Alvin Bayliss - US grants

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences of the National Science Foundation. This program supports the purchase of special purpose computing equipment dedicated to the conduct of research in the mathematical sciences. This equipment is required for several research projects and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers and with substantial cost-sharing from the institutions submitting the proposal. This program is an example of academic, corporate, and government cooperation in the support of the basic research in the mathematical sciences. This equipment will be used to support four research projects in the Department of Engineering Science and Applied Mathematics of Northwestern University. The names of the investigators and the titles of their projects follow: Numerical Study of Non-adiabatic Gaseous Fuel Combustion, Bayliss and Matkowsky; Numerical Study of Gasless Combustion, Bayliss and Matkowsky; Numerical Solution of Integro-partial Differential Equations in Stochastic Dynamics, Matkowsky; and Numerical Simulations of Activation in Stochastic Dynamics, Matkowsky.

This research program intends to develop techniques in adaptive pseudo-spectral methods for combustion problems, with special attention paid to bifurcation, chaoticity and two dimensional problems. The research results will have applications in combustion engineering and wave phenomena.

This proposal intends to take two fairly diverse areas, combustion and nonlinear dynamics of shells, and apply a similar numerical method (domain decomposition) to both areas. Each area is represented by a well known researcher, Belytschko- shells & Bayliss-combustion, and in the application of the method an expert in numerical optimization, Nocedal, will contribute to the effort. This proposal is a nice example of interdisciplinary research within a university. Using domain decomposition as a connection the three researchers will cooperate to make advances not only in their individual areas, but also in parallel processing. The original proposal was for a three year effort. Due to the uncertain program funding, and the heavy future commitment, this proposal is recommended for only two years as a standard grant. As discussed with the PI's, a number of other cost reductions have been included: the visiting professor has not been approved; the disc upgrade has not been included; the networking request has been denied; most important, the number of graduate students has been reduced from three to two. It is felt that by this last action that the three faculty members will have to coordinate their research in order to best occupy these two students.

This project involves the use of an adaptive pseudo- spectral method to compute solutions of models of solid and gaseous combustion that display spatio-temporal patterns. In particular, the principal investigator will study how flames undergo transitions to increasingly complex states through the mechanism of pattern formation. The study of how flames propagate is an important area that has attracted the attention of scientists, mathematicians and engineers. In this proposal the principal investigator will find numerical solutions of combustion problems that model the evolution in time of pulsating flames, cellular flames and travelling flame fronts.

Spectral methods have been shown to be effective in approximating the high gradient bands which occur during failure processes. These methods are effective in solid mechanics when applied in small patches or domains, which correspond to the high gradient regions such as shear bands, coupled with a finite element discretization of the rest of the problem. Here, adaptive procedures will be developed to determine the location and extent of these spectral domains from properties of the solution, for example the strain field. These numerical methods will be employed in large scale computations to study the structure of shear bands and factors influencing their growth. The use of spectral patches or subdomains provides a high degree of resolution of high gradient bands. Mappings within the subdomains and patches will be employed to adaptively enhance the resolution. These methods, because of their employment of subdomains, lend themselves naturally to MIMD parallel computers. Implementations designed to exploit this parallelism, will be developed. These methods will be implemented and applied in large scale computational studies relating to materials forming processes. In particular the role of imperfections and thermal coupling in the development of shear bands will be studied. Additionally, these methods will be applied in computational studies of the dynamics of gaseous combustion with multiple stage reaction mechanisms.

The Department of Engineering Sciences and Applied Mathematics and the Department o Electrical Engineering and Computer Science of Northwestern University propose to purchase a Silicon Graphics IRIS Crimson/VGXT graphics workstation together with peripheral equipment to allow for hard copy and video output. The equipment will be used to provide graphical analysis and animations for the following research projects: . Visualization of the complex spatial and temporal patterns that occur in combustion. . Visualization of the development of microstructures in solidification. . Visualization of complex spatial patterns an temporal dynamics governed by Ginzburg-Landau, Kuramoto-Sivashinsky and other generic amplitude equations. . Visualization of both the spatial structure and the evolution of shear bands in viscoplasticity.

