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Wavelet transforms as solutions of partial differential equations

Description: This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). Wavelet transforms are useful in representing transients whose time and frequency structure reflect the dynamics of an underlying physical system. Speech sound, pressure in turbulent fluid flow, or engine sound in automobiles are excellent candidates for wavelet analysis. This project focused on (1) methods for choosing the parent wavelet for a continuous wavelet transform in pattern recognition applications and (2) the more efficient computation of continuous wavelet transforms by understanding the relationship between discrete wavelet transforms and discretized continuous wavelet transforms. The most interesting result of this research is the finding that the generalized wave equation, on which the continuous wavelet transform is based, can be used to understand phenomena that relate to the process of hearing.
Date: October 1, 1997
Creator: Zweig, G.
Partner: UNT Libraries Government Documents Department

Sinuous oscillations and steady warps of polytropic disks

Description: In an asymptotic development of the equations governing the equilibria and linear stability of rapidly rotating polytropes we employed the slender aspect of these objects to reduce the three-dimensional partial differential equations to a somewhat simpler, ordinary integro-differential form. The earlier calculations dealt with isolated objects that were in centrifugal balance, that is the centrifugal acceleration of the configuration was balanced largely by self gravity with small contributions from the pressure gradient. Another interesting situation is that in which the polytrope rotates subject to externally imposed gravitational fields. In astrophysics, this is common in the theory of galactic dynamics because disks are unlikely to be isolated objects. The dark halos associated with disks also provide one possible explanation of the apparent warping of many galaxies. If the axis of the highly flattened disk is not aligned with that of the much less flattened halo, then the resultant torque of the halo gravity on the disk might provide a nonaxisymmetric distortion or disk warp. Motivated by these possibilities we shall here build models of polytropic disks of small but finite thickness which are subjected to prescribed, external gravitational fields. First we estimate how a symmetrical potential distorts the structure of the disk, then we examine its sinuous oscillations to confirm that they freely decay, hence suggesting that a warp must be externally forced. Finally, we consider steady warps of the disk plane when the axis of the disk does not coincide with that of the halo.
Date: May 1, 1995
Creator: Balmforth, N.J. & Spiegel, E.A.
Partner: UNT Libraries Government Documents Department

Modeling Solution of Nonlinear Dispersive Partial Differential Equations using the Marker Method

Description: A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details.
Date: January 25, 2005
Creator: Lewandowski, Jerome L.V.
Partner: UNT Libraries Government Documents Department

Users' Guide to ADIC 1.1.

Description: This guide describes the use of the Automatic Differentiation in C (ADIC) system. ADIC is a suite of tools and libraries that automates the process of generating derivatives for scientific programs. In the context of solving PDEs, optimizations, sensitivity analysis, and inverse problems, researchers often require the derivatives {partial_derivative}f/{partial_derivative}x of a function f expressed as a program with respect to some input parameter(s) x. Automatic differentiation (AD) techniques augment the program with derivative computation by applying the chain rule of calculus to elementary operations in an automated fashion. ADIC uses sophisticated compiler techniques to augment the input C programs with derivative computation capability in an automatic fashion. It also provides a finer control of derivative code generation process via control scripts and pragmas. Another significant capability of ADIC is its component architecture, AIF, that allows ADIC's capability to be extended via plug-in modules.
Date: August 31, 2004
Creator: Hovland, P. D. & Norris, B.
Partner: UNT Libraries Government Documents Department

Spectral Representations of Uncertainty: Algorithms and Applications

Description: The objectives of this project were: (1) Develop a general algorithmic framework for stochastic ordinary and partial differential equations. (2) Set polynomial chaos method and its generalization on firm theoretical ground. (3) Quantify uncertainty in large-scale simulations involving CFD, MHD and microflows. The overall goal of this project was to provide DOE with an algorithmic capability that is more accurate and three to five orders of magnitude more efficient than the Monte Carlo simulation.
Date: April 24, 2005
Creator: Karniadakis, George Em
Partner: UNT Libraries Government Documents Department

An object-oriented approach to development and testing of parallel solution algorithms for nonlinear PDES

Description: An object-oriented design that provides flexibility in simulation codes is presented. This flexibility allows programmers freedom to easily change solution algorithms and discretization schemes as well as add new solver packages as they become available. Careful attention is paid to separating algorithm, data, and specific problem classes to provide for ease in changing any of these components. Furthermore, data structures are chosen so that each component works with data in a form best suited to its needs. Lastly, we present some experiences and comments on the tradeoffs involved with this design.
Date: September 17, 1998
Creator: Hornung, R & Woodward, C
Partner: UNT Libraries Government Documents Department

Reducing the memory requirement in reverse mode automatic differentiation by solving TBR flow equations.

