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Ricardo Cortez, Professor
Mathematics Department
Tulane University

"The Method of Regularized Stokeslets with Applications to Biological Fluid Flows"

Biological flows, such as those surrounding swimming microorganisms or beating cilia, are often modeled using the Stokes equations due to the small length scales. The organism surfaces can be viewed as flexible interfaces imparting force on the fluid. I will present the Method of Regularized Stokeslets and some extensions of it that are used to compute Stokes flows interacting with immersed flexible bodies or moving through obstacles. The method treats the flexible bodies as sources of force or torque in the equations and the resulting velocity is the superposition of flows due to all the elements. Exact flows are derived for forces that are smooth but supported in small spheres, rather than point forces. I will present the idea of the method, some of the known results and several examples from biological applications.

Prof. Juan M. Restrepo
Mathematics Department and Physics Department
University of Arizona
Tucson AZ, USA

“Improving Predictions Using Estimation Techniques”

Significant improvements in predictions and control have been possible by blending data and models.  The improvements can be  dramatic when the models and the data have inherent errors. The model errors may arise due to uncertainty in parameters, unresolved processes.

The data error may be inherent in the measurement or in measurement coverage.
In problems with linear dynamics and Gaussian error statistics, there is a very well
known optimal solution to this estimation problem, namely the prediction of a mean
history and its uncertainty conditioned on observations: it is called the Kalman Filter. It
is presently used in weather prediction, navigation systems, roboting control.
When the dynamics are nonlinear and/or the statistics non-Gaussian the prediction problem may no longer be amenable to linearized methods based on least-squares methods. New methods are required and we will present some of these methods, developed by our group.

The workshop will include a brief introduction on maximum likelihood methods for parameter estimation, global and sequential least-squares (Kalman filter/smoothers), and 2 nonlinear non-Gaussian methods, one based on the path integral and one on the diffusion kernel, which arises in the small diffusion limit of a particle-filter recasting of the filtering/smoother problem.

Dr.  Pablo V. Negron
University of Puerto Rico-Humacao
Department of Mathematics

“The Numerical Computation of the Critical Boundary Displacement for Radial Cavitation”

The phenomenon of void formation in bodies under tension has been observed in laboratory experiments by Gent and Lindley (1958} and others. Ball (1982) showed, in the context of nonlinear elasticity, that void formation or “cavitation" can decrease the (potential) energy of a body in tension when the tension is sufficiently large. As cavitation can point to the initiation of fracture or rupture in a body, the computation or characterization of the critical boundary displacement at which cavitation occurs is important from the point of view of design. The problem of characterizing the critical boundary displacement has been studied extensively in the past. However, the emphasis has been mostly in deriving exact, closed form solutions for the cavitation solution for specific materials from which the critical boundary displacement can then be obtained. In this talk we describe a numerical scheme for computing the critical boundary displacement and bifurcation diagram for cavitation that applies to a very general class of compressible materials. The method is based on the solution of a sequence of initial value problems, thus allowing for the use of commercially available software that can solve these types of problems very efficiently. We give examples for specific materials and compare our numerical computations with some previous analytical results.

 

In this workshop we give an introduction to the theory proposed by Ball [1] to model cavitation, which based on nonlinear elasticity and the calculus of variations. We discuss some of the theoretical aspects of this model like existence and stability of minimizers. We then go over some of the numerical aspects of computing cavitated solutions both radial and non{radial. We will experiment with implementations of some of these methods in MATLAB to get a hands on experience on their computational complexity and usefulness.