9301635 Bayliss The investigator numerically studies the development of complex spatial and temporal patterns in combustion. These patterns are associated with narrow reaction zones in which rapid changes in the dependent variables occur across narrow spatial regions. The solutions are obtained using an adaptive pseudo-spectral method in which coordinate transformations are introduced during the course of the computation to better resolve the reaction zone and the associated patterns. He considers the formation of modulated traveling waves (MTWs) in cellular flames, transitions between different types of MTWs and transitions between MTWs and chaotic patterns, one-step and multiple reaction mechanisms, the role of thermal expansion of the gas on the formation and evolution of the patterns, and the formation of spatial and temporal patterns in condensed phase gasless combustion. He also further develops the numerical method to account for problems where there are multiple regions of rapid variation. It has been observed that many combustible mixtures do not necessarily burn in a unifrom pattern. Rather, ripples and other patterns can occur along the flame. The formation of such patterns is believed to be an important stage in the transition from laminar combustion (in which the burning occurs in a regular, predictable manner) to turbulent combustion (in which the burning occurs in an erratic, seemingly random fashion). An understanding of the transition process is an essential prerequisite to control of the transition process to lead to more efficient burning. In many instances these patterns can not be predicted by analytic means and experiments are much less advanced than for nonreacting flows. In the project, the investigator studies the problem of the formation of patterns and the transition to turbulence in combustion. The project uses supercomputer computations to study the formation of these patterns and the physical mechanisms underly ing them. The study relates both to gaseous combustion, i.e., the burning of fossil fuels, and gasless combustion, in which a solid burns directly. The last process is employed in the synthesis of materials, in particular of advanced ceramics and metallic alloys.

9422696 Belyschko Health monitoring and assessment of steel bridges and other steel facilities offers tremendous potential benefits if used to asses the safety and reliability and to extend the remaining useful life by timely repairs. This project involves the development of high performance computing tools for identifying flaws in critical bridge components form the output of nondestructive evaluation (NDE) sensing devices and assessing the impact of these flaws on the useful life of the component. High performance computing tools will be developed to provide a characterization of flaws from the output of NDE sensors and asses the reliability of the component. The proposed research addresses this objective in an integrated approach which will involve the following tasks: 1. adapting ultrasonic QNDE methods to in situ health monitoring of bridges; 2. the solution of the inverse problem of identifying and characterizing the crack, which will be accomplished with the aid of neural networks; 3. development of high fidelity forward solutions of the wave equation for detailed models with one or more cracks, which are needed for the identification process; because of the large size of these problems, parallel computations will be used; 4. estimating the reliability of a component in terms of the crack which is identified. Testing of the overall model will take place on test prototypes and on actual bridges. Working groups form various states and several interested companies will be set up to facilitate dissemination of the research results to practice.

9530937 Bayliss The effectiveness of non-reflecting boundary conditions for wave propagation problems which are based on far field expansions of the solution will be developed, implemented and assessed. Primarily transient wave propagation problems will be considered, although many of the techniques will also be applicable in the frequency domain. The boundary conditions are based on first deriving convergent or asymptotic expansions of the solution valid in the far field and then constructing differential operators which annihilate the leading order terms in the expansions. Generally these operators require specification of an origin from which outgoing waves are assumed to emanate. Techniques will be developed in which interior information is used to improve the performance of the boundary condition. In particular, adaptive boundary conditions will be developed in which the origin is estimated dynamically and locally from properties of the solution in the interior. Techniques to increase the order of the boundary condition without increasing the order of the differential operator will also be developed. This will be done by incorporating inhomogeneous terms into the boundary operator, again using properties of the computed solution in the interior. Finally, far field expansions and associated boundary conditions suitable for damping layers will be developed, thereby allowing a reduction in size of the layers. The numerical computation of waves, for example, acoustic and electromagnetic waves, will be considered. These problems are typically governed by equations specified in regions which are unbounded in at least one direction. In order for the problem to be formulated on a computer it is necessary that the unbounded region be replaced by a bounded region. Furthermore, it is necessary that some boundary condition be imposed at the edge (boundary) of the bounded region. This boundary condition should ideally simulate the ori ginal unbounded problem. However, in general the boundary treatment will not be a perfect simulation of the unbounded problem and there will be spurious reflections emanating from the boundary. These reflections will propagate into the interior region and seriously degrade the accuracy of the computed solution. Thus, it is essential for accurate computation that these spurious reflections be minimized. Boundary conditions will be considered based on properties of the waves in the far field, i.e., far from the sources where the waves are generated. Currently for any given computation the boundary conditions are generally determined in advance, i.e., at the beginning of the computation, and are not changed as the solution evolves in time. Thus, the boundary conditions make no use of information about the waves in the interior. Methodologies will be developed in which interior information can be used to improve the performance of the boundary conditions. In this approach, the boundary conditions will change as time evolves, adapting to the nature of the solution by making use of interior information. It is anticipated that the resulting boundary treatment will allow a significant reduction in spurious reflections. Thus, the proposed new boundary conditions offer the prospect of significant improvement in efficiency and accuracy for the computation of wave propagation problems. Applications to acoustic, electromagnetic and elastic wave propagation will be considered.

