JOHN TUCKER: It is my pleasure to introduce the individuals who will moderate the open discussion: Art Jordan from the Naval Research Laboratory and Tom Hughes from Stanford University.

THOMAS HUGHES: This moderated open discussion is intended as a period of fermentation where issues are put on the table, the most important of which will be summarized in the concluding panel discussion. In this spirit, we would like the audience to take the lead and raise issues. Some questions and comments have been proposed in writing, but we will introduce those after people express spontaneous perspectives.

RICHARD ZIOLKOWSKI: I have been struck these two days by how much overlap there appears to be between the acoustics community and the electromagnetics community on the type of modeling techniques applied to essentially the same kinds of wave propagation problems. There are a number of Helmholtz approaches at which both communities are looking. In the time domain, I believe there are similar approaches in both communities dealing strictly with waves.

For example, one of the outstanding electromagnetics problems is to get better absorbing boundary conditions. Antennas and propagation annual meetings have three or four parallel sessions each afternoon on nothing but absorbing boundary conditions and their similarities. The Enquist and Majda type of boundary conditions are well known. We started out using those but have developed other techniques, such as ion surface radiation conditions. There are new Berringer and absorbing loci electric materials that one can now put around the problem. From the acoustics standpoint, what kinds of techniques are being used for which there might be direct applicability to the electromagnetics regime, especially with regard to things such as absorbing boundary conditions?

I was also struck this morning by the idea of gridless methods. They would be wonderful. The biggest problem with three-dimensional modeling techniques right now is grid generation. Although the grids are sometimes good for the techniques being used, there are a number of unstructured grid capabilities for which, depending on the modeling technique being used, the generated grids sometimes cannot be used. These are merely a couple of representative issues of similarity between acoustics and electromagnetics that it would be useful to discuss.

HUGHES: The first question raised from the electromagnetics community concerns the status and thinking regarding absorbing boundary conditions in current use and being explored in acoustics, and what individuals in this allied field should consider: Where are the opportunities and current thinking?

LONNY THOMPSON: Some of the radiation boundary conditions that are used for electromagnetics and acoustics are actually the same for the lower-order boundary conditions. Some people have developed the same boundary conditions in both fields independently. If they were to look at each other's work, they would realize the conditions are very similar, if not the same.

I would stress that one should probably be using higher-order boundary conditions, instead of stopping at first- and second-order accurate boundary conditions as is typically done. The reason is that one would like to move the computational domain for exterior problems that arise in acoustic problems, as well as in

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11
Discussion
JOHN TUCKER: It is my pleasure to introduce the individuals who will moderate the open discussion: Art Jordan from the Naval Research Laboratory and Tom Hughes from Stanford University.
THOMAS HUGHES: This moderated open discussion is intended as a period of fermentation where issues are put on the table, the most important of which will be summarized in the concluding panel discussion. In this spirit, we would like the audience to take the lead and raise issues. Some questions and comments have been proposed in writing, but we will introduce those after people express spontaneous perspectives.
OPEN DISCUSSION
RICHARD ZIOLKOWSKI: I have been struck these two days by how much overlap there appears to be between the acoustics community and the electromagnetics community on the type of modeling techniques applied to essentially the same kinds of wave propagation problems. There are a number of Helmholtz approaches at which both communities are looking. In the time domain, I believe there are similar approaches in both communities dealing strictly with waves.
For example, one of the outstanding electromagnetics problems is to get better absorbing boundary conditions. Antennas and propagation annual meetings have three or four parallel sessions each afternoon on nothing but absorbing boundary conditions and their similarities. The Enquist and Majda type of boundary conditions are well known. We started out using those but have developed other techniques, such as ion surface radiation conditions. There are new Berringer and absorbing loci electric materials that one can now put around the problem. From the acoustics standpoint, what kinds of techniques are being used for which there might be direct applicability to the electromagnetics regime, especially with regard to things such as absorbing boundary conditions?
I was also struck this morning by the idea of gridless methods. They would be wonderful. The biggest problem with three-dimensional modeling techniques right now is grid generation. Although the grids are sometimes good for the techniques being used, there are a number of unstructured grid capabilities for which, depending on the modeling technique being used, the generated grids sometimes cannot be used. These are merely a couple of representative issues of similarity between acoustics and electromagnetics that it would be useful to discuss.
HUGHES: The first question raised from the electromagnetics community concerns the status and thinking regarding absorbing boundary conditions in current use and being explored in acoustics, and what individuals in this allied field should consider: Where are the opportunities and current thinking?
LONNY THOMPSON: Some of the radiation boundary conditions that are used for electromagnetics and acoustics are actually the same for the lower-order boundary conditions. Some people have developed the same boundary conditions in both fields independently. If they were to look at each other's work, they would realize the conditions are very similar, if not the same.
I would stress that one should probably be using higher-order boundary conditions, instead of stopping at first- and second-order accurate boundary conditions as is typically done. The reason is that one would like to move the computational domain for exterior problems that arise in acoustic problems, as well as in

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electromagnetics. The radiation boundary should be brought as close to the object or the scatter as possible. If one of these lower-order boundary conditions is used, then the needed accuracy is unavailable when the boundary is brought very close. So one must go to a higher-order boundary condition. A number of papers have looked at and developed higher-order boundary conditions. The challenge is how to implement those in an efficient way either into finite difference, finite element, or other discrete techniques for basic computational methods.
LESZEK DEMKOWICZ: I have two comments. First, I believe those high-order absorbing boundary conditions are nothing other than a truncated form of an infinite element. In a theory of infinite elements, it turns out that all one really needs is an extra infinite expansion in the fight direction. When second- or third-order truncated, nonreflecting boundary conditions are used, they actually correspond to a type of approximation in the real direction.
Second, when one looks at the classical separation-of-variables argument and the form of the exact solution, say, for the sphere problem under the Helmholtz equation, it becomes evident that the terms corresponding to a higher-order frequency on a sphere have simultaneously a larger exponent N in the denominator there. As one moves away from the body, the practical effect of those terms disappears, because having the denominator R raised to the power N for larger N makes the fraction converge to zero faster. For that reason and without any calculations, one can probably anticipate that there must be a tradeoff between the number of finite elements put in between the scatter and that artificial boundary (on which are positioned the infinite elements or nonreflecting boundary condition), and the numbers of terms in the real direction for the infinite element. The closer one gets to the scatter, the smaller will be the number of elements in between; but the number of terms in the infinite element will grow, and there is nothing that can be done about that. It is just the nature of the problem. Analogously, of course, as one gets further away from the scatter, then the number of the terms in the expansion one can use will probably be smaller, but that is compensated for by the number of elements that are positioned in between the two.
THOMPSON: I wish to respond to that. The infinite element and local radiation boundary conditions are both very similar in that they approximate the exact impedance on the radiation boundary. However, when one uses infinite elements and goes closer to the scatter, the fact that more terms or more layers in the infinite element must be used is offset by the cost [of adding extra layers for the infinite element] being much less than it is for having to discretize with regular finite elements in the domain that has been eliminated.
Also, the alternative to going to local boundary conditions and local infinite elements is, of course, to look at nonlocal boundary conditions that are theoretically exact. I am referring here to the Dirichlet-to-Neumann (DtN) boundary condition. It is nonlocal in this setting. So, if a direct solver were used, it would have disadvantages in storage costs. However, with some of the techniques for iterative solvers and some that Professor Pinsky presented—with matrix-free iterative solvers—one does not have to assemble a global matrix. There, some of the storage costs that arise from using nonlocal boundary conditions can be minimized or eliminated. So nonlocal boundary conditions require a very good approximation in order to go much higher up in the order of approximations that can be achieved.
HUGHES: I know that a number of acoustics techniques are already used in electromagnetics. Have infinite elements already been used also?
ZIOLKOWSKI: Yes, infinite elements have been used for a number of years.
HUGHES: It seems as though everything is used.
ZIOLKOWSKI: Yes, everything is used. What each technique is called for each community is going to be important with regard to interpretation. We in electromagnetics also call them infinite elements. My first project at Livermore was to do a global look-back scheme for the finite-difference time domain. It used Huygens' representation over a closed surface to provide the next point outside the surface in order to get an

