## NCSU Workshop Gives Students Hands-on Industrial Modeling Experience

**November 27, 1998**

The Aerospace team, working with problem presenter Tien Nguyen, considered the application of HPA linearization in a global broadcasting satellite.

"The problem is ill-defined, ill-posed, overconstrained, underdetermined, different from what it was three weeks ago, and we got the data we need immediately before the deadline-welcome to industrial math."

Addressing 36 graduate students, six of whom would be working on the reciprocating piston pump design problem he was about to present, Jeff Sachs* was one of six industrial mathematicians who spent ten days, July 27 to August 4, at North Carolina State University, at the 1998 Industrial Mathematics Modeling Workshop for Graduate Students.

The six problems that would be occupying the six-student teams for the next ten days, H.T. Banks told the group, "are all important to someone in the real world." To participate in the workshop, he pointed out, the presenters had to convince their companies not only that the problems were worth solving, but also that they were worth ten days of their employees' time.

Banks is director of the NCSU Center for Research in Scientific Computation, which, along with the university's mathematics department and the National Science Foundation, sponsored the workshop, the fourth to be held at NCSU. (The National Security Agency was a sponsor of the first three workshops.) The organizers were Hien Tran, Ralph Smith, and Pierre Gremaud, all of CRSC and the NCSU mathematics department; the students, selected from a pool of applicants that grows larger (and, according to the organizers, more qualified) every year, were from universities throughout the U.S., with six from NCSU (one per team). Also working with each team was a faculty member from NCSU who, in most cases, had recruited the problem presenter for the workshop.

**Two Employers, Three Problems**One problem presenter who needed very little recruiting was Tien Nguyen of The Aerospace Corporation, whose involvement in the workshop dates back to 1994. Then working at Caltech's Jet Propulsion Laboratory, he was simultaneously pursuing a second PhD (in applied mathematics) at Claremont Graduate University, where his thesis adviser was Ellis Cumberbatch, one of the workshop founders. Hien Tran, invited by Cumberbatch to Claremont for a workshop, invited Nguyen to present a problem in 1995, at what would be the first of the workshops held at NCSU.

Nguyen's good experience with the mobile receiver problem he presented in 1995 for JPL---"I was able to apply the model derived directly to the AMT (ACTS-Advanced Communications Technology Satellite-Mobile Terminal) program that I was working on"---led him to convince his current employer, Aerospace, to have him take two of its problems to subsequent workshops.

This year, Nguyen was asking the students to devise techniques for mitigating the effects of nonlinear power amplifiers in non-constant envelope signals. The problem, he told the group on the first morning of the workshop (in the course of the morning, all six industrial presenters described their problems to the entire group of 36 students), has global broadcasting service applications-in direct TV, for example, because the DVB waveform used in direct TV has very narrow bandwidth.

Briefly, he explained, a filtered audio/video signal that has been modulated on a carrier and amplified by an HPA operating at saturation will be distorted. The problem is not a new one, and the usual solution is to adjust the amplitude at RF before the signal passes through the HPA, although spectral regrowth and signal-to-noise power ratio degradation will occur. The attack on the problem that he suggested to the students is a predistortion of the original signal, at the baseband, that will compensate for the distortion wrought by the HPA.

The existing (Saleh's) model is not universal, Nguyen explained, and lookup tables are needed with a simple linear-log model. What the team needed to do, then, was "Extend the current algorithms to the general case, in which a priori knowledge of the HPA's characteristics is not required; with no lookup tables, this leads to an adaptive linearizer!"

How did the team do? During the workshop, Nguyen reports, the six students, working with him and faculty adviser Hien Tran, "came up with a simple adaptive linearizer to mitigate the effects of AM-AM and AM-PM [AM denoting amplitude modulation and PM, phase modulation] for any HPA." The team was able to show that "with an ideal adaptive linearizer, the power spectral density (PSD) remains the same after the signal passes through a typical HPA." The proposed adaptive linearizer, he says, is very easy to implement. The next step-to show good performance (both PSD and bit-error-rate) for nonideal adaptive linearizers-"appears very promising."

**Absolutely First-rate Work**"Equations are good," Jeff Sachs told the students, "but when you go out to sell your services to people, you need to be able to communicate with non-mathematicians." Like most of the problem presentations, his talk was a demonstration by example of the importance of effective communication in industrial settings. Asked how he came to participate in the workshop, Sachs looks back twenty years to his first class in applied math, at Brown University with Banks. This spring, he says, he gave a seminar at NCSU, after which Tran and Banks invited him to present a problem at the workshop.