DMS Award Abstract
Award #: 0202485
PI: Bayliss, Alvin
Institution: Northwestern University
Program: Applied Mathematics
Program Manager: Catherine Mavriplis

Title: A Model for Combustion in Strongly Stratified Environments

We develop and solve numerically a low Mach number model for deflagrations in stellar envelopes. For many stellar environments the Mach number, the ratio of flame or convection speed to the sound speed, is very small (order .01 or less) so that for explicit computations based on full hydrodynamics timesteps are limited by the acoustic speeds, thus requiring excessive computation time to simulate the relatively long timescales associated with the deflagrations and associated convection flows. Our model filters out the sound waves while accounting for strong vertical pressure stratification (of the order of ten orders of magnitude). The model is based on initially pre-computing a strongly stratified hydrostatic solution by solving a boundary value problem in the vertical coordinate. The hydrodynamic equations are then expanded about this solution. The resulting system for the velocities and the non-hydrostatic pressure is then expanded in Mach number, effectively filtering out sound waves and yielding a system where timesteps are restricted by fluid and flame speeds rather than the sound speed. The new model will be applied to the study of convection in the envelope of an accreting white dwarf during a nova outburst. The model will allow simulation over convective timescales and thus allow a theoretical description of the multi-dimensional nuclear burning development of novae.

We propose to simulate the hydrodynamics of nuclear fluids in a stellar envelope in which sound waves are filtered out. It is typical of such environments that energy is released via nuclear reactions in a manner similar to terrestrial combustion (nuclear combustion). In many such configurations the flame speed and associated fluid speeds are much smaller than the sound speed. This limits the length of time for which these models can be solved and effectively precludes a simulation of phenomena associated with the burning and convection. For example, a simulation of burning in a neutron star envelope required approximately 30,000 hours net processor time, i.e., the time for all processors on a parallel Silicon Graphics computer, for approximately 200 milliseconds of simulated time. For this problem, the burning and convection timescales were of the order of minutes, e.g., about 5 minutes. Thus, a simulation and computer analysis of such phenomena would be prohibitive using current models. We anticipate that such analyses would be feasible with the proposed new model, because acoustic effects would be eliminated. The model will be used to study the nature of white dwarf novae, and in particular to resolve competing theories regarding the role of mixing of different species in promoting nova outbursts. The numerical solution of the model will employ parallel algorithms and computers and thus the project will entail the use of high performance computing to explain observed astrophysical phenomena.

Date: June 21, 2002

This project incorporates four different applications, which are
related through their common need for high-speed computational
capabilities. The first project involves the simulation of growing
bacterial biofilms. The simulation will couple the extended finite
element method with the level set method in order to capture the
interaction between the growing biofilm and the surrounding fluid
flow. The second project will solve a low Mach number model for
deflagrations in stellar envelopes. By filtering out sound waves from
the model, significantly larger time steps can be achieved than are
possible for full hydrodynamical models, thus making long time
computations feasible. The third project will study the role of
defects in the break-down of ordered structures. Parallelized
Monte-Carlo and Fourier spectral methods will be employed to
investigate defect trajectories in two paradigmatic systems, a
magnetic system and a dynamical system of coupled oscillators in two
dimensions. They exhibit spatially ordered as well as disordered
states. The goal is to identify to what extent various statistical
measures of the trajectories follow universal power laws. The fourth
project will use recently developed importance-sampling methods for
simulating rare events that set the performance of optical fiber
communication systems.