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exact boundary condition. The problem with it is that, even though one can do it—in some sense—better and better with larger computers, one still has to back store for retarded time effects. It turned out it is not very efficient to do that. In a setting with extremely short pulses, one can get away with it. However, if one is working with very long pulses, then the back storage is a tremendous liability.
With regard to the techniques, something should be done to find out exactly what everyone knows and how well each approach works. For instance, we at the University of Arizona have difficulties with a lot of these conditions because we have evanescent wave problems. I do not know exactly what happens in applying boundary conditions for evanescent waves, but evanescent waves cancel out the use of many of the absorbing boundary conditions that people might try to use for certain problems.
PETER PINSKY: Concerning DtN versus infinite elements, the comment made by Leszek Demkowicz was a very sensible one in that there is an underlying relationship in the formulation between DtN and infinite elements. In fact, we have recently been performing numerical convergence studies at Stanford between these two types of boundary conditions, the local infinite element and the non-local DtN. We see numerically some very interesting asymptotic characteristics of these methods and, indeed, believe that fundamentally there are some similarities. In terms of the numerical implementation and the frequency domain for large-scale problems, as was pointed out by Lonny Thompson, maybe the differences between these things tend to diminish. I think they both represent very good opportunities.
I believe they can treat general complex waves, propagating waves, or evanescent waves. I see no difficulty in terms of absorbing complex wave numbers in these boundary conditions. The situation is very different, however, in the time domain. The issue of representation of boundary conditions in the time domain is an open question that deserves a lot of attention. To my knowledge, there is no notion of an infinite element in the time domain that is truly effective. One needs to create a class of methods and implement them in a way that provides a framework that will allow these boundary conditions to be placed close to the structure.
J. TINSLEY ODEN: In some work that I discussed yesterday on implementing Enquist-Majda-type local boundary conditions and nonabsorbing boundary conditions, these are locally applied. They work in the time domain, but a key feature in these considerations is the availability of an a posteriori error estimator that can be used to guide an adaptive process. If one has a robust error estimator, one can assess to what extent the boundary conditions are being correctly imposed. Such an estimation also allows decisions to be made on whether or not a very low order approximation at the boundary is sufficient, or whether one may go up to second-, third-, or in some cases even fourth-order approximations at the boundary.
HUGHES: Richard Ziolkowski raised the issue of gridless methods and their potential applicability in the electromagnetics area. Since Ted Belytschko is the most gridless person I know, I would like to ask him to make some remarks on the applicability of such methods.
TED BELYTSCHKO: The issue of gridless methods is something that still needs to be resolved. There are definitely computational penalties that are paid for using gridless methods in smooth problems, and consequently I am not sure that they will supplant methods that do have a grid. On the other hand, they do avoid the entire problem of meshing, which in three-dimensional problems or in problems with complex geometries is often an important consideration. Furthermore, one area where I see gridless methods as being particularly attractive is in adaptivity, because remeshing for adaptivity can generally be quite onerous; with gridless methods this burden is diminished considerably. However, I think the main application area of gridless methods is now in discontinuous problems. There, it is very easy to track fronts via gridless methods. That is exemplified to some degree in the dynamic fracture problems on which I have worked. There are also other problems that have clear fronts where gridless methods may be of considerable advantage.

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LAKSHMAN TAMIL: Could somebody shed some light on the nonlinear problems encountered in acoustics?
IRA DYER: I would say the light to be shed is no light at all because acousticians always assume the problem to be linear, both in the fluid medium and in the structure. Some structures can behave nonlinearly in the structural acoustics problem, but we ignore that in favor of other things that seem currently to be of greater importance.
HUGHES: I am beginning to see some industry interest in nonlinear mechanisms in acoustics. One might model the acoustical field in a traditional way with a Helmholtz equation, but there are important sources of noise that are nonlinear, such as time-dependent turbulent phenomena and flows. There have been recent examples in which if you do not model that nonlinearity, you cannot possibly model the source of noise. Also, some nonlinear structural response is easy to incorporate in a general coupled acoustical structural problem. So some introduction of nonlinearity is beginning, although I do not believe it is yet widespread.
BELYTSCHKO: If people are going to come to grips with the interaction problem of structures, they will undoubtedly have to start looking at nonlinearities in structures. The more complex acoustical models may perhaps already exhibit some nonlinearities, and if one starts talking about a practical, real structure, one finds many joints and other, let us say, structure classes of that type that are inherently nonlinear. In order to truly assess their effect on the general behavior, nonlinear modeling will have to be included.
DYER: Indeed, nonlinearities ultimately will have to be included, but at this point they have not been. Many things in acoustics such as bells and gongs are basically structures that respond nonlinearly. They owe their musical qualities to the fact that they possess a nonlinear response. Although nobody bothered to find the differential equations in antiquity, nonetheless we are able to transmit the designs of bells and gongs. But modem-day acousticians seem to be focused on other problems. This is not to suggest that nonlinearities do not exist in structural systems, but acoustics, as it is usually defined, nowadays tends to ignore them.
TAMIL: Are inverse problems of interest to the acoustics community?
DYER: Half of the acoustics community solves inverse problems, and the other half solves direct problems. There are many inverse acousticians. For example, in a class of ocean acoustics called acoustical oceanography, sound waves in the ocean are used not to detect foreign objects, but rather to interact with the basic oceanic properties. Sound waves in the ocean have become a major oceanographic and geophysical measuring tool. For a long time, and still to this day, Offshore exploration geophysicists have used acoustics to find oil. That is an old problem in acoustics, and there is an enormous field of activity in inversions.
ADRIANUS DE HOOP: One of the symposium goals that John Tucker put forward is to evaluate frequency domain versus time domain. As far as I know, one problem has been solved where the frequency-domain answer cannot predict the time-domain answer. This is the scalar wave scattering by an object of compact support'. If one does the Neumann expansion of the relevant integral equation and looks for the convergence condition in the frequency domain, there is a combined convergence criterion in which the maximum contrast of the properties of the object with respect to its embedding occurs. One gets the square of the maximum diameter of the wavelength with another 2π, and so on; it must be less than one for convergence. Under that condition, the frequency-domain Neumann expansion of the integral equation converges. If you approach the same problem in the time domain, you end up with only a very simple restriction on maximum contrast. The maximum value of the absolute relative contrast must be less than one, and that is all. There is no condition on the size of the object or the convergence of the Neumann expansion of the time-domain integral equation.

