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A parallel multigrid method for data-driven multiprocessor systems

Description: The multigrid algorithm (MG) is recognized as an efficient and rapidly converging method to solve a wide family of partial differential equations (PDE). When this method is implemented on a multiprocessor system, its major drawback is the low utilization of processors. Due to the sequentiality of the standard algorithm, the fine grid levels cannot start relaxation until the coarse grid levels complete their own relaxation. Indeed, of all processors active on the fine two dimensional grid level only one fourth will be active at the coarse grid level, leaving full 75% idle. In this paper, a novel parallel V-cycle multigrid (PVM) algorithm is proposed to cure the idle processors` problem. Highly programmable systems such as data-flow architectures are then applied to support this new algorithm. The experiments based on the proposed architecture show that the convergence rate of the new algorithm is about twice faster than that of the standard method and twice as efficient system utilization is achieved.
Date: December 31, 1989
Creator: Lin, C. H.; Gaudiot, J. L. & Proskurowski, W.
Partner: UNT Libraries Government Documents Department

Solving Partial Differential Equations in a data-driven multiprocessor environment

Description: Partial differential equations can be found in a host of engineering and scientific problems. The emergence of new parallel architectures has spurred research in the definition of parallel PDE solvers. Concurrently, highly programmable systems such as data-how architectures have been proposed for the exploitation of large scale parallelism. The implementation of some Partial Differential Equation solvers (such as the Jacobi method) on a tagged token data-flow graph is demonstrated here. Asynchronous methods (chaotic relaxation) are studied and new scheduling approaches (the Token No-Labeling scheme) are introduced in order to support the implementation of the asychronous methods in a data-driven environment. New high-level data-flow language program constructs are introduced in order to handle chaotic operations. Finally, the performance of the program graphs is demonstrated by a deterministic simulation of a message passing data-flow multiprocessor. An analysis of the overhead in the data-flow graphs is undertaken to demonstrate the limits of parallel operations in dataflow PDE program graphs.
Date: December 31, 1988
Creator: Gaudiot, J. L.; Lin, C. M. & Hosseiniyar, M.
Partner: UNT Libraries Government Documents Department

Singularities and symmetries of nonlinear ordinary and partial differential equations. Technical report, May 15, 1993--May 14, 1994

Description: In this project we have studied singularities as both fundamental mathematical objects in their own right determining, for example, integrable and nonintegrable properties of nonlinear differential equations; and the determining mechanism of crucial physical processes such as self-focusing singularities and current sheet formation. This approach defines broad based, interdisciplinary research program relevant to the DOE/AMS mission.
Date: October 1, 1994
Partner: UNT Libraries Government Documents Department

Performance of asynchronous algorithms in multi-level data-driven systems

Description: Asynchronous algorithms are efficient methods in solving scientific and engineering problems. Much research has been devoted to the study of asynchronous algorithms in different areas. This paper will show asynchronous algorithms applied to logic circuit simulation, communication networks, partial differential equations (PDE) and artificial neural networks, and as well as implementations of these asynchronous algorithms on a special class of multiprocessor systems, namely Multi-level Tagged-token Data-flow (MTD) architectures.
Date: December 31, 1989
Creator: Gaudiot, J. L. & Lin, C. M.
Partner: UNT Libraries Government Documents Department

