# Projects

# Topics

#### Condensed Matter Theory of Materials

**Modeling, Numerical Approximation, and Control of Aeroacoustic Noise**

This project centers on the modeling and control of high intensity cavity pressure fields generated by adjacent flow fields. Flow-induced acoustic fields in cavities such as wheel wells or bomb bays can have sufficient magnitude (140-170 dB) to produce structural damage if unaccommodated. The basic physical mechanisms which generate the acoustic fields and the utilization of these mechanisms for control design are under investigation.

The adjacent flow fields are inherently nonlinear and are typically modeled by either Navier-Stokes or Euler equations. The acoustic field is inherently compressible and also exhibits nonlinear dynamics at the decibel levels which produce damage. The modeling component focuses on issues concerning the coupling of the fields through unsteady shear layers and the development of boundary conditions which quantify the energy losses measured in experiments. Large scale numerical simulations are being run to determine the accuracy and limitations of the models and to ascertain the open loop dynamics of the system. To facilitate both simulations and control design, it is necessary to develop reduced-order approximation methods. One method under consideration entails the representation of system dynamics through the construction of appropriate POD basis elements. Once this phase is completed, the investigation will focus on the design and implementation of open and closed loop control methods which utilize the reduced-order basis.

CRSC researchers participating in this project include H.T. Banks, D. Bortz, A. Cain and R.C. Smith working in collaboration with scientists at the Boeing Company and at Innovative Technology Applications Company, LLC.

#### Elastomer Modeling

Collaborators/Sponsors:

Lord Corporation

The elastomer modeling project started in September, 1994, in cooperation with scientists at the Thomas Lord Research Center, Lord Corporation. Lord Corporation, based in Cary, NC, produces many products with rubber--like (elastomer) components. Many of these products are used as vibration control devices, such as engine mounts for buses and airplanes. Thus models which accurately predict the dynamic mechanical behavior of elastomers could be used to aid engineers in the design of components.

Many models have been developed which predict the behavior of rubber under static conditions. While these models can do an excellent job of fitting static data for lightly filled (non-hysteretic) samples, they do not include damping or hysteresis terms that are necessary for dynamic models or for highly filled samples. The goal of this project has been to develop more general models that encompass the nonlinear constitutive laws, damping, and hysteresis which are common to elastomers.

Two deformations have been studied thus far: a rod in simple extension, and a block in simple shear. Using nonlinear partial differential equation models for lightly filled (non-hysteretic) materials, good results have been achieved using inverse problem methodologies with dynamic data to identify material dependent parameters. Integral equations have been developed as a model of hysteresis. Inverse methodologies with quasi-static data for a rod in extension have been used to validate this model. Most recently, hysteresis has been incorporated in the dynamic model, yielding satisfactory results for highly filled elastomers.

This project involves a wide variety of scientific skills. Theoretical results regarding existence and uniqueness of solutions, as well as convergence of approximation schemes, have been developed. Numerical techniques have been used to solve the partial differential equation systems within the parameter identification, which is itself a numerical optimization problem. Finally, engineering methods have been used to develop and perform experiments which will lead to the best results in the identification problem.

Project members are: H. T. Banks, Gabriella Pinter and Laura K. Potter of the CRSC, along with Mike Gaitens, and Lynn Yanyo of Lord Corporation.

#### Estimation and Control of Electromagnetic Energy

Collaborators/Sponsors:

Tyndall AFB, Brooks AFB, and UNC-Chapel Hill

The focus of this research project is the understanding and control of the interaction of electromagnetic fields with various media including human tissue, underground contaminants and electromechanical materials. It is well known that such media are dispersive in their response to short microwave pulses. What is not well understood are the mechanisms for dispersion.

CRSC research efforts are focused on the development and validation of dispersion models using measured scattered electromagnetic field intensity. This entails attenuation mechanisms in the context of Maxwell's equations for the propagation of electromagnetic fields in complicated dielectric media. The desired end product of the research effort are efficient software packages for the identification of dielectric constants, conductivity, and polarization in biological and other materials such as peizoelastomers. Coupled with this effort on modeling and identification are the development of theoretical and computational methods for localization and control of incident electromagnetic energy in such materials.