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I have not been able to generalize this to the basic, let us say, electromagnetic, electrodynamic problem because the bounds I need for the Green's function are not as simple as in the scalar wave scattering theory. On the other hand, someone here might know similar theorems, generalizations, or things of that kind.
ODEN: I do not know whether I can address the question in the particular framework you describe, but I made some very quick comments in my presentation that seem to be directly related to this feature. If one looks at certain frequency-domain formulations of the coupled problem, particularly with boundary element methods, finite element methods, where there is no damping—neither structural damping nor damping of any kind—then one can show that the governing operators are well behaved. They are strongly elliptic. Bilinear forms are strongly elliptic provided one is not close to resonant frequencies of the coupled system.
As the wave numbers get higher, these resonant frequencies stack up, and it becomes increasingly difficult to find a gap between the eigenvalues of the system and the natural frequencies in which a solution exists, and you experience this deterioration in stability and conditioning of the system numerically, as well as theoretically. A little damping changes the whole situation. If there is damping in the system, then at least from the theoretical point of view the associated sesquilinear forms are strongly elliptic, and one can infer that solutions exist.
If one takes an integral of these equations in time, and goes to the time-domain formulation, under suitable conditions on the initial data described in my presentation, one gets an abstract Cauchy problem with a self-adjoint operator on a complex Hilbert space that is completely well defined. There is no question of the existence of solutions, and then various methods of approximation essentially revert to ways of approximating the spectrum of that operator.
This recent observation would lend some weight to Professor de Hoop's comments.
ARTHUR JORDAN: A written question asks: ''In what ways are methods that alternate implicit and explicit schemes to improve stability while maintaining efficiency analogous to those that alternate time-domain and frequency-domain methods for the same purposes? Can these parallels be used to further beneficial ends here?''
HUGHES: I would say, "In what ways, if any?" I am not sure they are related at all.
JORDAN: If the author of the question is here, maybe he could clarify it.
TUCKER: When I was speaking with Hermann Haus the other day, one thing he described was a way in which the time-domain and frequency-domain methods were alternated as a means of maintaining stability of the computational scheme, and it struck me that this parallels what was done about 15 years earlier to achieve stability without sacrificing efficiency for the solution of stiff systems of ordinary differential equations. Are there any other parallels that could beneficially improve what this symposium addresses?
THOMPSON: Implicit-explicit methods partition the domain into different regions where one uses different techniques depending on the character of the subdomains. I am not sure there is any parallel, but that brings up an interesting idea. Can one somehow use a time-dependent or time-domain technique for a certain domain and then use a frequency-domain technique for a different domain, and then use some transformation between the two to hook them up between different domains? I am dreaming, but perhaps something like that could be done.
ZIOLKOWSKI: At the University of Arizona, we actually do something like that; people are now looking at domain decomposition techniques to decompose the region in which the problem is to be solved so that in one particular region, say, one frequency-domain-type technique can be used, and one takes another domain in the same region and solves the problem there with some other frequency-domain technique, and provides the means of joining the regions together. That is something people have been working on. I do not know of anyone who has done it in one region in the time domain and in another region in the frequency domain, but if you can do one, you could probably do the other.

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HUGHES: I am gratified that you did not say, "We have already done that. So, it is not Thompson's method." I would also like to mention that there are lots of techniques of domain decomposition in the structural and fluid area where one uses very different algorithms in different subdomains. That is commonly done nowadays in multiphysics problems.
HERMANN HAUS: To elaborate on that conversation with Dr. Tucker, my students solved a nonlinear Schrödinger equation. It turns out that if you stay solely in the time domain, the solution becomes unstable. So, if you go to a frequency-domain approach, there is a smoothing for the d2 / dt2 operator in the nonlinear Schrödinger equation, and the instability is avoided.
DEMKOWICZ: I want to say a little more on the argument Tinsley Oden put forward. The essential difference between the transient- and frequency-domain formulations is that the solution in the transient domain is regular. It is very nice; it has no singularities in it. The original physical problem is stable. If energy is not pumped in, the energy actually stays constant. If stable schemes are used, both explicit and implicit, in principle it should be no problem to solve the transient formulation except, of course, for higher-order theories. With no explicit damping mechanism in the model then, from a purely formal mathematical point of view, when one takes the Laplace transform of the fully coupled problem, one ends up with a solution that has at least a single pulse at resonance frequencies.
Now, if one continues with an infinite time signal, those single pulses become double pulses. Furthermore, when it comes to the Fourier inverse transform of such a function involving double pulses, it is no surprise that some difficulties arise because one cannot numerically integrate a function even with a single pulse; actually, one can do it by using the Cauchy principal value integral sum, but it is not simple, and it is almost hopeless with a double-order pulse. That double pulse shows up in practical computations. Put simply, if one hits or gets too close to one of the numerical resonance frequencies, a double pulse shows up in that Fourier inverse transformed solution of the transient problem. However, this does not happen when one simulates the problem by treating it as a transient problem from the very beginning.
DYER: Those who work in structural acoustics never have any problems with damping because at a minimum, even if we are perhaps ignorant of what the energy loss mechanism is in the structure, we always have the energy carded to, let us say, infinity through the coupling to the external medium. So the practical answer to the question is that frequency-domain methods ought not to be difficult because there is always a way in which energy is pumped out of the system, and so out of the computation, that is, at the boundary conditions or at infinity. To the extent one does a computation that couples to an external fluid, there should be no problem. However, I do not perform these numerical calculations, and so I cannot be certain. Every problem that I know of has a coupling to an external fluid.
DEMKOWICZ: This was exactly the problem I was puzzled by when I started looking at dependence of Ladyzhenskaya-Babuška-Brezzi (LBB) constants on the wave number and, in particular, on the radiation damping. Of course, we really have only one problem in three dimensions with the exact solution (and please correct me if I am wrong): the sphere problem. For that particular problem I managed to evaluate the LBB constant that governs the stability. It turns out that, except for the first two or three eigenmodes, the LBB constant drops below 10 to roughly minus 8 for typical data for steel shelves sitting in water. For a computer, that means this is a full resonance. The computer does not see merely the radiation density. It simply cannot solve those problems.
DYER: Then one should treat a more realistic structure.
DEMKOWICZ: That is correct; stop playing with just the sphere.
PINSKY: Using domain-based methods to solve frequency-domain problems does not seem to introduce any difficulties provided that there is appropriate resolution of the wavelengths. We have experienced no difficulties for any kind of coupled problem. Indeed, the systems do experience damping in the form of radiation damping as was suggested. I am wondering if some of the problems that have been

