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Research in applied mathematics, numerical analysis, and computer science

Research conducted at the Institute for Computer Applications in Science and Engineering (ICASE) in applied mathematics, numerical analysis, and computer science is summarized and abstracts of published reports are presented. The major categories of the ICASE research program are: (1) numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; (2) control and parameter identification; (3) computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and (4) computer systems and software, especially vector and parallel computers.

Source record↗

Model description document for a computer program for the emulation/simulation of a space station environmental control and life support system (ESCM)

Emulation/Simulation Computer Model (ESCM) computes the transient performance of a Space Station air revitalization subsystem with carbon dioxide removal provided by a solid amine water desorbed subsystem called SAWD. This manual describes the mathematical modeling and equations used in the ESCM. For the system as a whole and for each individual component, the fundamental physical and chemical laws which govern their operations are presented. Assumptions are stated, and when necessary, data is presented to support empirically developed relationships.

Yanosy, James L.↗

A Meteorological Model's Dependence on Radiation Update Frequency

Numerical weather models are used to simulate circulations in the atmosphere including clouds and precipitation by applying a set of mathematical equations over a three-dimensional grid. The grid is composed of discrete points at which the meteorological variables are defined. As computing power continues to rise these models are being used at finer grid spacing, but they must still cover a wide range of scales. Some of the physics that must be accounted for in the model cannot be explicitly resolved, and their effects, therefore, must be estimated or "parameterized". Some of these parameterizations are computationally expensive. To alleviate the problem, they are not always updated at the time resolution of the model with the assumption being that the impact will be small. In this study, a coupled land-atmosphere model is used to assess the impact of less frequent updates of the computationally expensive radiation physics for a case on June 6, 2002, that occurred during a field experiment over the central plains known as International H20 Project (IHOP). The model was tested using both the original conditions, which were dry, and with modified conditions wherein moisture was added to the lower part of the atmosphere to produce clouds and precipitation (i.e., a wet case). For each of the conditions (i.e., dry and wet), four set of experiments were conducted wherein the model was run for a period of 24 hours and the radiation fields (including both incoming solar and outgoing longwave) were updated every 1, 3, 10, and 100 time steps. Statistical tests indicated that average quantities of surface variables for both the dry and wet cases were the same for the various update frequencies. However, spatially the results could be quite different especially in the wet case after it began to rain. The near-surface wind field was found to be different most of the time even for the dry case. In the wet case, rain intensities and average vertical profiles of heating associated with cloudy areas were found to differ for the various radiation update frequencies. The latter implies that the mean state of the model could be different as a result of not updating the radiation fields every time step and has important implications for longer term climate studies

Eastman, Joseph L.↗

Structural optimization - Challenges and opportunities

A review of developments in structural optimization techniques and their interface with growing computer capabilities is presented. Structural design steps comprise functional definition of an object, an evaluation phase wherein external influences are quantified, selection of the design concept, material, object geometry, and the internal layout, and quantification of the physical characteristics. Optimization of a fully stressed design is facilitated by use of nonlinear mathematical programming which permits automated definition of the physics of a problem. Design iterations terminate when convergence is acquired between mathematical and physical criteria. A constrained minimum algorithm has been formulated using an Augmented Lagrangian approach and a generalized reduced gradient to obtain fast convergence. Various approximation techniques are mentioned. The synergistic application of all the methods surveyed requires multidisciplinary teamwork during a design effort.

Sobieszczanski-Sobieski, J.↗

Structural optimization: Challenges and opportunities

A review of developments in structural optimization techniques and their interface with growing computer capabilities is presented. Structural design steps comprise functional definition of an object, an evaluation phase wherein external influences are quantified, selection of the design concept, material, object geometry, and the internal layout, and quantification of the physical characteristics. Optimization of a fully stressed design is facilitated by use of nonlinear mathematical programming which permits automated definition of the physics of a problem. Design iterations terminate when convergence is acquired between mathematical and physical criteria. A constrained minimum algorithm has been formulated using an Augmented Lagrangian approach and a generalized reduced gradient to obtain fast convergence. Various approximation techniques are mentioned. The synergistic application of all the methods surveyed requires multidisciplinary teamwork during a design effort.

