Paul Barbone PhD Mechanical Engineering, Stanford University, 1992 

Department of Aerospace & Mechanical Engineering 110 Cummington Street Boston MA 02215 (617) 3536063 (617) 3535866 (FAX) barbone@bu.edu 
Subsystem Representation in Dynamics Simulations
We are interested in the problem of simulating the dynamic response of a large system comprised of many subsystems. It is often the case that a simulation is performed in order to determine the response in a small subset of the entire system. Because all the subsystems are coupled together, however, one must account for all parts of the whole system even if one is interested in the response of just a small portion of it. The goal of our work is to eliminate extra subsystems from dynamical simulations, but to account for their effect on the overall dynamics in the region of interest.
For example, in the simulation of a frontend automobile collision, one may not be interested in the detailed response of the rear trunk. Thus, one might wish to replace the detailed model of the trunk with a simpler model, but one that correctly accounts for the presence of trunk on the rest of the automobile.
A particular emphasis of our work is the asymptotic limit of high spectral density ("infinitely complicated"). We find the result that each linear complicated substructure is asymptotically equivalent to a low order dynamical system. This equivalence is valid for time of o(1/e), where e represents the mean modal spacing. We have been exploring the asymptotic limit e=0 in detail, and seek to exploit this limit in developing computational representations of complicated subsystems for large scale dynamical simulations.
Optimal Finite Element Methods
A discrete solution of a given boundary value problem represents a finite dimensional approximation of the exact solution. In some cases of practical interest (primarily for elliptic problems), finite elements can provide the optimal approximation on the given mesh in some inner product. In most other cases, and especially in dynamics problems, standard finite element methods fail to give optimal approximations. This failure often leads to large pointwise error and spurious oscillation, (even for "stable" methods).
Our approach is simple to state: to design a method that produces the best approximation possible on a given finite element mesh, as measured in a user defined norm. One view of this goal is to correctly account for the effects of the fine (unresolved) scales on the evolution of the coarse (resolved) scales. Our approach is built on a basis of projection, however, rather than on an explicit assumption of a separation of scales which may or may not exist in practice.
Stress Waves in Biological Media with Microstructure
Ultrasonic wave propagation through soft and hard tissues are
examples of waves in media with microstructure. In both cases, the
interaction of the microstructure with the propagating wave can give
information related to the health state of the tissue. Though the
medium can be modeled as random, it's important to realize that a
single measurement on an individual gives a measurement through just
one realization of the medium, and decisions regarding the health of
that tissue must be made on the basis of that single measurement. For
that reason, we study waves in media with microstructure by exploiting
the implicit separation of scales that exists. We use the asymptotic
method of homogenization to determine the precise averaging relation
between scattered field structure and tissue microstructure.
Further information is available at the CBD ongoing research pages.
Igor Patlashenko, Paul Barbone and Dan Givoli (1999) Numerical Solution of Systems of Integrodifferential Equations. BU Dept. Aerospace & Mechanical Eng. Technical Report No. AM99008.
Paul E. Barbone and Isaac Harari (1999) Nearly H^{1}optimal Finite Element Methods. Computer Methods in Applied Mechanics and Engineering. Submitted.
Paul E. Barbone, Aravind Cherukuri and Daniel Goldman (1999) Canonical representations of complex vibratory subsystems: Time domain Dirichlet to Neumann maps. Internat. J. of Solids and Struct. To appear.
Aravind Cherukuri and Paul E. Barbone (1998) High Modal Density Approximations for Equipment in the Time Domain. Journal of the Acoustical Society of America 104(4): 20482053.
Paul E. Barbone (1998) Effective Dynamical Properties. Proc. ASME Noise Control and Acoustics Division, No. G0 1089, 1998 International Mechanical Engineering Congress, Anaheim, CA, November 1520, 1998. ASME Press, New York, 333339.
Paul E. Barbone and Isaac Harari (1998) Dispersion Free Finite
Element Methods for Helmholtz Equation. Proc. 16th International
Congress on Acoustics and 135th Meeting Acoustical Society of
America, Vol. 1, pp. 199200, Acoustical Society of America,
1998. P.K. Kuhl and L.A. Crum, eds.