The impact of these projects will be felt across a broad spectrum of
disciplines, and will aid in the training and education of several
graduate students and postdoctoral research associates. In addition
to the scientific areas described above, the training will also
include the efficient use of high-performance parallel computing
architectures. The study of biofilms will improve our understanding
of quourum sensing organisms, and how to treat them. Such organisms
are responsible for a number of diseases, including diseases
associated with cystic fibrosis and deep burn wounds. The study of
deflagrations in stellar envelopes will improve our understanding of
the dynamics of novae of white dwarfs and the nature of X-ray bursts
from neutron stars. Transitions from ordered patterns to disordered
states are observed in many physical systems undergoing phase
transitions as well as in dynamical systems like arrays of coupled
oscillators, various kinds of fluid flow, and optical systems. Often
defects in the patterns are striking features of the disordered
states. The research will elucidate the role the defect dynamics play
in the break-down of the ordered states. The study of rare events in
optical fiber communication systems will lead to the construction of a
set of simulation tools capable of predicting the performance of
lightwave communication systems.

This award supports the acquisition of high-performance computing equipment that will enable research on the theory of gravitational-wave (GW) sources. The work will involve both research and technology training of undergraduate and graduate students. The equipment consists of a large computer cluster (56 nodes, each with two dual-core CPUs) with gigabit networking, designed for high computational efficiency-to-cost ratio for GW source simulations. In addition, 16 of the nodes will be equipped with specialized hardware (GRAPE boards) for direct N-body simulations in stellar dynamics. A major storage component is also part of the proposed system. Visualizations from the simulation results will be developed for use in both research analysis and training as well as for public outreach presentations at the nearby Adler Planetarium. For this development effort will be enabled by the acquisition of a Tiled Wall Display (almost 25 million pixels) operated by a small cluster of 7 nodes with high-level graphics capabilities. All the acquired instrumentation will be used in a wide range of computational astrophysics projects on binary star evolution and compact object formation, stellar dynamics, and hydrodynamics. The results will greatly improve the understanding of the most important GW sources for current laser interferometer detectors, and they will also allow for better physical interpretation of future GW observations. Examples of specific projects include: (i) modeling populations of binary compact objects driven to inspiral by GW emission; the primary goal is the application of empirical constraints to theoretical models and the derivation of reliable predictions for binary inspiral detection rates and for measurements of source properties. (ii) dynamical simulations of black holes in dense star clusters; frequent dynamical interactions in these systems can lead to the formation of large numbers of merging black hole binaries and may be the dominant source of detectable black hole mergers for ground-based interferometers such as LIGO. (iii) hydrodynamic calculations of the final mergers of compact binaries containing a black hole and a neutron star; these will be performed using a sophisticated 3-D relativistic hydrodynamics code and will represent a significant improvement over previous, more approximate treatments based on Newtonian gravity. The study of GW sources is also important in many other areas of astrophysics. For example, coalescing compact binaries may also be sources of gamma-ray bursts, and they may play a key role for the production of many heavy elements in galaxies. The research activities enabled by this instrumentation will involve the training of several undergraduate and graduate students, including students from under-represented minorities. A graduate student studying high performance computing and a team of up to about ten undergraduates will also be trained in all phases of the instrument acquisition, set-up, and commissioning.

This project establishes a flexible, modern training program in applied mathematics within the Research Training Group program.

Mathematics is a central component in physical sciences, biological sciences, and engineering, and many diverse disciplines exhibit common mathematical structures. Work in these fields profits from the involvement of applied mathematicians with broad backgrounds. This program will train such young applied mathematicians at all levels, from undergraduates through postdoctoral researchers.

A significant feature of the program is its interdisciplinary nature -- the group's activities involve not just mathematicians, but engineers and scientists as well. Thus, group members, while trained as applied mathematicians, will be comfortable interacting with non-mathematicians in research teams. The research activities will emphasize breadth and flexibility; trainees will be able to tackle problems involving a wide variety of mathematical techniques, ranging from analytical and computational methods to the development of suitable models of physical processes, and they will be able to adapt their research to diverse areas as opportunities arise.

The group's activities will involve mathematical research ranging from analytical to computational, in application areas including life sciences (particularly microbiology and biological fluids), fluid mechanics, materials science, and combustion. A special focus is on interfacial phenomena and phenomena involving multiple scales.

The project will produce a group of young applied mathematicians who are multifaceted in their scientific knowledge, able to grasp common mathematical features in problems from a diverse range of application areas, comfortable working and interacting with scientists and engineers in an interdisciplinary environment, and sufficiently flexible to pursue productive research in other areas as national priorities evolve.