Description: The fast computation of gradients in reverse mode Automatic Differentiation (AD) requires the generation of adjoint versions of every statement in the original code. Due to the resulting reversal of the control flow certain intermediate values have to be made available in reverse order to compute the local partial derivatives. This can be achieved by storing these values or by recomputing them when they become required. In any case one is interested in minimizing the size of this set. Following an extensive introduction of the ''To-Be-Recorded'' (TBR) problem the authors present flow equations for propagating the TBR status of variables in the context of reverse mode AD of structured programs.
Date: January 11, 2002
Creator: Naumann, U.
Partner: UNT Libraries Government Documents Department

ROSE: Compiler Support for Object-Oriented Frameworks

Description: ROSE is a preprocessor generation tool for the support of compile time performance optimizations in Overture. The Overture framework is an object-oriented environment for solving partial differential equations in two and three space dimensions. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures. Overture can be used to solve problems in complicated, moving geometries using the method of overlapping grids. It has support for grid generation, difference operators, boundary conditions, database access and graphics. In this paper we briefly present Overture, and discuss our approach toward performance within Overture and the A++P++ array class abstractions upon which Overture depends, this work represents some of the newest work in Overture. The results we present show that the abstractions represented within Overture and the A++P++ array class library can be used to obtain application codes with performance equivalent to that of optimized C and Fortran 77. ROSE, the preprocessor generation tool, is general in its application to any object-oriented framework or application and is not specific to Overture.
Date: November 17, 1999
Creator: Qainlant, D.
Partner: UNT Libraries Government Documents Department

Pressure Waves Induced by Megasonic Agitation in a LIGA Development Tank

Description: Megasonic agitation is used to improve the uniformity of the LIGA{sup 1} development process. To investigate the acoustic wave fields induced by megasonic agitation, we compute wave fields for a development tank containing a submerged wafer and for a typical trench-like feature on the wafer face. This separate treatment of these two problems is advantageous, because the length scales of the tank and the feature differ by three to four orders of magnitude. A spectral method based on Green's functions is used to construct the acoustic wave field, avoiding the alternative of solving partial differential equations over the entire domain. The total acoustic wave field is obtained by superposing of the primary wave field and the first reflected wave field, which are computed in sequence without any need for iterations. The wafer interference to the wave field is treated directly by a priori recognition of shadow regions in the primary field and a concept of boundary of dependence in the reflected field. Unlike a divergent wave field produced by ultrasonic agitation, results show that the wave field in the tank becomes narrowly focused at megasonic frequencies such that the most effective agitation is confined in a region directly above the acoustic source; this numerical expectation has been verified analytically and further confirmed experimentally by Sandia's LIGA Group.{sup [13]} The amplitude of the focused wave pressure is proportional to square root of the wave frequency. The wave pattern in a feature cavity also depends strongly on the orientation of the wafer and the aspect ratio of the cavity. It is concluded that the LIGA development process will be greatly accelerated, if the orientation and the location of the immersed wafer is arranged so that the wafer spends more time in the focused wave field of high frequency agitation.
Date: August 1, 2002
Creator: Ting, Aili
Partner: UNT Libraries Government Documents Department

Overture: Object-Oriented Tools for Application with Complex Geometry

Description: The Overture framework is an object-oriented environment for solving partial differential equations in two and three space dimensions. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures. Overture can be used to solve problems in complicated, moving geometries using the method of overlapping grids. It has support for grid generation, difference operators, boundary conditions, data-base access and graphics. Short sample code segments are presented to show the power of this approach.
Date: May 31, 1999
Creator: Brown, D.; Henshaw, B. & Quinlan, D.
Partner: UNT Libraries Government Documents Department

