Georgia Institute of Technology College of Sciences

Finite element approximations for the eigenvalue problem of the Laplace  operator are discussed. A gradient recovery scheme is proposed to enhance  the finite element solutions of the eigenvalues. By reconstructing the  numerical solution and its gradient, it is possible to produce more accurate  numerical eigenvalues. Furthermore, the recovered gradient can be used to  form an a posteriori error estimator to guide an adaptive mesh refinement.  Therefore, this method works not only for structured meshes, but also for  unstructured and adaptive meshes. Additional computational cost for this  post-processing technique is only O(N) (N is the total degrees of freedom),   comparing with O(N^2) cost for the original problem.

Using the technique of intertwining multiresolution analysis piecewise linear, continuous, orthogonal, wavelets on a regular hexagon are constructed. We will review the technique of intertwining multiresolution analysis in the one variable case then indicate the modifications necessary for the two variable construction. This is work with George Donovan and Doug Hardin.

Medical imaging is the application of mathematical and engineering models to create images of the human body for clinical purposes or medical science by using a medical device. One of the main objectives of medical imaging research is to find the boundary of the region of the interest. The procedure to find the boundary of the region of the interest is called a segmentation. The purpose of this talk is to present a variational region based algorithm that is able to deal with spatial perturbations of the image intensity directly. Image segmentation is obtained by using a Gamma-Convergence approximation for a multi-scale piecewise smooth model. This model overcomes the limitations of global region models while avoiding the high sensitivity of local approaches. The proposed model is implemented efficiently using recursive Gaussian convolutions. The model is applied to magnetic resonance (MR) images where image quality depends highly on the acquisition protocol. Numerical experiments on 2-dimensional human liver MR images show that our model compares favorably to existing methods.This work is done in collaborated with Mikael Rousson and Chenyang Xu.

Biomimetic synthetic cilia can be effectively utilized for regulating microscale transport processes at interfaces. Using computer simulations, we examine how polymeric cilia can be harnessed to control the motion of microscopic particles suspended in a viscous fluid. The cilia are modeled as deformable, elastic filaments and our simulations capture the complex fluid-structure interactions among these filaments, channel walls and surrounding solution. We show that non-motile cilia that are tilted with respect to the surface can hydrodynamically direct solid particles towards channel walls, thereby, inducing their rapid deposition. When synthetic cilia are actuated by a sinusoidal force that is applied at the free ends, the beating cilia can either drive particles downwards toward the substrate or expelled particles into the fluid above the actuated cilial layer. This dynamic behavior can be regulated by changing the driving frequency. The findings uncover new routes for controlling the deposition of microscopic particles in microfluidic devices.