Sobieszczanski-Sobieski, J.↗

An investigation of a mathematical model of an optically pumped Ti(3+):Al2O3 laser system

During the last several years, solid state lasers were developed that have the potential for meeting rigorous performance requirements for space-based remote sensing of the atmosphere. In order to design a stable and efficient laser and to understand the effect on laser output of changes in the physical and design parameters, an understanding of the development of the dynamical processes of the laser is necessary. Typically, the dynamical processes in a laser system are investigated via rate equations describing the evolution of the occupancy in the electronic levels and of the photon density in the laser cavity. There are two approaches to this type of study. Most often, for the sake of simplicity, the spatial variations of the dynamic variables in the laser system are disregarded and the mathematical model consists of a system of first order nonlinear ordinary differential equations (ODE). The second approach is to take into account both spatial and temporal variations in the dynamic variables in the laser cavity. The resulting model consists of a first order semilinear system of partial differential equations (PDE). The model which was studied was studied was generic in the sense that it was a four-level laser system, but the parameters used in the numerical study were specific to Titanium-doped sapphire. For simplicity, a constant, spatially uniform pumping scheme was considered. In addition, a simplification of the model was made so that it treats a single lasing wavelength with a narrow bandwidth. The purpose was to investigate both versions of the mathematical model and to determine whether the numerical solutions are similar both qualitatively and quantitatively. The systems of ordinary differential equations were solved numerically using a Runge-Kutta-Fehlberg algorithm which was very efficient for typical values of the physical parameters. A numerical scheme, based on the Modified Euler method, for computing solutions to the system of partial differential equations was developed and implemented. The PDE model was solved numerically at the expense of greatly increased computer time.

Roberts, Lila F.↗

Robust control with structured perturbations

This semi-annual report describes continued progress on the research. Among several approaches in this area of research, our approach to the parametric uncertainties are being matured everyday. This approach deals with real parameter uncertainties which other techniques such as H (sup infinity) optimal control, micron analysis and synthesis, and l(sub 1) optimal control cannot deal. The primary assumption of this approach is that the mathematical models are well obtained so that the most of system uncertainties can be translated into parameter uncertainties of their linear system representations. These uncertainties may be due to modeling, nonlinearity of the physical system, some time-varying parameters, etc. In this report period of research, we are concentrating on implementing a computer aided analysis and design tool based on new results on parametric robust stability. This implementation will help us to reveal further details in this approach.

Keel, Leehyun↗

Space proton transport in one dimension

An approximate evaluation procedure is derived for a second-order theory of coupled nucleon transport in one dimension. An analytical solution with a simplified interaction model is used to determine quadrature parameters to minimize truncation error. Effects of the improved method on transport solutions with the BRYNTRN data base are evaluated. Comparisons with Monte Carlo benchmarks are given. Using different shield materials, the computational procedure is used to study the physics of space protons. A transition effect occurs in tissue near the shield interface and is most important in shields of high atomic number.

NASA Discipline Number 45-10↗

Recent advances in the solution of three-dimensional flow over wings with leading edge vortex separation

Recent advances in a panel method for the solution of three-dimensional flow about wing and wing-body combinations with leading-edge vortex separation are presented. These advances were achieved as part of an ultimately successful assault on two shortcomings of the method, namely convergence failures in seemingly random cases, and overprediction of lift coefficient for high aspect-ratio wings. Advances include the implementation of improved panel numerics for the purpose of eliminating the highly non-linear effects of ring vortices around doublet panel edges, and the development of a least squares procedure for damping vortex sheet geometry update instabilities. A variety of cases generated by the computer program implementing the method are presented. These cases are of two types. The first type consists of numerical studies, which verify the underlying mathematical assumptions of the method and moreover show that the results are strongly invariant with respect to such user dependent input as wing panel layout, initial sheet shape, sheet rollup, etc. The second type consists of cases run for the purpose of comparing computed results with experimental data, and these comparisons verify the underlying physical assumptions made by the method.