Multigrid Methods for Nonlinear Problems: An Overview

Description: Since their early application to elliptic partial differential equations, multigrid methods have been applied successfully to a large and growing class of problems, from elasticity and computational fluid dynamics to geodetics and molecular structures. Classical multigrid begins with a two-grid process. First, iterative relaxation is applied, whose effect is to smooth the error. Then a coarse-grid correction is applied, in which the smooth error is determined on a coarser grid. This error is interpolated to the fine grid and used to correct the fine-grid approximation. Applying this method recursively to solve the coarse-grid problem leads to multigrid. The coarse-grid correction works because the residual equation is linear. But this is not the case for nonlinear problems, and different strategies must be employed. In this presentation we describe how to apply multigrid to nonlinear problems. There are two basic approaches. The first is to apply a linearization scheme, such as the Newton's method, and to employ multigrid for the solution of the Jacobian system in each iteration. The second is to apply multigrid directly to the nonlinear problem by employing the so-called Full Approximation Scheme (FAS). In FAS a nonlinear iteration is applied to smooth the error. The full equation is solved on the coarse grid, after which the coarse-grid error is extracted from the solution. This correction is then interpolated and applied to the fine grid approximation. We describe these methods in detail, and present numerical experiments that indicate the efficacy of them.
Date: December 23, 2002
Creator: Henson, V E
Partner: UNT Libraries Government Documents Department

Portent of Heine's Reciprocal Square Root Identity

Description: Precise efforts in theoretical astrophysics are needed to fully understand the mechanisms that govern the structure, stability, dynamics, formation, and evolution of differentially rotating stars. Direct computation of the physical attributes of a star can be facilitated by the use of highly compact azimuthal and separation angle Fourier formulations of the Green's functions for the linear partial differential equations of mathematical physics.
Date: October 12, 2002
Creator: Cohl, H S
Partner: UNT Libraries Government Documents Department

Graphics development of DCOR: Deterministic combat model of Oak Ridge

Description: DCOR is a user-friendly computer implementation of a deterministic combat model developed at ORNL. To make the interpretation of the results more intuitive, a conversion of the numerical solution to a graphic animation sequence of battle evolution is desirable. DCOR uses a coarse computational spatial mesh superimposed on the battlefield. This research is aimed at developing robust methods for computing the position of the combative units over the continuum (and also pixeled) battlefield, from DCOR`s discrete-variable solution representing the density of each force type evaluated at gridpoints. Three main problems have been identified and solutions have been devised and implemented in a new visualization module of DCOR. First, there is the problem of distributing the total number of objects, each representing a combative unit of each force type, among the gridpoints at each time level of the animation. This problem is solved by distributing, for each force type, the total number of combative units, one by one, to the gridpoint with the largest calculated number of units. Second, there is the problem of distributing the number of units assigned to each computational gridpoint over the battlefield area attributed to that point. This problem is solved by distributing the units within that area by taking into account the influence of surrounding gridpoints using linear interpolation. Finally, time interpolated solutions must be generated to produce a sufficient number of frames to create a smooth animation sequence. Currently, enough frames may be generated either by direct computation via the PDE solver or by using linear programming techniques to linearly interpolate intermediate frames between calculated frames.
Date: October 1, 1992
Creator: Hunt, G. & Azmy, Y. Y.
Partner: UNT Libraries Government Documents Department

A multi-level data-flow architecture for signal and data processing applications. Final report

Description: A grant was awarded to us by the Department of Energy, Office of Energy Research, in May 1987 to support the design and performance analysis of a large grain data-driven multiprocessor system for numerical applications. The basic idea of the work is to apply the data-driven principles of execution at a more appropriate level than conventional ``atomic`` instructions. For this purpose, a level such as that of vector operations was under study. This document represents the final report concerning the results of the research supported by this grant. The goals of the project entailed an analysis of Partial Differential Equation solvers on data-driven environments, a preliminary design of our multi-level architecture, an in-depth study of some of the mechanisms of execution, and a design of the software environment. As enumerated in the original proposal, our work has yielded results in three different domain: Specifications of the application programs; design of the general concepts of the architecture and simulation; implementation of a translating environment; and we discuss each of the above items and examine specific research results.
Date: September 2, 1993
Creator: Gaudiot, J. L.
Partner: UNT Libraries Government Documents Department