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alluded to here do not really pertain to resonance. Are they simply manifestations of the lack of uniqueness in the integral representation that has been used in the boundary element method as opposed to real resonances that are somehow causing difficulty in the numerical solution?
EDWARD NEWMAN: In electromagnetics, the resonance corresponds to physical cavity modes. If the setting is scattering by a sphere, there is a cavity mode. When one is at a frequency where one of those cavity modes can exist, then depending on the integral formulation, the solution is nonunique.
DEMKOWICZ: First, I want to defend boundary integral formulations. As you know, some of them do have the defect of forbidden fictitious frequencies; some do not. The boundary obviously does not, and it is uniformly equivalent to the original formulation with respect to wave number. So, it is not a problem with the formulation. The entire problem of the existence of eigenvalues for the coupled problem of the scattering frequencies, as some people prefer to call them (people differ in the way they name these frequencies), is very much dependent on the geometry of the problem.
If the domain is concave, and you experience what I call a bay phenomenon, by which I mean a cavity, then even when solving the Helmholtz problem you do end up with a resonance. The corresponding eigenvalue, the scattering frequency, is going to have a small negative real contribution resulting in decay in time, some damping in time, but it is going to be very small; and the more concave the domain is, the more that particular resonance phenomenon may show up in the simulation.
JORDAN: The following written question was prompted by Tom Hughes' talk: "Does the poor estimation observed for the Galerkin least squares with adaptivity indicate a need for straightforward and more accurate upper bounds?"
HUGHES: First, what we observed was that the Galerkin least-squares (GLS) method on a mesh of an equal number of nodes compared with Galerkin was more accurate whether the solution is computed using a uniform approach or an adaptive one. So GLS is a more accurate method. On the other hand, the particular a posteriori error estimator that we derived from Galerkin least squares was indicating a larger error. So, my answer would be: Absolutely! One should endeavor to use a better error estimation to account for the improvement that is inherent in that method. Such a method would automatically fall into the category of implicit techniques that solve a local problem, because this explicit procedure simply has some shortcomings. It is an error estimation procedure based on very crude functional analysis in which inequalities start to concatenate, and every additional term that is actually doing something beneficial ends up appearing on the right-hand side. As error is currently estimated, the right-hand side, which is your bound, just piles up. Something that is more implicit, where the value of the method also appears in the left-hand side in the local problem, would definitely improve the error estimation.
PINSKY: To add a comment, we have been looking at that implicit error estimation through residual-based approaches for the scalar advection diffusion equations, steady state, in which case we have been using GLS to stabilize the basic discretization methodology. Indeed, the residual-based formulation does lend itself very naturally to stabilization of GLS-type methods even in the local problem for the estimation of the error. Therefore, I believe that one can, in fact, apply implicit-type error estimation methods directly to GLS-type formulations.
JORDAN: The next written question is: "What are the effects of stretched grid on transmissivity of acoustic waves through a medium? How does the grid affect amplitude and frequency of the signal? In predictions for finite difference methods, what are the effects?"
BELYTSCHKO: I do not believe we have studied the problem for stretched grids, but the transmissivity and the wave propagation characteristics through grids of both finite element and finite difference types have been studied for plane waves. I wrote a paper on that with Bob Mullin, and several other dispersion analyses have appeared. There are analogous results in the finite difference literature, too.

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In general, these results seem to indicate that waves that are not traveling along directions of the grid lines suffer much more severe dispersion than waves that go along the grid lines. In fact, it is interesting that in several recent papers the authors have in essence tried to get dispersion-minimizing schemes by combining several finite difference schemes or finite element schemes. I believe someone has combined Galerkin least squares in order to minimize dispersion. There is considerable research in this area, and it is a fruitful area that does need further study.
PINSKY: We have been using the techniques of complex dispersion analysis to understand the dispersion that is introduced, for example, for a plane wave propagating obliquely to mesh lines. We have sought to enhance the accuracy of basic Galerkin methods by appending Galerkin least-squares terms and then designing these terms in such a way that the dispersion error is minimized in some rather general way to optimize the accuracy of the method of arbitrary wave directions. In fact, we have extended this to higher-order elements as well.
ZIOLKOWSKI: In the finite-difference time-domain community in electromagnetics, people have been trying to use stretched grids for many years and essentially the same phenomenon occurs. One has to be very careful about not proceeding too quickly so that artificial numerical reflections within the grid do not arise, and be careful of the phase front propagation errors that can result from dispersion errors. These have been similar experiences in the electromagnetics community.
THOMPSON: In a paper by, I believe, Zaunt, who did a series of papers looking at stretched elements, he had the domain spread into small elements and then he abruptly changed to a domain that had large elements, and thus addressed the effect of any numerical transmission or reflection that occurs there. He did find additional discretization there due to big changes in mesh sizing. So it is preferable to keep the mesh sizing relatively uniform or at least graded from a small to a large mesh.
ZIOLKOWSKI: A number of people on the electromagnetic side are interested in so-called hybrid techniques, in which discretized methods are coupled with other things. An example we saw was Newman's presentation of the method of moments coupled with the Geometrical Theory of Diffraction. I described approaches with FDTD, with integral formulations to do near-field, far-field types of things, and attempts to combine FDTD with ray asymptotics. There are a number of hybrid approaches, and for the last couple of years using hybrid approaches has seemed to be the big push in electromagnetics modeling. Yet in all the talks from the structural acoustics side, I did not hear about anyone trying hybrid approaches. Is that not done, or is it done in just this community, or is it something of a new interest?
PINSKY: First, I would suggest that the DtN formulation itself, the Dirichlet-to-Neumann boundary condition, is in a sense an example, at least, of combining an exact analytic boundary condition within the framework of a numerical approximation. So DtN can be viewed as something of a step in a hybrid direction. We have thought about trying to extend some of the analytic solutions, asymptotic solutions that describe near-field phenomena, around discontinuities. Essentially, such an approach creates a kind of bubble around these fine features in which analytical solutions are suitably embedded in the bubble boundary. That extension is then coupled with more global techniques where the solution is varying in a smooth way. Some of the work done at Stanford by Joe Keller in the geometric theory of diffraction for underwater acoustics would certainly be applicable here. It again goes back to solving certain canonical problems for certain kinds of features.
We certainly do have a good framework now for pursuing this approach, but for some reason it has, to my knowledge, never really been pursued in a vigorous way in the context of underwater acoustics. It probably should be.
DYER: From a perspective in structural acoustics—underwater acoustics—there is an interest in hybrid methods, but I believe of a rather different kind than Peter has mentioned. One motivation for a kind of experimental research that MIT has been involved with is identifying processes rather than domains

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within which analytical techniques then might be coupled domain to domain. It was initially hoped that some people in the numerical computation community and the mathematics community would be motivated to find a way to combine processes that cut across domains—spatial domains, say, or time domains. To some degree, I am not discontented that I see no activity in that direction. However, there is a distinction between an analytical solution and a spatial domain that might be coupled to a set of processes.
LOUISE COUCHMAN: A number of areas in which hybrid solutions are being attempted happen not to be represented today. The problem is somewhat more difficult in acoustoelasticity than it is in electrodynamics because the elasticity has to be dealt with. Nevertheless, hybrid solutions are being attempted in which rays are traced on shelves and the elasticity due to the shelf is included in the formulation of the ray tracing; that is, coefficients of coupling of energy onto and off the shelf are computed from canonical solutions.
There are also examples of solutions in which discontinuities in structures are being treated using canonical solutions and coupled to a more global solution. Those are at least two instances of hybrid coupling-type approaches being studied.
I.C. MATHEWS (Imperial College): Every U.S. submarine for the last, say, 20 years has been designed for shock loading using, effectively, asymptotic-type numerical techniques and the USA codes. Almost everyone here must have come across that work, and it should also be represented. John DeRhams pushed it into the frequency domain, and I think that approach could be used for higher frequencies. Boundary integral techniques were very good because they were in the sort of midfrequency range with which people had trouble. To go to higher frequency, one would use some asymptotic technique; the point is that the scattering response at the higher frequencies is generally quite smooth. It is in the midfrequency range, where it is terribly peaky, that one wants the accuracy.
THOMPSON: Regarding the USA code and the use of the doubly asymptotic approximation, it too is a combining of numerical techniques. For the radiation boundary, an asymptotic-type boundary condition is used that is basically exact for very high frequencies. So at infinite frequency, where that boundary condition would be exact, the ultimate goal at the low end is also exact. However, the doubly asymptotic approximation presents difficulties for structural acoustics problems because it has a large gap in the midfrequency range, which Professor Mathews just suggested was where you really want to capture the accuracy. This is an area where one has to be careful when using asymptotic methods. One needs to be aware in what part of the frequency range is the accuracy valid for the methods.
MATHEWS: We have been using integral techniques, the Burton-Miller approach, for the last 10 years, and they do work. Within this frequency regime they give extremely accurate results for full three-dimensional structures and for realistic sorts of wave numbers.
DE HOOP: Let me return to the question of whether there are any similarities of combined techniques—hybrid techniques used in elastodynamics—that more or less parallel the ones in electromagnetics. In work done roughly 10 years ago, S.K. Datta of the University of Colorado performed matched asymptotic experiments. For a part of the regime he used analytic techniques of an asymptotic nature, and in the remaining part of the structure he used separation-of-variable techniques and other techniques. Everything went fine for a couple of situations.
HUGHES: Another example, in shell analysis rather than in acoustics, is work that Charles R. Steele at Stanford has done over the years involving asymptotic solutions around boundary conditions that are highly oscillatory in shell response, typically with fixed boundary conditions, and also around junctures in shells. He has, further, used asymptotic methods in dynamic wave propagation shell problems and combined some of this technology into what he calls a big-element finite element approach that is based on the asymptotic solution. He can model a complex intersecting shell such as that associated with the North Shore problem with a very small number of elements, because all of the critical asymptotic features are built into the basis