Johnson, F. T.↗

Optimization of Microelectronic Devices for Sensor Applications

The NASA/JPL goal to reduce payload in future space missions while increasing mission capability demands miniaturization of active and passive sensors, analytical instruments and communication systems among others. Currently, typical system requirements include the detection of particular spectral lines, associated data processing, and communication of the acquired data to other systems. Advances in lithography and deposition methods result in more advanced devices for space application, while the sub-micron resolution currently available opens a vast design space. Though an experimental exploration of this widening design space-searching for optimized performance by repeated fabrication efforts-is unfeasible, it does motivate the development of reliable software design tools. These tools necessitate models based on fundamental physics and mathematics of the device to accurately model effects such as diffraction and scattering in opto-electronic devices, or bandstructure and scattering in heterostructure devices. The software tools must have convenient turn-around times and interfaces that allow effective usage. The first issue is addressed by the application of high-performance computers and the second by the development of graphical user interfaces driven by properly developed data structures. These tools can then be integrated into an optimization environment, and with the available memory capacity and computational speed of high performance parallel platforms, simulation of optimized components can proceed. In this paper, specific applications of the electromagnetic modeling of infrared filtering, as well as heterostructure device design will be presented using genetic algorithm global optimization methods.

Cwik, Tom↗

Genetically Engineered Microelectronic Infrared Filters

A genetic algorithm is used for design of infrared filters and in the understanding of the material structure of a resonant tunneling diode. These two components are examples of microdevices and nanodevices that can be numerically simulated using fundamental mathematical and physical models. Because the number of parameters that can be used in the design of one of these devices is large, and because experimental exploration of the design space is unfeasible, reliable software models integrated with global optimization methods are examined The genetic algorithm and engineering design codes have been implemented on massively parallel computers to exploit their high performance. Design results are presented for the infrared filter showing new and optimized device design. Results for nanodevices are presented in a companion paper at this workshop.

Cwik, Tom↗

Resolving Wave-Particle Duality Could Accelerate the Mass Production of Quantum Computers