Investigations on detonation shock dynamics and related topics. Final report

Description: This document is a final report that summarizes the research findings and research activities supported by the subcontract DOE-LANL-9-XG8-3931P-1 between the University of Illinois (D. S. Stewart Principal Investigator) and the University of California (Los Alamos National Laboratory, M-Division). The main focus of the work has been on investigations of Detonation Shock Dynamics. A second emphasis has been on modeling compaction of energetic materials and deflagration to detonation in those materials. The work has led to a number of extensions of the theory of Detonation Shock Dynamics (DSD) and its application as an engineering design method for high explosive systems. The work also enhanced the hydrocode capabilities of researchers in M-Division by modifications to CAVEAT, an existing Los Alamos hydrocode. Linear stability studies of detonation flows were carried out for the purpose of code verification. This work also broadened the existing theory for detonation. The work in this contract has led to the development of one-phase models for dynamic compaction of porous energetic materials and laid the groundwork for subsequent studies. Some work that modeled the discrete heterogeneous behavior of propellant beds was also performed. The contract supported the efforts of D. S. Stewart and a Postdoctoral student H. I. Lee at the University of Illinois.
Date: November 1, 1993
Creator: Stewart, D. S.
Partner: UNT Libraries Government Documents Department

Final Report for LDRD Project on Rapid Problem Setup for Mesh-Based Simulation (Rapsodi)

Description: Under LLNL Exploratory Research LDRD funding, the Rapsodi project developed rapid setup technology for computational physics and engineering problems that require computational representations of complex geometry. Many simulation projects at LLNL involve the solution of partial differential equations in complex 3-D geometries. A significant bottleneck in carrying out these simulations arises in converting some specification of a geometry, such as a computer-aided design (CAD) drawing to a computationally appropriate 3-D mesh that can be used for simulation and analysis. Even using state-of-the-art mesh generation software, this problem setup step typically has required weeks or months, which is often much longer than required to carry out the computational simulation itself. The Rapsodi project built computational tools and designed algorithms that help to significantly reduce this setup time to less than a day for many realistic problems. The project targeted rapid setup technology for computational physics and engineering problems that use mixed-element unstructured meshes, overset meshes or Cartesian-embedded boundary (EB) meshes to represent complex geometry. It also built tools that aid in constructing computational representations of geometry for problems that do not require a mesh. While completely automatic mesh generation is extremely difficult, the amount of manual labor required can be significantly reduced. By developing novel, automated, component-based mesh construction procedures and automated CAD geometry repair and cleanup tools, Rapsodi has significantly reduced the amount of hand crafting required to generate geometry and meshes for scientific simulation codes.
Date: February 7, 2003
Creator: Brown, D L; Henshaw, W; Petersson, N A; Fast, P & Chand, K
Partner: UNT Libraries Government Documents Department

Portent of Heine's Reciprocal Square Root Identity

Description: Precise efforts in theoretical astrophysics are needed to fully understand the mechanisms that govern the structure, stability, dynamics, formation, and evolution of differentially rotating stars. Direct computation of the physical attributes of a star can be facilitated by the use of highly compact azimuthal and separation angle Fourier formulations of the Green's functions for the linear partial differential equations of mathematical physics.
Date: January 13, 2003
Creator: Cohl, H W
Partner: UNT Libraries Government Documents Department

Final Report for''Numerical Methods and Studies of High-Speed Reactive and Non-Reactive Flows''

Description: The work carried out under this subcontract involved the development and use of an adaptive numerical method for the accurate calculation of high-speed reactive flows on overlapping grids. The flow is modeled by the reactive Euler equations with an assumed equation of state and with various reaction rate models. A numerical method has been developed to solve the nonlinear hyperbolic partial differential equations in the model. The method uses an unsplit, shock-capturing scheme, and uses a Godunov-type scheme to compute fluxes and a Runge-Kutta error control scheme to compute the source term modeling the chemical reactions. An adaptive mesh refinement (AMR) scheme has been implemented in order to locally increase grid resolution. The numerical method uses composite overlapping grids to handle complex flow geometries. The code is part of the ''Overture-OverBlown'' framework of object-oriented codes [1, 2], and the development has occurred in close collaboration with Bill Henshaw and David Brown, and other members of the Overture team within CASC. During the period of this subcontract, a number of tasks were accomplished, including: (1) an extension of the numerical method to handle ''ignition and grow'' reaction models and a JWL equations of state; (2) an improvement in the efficiency of the AMR scheme and the error estimator; (3) an addition of a scheme of numerical dissipation designed to suppress numerical oscillations/instabilities near expanding detonations and along grid overlaps; and (4) an exploration of the evolution to detonation in an annulus and of detonation failure in an expanding channel.
Date: November 20, 2002
Creator: Schwendeman, D W
Partner: UNT Libraries Government Documents Department

