Research in Mathematics & Engineering

Macro Stokes elements

We construct a pair of conforming and inf-sup stable finite element spaces for the two– dimensional Stokes problem yielding divergence-free approximations on general convex quadrilateral partitions. The velocity and pressure spaces consist of piecewise quadratic and piecewise constant polynomials, respectively. We show that the discrete velocity and a locally post-processed pressure solution are second-order convergent.

Stokes elements on cubical meshes

We construct conforming piecewise polynomial spaces with respect to cubic meshes for the Stokes problem in arbitrary dimensions yielding exactly divergence-free velocity approximations. The derivation of the finite element pair is motivated by a smooth de Rham complex that is well-suited for the Stokes problem. We derive the stability and convergence properties of the new elements as well as the construction of reduced elements with less global unknowns.

The Derivation of Dielectrical Breast Models for MRI Images for Breast Cancer Detection

In this work, we investigate the electrical properties’ variations in breast tissues. We construct MRI-based breast models for use in the breast cancer research carried out at microwave frequencies. We utilize several image processing tools such as smoothing and edge detection filters, and Gaussian mixture models. We model the dielectric value distribution via piecewise-linear and cubic-spline interpolation techniques.

Genetic Algorithms Applied to Floor-plan Area Optimization Problem

In this work, the representations of the floor-plan area designs are set forth with their effectiveness, a detailed analysis of the structure of genetic algorithms is made and the applications of genetic algorithms to the floor-plan area optimization problem are discussed comparatively.

Time Series Analysis Applied to Ionospheric Data

In this work, we use the source data provided by the Space Physics Interactive Data Resource (SPIDR). We build a regression model of the monthly-smoothed sunspot number data, R12, and search for the dominant periodicity for R12 using the sliding window technique.


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