Quantum computers, hypothesized in 1980s, use concepts of superposition and entanglement phenomena. Although theoretical propositions and associated search algorithms for accurate measurements are being generated, the development of practical quantum computers themselves are advancing very slowly requiring enormous time and investments. The underlying concepts of a quantum computer are not new to the optical domain. However, the crucial enabling concepts of Entanglement and Superposition Principle are remaining clouded under the unresolved postulates, Wave-Particle Duality (WPD), and Wave Packet Reduction (WPR), implicating incompleteness in the interpretations of the mathematical formalism behind Quantum Mechanics. The WPD debate started during late1600 between Newton and Huygens. Young’s resolution of WPD through his double-slit experiment in 1802 was effectively overturned by Einstein’s interpretation of photoelectric effect as due to “indivisible light quanta”. However, Einstein disowned his “light quanta” postulate shortly before his death in1955, even though it had earned him the Nobel Prize. We resolve WPD by synthesizing Newton’s and Maxwell’s concepts and assume atoms do emit quanta but propagate as time-finite exponential pulses. This assumption also resolves WPR for light-matter interaction with the assumption that Schrodinger’s ψ represents atom’s internal dipolar amplitude stimulations. This over-turns Born’s interpretation that ψ only represents the abstract mathematical probability amplitude, rather than the physical “internal amplitude stimulation” of the quantum entity. However, our concept of atomic pulse emission forces us to re-derive the expression for the N-slit grating-spectrometer response since the classical derivation uses CW light, which does not exist. This pulsespectrometric response function strengthens our postulate since the grating response to the exponential pulse appears to be the convolution of a Lorentzian spectrum with the classical CW response function of the grating. The Fourier Transform of an exponential function is Lorentzian and QM predicts spontaneous emission line width to be Lorentzian. Then, conceptually one can extend the grating-expression (with N=2) to get the double-slit pattern. This approach preserves the classical causality that each of the two slits, like the N-signals out of a grating, are physically real and jointly stimulate the quantum detector array at the far field to generate the “Local” cosine fringes. The detector array executes the square modulus operation on its imposed dipolar amplitude stimulation and absorbs the necessary energy to fill up their quantum cups. Hence the double-slit pattern must also be “Local”, just as the N-slit grating spectrum is generated locally at the exit spectral-plane of the spectrometer. This removes the need to believe that “single photons” mysteriously generate the double slit pattern. Quantum computers, hypothesized in 1980s, use concepts of superposition and entanglement phenomena. Although theoretical propositions and associated search algorithms for accurate measurements are being generated, the development of practical quantum computers themselves are advancing very slowly requiring enormous time and investments. The underlying concepts of a quantum computer are not new to the optical domain. However, the crucial enabling concepts of Entanglement and Superposition Principle are remaining clouded under the unresolved postulates, Wave-Particle Duality (WPD), and Wave Packet Reduction (WPR), implicating incompleteness in the interpretations of the mathematical formalism behind Quantum Mechanics. The WPD debate started during late1600 between Newton and Huygens. Young’s resolution of WPD through his double-slit experiment in 1802 was effectively overturned by Einstein’s interpretation of photoelectric effect as due to “indivisible light quanta”. However, Einstein disowned his “light quanta” postulate shortly before his death in1955, even though it had earned him the Nobel Prize. We resolve WPD by synthesizing Newton’s and Maxwell’s concepts and assume atoms do emit quanta but propagate as time-finite exponential pulses. This assumption also resolves WPR for light-matter interaction with the assumption that Schrodinger’s ψ represents atom’s internal dipolar amplitude stimulations. This over-turns Born’s interpretation that ψ only represents the abstract mathematical probability amplitude, rather than the physical “internal amplitude stimulation” of the quantum entity. However, our concept of atomic pulse emission forces us to re-derive the expression for the N-slit grating-spectrometer response since the classical derivation uses CW light, which does not exist. This pulsespectrometric response function strengthens our postulate since the grating response to the exponential pulse appears to be the convolution of a Lorentzian spectrum with the classical CW response function of the grating. The Fourier Transform of an exponential function is Lorentzian and QM predicts spontaneous emission line width to be Lorentzian. Then, conceptually one can extend the grating-expression (with N=2) to get the double-slit pattern. This approach preserves the classical causality that each of the two slits, like the N-signals out of a grating, are physically real and jointly stimulate the quantum detector array at the far field to generate the “Local” cosine fringes. The detector array executes the square modulus operation on its imposed dipolar amplitude stimulation and absorbs the necessary energy to fill up their quantum cups. Hence the double-slit pattern must also be “Local”, just as the N-slit grating spectrum is generated locally at the exit spectral-plane of the spectrometer. This removes the need to believe that “single photons” mysteriously generate the double slit pattern.

Quantum Computer↗

Nonoscillatory solution of the steady-state inviscid Burgers' equation by mathematical programming

In order to obtain the physically relevant discontinuous numerical solution, the steady-state inviscid Burgers' equation is singularly perturbed through the addition of a small amount of viscosity. A 'cell-centered' finite-difference scheme is proposed which employs two points for the inviscid part and four points for the viscous one. While difficulties are experienced in the capture of interior layers centered at node points, computational results for interior layers centered between node points, and for boundary layers, exhibit accurate nonoscillatory solutions whose discontinuities are captured in one cell on both coarse and fine grids.

Lavery, John E.↗

Performance of a parallel code for the Euler equations on hypercube computers

The performance of hypercubes were evaluated on a computational fluid dynamics problem and the parallel environment issues were considered that must be addressed, such as algorithm changes, implementation choices, programming effort, and programming environment. The evaluation focuses on a widely used fluid dynamics code, FLO52, which solves the two dimensional steady Euler equations describing flow around the airfoil. The code development experience is described, including interacting with the operating system, utilizing the message-passing communication system, and code modifications necessary to increase parallel efficiency. Results from two hypercube parallel computers (a 16-node iPSC/2, and a 512-node NCUBE/ten) are discussed and compared. In addition, a mathematical model of the execution time was developed as a function of several machine and algorithm parameters. This model accurately predicts the actual run times obtained and is used to explore the performance of the code in interesting but yet physically realizable regions of the parameter space. Based on this model, predictions about future hypercubes are made.

