difference equations

Abstract: A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.

Simulation of ultrasonic pulse propagation through the abdominal wall

Abstract: Ultrasonic pulse propagation through the human abdominal wall has been simulated using a model for two-dimensional propagation through anatomically realistic tissue cross sections. The time-domain equations for wave propagation in a medium of variable sound speed and density were discretized to obtain a set of coupled finite-difference equations. These difference equations were solved numerically using a two-step MacCormack scheme that is fourth-order accurate in space and second-order accurate in time. The inhomogeneous tissue of the abdominal wall was represented by two-dimensional matrices of sound speed and density values. These values were determined by processing scanned images of abdominal wall cross sections stained to identify connective tissue, muscle, and fat, each of which was assumed to have a constant sound speed and density. The computational configuration was chosen to simulate that of wavefront distortion measurements performed on the same specimens. Qualitative agreement was found betwee...

On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs

Abstract: We present a new, naturally parallelizable, accurate numerical method for the solution of transport-dominated diffusion processes in heterogeneous porous media For the discretization in time of one of the governing partial differential equations, we introduce a new characteristics-based procedure which is mass conservative, the modified method of characteristics with adjusted advection (MMOCAA) Hybridized mixed finite elements are used for the spatial discretization of the equations and a new strip-based domain decomposition procedure is applied towards the solution of the resulting algebraic problems We consider as a model problem the two-phase immiscible displacement in petroleum reservoirs A very detailed description of the numerical method is presented Following that, numerical experiments are presented illustrating the important features of the new method and comparing computed results with ones derived from previous, related techniques

Nonuniform dynamic discretization in hybrid networks

Abstract: We consider probabilistic inference in general hybrid networks, which include continuous and discrete variables in an arbitrary topology. We reexamine the question of variable discretization in a hybrid network aiming at minimizing the information loss induced by the discretization. We show that a nonuniform partition across all variables as opposed to uniform partition of each variable separately reduces the size of the data structures needed to represent a continuous function. We also provide a simple but efficient procedure for nonuniform partition. To represent a nonuniform discretization in the computer memory, we introduce a new data structure, which we call a Binary Split Partition (BSP) tree. We show that BSP trees can be an exponential factor smaller than the data structures in the standard uniform discretization in multiple dimensions and show how the BSP trees can be used in the standard join tree algorithm. We show that the accuracy of the inference process can be significantly improved by adjusting discretization with evidence. We construct an erative anytime algorithm that gradually improves the quality of the discretization and the accuracy of the answer on a query. We provide empirical evidence that the algorithm converges.