Development and application of the quasi-potential transformation

Description: The quasi-potential transformation, based on the Kirchhoff transformation, reduces the equations governing mass-transfer in a steady-state, nonconvective electrolytic system into two independent parts. The geometry-specific part involves the solution of Laplace`s equation subject to the relevant boundary conditions. The system-specific part involves the solution of a set of coupled first-order, nonlinear, ordinary differential equations. We develop a theoretical basis for the quasi-potential transformation using potential theory. The major assumption on which the quasi-potential transformation is based is that the concentrations can be written as single-valued functions of the electrostatic potential. We see how the system-specific part of the calculation is developed. Boundary conditions are outlined, and the geometry-specific calculations for the disk and hemisphere electrodes are developed. We combine the system-specific calculations for the binary and acidic copper sulfate solutions with these geometry-specific calculations to obtain complete concentration profiles, potential distributions, and current density distributions for these systems. We also investigate the effect of migration on limiting currents.
Date: August 1, 1992
Creator: Pillay, B.
Partner: UNT Libraries Government Documents Department

The ignition temperature of solid explosives exposed to a fire

Description: When a system containing solid explosive is engulfed in a fire it receives a heat flux that causes the temperature of the system to rise monotonically. The temperature rise can often be approximated by a linear rise for extended periods of time. When some portion of the explosive, usually near the surface, reaches its ignition temperature it will begin to burn. If the explosive is unconfined, or can breach its confinement at low pressure, it will burn, not explode. Typically the burn front will propagate through a slab or shell at speeds on the order of a centimeter a minute. If the explosive is confined, the gas resulting from its burning will generate pressures high enough to rupture the confinement, but the peak pressure will generally be only a fraction of the pressure from a true detonation. When a system is not engulfed in the fire, but is close enough to be heated slowly by the fire, the behavior will be different. If the explosive is heated slowly it will have a nearly uniform temperature and ignition will occur inside the explosive. This almost always causes an explosion, even when the explosive as a whole is unconfined. The reason for this behavior is not well understood but slow heating of an explosive generally results in a more violent explosion than fast heating. These two situations are recognized by fast and slow cookoff tests used with munitions. Many munitions pass the fast cookoff test with heating rates around 2 K/min. Slow cookoff tests with heating rates around 4 K/hr generally result in an explosion. (The equations in this paper assume absolute temperatures in Kelvins, equal to Celsius + 273.16.) Mathematical models predicting the time to explosion are usually based on the assumption that the explosive has a uniform initial temperature and ...
Date: September 1, 1993
Creator: Creighton, J. R.
Partner: UNT Libraries Government Documents Department

Group-invariant solutions of hydrodynamics and radiation hydrodynamics

Description: Using the property of invariance under Lie groups of transformations, the equations of hydrodynamics are transformed from partial differential equations to ordinary differential equations, for which special analytic solutions can be found. These particular solutions can be used for (1) numerical benchmarks, (2) the basis for analytic models, and (3) insight into more general solutions. Additionally, group transformations can be used to construct new solutions from existing ones. A space-time projective group is used to generate complicated solutions from simpler solutions. Discussion of these procedures is presented along with examples of analytic of 1,2 and 3-D hydrodynamics.
Date: August 1, 1993
Creator: Coggeshall, S. V.
Partner: UNT Libraries Government Documents Department