Barszcz, Eric↗

Monitoring by Use of Clusters of Sensor-Data Vectors

The inductive monitoring system (IMS) is a system of computer hardware and software for automated monitoring of the performance, operational condition, physical integrity, and other aspects of the health of a complex engineering system (e.g., an industrial process line or a spacecraft). The input to the IMS consists of streams of digitized readings from sensors in the monitored system. The IMS determines the type and amount of any deviation of the monitored system from a nominal or normal ( healthy ) condition on the basis of a comparison between (1) vectors constructed from the incoming sensor data and (2) corresponding vectors in a database of nominal or normal behavior. The term inductive reflects the use of a process reminiscent of traditional mathematical induction to learn about normal operation and build the nominal-condition database. The IMS offers two major advantages over prior computational monitoring systems: The computational burden of the IMS is significantly smaller, and there is no need for abnormal-condition sensor data for training the IMS to recognize abnormal conditions. The figure schematically depicts the relationships among the computational processes effected by the IMS. Training sensor data are gathered during normal operation of the monitored system, detailed computational simulation of operation of the monitored system, or both. The training data are formed into vectors that are used to generate the database. The vectors in the database are clustered into regions that represent normal or nominal operation. Once the database has been generated, the IMS compares the vectors of incoming sensor data with vectors representative of the clusters. The monitored system is deemed to be operating normally or abnormally, depending on whether the vector of incoming sensor data is or is not, respectively, sufficiently close to one of the clusters. For this purpose, a distance between two vectors is calculated by a suitable metric (e.g., Euclidean distance) and "sufficiently close" signifies lying at a distance less than a specified threshold value. It must be emphasized that although the IMS is intended to detect off-nominal or abnormal performance or health, it is not necessarily capable of performing a thorough or detailed diagnosis. Limited diagnostic information may be available under some circumstances. For example, the distance of a vector of incoming sensor data from the nearest cluster could serve as an indication of the severity of a malfunction. The identity of the nearest cluster may be a clue as to the identity of the malfunctioning component or subsystem. It is possible to decrease the IMS computation time by use of a combination of cluster-indexing and -retrieval methods. For example, in one method, the distances between each cluster and two or more reference vectors can be used for the purpose of indexing and retrieval. The clusters are sorted into a list according to these distance values, typically in ascending order of distance. When a set of input data arrives and is to be tested, the data are first arranged as an ordered set (that is, a vector). The distances from the input vector to the reference points are computed. The search of clusters from the list can then be limited to those clusters lying within a certain distance range from the input vector; the computation time is reduced by not searching the clusters at a greater distance.

Iverson, David L.↗

The design, analysis and experimental evaluation of an elastic model wing

An elastic orbiter model was developed to evaluate the effectiveness of aeroelasticity computer programs. The elasticity properties were introduced by constructing beam-like straight wings for the wind tunnel model. A standard influence coefficient mathematical model was used to estimate aeroelastic effects analytically. In general good agreement was obtained between the empirical and analytical estimates of the deformed shape. However, in the static aeroelasticity case, it was found that the physical wing exhibited less bending and more twist than was predicted by theory.

Cavin, R. K., III↗

Rigorous Justification of the Localized Approximation to the Beam-Shape Coefficients in Generalized Lorenz-Mie Theory: On-Axis Beams - Part 1

Generalized Lorenz-Mie theory describes electromagnetic scattering of an arbitrary light beam by a spherical particle. The computationally most expensive feature of the theory is the evaluation of the beam-shape coefficients, which give the decomposition of the incident light beam into partial waves. The so-called localized approximation to these coefficients for a focused Gaussian beam is an analytical function whose use greatly simplifies Gaussian-beam scattering calculations. A mathematical justification and physical interpretation of the localized approximation is presented for on-axis beams.

Lock, James A.↗

Computing Propagation Of Sound In Engine Ducts

Frequency Domain Propagation Model (FREDOM) computer program accounts for acoustic loads applied to components of engines. Models propagation of noise through fluids in ducts between components and through passages within components. Used not only to analyze hardware problems, but also for design purposes. Updated version of FREQPL program easier to use. Devised specifically for use in analyzing acoustic loads in rocket engines. Underlying physical and mathematical concepts implemented also applicable to acoustic propagation in other enclosed spaces; analyzing process plumbing and ducts in industrial buildings with view toward reducing noise in work areas.

Saylor, Silvia↗