CRSC researchers participating in these projects include H.T. Banks and K. Ito and their graduate students in collaboration with scientists at Brooks AFB and at Lord Corporation.

#### Estimation and Inference for Nonlinear Dynamical Systems

Collaborators/Sponsors:

Brooks AFB and Lord Corporation

Nonlinear dynamical systems can behave in ways that look random but aren't, a phenomenon called ``chaos''. Claims of evidence for chaotic vs. random dynamics in ecology (population dynamics), economics (stock market prices, exchange rates, unemployment rates), geology (earthquake times and magnitudes) and medicine (EEG and ECG), are active scientific controversies. It is now recognized that the conventional methods of analyzing such data, largely developed within theoretical physics, require unrealistically large amounts of data for many of these application areas, and moreover require a completely noise-free system, or they may give unreliable results.

CRSC investigators have developed methods based on nonparametric function estimation, which require 80-90% less data than conventional methods, and remain accurate at much higher levels of noise. Current research is focusing on: (1) increasing the reliability of numerical methods for data-based model selection; (2) estimation of short-term predictability as a function of current state; and (3) applications to global climate modeling, using our methods as tools for summarizing the behavior of complex models and for model validations. Efficient computational methods are essential for simulation studies to validate proposed methods, and we are interested in implementations on parallel machines, and the use of regularization methods to accelerate model fitting. An important direction for future research is methods for multivariate rather than scalar data, in particular situations where spatially replicated measurements are available, such as satellite monitoring of terrestrial climate and environmental variables.

CRSC investigators include S. Ellner, in collaboration with D. Nychka and A.R. Gallant in the Statistics Department at NCSU, and with H. Abarbanel (UC San Diego) and S. Koonin (Cal Tech) on climatology applications.

#### Financial Mathematics

This project concerns the derivation of formulas for pricing options in the context of stochastic volatility models. Option pricing consists in finding the fair price of a contract written on the future random evolution of a risky asset. These options represent now a large part of the market, largely due to growing hedging funds, and their pricing is an extremely important problem. This problem is mathematically modeled by second order partial differential equations with random coefficients which require heavy computations. Our study consists in deriving simple and practical formulas which approximate their solutions using a separation of scales. Free boundary value problems associated to American style options are also investigated as well as models for interest rates. The original techniques that we propose to use are general and relevant to the theory of stochastic processes and their numerous applications.

#### First Passage Time Problems in Biology

When the voltage at a particular place on a neuron reaches a threshold, an action potential (nerve impulse) is produced. Many point processes in biology have similar origins as ``first passage times''; that is, they occur when some underlying process first reaches a critical level or threshold. In an environmental context, the level of pollutant in a lake may be considered as a recovery process with inputs at variable times; the threshold here may be a biological one, or an artificial one such as a government standard. In a growth process context, the threshold may be the optimal time to market a particular crop or animal.

Even for simple models of the underlying process (1-dimensional stochastic differential equations), very few analytical results are available for first passage times. Through simulation and heuristic approximation methods, several different types of behavior have been identified. The main current research activities are further development of approximation methods, and new application areas for the basic methods, including reliability theory and survival analysis. For example, of the common distributions used to model time to death (of an organism) or failure (of a manufactured product), what underlying stochastic process could produce them if failure is viewed as a first passage time?

CRSC research on this topic is mainly by C.E. Smith, in collaboration with graduate students and experimental neurobiologists.

#### Inverse Problems in Nondestructive Evaluation

Collaborators/Sponsors:

NASA Langley Research Center

The detection of structural flaws in flexible structures is a very important problem in the aircraft industry. One idea for flaw detection involves embedding piezoceramic patches in composite viscoelastic members (in a so-called "smart material'' configuration). To test for cracks or delaminations, one can introduce a heat source on one side of the member. The temperature difference induces a deformation in the member, and this deformation generates a strain in the patch. An alternative test could involve production of strain via vibrations induced by the patches themselves. The polarized patch, in response to this strain, produces a voltage which can be monitored. Cracks and delaminations will of course produce different deformational behavior, and a question of great practical importance is whether the differences in deformation can be detected in a manner which characterizes the flaw. A third approach involves the use of SQUIDs (superconducting quantum interference devices) in detection of eddy current flow disruptions by internal damages.