The Distinct Element Method - Application to Structures in Jointed Rock

Description: The Distinct Element Method (DEM) is a meshfree method with applications to rock mechanics, mining sciences, simulations of nuclear repositories, and the stability of underground structures. Continuum mesh-based methods have been applied successfully to many problems in geophysics. Even if the geology includes fractures and faults, when sufficiently large length scales are considered a continuum approximation may be sufficient. However, a large class of problems exist where individual rock joints must be taken into account. This includes problems where the structures of interest have sizes comparable with the block size. In addition, it is possible that while the structure may experience loads which do no measurable damage to individual blocks, some joints may fail. This may launch smaller blocks as dangerous projectiles or even cause total failure of a tunnel. Traditional grid-based continuum approaches are wholly unsuited to this class of problem. It is possible to introduce discontinuities or slide lines into existing grid-based methods, however, such limited approaches can break down when new contacts form between blocks. The distinct element method (DEM) is an alternative, meshfree approach. The DEM can directly approximate the block structure of the jointed rock using arbitrary polyhedra. Using this approach, preexisting joints are readily incorporated into the DEM model. In addition, the method detects all new contacts between blocks resulting from relative block motion. We will describe the background of the DEM and review previous application of the DEM to geophysical problems. Finally we present preliminary results from a investigation into the stability of underground structures subjected to dynamic loading.
Date: November 30, 2001
Creator: Morris, J.P.; Glen, L.; Blair, S. & Heuze, F.
Partner: UNT Libraries Government Documents Department

Physical Motivation and Methods of Solution of Classical Partial Differential Equations

Description: We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.
Date: August 1995
Creator: Thompson, Jeremy R. (Jeremy Ray)
Partner: UNT Libraries

Overture: object-oriented tools for overset grid applications

Description: The Overture framework is an object-oriented environment for solving partial differential equations in two and three space dimensions. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures. Overture can be used to solve problems in complicated, moving geometries using the method of overlapping grids. It has support for grid generation, difference operators, boundary conditions, data-base access and graphics. Short sample code segments are presented to show the power of this approach.
Date: April 28, 1999
Creator: Brown, D L; Henshaw, W D & Quinlan, D J
Partner: UNT Libraries Government Documents Department

A Note on Equations for Steady-State Optimal Landscapes

Description: Based on the optimality principle (that the global energy expenditure rate is at its minimum for a given landscape under steady state conditions) and calculus of variations, we have derived a group of partial differential equations for describing steady-state optimal landscapes without explicitly distinguishing between hillslopes and channel networks. Other than building on the well-established Mining's equation, this work does not rely on any empirical relationships (such as those relating hydraulic parameters to local slopes). Using additional constraints, we also theoretically demonstrate that steady-state water depth is a power function of local slope, which is consistent with field data.
Date: June 15, 2010
Creator: Liu, H.H.
Partner: UNT Libraries Government Documents Department

Optimization and geophysical inverse problems

Description: A fundamental part of geophysics is to make inferences about the interior of the earth on the basis of data collected at or near the surface of the earth. In almost all cases these measured data are only indirectly related to the properties of the earth that are of interest, so an inverse problem must be solved in order to obtain estimates of the physical properties within the earth. In February of 1999 the U.S. Department of Energy sponsored a workshop that was intended to examine the methods currently being used to solve geophysical inverse problems and to consider what new approaches should be explored in the future. The interdisciplinary area between inverse problems in geophysics and optimization methods in mathematics was specifically targeted as one where an interchange of ideas was likely to be fruitful. Thus about half of the participants were actively involved in solving geophysical inverse problems and about half were actively involved in research on general optimization methods. This report presents some of the topics that were explored at the workshop and the conclusions that were reached. In general, the objective of a geophysical inverse problem is to find an earth model, described by a set of physical parameters, that is consistent with the observational data. It is usually assumed that the forward problem, that of calculating simulated data for an earth model, is well enough understood so that reasonably accurate synthetic data can be generated for an arbitrary model. The inverse problem is then posed as an optimization problem, where the function to be optimized is variously called the objective function, misfit function, or fitness function. The objective function is typically some measure of the difference between observational data and synthetic data calculated for a trial model. However, because of incomplete and inaccurate data, the ...
Date: October 1, 2000
Creator: Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F. et al.
Partner: UNT Libraries Government Documents Department

Final Report: Symposium on Adaptive Methods for Partial Differential Equations

Description: Complex physical phenomena often include features that span a wide range of spatial and temporal scales. Accurate simulation of such phenomena can be difficult to obtain, and computations that are under-resolved can even exhibit spurious features. While it is possible to resolve small scale features by increasing the number of grid points, global grid refinement can quickly lead to problems that are intractable, even on the largest available computing facilities. These constraints are particularly severe for three dimensional problems that involve complex physics. One way to achieve the needed resolution is to refine the computational mesh locally, in only those regions where enhanced resolution is required. Adaptive solution methods concentrate computational effort in regions where it is most needed. These methods have been successfully applied to a wide variety of problems in computational science and engineering. Adaptive methods can be difficult to implement, prompting the development of tools and environments to facilitate their use. To ensure that the results of their efforts are useful, algorithm and tool developers must maintain close communication with application specialists. Conversely it remains difficult for application specialists who are unfamiliar with the methods to evaluate the trade-offs between the benefits of enhanced local resolution and the effort needed to implement an adaptive solution method.
Date: December 8, 1998
Creator: Pernice, Michael; Johnson, Christopher R.; Smith, Philip J. & Fogelson, Aaron
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