Arbitrarily high order nodal and characteristic methods

Description: The quest for higher computational efficiency initially led researchers in the neutron transport area to develop and implement high-order approximations for solving the linear Boltzmann equational. This drive aimed at achieving higher accuracy on coarse meshes, thereby resulting in a net savings of computational resources represented by execution time and memory. Many endeavors succeeded in reaching this goal, producing a variety of elegent, albeit complicated, formalisms, that proved extremely accurate and efficient in solving test, as well as practical applications, problems. The two main classes of high order transport methods that recieved the most attention are the Nodal and Characteristic methods. A de facto linear order standard for the spatial approximation (even though Quadratic Nodal Methods were also considered) was dictated by the algebraic complexity of the derivation of the discrete variable equations, the programming complexity of implementing and verifying them in codes, and limitations on computational resources available to run such codes. The significant advances in computational resources in terms of hardware capacity and speed, as well as architectural innovations such as vector and parallel processing, all but eliminated the third (above) obstacle towards the development and implementation of even higher order methods. The algebraic and programming complexities, on the other hand, were alleviated to some extent by the development of Arbitrarily High Order Transport methods of the Nodal and the Characteristic types, which are discussed in this report.
Date: September 1, 1994
Creator: Azmy, Y. Y.
Partner: UNT Libraries Government Documents Department

A comparison of iterative methods for a model coupled system of elliptic equations

Description: Many interesting areas of current industry work deal with non-linear coupled systems of partial differential equations. We examine iterative methods for the solution of a model two-dimensional coupled system based on a linearized form of the two carrier drift-diffusion equations from semiconductor modeling. Discretizing this model system yields a large non-symmetric indefinite sparse matrix. To solve the model system various point and block methods, including the hybrid iterative method Alternate Block Factorization (ABF), are applied. We also employ GMRES with various preconditioners, including block and point incomplete LU (ILU) factorizations. The performance of these methods is compared. It is seen that the preferred ordering of the grid variables and the choice of iterative method are dependent upon the magnitudes of the coupling parameters. For this model, ABF is the most robust of the non-accelerated iterative methods. Among the preconditioners employed with GMRES, the blocked ``by grid point`` version of both the ILU and MILU preconditioners are the most robust and the most time efficient over the wide range of parameter values tested. This information may aid in the choice of iterative methods and preconditioners for solving more complicated, yet analogous, coupled systems.
Date: August 1, 1993
Creator: Donato, J. M.
Partner: UNT Libraries Government Documents Department

Turbulent dispersion of particles: The STP model

Description: A mathematical description of the stochastic transport of particles (STP) model for particle dispersion in turbulent flows is presented. The STP model is based on established theories of stochastic process modeling. The parameters of the model include physical properties of the particle (diameter and mass) and a description of the turbulent characteristics (mean velocities with rms fluctuations and their residence-time correlations) of the fluid phase in which the particles are dispersed. The model includes no adjustable parameters in the sense of calibration factors. It is independent of any particular turbulence model, but estimates of its parameters from information available from the common k-{var_epsilon} turbulence model are presented. Elements of the STP model are compared with exact solutions of the diffusion equation, alternative dispersion models, and experimental data collected under well-defined conditions. The STP model reproduces the exact solutions when both are based on consistent assumptions. Suggested approximated terms in the model also reproduce the experimental data nearly within its error under simple flow conditions. The STP model describes the origin of more complex behavior (counter-gradient diffusion, for example) and has the potential of describing it if sufficient detail about the gas-phase turbulence is known.
Date: August 1, 1994
Creator: Baxter, L. L. & Smith, P. J.
Partner: UNT Libraries Government Documents Department

The paradigm of Pseudodual Chiral Models

Description: This is a synopsis and extension of Phys. Rev. D49 5408 (1994). The Pseudodual Chiral Model illustrates 2-dimensional field theories which possess an infinite number of conservation laws but also allow particle production, at variance with naive expectations-a folk theorem of integrable models. We monitor the symmetries of the pseudodual model, both local and nonlocal, as transmutations of the symmetries of the (very different) usual Chiral Model. We refine the conventional algorithm to more efficiently produce the nonlocal symmetries of the model. We further find the canonical transformation which connects the usual chiral model to its fully equivalent dual model, thus contradistinguishing the pseudodual theory.
Date: August 1, 1994
Creator: Zachos, C. K. & Curtright, T. L.
Partner: UNT Libraries Government Documents Department

