Real world problems can be formulated in mathematical terms (modeling), as the models have grown complex, a great deal of computing power is required to study these models. Scientific computation involves investigation of robust and efficient methods for studying problems that require heavy computations. Dr. Sial’s has been working on using finite element methods, on problems form material science. One current project looks at optimization of functionals arising in physics, control theory, and differential equations. This often corresponds to finding a lowest energy, least error, or minimum cost. I look at the construction of Sobolev spaces in which numerical optimization occurs, taking into account the essential features of the analytic formulation of the problem. A second current project looks at machine learning for materials science. One seeks to find the essential features that determine the properties or characteristics of materials.
Dr. Amer Rasheed’s research interest focuses on investigating crystal growth in metals using finite element methods. The solidification (or freezing) is process wherein a pure or a mixture of metals in the liquid form turns into solid when the temperature (or pressure in some cases) decreases. In the presence of impurities in the metals, when the process of solidification commences, the star like crystals, called dendrites, start evolving around the impurities. The microstructure of the dendrites growing in this process play vital role in the properties of the solidified material and have great interest to the material scientists. In order to ameliorate the quality and other properties of the mixtures, the major industrial challenges lie in the possibility to control the metal structure and blemishes (that occur during the solidification process). Currently,he is working on the phase field method which represents the effect of magnetic field on the dendritic evolution of a binary alloy during solidification process in an isothermal environment. He is also working to develop the optimal control of the model wherein the magnetic field would be the target variable in order to procure the control over the dynamics and other properties of the final solidified alloy.