Variational particle scheme for the porous medium equation and for the system of isentropic Euler equations

Description: We introduce variational particle schemes for the porous medium equation and the system of isentropic Euler equations in one space dimension. The methods are motivated by the interpretation of each of these partial differential equations as a 'steepest descent' on a suitable abstract manifold. We show that our methods capture very well the nonlinear features of the flows.
Date: December 10, 2008
Creator: Westdickenberg, Michael & Wilkening, Jon
Partner: UNT Libraries Government Documents Department

Some free boundary problems in potential flow regime usinga based level set method

Description: Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover the fluid front will possibly be carrying some material substance which will diffuse in the front and be advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.
Date: December 9, 2008
Creator: Garzon, M.; Bobillo-Ares, N. & Sethian, J.A.
Partner: UNT Libraries Government Documents Department

Peridynamics with LAMMPS : a user guide.

Description: Peridynamics is a nonlocal extension of classical continuum mechanics. The discrete peridynamic model has the same computational structure as a molecular dynamics model. This document provides a brief overview of the peridynamic model of a continuum, then discusses how the peridynamic model is discretized within LAMMPS. An example problem is also included.
Date: November 1, 2011
Creator: Lehoucq, Richard B.; Silling, Stewart Andrew; Seleson, Pablo (University of Texas at Austin, Austin, TX); Plimpton, Steven James & Parks, Michael L.
Partner: UNT Libraries Government Documents Department

Dynamic crack initiation toughness : experiments and peridynamic modeling.

Description: This is a dissertation on research conducted studying the dynamic crack initiation toughness of a 4340 steel. Researchers have been conducting experimental testing of dynamic crack initiation toughness, K{sub Ic}, for many years, using many experimental techniques with vastly different trends in the results when reporting K{sub Ic} as a function of loading rate. The dissertation describes a novel experimental technique for measuring K{sub Ic} in metals using the Kolsky bar. The method borrows from improvements made in recent years in traditional Kolsky bar testing by using pulse shaping techniques to ensure a constant loading rate applied to the sample before crack initiation. Dynamic crack initiation measurements were reported on a 4340 steel at two different loading rates. The steel was shown to exhibit a rate dependence, with the recorded values of K{sub Ic} being much higher at the higher loading rate. Using the knowledge of this rate dependence as a motivation in attempting to model the fracture events, a viscoplastic constitutive model was implemented into a peridynamic computational mechanics code. Peridynamics is a newly developed theory in solid mechanics that replaces the classical partial differential equations of motion with integral-differential equations which do not require the existence of spatial derivatives in the displacement field. This allows for the straightforward modeling of unguided crack initiation and growth. To date, peridynamic implementations have used severely restricted constitutive models. This research represents the first implementation of a complex material model and its validation. After showing results comparing deformations to experimental Taylor anvil impact for the viscoplastic material model, a novel failure criterion is introduced to model the dynamic crack initiation toughness experiments. The failure model is based on an energy criterion and uses the K{sub Ic} values recorded experimentally as an input. The failure model is then validated against one class of ...
Date: October 1, 2009
Creator: Foster, John T.
Partner: UNT Libraries Government Documents Department

Numerical study of a matrix-free trust-region SQP method for equality constrained optimization.

Description: This is a companion publication to the paper 'A Matrix-Free Trust-Region SQP Algorithm for Equality Constrained Optimization' [11]. In [11], we develop and analyze a trust-region sequential quadratic programming (SQP) method that supports the matrix-free (iterative, in-exact) solution of linear systems. In this report, we document the numerical behavior of the algorithm applied to a variety of equality constrained optimization problems, with constraints given by partial differential equations (PDEs).
Date: December 1, 2011
Creator: Heinkenschloss, Matthias (Rice University, Houston, TX); Ridzal, Denis & Aguilo, Miguel Antonio
Partner: UNT Libraries Government Documents Department

Filtering Algebraic Multigrid and Adaptive Strategies

Description: Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically.
Date: January 31, 2006
Creator: Nagel, A; Falgout, R D & Wittum, G
Partner: UNT Libraries Government Documents Department

A Marker Method for the Solution of the Damped Burgers' Equatio

Description: A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations.
Date: November 1, 2005
Creator: Lewandowski, Jerome L.V.
Partner: UNT Libraries Government Documents Department