Automatic differentiation for PDES: Unsaturated flow case study

Description: The techniques of automatic differentiation are applied to an example partial differential equation arising from the modeling of unsaturated flow. One common paradigm for the numerical solution to some classes of two-, three-, or higher-dimensional partial differential equations is as follows: Given a PDE and boundary conditions, apply finite difference or finite element approximations on some appropriate (frequently nonuniform) grid, and enforce an approximate solution by solving a nonlinear system F(u)=0 for the residual by Newton`s method. The dimension of the nonlinear system F(u)=0 is proportional to the number of grid points. In current algorithms, the Jacobian J required by Newton`s method is computed by some combination of hand coding, divided differences, matrix coloring, and partial separability. We present a case study documenting the steps we took in analyzing a code provided by Robey for modeling unsaturated flow in porous media. Our purpose was to compute J by automatic differentiation using ADOL-C, a tool for automatic differentiation using overloaded operators in C++.
Date: July 1, 1992
Creator: Corliss, G. F.; Bischof, C.; Griewank, A.; Wright, S. J. & Robey, T.
Partner: UNT Libraries Government Documents Department

A one dimensional numerical model, with an application to the vertical distribution of dissolved oxygen in the ocean. Manual: M1D

Description: This manual describes the one dimensional model M1D, and its application to the vertical distribution of dissolved oxygen in the ocean. Section 2 describes the partial differential equation upon which the model is based, and the required boundary conditions. Section 3 gives the finite difference equations used to approximate the partial differential equations, and the scheme used for their solution. In Section 4 a linear stability analysis of the finite difference equations is given. Section 5 describes the program M1D that implements the solution of the finite difference equations. The program description is done from a programmer`s point of view, especially with regard to parameters that control the simulation, location of variables on the grid, and the output file. A flow chart of the solution algorithm is also provided in this section. Section 6 presents some results of using the model to simulate the vertical distribution of oxygen in the ocean. Comparisons of model results to an analytical solution and to measured data from the North Eastern Tropical Pacific are given in this section. Finally, a summary and conclusions are presented in Section 7.
Date: May 1, 1993
Creator: Eliason, D. E.
Partner: UNT Libraries Government Documents Department

Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers

Description: We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK`s symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.
Date: September 1, 1992
Creator: Nataf, J. M. & Winkelmann, F.
Partner: UNT Libraries Government Documents Department

Analysis of hydraulic instability of ANS involute fuel plates

Description: Curved shell equations for the involute Advanced Neutron Source (ANS) fuel plates are coupled to two-dimensional hydraulic channel flow equations that include fluid friction. A complete set of fluid and plate boundary conditions is applied at the entrance and exit and along the sides of the plate and the channel. The coupled system is linearized and solved to assess the hydraulic instability of the plates.
Date: November 1, 1991
Creator: Sartory, W. K.
Partner: UNT Libraries Government Documents Department

Parallel fast Fourier transforms for non power of two data

Description: This report deals with parallel algorithms for computing discrete Fourier transforms of real sequences of length N not equal to a power of two. The method described is an extension of existing power of two transforms to sequences with N a product of small primes. In particular, this implementation requires N = 2{sup p}3{sup q}5{sup r}. The communication required is the same as for a transform of length N = 2{sup p}. The algorithm presented is intended for use in the solution of partial differential equations, or in any situation in which a large number of forward and backward transforms must be performed and in which the Fourier Coefficients need not be ordered. This implementation is a one dimensional FFT but the techniques are applicable to multidimensional transforms as well. The algorithm has been implemented on a 128 node Intel Ipsc/860.
Date: September 1, 1994
Creator: Semeraro, B. D.
Partner: UNT Libraries Government Documents Department