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functions of the elements. These codes are actually used quite extensively by designers in industry. He has made the comment that, due to the fine scale features of shells that appear when one does an exact or an asymptotic analysis, he has never seen a finite element mesh for a shells problem that was sufficiently resolved to actually calculate a good solution.
JORDAN: With regard to using various asymptotic methods for inverse scattering, in inverse scattering the ground rule is that one wants all the scattering information that can be obtained. People will start with volume scattering to get the rough volume of the unknown object, but then, to get finer detail, go to higher frequencies, physical optics. Those give all the fine details, all the cracks and comers and such, but the combination of the two gives the whole picture for the unknown object. This is all done in the frequency domain and so is an example of a hybrid combining technique in the frequency domain for inverse scattering.
Philip Abraham of the Office of Naval Research has submitted a list of fairly general questions that I have held off addressing until last because it provides a type of summary for this discussion. I would like Phil to pose these questions in some prioritized order that he thinks is most effective.
PHILIP ABRAHAM: Of the topics listed [see box], some have already been covered in the preceding discussion. One that has not been discussed is the physics underlying these problems, although some people have referred to it and stated differential equations for a particular system. Similarly the question regarding the desired end product has not been formally addressed. What do you want from a solution? Do you want the pressure or the radiation? Do you want the displacement of a structure, response of the structure, response to the fluid if it is a structure acoustic? What I mean is that the calculations or computations to be done depend entirely on what information is sought about a particular system.
Other issues to be addressed include how computational methods apply to each category of linear problems and nonlinear problems. There are common obstacles to accuracy and efficiency in the various methods. There is a challenge in dealing with real-time applications. If the context is surveillance problems, sonar, radar, and control problems for the Navy and Air Force, one needs to obtain a solution in real time. One cannot wait for the computational method to work in 6,000 hours. We just heard comments that there is damping in structures, but it may not influence the results.
In systems that are built up from many subsystems, there are various components to be considered. For example, in a microstructure, we have seen that one must faithfully model what is happening at the junction of the subsystems, and do so not just computationally. There is a prior setting in which that faithful modeling must be done; it has to be done experimentally and perhaps physically. One must attempt to model what is happening at the physical level, and then give this result to those who perform the computations. This experimental and fundamental modeling issue has not been discussed here. While it may not necessarily be part of this symposium's discussions, it is an important aspect.
What are the limitations of the computational tools? In the computational methods described at this symposium, are the present limitations due to hardware or software? Can these methods be adapted from one area to another? These are some of the topics on which further comments could be useful.
TAMIL: To address the first question, with regard to optics, the time-domain technique has the advantage of bringing out some of the physics. For example, when a nonlinear medium with a short pulse and with Raman scattering is analyzed, these aspects of the physics can be obtained out of the computation. However, extreme care should be taken not to mix up the physics with erroneous outcomes that are due to inexact computations. By doing nice filtering, the physics can definitely be brought out.
Concerning the last question, the methods can definitely be adapted from one area to another. Any equation that is normalized can be used irrespective of the originating area of science. As we have seen, most of the equations of acoustics and electromagnetics are of the same type so that the same methodology can be used. I am sure this generalization extends to other areas.

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PHILIP ABRAHAM'S TOPICS FOR DISCUSSION
Delineate domains of applicability of frequency and time domain, in terms of the
Underlying Physics (what is the desired end product),
Linear vs. Nonlinear problem under consideration,
Common problems of Accuracy and Efficiency (cost), and
Real-time applications, such as surveillance and control, which 'impose different demands on the two computational methods (in all of their variants).
The need for good, physically sound, descriptive models of large-scale built-up Systems; e.g., the actual modeling of damping resulting from connections between subsystems, and of prestresses and residual stresses present in a system, is a vitally needed input to any computational effort.
Present limitations of computational tools:
Hardware, and
Software, i.e., the computational methods presented.
Cross-fertilization: Can methods from one area of application be adapted to other areas, or can different variants of the same method be somehow joined (e.g., frequency-domain or time-domain method).
DE HOOP: Let me comment on the problem of built-up systems composed of subsystems for which, for instance, we have determined scattering properties and so on. One of the most fruitful methods is the no-field or T-matrix method in which, within a certain set of basis functions that are selected for a particular three-dimensional object of compact support, you essentially make a signature of scattering properties in terms of what I call the T matrix. In combining these with an adjacent part, in building up the structure—again, with its own signature—if you have used basic functions that for the most part have a common domain exterior to what is being built up, easy combinations are possible, because if there is an orthogonal basis one uses the orthogonality properties. The combined signature of two parts can be easily constructed by matrix multiplication, and then one can go on with the third.
None of these computations has singular integrals to evaluate as in standard boundary element techniques simply because the integrals that are evaluated are easily done. One either can take, in the no-field point-source case, solutions for which the source points are not in the domain of interest, or can take wave function expansions in the frequency domain, such as spherical waves or cylindrical waves or any other type of waves convenient to the problem. Based on what I have seen, this signature-combination approach is the only technique involving a specific strategy for combining properties in the acoustics or electrodynamics problems, and also electromagnetic problems, so that once the signatures of the separate elements have been built up, one can combine them in a systematic mathematical manner.
JORDAN: I have had a general question in mind during these two days. We have discussed acoustics, electromagnetics, and electrodynamics, and, let us say, the differential equations for each of these three disciplines. These differential equations have certain symmetry properties. Are these properties the same for