The system of interest to Wagner's client, Sachs told the group, consists of two independently controlled piston pumps, reciprocating so as to produce a constant flow rate. With two motors and two pressure transducers, the system is pulling in and then pumping out a dynamically changing fluid. The fluid could be drugs, reagents, cleaning fluid. . . . (Sachs's earlier description of the setting for the problem---consisting in its entirety of the sentence "It's a biotech application from a national corporation"---drew a laugh as it told the students something about the need for confidentiality in industrial work.)

Summarizing the (admittedly ambitious) goals for the week, Sachs told the students that, using the requirements specified by the client, they were to create a mathematical model relating fluid properties, motor motion control signals, pressure transducer output, and output fluid flux and pressure. While looking at the ideal working of the system, the team would also have to consider reality: compressibility of the fluid, acceleration limits of the motors (at the corners), pressure pulses, and valve leakage, all of which are changing dynamically, along with fluid and load properties.

What the students needed to strive for, Sachs concluded, was a *feasible* solution that would meet the client's requirements. "Think algebra," he counseled; "we can't solve these differential equations in real time." "We don't want pressure pulses, which could damage the biological system of interest, and we need to meet the stringent outlet flow-rate accuracy specifications."

Ten days later, the team had produced a simple model (a compressible, viscous fluid in a pipe, with linearized conservation of mass and momentum), followed by an ultrasimple model (lumping many effects together and ignoring the effects of several components) to be used for the control analysis. One feature of the model is the component-wise construction of the solution matrix---modeling the addition of a component to the system is accomplished by the addition of a 2 x 4 submatrix to the matrix representing the coupled differential-algebraic equations. Numerically, the team used backward Euler time-integration to solve the equations.

Future work on the problem, as recommended by the team, would entail continued improvements to the model, development of a step motor model, and implementation of their estimator, Kalman filter, and main controller.

The "absolutely first-rate" team, says Sachs, worked each day from 8:30 A.M. to (at least) midnight. "Even when I bought them baseball tickets to take a night off, they decided to stay and work because we were behind the schedule we had set for ourselves."

The team's hard work paid off not only in being a superb learning experience for the students. Sachs, at the client's request, is continuing to pursue several of the group's ideas for analyzing and controlling the system. "In short," he tells *SIAM News*, "the analytical thought processes that we mathematicians (and control theory engineers) bring to this kind of problem are valuable and complementary to the kind of (also valuable) skills that the client's system/software/hardware/etc. engineers have."

**Getting Started**

*SIAM News*, able to visit the workshop only for the first of the ten days, took in all six problem presentations, learning, along with the students, something about the very different industrial environments in which the presenters work. Represented, in addition to Aerospace and Wagner, were

- Battelle Memorial Institute, the largest independent R&D company in the world, which does work for government and industry;
- Jenike & Johanson, Inc., a specialized engineering firm that provides solutions to solids-handling problems;
- Ortho Clinical Diagnostics, a Johnson & Johnson company involved in pharmaceutical diagnostics; and
- FACE International Corporation, the manufacturer of an advanced piezoelectric technology developed at NASA Langley.

At the conclusion of the morning presentations, the students separated into teams, each with the industrial presenter and an NCSU faculty adviser, to begin planning their attacks on the problems.

Dropping in on the Battelle team---made up, like all the teams, of a mix of students in master's and PhD programs in a wide range of areas---*SIAM News* found the students eager to absorb some basics from the problem presenter, Chad Bouton, and the NCSU faculty adviser, Ralph Smith. Smith, whose recent work has been in control theory, had invited Bouton to present a problem at the workshop after getting to know him in the course of some work on piezoelectric problems. The problem (which, Bouton explained, was of immediate interest to one Battelle client and potentially useful for several others) was to model a circular piezoceramic actuator, maximizing the displacement of the actuator while minimizing power consumption.

Gently guided by Bouton and Smith, the students put off looking at the final goal, a model of a circular piezoceramic actuator mounted on a plate with larger radius, and began by reviewing some simple elasticity theory, looking at the simplest possible structure and starting to construct a model.

"What's an actuator?" asked one student, not too far into the discussion.

