Georgia Institute of Technology College of Sciences

Locomotion at the micro-scale is important in biology and in industrialapplications such as targeted drug delivery and micro-fluidics. Wepresent results on the optimal shape of a rigid body locomoting in 3-DStokes flow. The actuation consists of applying a fixed moment andconstraining the body to only move along the moment axis; this models theeffect of an external magnetic torque on an object made of magneticallysusceptible material. The shape of the object is parametrized by a 3-Dcenterline with a given cross-sectional shape. No a priori assumption ismade on the centerline. We show there exists a minimizer to the infinitedimensional optimization problem in a suitable infinite class ofadmissible shapes. We develop a variational (constrained) descent methodwhich is well-posed for the continuous and discrete versions of theproblem. Sensitivities of the cost and constraints are computedvariationally via shape differential calculus. Computations areaccomplished by a boundary integral method to solve the Stokes equations,and a finite element method to obtain descent directions for theoptimization algorithm. We show examples of locomotor shapes with andwithout different fixed payload/cargo shapes.

We are interested in the cubic to tetragonal phase transition in a shape memory alloy. We consider geometrically linear elasticity. In this framework, Dolzmann and Mueller have shown the following rigidity result:The only stress-free configurations are (locally) twins (i.e. laminates of just two of the three Martensitic variants).However, configurations with arbitrarily small elastic energy are not necessarily close to these twins: The formation of microstructure allows to mix all three Martensitic variants at arbitrary volume fractions. We take an interfacial energy into account and establish a (local) lower bound on elastic + interfacial energy in terms of the Martensitic volume fractions. The model depends on a non-dimensional parameter that measures the strength of the interfacial energy. Our lower, ansatz-free bound has optimal scaling in this parameter. It is the scaling predicted by a reduced model introduced and analyzed by Kohn and Mueller with the purpose to describe the microstructure near an interface between Austenite and twinned Martensite. The optimal construction features branching of the Martensitic twins when approaching this interface.

I will describe a sequence of models for predicting crack paths in brittlematerials, with each model based on some type of variational principleconcerning the energy. These models will cover the natural range ofstatics, quasi-statics, and dynamics. Some existence results will bedescribed, but the emphasis will be on deficiencies of the models and openquestions.

The over-abundance of remotely sensed data has resulted inthe realization that we do not have nor could ever acquire asufficient number of highly trained image analysts to parse theavailable data.  Automated techniques are needed to perform low levelfunctions, identifying scenarios of importance from the availabledata, so that analysts may be reserved for higher level interpretativeroles. Data fusion has been an important topic in intelligence sincethe mid-1980s and continues to be a necessary concept in thedevelopment of these automated low-level functions. We propose anapproach to multimodal data fusion to combine images of varyingspatial and spectral resolutions with digital elevation models.Furthermore, our objective is to perform this fusion at the imagefeature level, specifically utilizing Gabor filters because of theirresemblance to the human visual system.