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each of the three types of equations? Are they all different? Are there some areas of common symmetries, and if there are common symmetries, can they be used to generate common solutions in these three areas? This is a very general mathematical question, but I hope it addresses the basic question that we have considered in this symposium.
DE HOOP: I pointed out in my presentation the general structure of all the differential equations we have considered [see equation 3.12]. It is a first-order coupled system of equations where the spatial differentiation appears as a square array of spatial differential operators. The time differentiation is a first-order convolution of the material properties of the wave function, with losses present. An interesting property, which was very surprising to me, is that this differential operator enfolds all the types of wave motions in a common structure. For the acoustic case, I have shown that its shape is that of a square array where, at the main diagonal, it is block zero. The elastodynamic case and the electromagnetic case have exactly the same form. In that respect all the wave motions that we know of in physics have this structure. This differential operator satisfies one additional property that leads automatically to what in my presentation I referred to as the reciprocity theorem. It surprised me that all these wave motions had this common structure, apart from reciprocity. One other thing: the acoustic waves in porous media also have the same structure. It is a more extensive array, but again, it has this same structure. So if one seeks either general mathematical theorems or particular numerical applications, this is one thing on which to focus.
HUGHES: To make an addendum to those observations, that structure is the structure of a symmetric hyperbolic system. Thus, all existence and uniqueness are governed by Friedrich's theory, for example, and in addition there are theorems due to Mach and Gudanov that point out the equivalence of a symmetric hyperbolic set of equations and the existence of an entropy function. So there is a convex function associated with each of these theories that in the linear and nonlinear cases takes on the interpretation of an entropy. It is the basic quantity one would look at not only when examining the nonlinear stability of nonlinear systems such as these, but for particular linear systems as well. They share many features, but the possibilities in structural mechanics are a bit greater. First, the tensorial nature of the theory is one level higher. There is the possibility of anisotropy that is somewhat precluded by the scalar nature of the acoustical problem.
DE HOOP: For the elastodynamic formulation of these equations, inhomogeneity and anisotropy are present.
HUGHES: Yes, one maintains the symmetry, but as to techniques to solve problems, there are decomposition theorems for at least the isotropic case where one can do some classical analysis by breaking up the elastodynamic problem into a set of scalar wave equations. We do not know whether one can do that for the anisotropic case.
DE HOOP: It is not only isotropy that counts, because the breaking up into waves applies only to homogeneous domains. On homogeneous and isotropic media, of course, one again goes to the eigenvalue. I was puzzled a little by the fact that the hyperbolic system generally does not require only the symmetry of that spatial differentiation to operate. It does not require the block of zeroes at the diagonal. But for any wave propagation problem, starting from the physics, it puzzled me why those blocks of zeroes occur on the diagonal. In a general hyperbolic system one would expect this matrix to be symmetrical and the like, but that is not indicated for wave propagation in general. Only for these differential equations for physical wave propagation examples in acoustics, elastodynamics, and electromagnetics do I get these block zero diagonal terms. That is not necessarily so for general wave point systems. So not every hyperbolic system is a wave motion. Each wave motion is a hyperbolic system, but it is not a one-to-one mapping, because there is an extra feature in wave propagation that, let us say, is a restriction on the total amount of operators possible in general hyperbolic systems. With a more restrictive operating context, one can prove more than can be proved for general hyperbolic systems. Of course, one can borrow the results from Cauchy problem

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stability, and so on, but it is even more puzzling then, knowing that there is this main diagonal block zero structure.
HUGHES: If you look, though, at a theory such as compressible flows, say for a perfect gas, and put it in a quasi-linear form where these matrix operators could be examined, I do not believe they would be diagonal.
DE HOOP: It still leaves open the mathematical problem: Is a subclass of hyperbolic systems included in this framework? When there are zeroes on the main diagonal, does it offer nicer, better, more extensive, or whatever properties than the general hyperbolic systems have?
HUGHES: I am unsure what the significance of that is.
DE HOOP: I do not know either. But the appearance of this common form suggests underlying structure and relationships.
PANEL DISCUSSION
TUCKER: There have been many thought-provoking observations in the open discussion. To now close the symposium, Ted Belytschko, Adrian de Hoop, Tinsley Oden, Peter Pinsky, Lakshman Tamil, and Richard Ziolkowski have kindly agreed to form a discussion panel. They will attempt to synthesize the presentations and the discussions into a summary that captures the most important points, issues, and ideas and indicates what are the best directions in which to focus research, and what approaches or ideas are the ones that merit emphasis for the future.
ODEN: No matter how objective one attempts to be, in the final analysis one sees the world through one's own, perhaps tinted, glasses. Consequently, when asked to identify areas that are important for future research, one cannot help but identify some of those areas that are specifically of interest to oneself. With that admitted bias, I hope the rest of the panel will help add balance to what I say.
In the lectures, I saw a number of intriguing and interesting activities under way that will ultimately have a significant impact on how large-scale structures are addressed in acoustics and electromagnetics. One area was that of adaptivity. I lump adaptive methods into a very large collection of techniques I call "smart" algorithms. What that means is that one attempts to endow the algorithm with some decision-making capability, based on some future aspect of the solution that is calculable, and which has some beating on the quality of the solution. Generally that aspect is the numerical error.
Only a decade ago it was a farfetched idea that one could actually calculate with any degree of confidence some estimate of error. Now, methods are emerging that lead one to believe otherwise, that one can in fact estimate the error, and on the basis of that estimated error, control the computational process. We have seen several examples in this symposium indicating that. Indeed, this is a viable approach to large-scale computation. I will warn those who are novitiates to adaptivity in error estimation that this is not necessarily a straightforward area of technology. Error estimates, at least the good ones, are expensive. If one develops a successful adaptive scheme based on parallel computing, multiprocess computing, and domain decomposition techniques, be prepared to include the error estimation package as part of the package that must be parallelized.
Hand in hand with error estimation and adaptive methods is the ability to control data at the boundary. If nonabsorbing, nonreflecting boundary conditions are going to be imposed, there are viable techniques described in these presentations. More work needs to be done on these methods, but the availability of error estimation and adaptivity will influence how well those boundary conditions are imposed.
During the open discussion the word "damping" was mentioned. Every physical system has damping. I believe that good models of acoustics and electromagnetism should also include damping, and that the

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models of damping should arise from the physics of the problem at hand, and not be included merely as a numerical artifact to control the stability of the method. That means research needs to be done on physical mechanisms that give rise to dissipation in a system, that is, material damping, structural damping, and so on. Structural damping is a very complex phenomenon. It is manifested by frictional effects, for example, and contact conditions in structural systems. Viscoelastic properties of materials give rise to material damping. I believe that a systematic look at physical mechanisms that produce damping is needed, and that these damping mechanisms need to be incorporated in successful models.
Finally, the behavior of all of these numerical techniques must be better understood, for frequency-domain approaches on mesh parameters and on frequencies and wave numbers. The interplay needs to be understood between mesh parameters and frequency wave number content and solution. These behaviors can have a dramatic influence not only on the rates of convergence of methods but also on their stability.
BELYTSCHKO: I agree wholeheartedly with Tinsley Oden's comments. The only area that seems rather underrepresented is that of experimental acoustics. Professor Dyer stated fairly categorically that acousticians do not believe in anything that is not linear. Here is an area where one could perhaps find significant payoffs if one were trying to match experiments. In structural dynamics, one finds many small components that have nonlinear behavior. Friction and the associated damping, to which Tinsley alluded, are a consequence of that. In joints, found in real structures, and in connections, one quickly finds nonlinear behavior. If one is trying to model experiments on the basis of first principles, this nonlinear behavior must be treated from those first principles.
This issue will open a tremendous amount of needed research because the entire area of acoustics has in a sense been built on a technology that is basically linear in character. How to study scattering when one has nonlinear components is an open question. However, if one is going to deal with real structures, experiments will be very difficult to match. This was already indicated in Professor Dyer's comment that nobody wants to deal with a submarine model that has some internal structure. When one deals with a real submarine, one has very complex internal structure. That problem should be investigated.
PINSKY: Echoing Tinsley Oden, we tend to concentrate on those parts of the problem with which we are immediately concerned. In that way, my comments pertain to computation. One outcome of this symposium is recognition of the potential significance of a confluence of ideas. Concerning the numerical treatment of the acoustic fluid and the electromagnetic problem, this is true, for example, in discretization techniques and in radiation boundary conditions. One essential difference that needs to be addressed, with regard to the acoustics problem versus the electromagnetics problem, is the feedback mechanism, the fluid loading on the structure.
Many of us would probably agree that the numerical approximation in the acoustic medium is quite good. We have heard reports in this meeting, and indeed in many other meetings, of the lack of ability of computational techniques to treat truly complex structures in the midfrequency range. Some blame for this has been placed on the representation not of the fluid, but of the structure. Though many of us have been working in the area of structural dynamics and have considerable experience in that, it may still be worthwhile to revisit the issue of thoroughly understanding the nature of the structural representation for the coupled acoustics problem, and completely understanding the role of the coupling.
At Stanford, we have attempted to do some of this when looking at the effects of point-loaded cylindrical shells in the context of finite elements, and making an exact representation of the pressure loading on a cylindrical shell. We are analyzing the ability of the finite element discretization to represent propagating waves emanating from the point drive as well as the decaying evanescent waves. We found that the finite element representation for that particular class of problems worked surprisingly well.
However, for the structural acoustics problem where interaction is a major consideration, attention may need be paid again to how one captures the dynamic properties of the shell as it is coupled to the fluid,