From these tenuous beginnings, Bouton tells *SIAM News*, the students successfully scaled the learning curve for piezoelectric actuators and were able to attack the mathematical core of the problem. To give the students an idea of what contract R&D is like and to prepare them for the industrial world if they choose to take that path, Bouton, as the Battelle representative of the actual client, "pretended to be a client part of the time." At other times, he says, "I worked with the students in my true role as a visiting Battelle research engineer and was very anxious to see them go deeper into the problem than I had gone previously." By the end of the workshop, he reports, "they had not only succeeded in extending my original model, they were also able to demonstrate that the model could be used to optimize the performance of the class of devices they were modeling."

As Smith pointed out during the morning presentations, the team working on the Battelle problem would do well to talk to the team working on the (complementary) problem from FACE Corporation-prediction of a radius of curvature of a piezoelectric composite in prestressed condition and its impact on the actuator's properties.

Visiting the FACE team during its first session, *SIAM News* found Karla Mossi from FACE beginning to map out a strategy with the students on the team, a group that included a mathematics/physics major, a student who had switched from pure mathematics to statistics, and one team member with some experience in finite elements. After deciding to dispatch some members of the group to the library to read up on the mechanics and thermodynamics of materials, and others to look into some basics on piezoceramics, the students tentatively began to define the problem, thinking about the data they had (on curvature) and the data they needed (on stress).

There's prestress in the structure, produced in the manufacturing process, Mossi explained; "but you don't know how much. You want to know the curvature produced in the active material by the stress."

Although the students initially had no familiarity with materials and mechanics, Mossi found them "fast learning, enthusiastic, and hard working." The solution they came up with, she says, "is a beginning of more projects to come."

The silence of deep concentration was emanating from the Johnson & Johnson team's initial working session when *SIAM News* stopped in. Zhong Ding, who presented the problem---mass transport and surface reactions in immunoassay---had given the students an initial, small, pre-one-dimensional problem, with no boundary and a homogeneous solution that changes as a function only of time.

The students soon started to act like a team. "Does anyone know what a stiff equation is?" asked one member of the group. "What about a Green's function . . . or is that only with PDEs?" asked another. "We better do a little research."

The Johnson & Johnson team was investigating immunoassay procedures, in which the existence or concentration of a particular invading antigen is determined, typically by means of a labelled antigen that competes for binding sites with the antigen in question on the antibodies produced by the body.

Starting "from zero knowledge of the immunoassay," says Ding Zhong, who was invited to the workshop by NCSU faculty member Zhilin Li, the students were able by the end of the workshop to describe, in their own language, the immunoassay and its applications. The team successfully used mathematical and numerical techniques to predict the behavior of the assay under different conditions. In the process, Ding points out, "they also learned the applications and limitations of different numerical techniques in solving a mathematical problem."

Working "hard and cooperatively," he tells *SIAM News*, the students also came to realize that a broad range of knowledge, including physics, is very important in solving real-world problems. "In short," he concludes, "I think they gained a lot in a very short period of time."

Also hard at work on a simplified problem at their first meeting was the Jenike & Johanson team, whose problem, presented by T. Antony Royal, involved the flow of granular material. "We're about a hundred years behind in understanding granular materials, which are very different from fluids," Royal told the group in the morning. Pierre Gremaud, a faculty adviser on the team, had recruited Royal for the workshop through a project on numerical methods for granular flows.

The team's charge was to model the time-dependent consolidation of compressible fine powders, with an eventual goal of predicting, for example, how air escapes from a powder when a bin or silo is filled and then allowed to deaerate with time. The students considered an axisymmetric problem so as to reduce the general three-dimensional problem to two spatial dimensions and one time dimension.

*Hien Tran of NCSU, one of the organizers of this year's workshop, worked with the Aerospace team, on the nonlinear power amplifier presented by Tien Nguyen, who, on Tran's invitation, has now participated in three NCSU modeling workshops.*

*The students "tried very hard for their result" on what turned out to be a "very challenging problem," says faculty adviser Hung Ly, who worked with Jeff Sach's team on the reciprocating pump problem. As a hard-working student himself not so long ago, Ly had participated in the 1995 workshop at NCSU, working on the "chimera domain decomposition problem." He subsequently received his PhD, from the University of California, Irvine, and is now an NSF industrial postdoc at NCSU.*

H.T. Banks, who with Ralph Smith served as faculty adviser to the FACE team, has long been convinced of the value to students of work on real industrial problems. He could have been describing the entire workshop experience as he told the students on the FACE team, during their first tentative, seemingly undirected steps, "This is a mini, compressed version of the way you'll get problems in industry---the only difference is that you won't have so many people working on the problem, and you won't have a helpful adviser."

*Then of D.H. Wagner Associates, and now an independent consultant.