The CRSC research program in this area focuses on mathematical and computational questions involved in developing reliable damage detection algorithms. In particular, researchers are currently developing differential equation models that describe the deformation of viscoelastic solids under high temperature gradients or due to vibrations induced by the piezoceramic patches themselves. The next step is to infer from the temperature or piezo inputs and measured viscoelastic responses the nature of the interior of the solid (that is, whether damage is present). For SQUID measurements, one must develop electromagnetic models for dielectrics with damage. For each of these, solving the differential equations and fitting the model to the observed data, computational methods play a crucial role. An important component of the research program is the development of efficient computer implementations, especially using distributed memory parallel machines and groups of workstations networked to allow parallel computation across the network. This research is a part of ongoing efforts in the CRSC on identification and control methodologies for ``smart material'' structures which are capable of self-excitation and self-sensing.

CRSC researchers on this project include H. T. Banks, K. Ito, M. Joyner and J. Peach. The efforts are in collaboration with NASA Langley Research Center scientists and scientists in Japan.

#### Magnetorheological Fluids

Collaborators/Sponsors:

Lord Corporation

We present particle dynamics simulations for the response of magnetorheological (MR) fluids upon application of a magnetic field. The particles motion is considered to be governed by magnetic, hydrodynamic and repulsive interactions. Fluid-particle interactions are accounted for via Stokes' drag while inter-particle repulsions are modeled through approximate hard-sphere rejections. In accordance with their greater significance, on the other hand, (linear) magnetic interactions are fully simulated. The time evolution is considered to be magnetically quasi-static and magnetostatic forces are derived from the solution of (steady) Maxwell's equations, recomputed at each instant in time. For this we use a potential theoretic formulation where the boundary integral equations are solved with a fast multipole approach. We show that the resulting numerical codes can be effectively used to study a number of experimental observables such as effective magnetic permeabilities and response time-scales which are of crucial importance in the design of MR fluids.

CRSC researchers on this project include H.V. Ly, F. Reitich, M. Jolly, H.T. Banks, and K. Ito

#### Modeling and Control Issues Concerning Smart Materials with Hysteresis

Collaborators/Sponsors:

Iowa State University and Lockheed Martin

This research program focuses on modeling and control issues concerning certain smart material actuators utilized in nonlinear regimes. Piezoelectric, electrostrictive and magnetostrictive materials all exhibit various degrees of hysteresis and nonlinear dynamics at high drive levels. The accurate and efficient quantification of these effects and their incorporation in control design are necessary to attain the full capabilities of the materials.

Our research efforts have centered on the development of unified modeling techniques and corresponding control methods for ferroelectric and ferromagnetic materials which exhibit hysteresis and nonlinear dynamics. The models are based on the characterization of energy losses due to pinning sites in the material which inhibit domain wall motion and subsequent changes in the magnetization or polarization. In applications which require high drive levels, linear control methods prove ineffective due to phase delays resulting from unaccommodated hysteresis. For such regimes both nonlinear optimal control formulations and linear methods which utilize model-based inverse compensators have been developed. An integral component of both the model development and control implementation involves the design and implementation of validation experiments.

CRSC researchers participating in this project include R.C. Smith and his graduate students working in collaboration with scientists at Iowa State University and Lockheed Martin.

#### Modeling and Control of Transport Processes in High Pressure Vapor Transport Reactors

Collaborators/Sponsors:

NCSU Department of Materials Science and Engineering

The theory of high pressure vapor transport (HPVT) processes is a broad area of research involving combined chemical reactions and transport phenomena. HPVT is an industrially important process used to deposit thin solid films on solid substrates. The most extensive application of this technique is in the microelectronics industry where it is used in the fabrication of transistors, integrated circuits, and computer chips such as DRAM. Other materials which have been fabricated by HPVT processes include optical devices such as solid-state and semiconductor lasers and high efficiency energy conversion solar cells.