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and it may be necessary to truly understand the role of the near fields and the effects of discontinuities in the complete coupled problem.
The other area that is certainly going to be fruitful, and in which enormous progress is already visible, is error estimation. For example, the work of Tinsley Oden and others in this field displays a clear trend that should be exploited to the greatest possible degree, both in electromagnetics and acoustics. I concur with Tinsley that this needs to be done in the context of parallel methods. There is additional cost to be borne in the cost of error estimation, and indeed the adaptive procedure itself creates additional difficulties in the solution of the problem. Therefore, one should be taking a systematic approach to the problem of developing error estimation for the fully coupled situation. However, the approach should also take into account the nature of the matrix problem that arises from this adaptive procedure, and should aim for tailored global solution methods that are somehow optimized for this class of formulations.
As a final remark, this meeting has been fruitful in confirming my suspicion that much of the work in electromagnetics is, in fact, directly applicable to computational structural acoustics. I would certainly encourage these two communities to continue to exchange ideas.
ZIOLKOWSKI: There are a number of areas that I believe would benefit from more work. In electromagnetics, an obvious need, considering the direction in which computing is going, is to move our co-development efforts into the parallel arena. That move is beginning, and it is also clearly now occurring in the acoustics area, as we observe so many overlaps in algorithms, strategies, and so on. There are many areas where electromagnetics could benefit, from both ends of the spectrum, especially with regard to parallel processing because of the difficulties involved with moving codes into parallel environments. If there are common numerical approaches in acoustics and electromagnetics, we should definitely be working together to minimize the amount of effort devoted to converting codes, as well as to rediscovering the various wheels that we travel on in the parallel environment.
From the physics standpoint, and particularly as the physics is driven to shorter time scales and smaller distance scales, one of the major concerns for electromagnetics is to develop better material models that are more completely integrated into Maxwell's equations. My presentation showed some examples of what we at Clemson are trying to do. The nature of coupling microscopic effects with macroscopic effects will come much more radically into play in commercialization, for example. Everybody would like to have things in smaller packages and still be able to behave in the fully macroscopic manner. To model those integrated systems effectively, we have to couple together and integrate better material models with the numerical approaches.
One direction that the electromagnetics community has decided to pursue is that of hybrid approaches. A number of different hybrid approaches need to be addressed further, including coupling the discrete techniques with integral techniques and with asymptotic techniques, and doing so in an effective manner for the particular types of problems of interest as well as for the different computer architectures that are coming on line.
Last but not least, we saw in this symposium efforts to couple basic device modeling with basic system modeling. As we move into larger integrated packages of different systems, we will have to address not merely how to model devices, but also how to scale up—how to put devices into basic system packages, and put those systems together into a larger complex system that works. As one moves up the scale from devices to integrated packages, there will be a number of issues; it is a very different kind of hybridization. It is not simply a wave propagation. A serious systems integration issue will be how to take modeling from one regime and couple it effectively with modeling in other regimes. That is another important issue on which future work will be needed.
TAMIL: Complex systems will be made up of subsystems of components. Once the components and subsystems are characterized and put together to form the complex system, there will be a perturbation to

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the overall specification of the desired complex system. So the systematic study of such perturbations will be a part of putting subsystems or components together. This is very close to what Professor Ziolkowski just mentioned. Methods will also need to be found for isolating the perturbations between subsystems. If that can be done, then building complex systems out of individual components will be easy.
In electronics, there are very sophisticated packages for very large scale integration; it is time that the integrated optics community developed one also. People may be doing that, or maybe a package for optics integration exists but is not on the open market. Nevertheless, it is very important to have a means whereby one can design and synthesize an integrated component, and then get it fabricated. If this can be done for electronics, then surely the field is mature enough to do at least a small type of integration in optics.
We saw commonality between acoustics and electromagnetics. I am sure that a similar commonality exists also among other branches of science. We need somebody like the Russian who first compiled the integration tables to compile all these equations—in a normalized form that, say, identifies what the important parameters are. If one could also learn what are the most efficient codes available for what kinds of structures, that would be very helpful. It is another means to avoid energy being wasted in repeating the same inquiries. It would help inform people in acoustics about what the folks in electromagnetics have already done. And if tomorrow the people in biology need such knowledge, they need not redo it. For example, the chaos studied in biology has also been studied by many people in other areas. Whatever commonality there is from the equations should be compiled.
As to modeling for nonlinear systems, if the system is integrable, then deriving conservation laws to check the accuracy of a numerical computation is easy. However, for nonintegrable systems, which includes most of the realistic systems, how would one check the results of numerical computation? In certain cases, one closes one's eyes and extrapolates from the integrable systems. Is there a mathematically justifiable method of doing it?
Lastly, many people are pursuing the study of chaos and instability. To increase the rate at which we communicate, the rate of transmitting information optically, may require looking into chaos and instabilities. One of the necessary conditions for chaos is nonlinearity, but I do not know whether that is a sufficient condition.
DE HOOP: I have very little to add to all these magnificent remarks and suggestions and will merely focus on what I call quasi-analytical issues that are promising subjects of further research. The first, mentioned by Professor Oden, is the element of damping. Damping is present in any physical system, and it needs to be modeled in the computational scheme in accordance with the physics. In the frequency domain, this means essentially that the real and imaginary parts of the constitutive coefficient should obey the Kramers-Konig causality relations. In the time domain, they should be of the convolution type, at least within the linear framework of the Boltzmann type.
This may seem a completely superfluous remark for anybody who has studied elementary physics. However, my experience is with the inclusion of damping phenomena in seismic prospecting. Roughly 90 percent of all seismic prospecting papers interpret particular experimental results and try to match the results to a frequency dependence that can be easily shown to be noncausal. The better an author succeeds in matching the experimental results to that kind of frequency behavior, the better the damping phenomenon is believed to have been covered. It is essential that damping be included in the model, but it has to be done carefully.
Another promising subject is that of no-field methods as a possibility for modeling acoustic, elastodynamic, and electromagnetic wave fields. This was a very popular approach some 10 years ago, but. since then, it has completely died out. Almost all of the papers that have been published, mainly in the Journal of the Acoustical Society of America and on related areas such as wave motion, deal exclusively with the application of frequency-domain methods.

