Sunday, May 29, 2005

The Things We Do

I'm back. I spent one week at the Householder meeting in Pennsylvania, where I spoke with interesting people and with them mourned the lack of network access and good coffee. My talk went fine, and I listened to various other talks, some good and some not-so-good, but (as usual) all the most useful work and conversation happened between sessions, or in the hall outside the sessions. I flew to Boston on Friday, spent the night, and came back to California on Saturday. It has been a good two weeks, but I'm glad to be home, with my own bed and my own cooking.

Based on experiences on this trip, I think I may make three investments this summer. First, I should probably get a new pair of glasses, so that if I'm faced with a slide full of equations in a somewhat-too-small font, I will still be able to read them. Toward the end of last week, I would try to read an equation only to find that the fuzzy-looking screen split into two screens, each of them even less distinct than the original. This is suboptimal. Second, I should probably get a new laptop. This one is five years old: it crashes sporadically, the screen fades out at times, the battery lasts about 20 minutes, the plug at the back has to be propped up in order to deliver power, and from time to time the usual high-pitch whine of normal operation turns into a sort of warble. When I worked on my slides in my room, I was constantly worried that the laptop would quit working and leave me in the lurch. This is also suboptimal. Third, I am thinking (again) of getting a cell phone, mostly for when travel mishaps occur.

Comments on any of these things are welcome. (Scott, you told me that not having comments was an offense -- now you can use them to advise me on phones, if any of the advice has changed since last I thought about getting one.)

Now, to the point which led to the title of this post. One of the things I discovered in the past two weeks is that people find it a bit confusing when they try to find the common themes in the things I've worked on. In part, this is because many of the things I've thought about have come directly from questions that I've been asked. Since I know lots of different people in lots of different areas, it makes sense that the collaborations that arise from good questions go in lots of different directions. I am not picky, and am usually happy to spend at least a few minutes to entertain any problem that sounds -- well -- entertaining. However, there are a few directions where I know enough that someone is likely to seek my advice, and there are some directions where I've thought deeply. So there is a pattern.

My undergraduate degrees were mathematics and computer science. My graduate department is computer science, but I could just as well be called an applied mathematician. Fortunately, Berkeley is a good place for interdisciplinary work, and so nobody objects to me being both an applied mathematician and a computer scientist. Through training since starting graduate school, I'm also reasonably well-versed in continuum mechanics and finite element simulation of continuum behavior, and in MEMS engineering. This puts me in a good position to do problem-driven research in the analysis of physical systems, since I can take descriptions of a variety of continuum-level models as posed (for example) by an colleague interested in engineering widgets; figure out what simplifications can be done to reduce the problem to something of reasonable size; build a discrete model and write or adapt a code to compute a solution to the model; and provide the results in a useful form. At each level, from model-building to discretization to computer analysis, I have some areas of particular expertise. Consequently, I often analyze the behavior of resonant MEMS; I look at locally linearized models of coupled mechanical and electrical equations for these devices in order to compute their behavior near different equilibria; I write finite element codes in order to approximate the partial differential equations in these models; and I write specialized eigenvalue solvers that take advantage of different types of structures that appear on the way. Consequently, my thesis title is Structured and Parameter-Dependent Eigenvalue Calculations for Resonant MEMS Design, and the body consists of sections describing mathematical structures that I use, simulation software that use those structures, and applications where it actually makes a difference.

To be more specific, the problems that I'm thinking about most right now have applications in the design of high-frequency MEMS resonators. I've been thinking about ways of studying damping in these devices, and how they might be tuned. The balance of different physical mechanisms for these problems isn't always well understood -- for example, we think that the MEMS literature largely misrepresents the relative magnitudes of fluid damping, damping from radiation of elastic waves away from a support, thermoelastic damping, and damping due to intrinsic material losses (e.g. due to defect dynamics). But we know how to write down continuum models for these effects. This leads to eigenvalue problems involving partial differential and integral equations; usually I know that the solutions to these problems are fairly smooth and are not wildly nonlinear. To approximate these with a not-too-large discrete system, I use scaling analyses or perturbation analyses and then integrate these analytic results into finite-element types of models. I typically code these models either using HiQLab (my own finite element code) or FEAPMEX (which provides MATLAB access to the guts of Bob Taylor's simulator, FEAP). The discrete systems often mimic structures of the continuous system, and I have worked on different types of numerical eigenvalue solvers that can exploit these structures: continuous dependence on some tuning parameter, locality of interactions in the discrete system (sparsity), complex symmetry of the linearized system matrices, separations of time and space scales associated with different physical effects, and special approximately low-rank structures that come from interactions taking place at domain boundaries. At the end of the day, I produce results that show my engineering collaborators useful things like plots of the damped resonant modes in their devices, graphs of how those resonances and damping factors change as the designs change, and programs that allow them to do such analyses for themselves (or at least allow me to do similar analyses in much less time in the future).

This seems to me a straightforward to approach research. But universities have departments, and many folks stay within the boundaries of one or two departments. Consequently, one of the things for which I seem most useful is in facilitating communication: pointing engineers and physicists at good codes and computing tools, and pointing numerical mathematicians at problems they might not have realized fell into the purview of their tools. And on the way I build theorems and models and codes, and a merry time is had by all.

  • Currently drinking: Green tea