The quality of these devices depends strongly on the uniformity in thickness and composition of the film structure. These, in turn, depend on gas flow, thermodynamics and mass transfer in the deposition chamber reactor. Thus, recently, much attention has been given to the area of modeling and numerical simulation studies for gas flow motions in reactors of various geometries. The transport phenomena is described by gas-dynamic equations (conservation of mass, momentum and energy) including species diffusion. These equations are nonlinear partial differential equations and require a tremendous amount of computer resources. CRSC efforts, in addition to the modeling of the deposition process, involves the application of optimization and control techniques with a goal to improve the growth processes. High performance computers such as supercomputers and parallel computers will be utilized. A unique feature of this work is that it brings together the theory, computation and experiment.

CRSC research on control of HPVT reactors is related to the emerging field of ``smart'' material structures such as piezoceramic embedded composite materials. These smart materials are capable of both distributed sensing and actuation and in the context of this research might be employed to control flow motions in the reactor cavity via control of the boundary layers as well as in other structural estimation and control applications.

CRSC researchers on this project include H. T. Banks, K. Ito, G. Kepler, I. Lauko, J. S. Scroggs and H. T. Tran; and the research involves a major interdisciplinary collaborative effort with K. Bachmann and his team of researchers in the Department of Materials Science and Engineering, and N. Dietz and students in the Physics Department.

#### Modeling in Granular Materials

Systems of nonlinear partial differential equations appear in continuum descriptions of the flow of granular materials. Properties of these equations that have important consequences for the deformation of materials include: wave propagation, the formation of shock waves, loss of hyperbolicity (leading to material failure), nonlinear stabilization. Specific applications studied include the deformation of granular materials, the flow of granular materials in hoppers, dynamic shearing of granular materials.

The basic approach is the use of mathematical analysis and scientific computation to understand the properties of granular materials, motivated by specific applications, and by physical experiments.

Industrial problems being studied concern the transition between funnel flow and mass flow, an important indicator of dangerous instabilities in the flow of granular materials in hoppers, and the design and positioning of flow corrective inserts in funnel flow hoppers.

CRSC researchers on this project include P. A. Gremaud and M. Shearer.

#### Modeling, Numerical Approximation, and Control of Aeroacoustic Noise

Collaborators/Sponsors:

Boeing Company

This project centers on the modeling and control of high intensity cavity pressure fields generated by adjacent flow fields. Flow-induced acoustic fields in cavities such as wheel wells or bomb bays can have sufficient magnitude (140-170 dB) to produce structural damage if unaccommodated. The basic physical mechanisms which generate the acoustic fields and the utilization of these mechanisms for control design are under investigation.

The adjacent flow fields are inherently nonlinear and are typically modeled by either Navier-Stokes or Euler equations. The acoustic field is inherently compressible and also exhibits nonlinear dynamics at the decibel levels which produce damage. The modeling component focuses on issues concerning the coupling of the fields through unsteady shear layers and the development of boundary conditions which quantify the energy losses measured in experiments. Large scale numerical simulations are being run to determine the accuracy and limitations of the models and to ascertain the open loop dynamics of the system. To facilitate both simulations and control design, it is necessary to develop reduced-order approximation methods. One method under consideration entails the representation of system dynamics through the construction of appropriate POD basis elements. Once this phase is completed, the investigation will focus on the design and implementation of open and closed loop control methods which utilize the reduced-order basis.

CRSC researchers participating in this project include H.T. Banks, D. Bortz, A. Cain and R.C. Smith working in collaboration with scientists at the Boeing Company and at Innovative Technology Applications Company, LLC.

#### Numerical Weather Prediction

Weather forecasting and prediction of global climate change utilize models of the circulation of the earth's atmosphere. These Global Circulation Models (GCMs) are a system of highly nonlinear partial differential equations that include terms to model the effects of the rotation of the earth, and the interactions of land masses and oceans with the atmosphere.

A common approach to the numerical simulation of solutions to this system of partial differential equations is to use a fixed reference frame (latitude and longitude) and compute the air flow as it passes this fixed frame. The number of data points of these simulations is quite large. For example, to obtain the resolution of 1 degree in latitude and longitude with 10 points in the vertical direction would require more than 10^6 points. Further difficulties occur due to the high speeds in the atmosphere, such as those in the jet stream. Due to these large velocities, it is necessary to take very small increments in time when using such models. A consequence of the large number of data points with the small time steps demands computing resources beyond even the most advanced parallel supercomputers.