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I believe, but am not 100 percent sure, that one can also set up the no-fields approach in the time domain. As I mentioned in earlier discussion, it is an ideal method for putting things together from elementary building blocks. You can determine the T matrix, the relevant signature matrix of that object, in a fairly elementary way and then put things together in a manner that is not a perturbation technique but is exact. Being able to combine things exactly would be very important, and so developing this no-fields time-domain method presents a very interesting research opportunity.
The third subject with promise involves looking at the complex frequency domain for positive real values of the complex frequency. One takes the time Laplace transform for any causal function and works with the positive real values of S to solve the problem. On the one hand, one wants to return to the time domain, for which one uses specific algorithms that have been developed in certain aspects of physics (mostly of the acoustic diffusive type used in rock fracturing for, say, geophysical reservoir enhancement). On the other hand, some particular aspect may be sought that can be considered independent of S. An example is in inverse problems where reconstruction of a particular profile of a constitutive parameter that is independent of frequency is desired, for instance, the mass density in the interior of the earth. If damping is neglected in this setting, then of course, the whole analysis and signal processing can be done for positive real values of S, because the quantity you are seeking has nothing to do with S. So, one can approach things in the frequency domain, in the time domain, and in the S domain and ask what is the cheapest, and best, way to do it. There are indications at the moment that using the positive real values of S gives the cheapest way, and if done in a clever way, one can easily reconstruct those profiles.
The first thing you have to specify is what are you after. First tell me: What do you really want? In doing university research for industrial companies, the difficulty one usually encounters is that the company tries to think for the researcher. The only role the company should play is providing support; let the researcher do the thinking. What may happen is that the company formulates the problem, and the researcher says, ''I can do that.'' A contract is set, and the researcher obtains the required result and writes an invoice. Then the company says, "Oh, now we see we actually would have preferred... ," and it really wants something else. Thus the first thing to dig out is what the customer is after. It can take much time and involve considerable difficulty to fully get the flavor of what they really want. Once that is identified as something that can be determined, then accuracy becomes an issue. The usual university perspective is that if one has accuracy to three decimal places, six would be much nicer but one must then beware of round-off errors, and so on. The company's message might be that a factor of two is not important; that is, if the result is correct to within a factor of two, it is good enough.
Given that, one has to rethink everything one has ever learned about computational modeling. One's thinking must be recalibrated so that only being within a factor of two is important, and nothing else. That means the speed of the computational scheme for that particular application can be tremendously increased, because there is no interest in obtaining so many decimal places. One still has to be very accurate, but only under the constraint that the result is accurate up to a factor of two. So the first thing one must find out in all these computational techniques is what one is after.
That brings me to my final remark. One goes to a customer and says, "Do you want to compute something?" If the customer answers, "Yes," one then says, "Okay, tell me your accuracy requirements; specify the domain in space where you want accurate results, and specify how accurate they should be." Then one devises an error criterion that satisfies certain conditions (for instance, positive definiteness). The ideal situation occurs when a computational scheme automatically comes out from the criterion and conditions, an iterative scheme that the computer executes by minimizing the errors after each step in an iterative procedure. I believe this would be a very nice future direction to pursue. First, an error criterion is specified, usually in L2 norm, but more general ones are possible. Then one shows that each iterative step decreases the error. One has no guarantee of the decrease to zero in a general system, but in an algebraic

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system of 10,000 equations with 10,000 unknowns, we have had accuracy to three decimal points after three iterations, whereas in an ordinary method that would be out of the question. This was done on a PC; the program was begun on a Friday afternoon, and on Monday morning with the last cup of coffee at nine o'clock, everything was finished.
ABRAHAM: When calculating wave propagation via time domain, one can focus on the specific end of the structure of concern in the calculation. The rest of the structure does not need to be addressed except up to the point where the wave is propagating. In the frequency domain, on the other hand, when one frequency is sought, it must be calculated for the whole structure. One cannot do just one part of the structure. These two things are separate. People say that they want to calculate in the frequency domain because they need to know about one particular frequency. That is fine, but when one has to calculate something in time, this is where the time dimension is useful and should be the preferred method. When computing in the time domain, one determines how long one must compute in time. It is not necessary to go to infinity. Perhaps when computing, for instance, in the frequency domain, one may want to go back to the time domain. It would be necessary to take a Fourier transform and then the time domain is an infinite domain. There are thus advantages in the time domain that do not exist in the frequency domain, yet some areas of frequency domain may be more useful.
DE HOOP: Perhaps I can comment on that from experience with geophysical prospecting. Relating it to one of your remarks, first tell me what you are after. In geophysical prospecting, for example, one always has an inverse problem. One is trying to characterize the structure of the earth by doing acoustic electromagnetic or elastodynamic measurements. It turns out that oil is always beneath a discontinuity in surface material properties. So the first thing one must determine is where the discontinuities are. Those you detect by arrival times of reflected waves, which means that part of the analysis can be done better in the time domain. You are interested only in whether the oil is 7 kilometers deep or 5 kilometers deep. It turns out that once one has decided to drill a hole and has taken fairly detailed measurements in it, one wants to look a little bit sideways. There one runs into trouble because the travel times across, say, a layer 5 millimeters thick with an observational time window of a few milliseconds produce almost an infinity of wiggles. If one looks only at the wiggles, the interpretation of what is just a little outside that hole will miss quite a lot of detail. The fine details in the wiggles tell, for instance, if something is rather flat or has another bend or is perhaps a small but sharp aspect. So the standard practice is to take a Fourier transform, after which, fortunately, doing many of these fairly detailed things becomes much easier. In the picture, details show up more easily in the frequency spectrum than in the time domain.
So, again, the major question is what is one after, and the kind of approach one intends to use. If the object of interest is an arrival time of a wave, in the time domain one can see when it is coming. It can be traced; the philosophy of how deep it is is known. But if details get washed out because of high oscillations when it is the tiny details that are sought, then in many instances the frequency domain is the way to go. Then, of course, the computational scheme has to be tailored to what is being sought.
ZIOLKOWSKI: In electromagnetics, people have been working for many years in the frequency domain. In fact, computational electromagnetics started in the frequency domain, particularly because people were first interested in antennas. Once that interest was satisfied, then the methods-of-moments calculations started in the 1960s, with the numerical electromagnetics, called NEC, as the first large code for calculating. As we have moved up in frequency into the resonance regime, there are areas with multiple resonances present, and so people have had to move into using the time domain. If one is doing nonlinear problems, there is no question. As one moves into different regimes, as was pointed out, most measurements are done with pulses. They are finite windowed sinusoids, but one never really has a true continuous wave situation. One always has some pulse effects present in a system. If the intent is to

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compare with experiments—something we all should eventually be trying to do then calculating solutions in the time domain can actually provide some information that is very useful.
ODEN: I do not see any other comments. On behalf of the audience and the panelists, I want to give a word of thanks first to Phil Abraham for having conceived the idea of holding this symposium, to Louise Couchman for providing the resources of her office toward making this event happen, and to John Tucker of the National Research Council for his excellent hospitality and organization.
TUCKER: And I want to express my thanks to all of you for coming. Without you, this event could not happen and the benefit that it provides would be lost. I feel sure the research directions described in these presentations and discussions will bear fruit for these scientific areas, and that this information will be viewed by others as very interesting and provocative with regard to research opportunities.

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