This CRSC project involves developing new methods that utilize a moving reference frame (called the Lagrangian formulation) to compute variations in the air along the paths taken by particles in the air. The methods take advantage of some aspects of both the fixed and the Lagrangian frames of reference; thus, they are called Semi-Lagrangian Global Circulation Models. These methods have been demonstrated for weather prediction, and CRSC researchers are currently studying how to modify these methods for climate change modeling, and how to utilize state-of-the art parallel supercomputers for their implementation.

Investigators include J. S. Scroggs and Nelson Settumba (CRSC), F. H. H. Semazzi and George Pouliot (MEAS), and Andy Smith (CS).

#### Rate Distribution Models for Size Structure Population Dynamics

Collaborators/Sponsors:

University of California -- Davis, University of Southern California, and Universität Graz, Austria

This project was inspired by the wide use of mosquitofish in commercial rice fields as a means to reduce mosquito populations. Field experiments have produced mixed results regarding the effectiveness of mosquitofish for this purpose, and scientists at the University of California -- Davis and the California Department of Game an fisheries have been conducting closely monitored experiments to develop an understanding of the mosquitofish population dynamics. The data collected from these experiments exhibited certain features that were inconsistent with the commonly used Sinko-Streifer population model, which predicts how a population of individuals of different sizes evolves in time. One of the model's weaknesses is that it requires all individuals to have the same growth and mortality rates. In recent years several researchers both in the US and Europe have worked to develop models which are based on the Sinko-Streifer model but contain a distribution of growth rates. More general models involve distributions of birth and mortality rates, as well as nonlinear dependence on the total population size.

CRSC investigators have recently developed some parallel algorithms for estimating these rate distributions, and current fits to the Davis mosquitofish data using these algorithms are quite promising. Avenues of further research include the development of statistical tests of fit to quantify how well the model compares to the data, the incorporation of the mosquitoes into the model (as a predator/prey system), the development of optimization strategies to study how best to use mosquitofish in rice fields, and the investigation and generalization of the rate distributions to other population biology applications.

CRSC researchers on this project include H. T. Banks, K. Gaston and L. Potter. The efforts involve collaboration with scientists at the University of California -- Davis, University of Southern California, and Universitat Graz, Austria.

#### Waves in Random Media

Collaborators/Sponsors:

Biochemical Modeling of Orthopedic Soft Tissues, Duke University Medical Center

This project concerns three problems related to waves propagating in random media: refocalization effect for time-reversed acoustic waves, propagation of pulses in the regime of parabolic and white noise approximation and the use of wave automata in the numerical simulations of these phenomena.

**Time-Reversal:**

Time-Reversal Mirrors (TRM's) are piezoelectric devices which can convert acoustical pressures into electric signals and vice-versa. They are coupled to memories which permit to monitor an acoustical pressure and send it back into the medium in the reverse direction of time. If the initial source is like a pulse, localized in time and space, it has been observed experimentally that few TRM's are enough to refocalize at the source in a coherent way. Surprisingly the presence of desorder (or randomness) in the medium "seen" by the wave improves this refocalization effect. Applications of this technique to inverse problems are very promising (medical imaging, nondestructive industrial control,...). Also mathematical understanding of this phenomenum is essential in domains like Geophysics where techniques such as "migration" are based on the same ideas.

**Pulse propagation in the parabolic approximation:**

The parabolic and white noise approximation is widely used in domains such as underwater acoustic waves or optics in atmospheric turbulence where the fluctuations of the index of refraction are small, the frequency is high and the distances of propagation are long. The monochromatic (or stationary) wave field, in this limit, satisfies a stochastic Schrodinger equation which leads to an open system of moment equations. This technique will be generalized to multi-frequency cases to derive the equations of propagation of a correctly rescaled pulse and use its shape modification in inverse problems.

**Simulations using wave automata:**

Wave automata give an efficient (parallel) way to simulate waves propagating on a lattice in the time domain. This is achieved for any type of waves and in any dimension of space. The introduction of varying coefficients (like random coefficients for random media) is very simple. The fact that this simulation is done in the time domain makes it well-adapted to the study of the refocalization effect by TRM's or to the study of the transmitted pulse.