My work makes use of a broad spectrum of ideas drawn from various areas of applied mathematics, physics, solid mechanics, materials science and high performance scientific computing. Of particular use to me are tools from mathematical quantum mechanics, solid state physics, group representation theory / abstract harmonic analysis, the theory of partial differential equations and numerical analysis.
Outlined below are some research problems that I have been working on.
1) Objective Density Functional Theory
Objective Structures are atomistic and molecular systems that can be thought of as natural generalizations of crystals. It appears that matter fundamentally prefers to arrange itself as objective structures, as a result of which, these structures are ubiquitously present in all of science and technology. The objective structures frame-work provides a strong mathematical foundation for the systematic discovery, synthesis and characterization of novel nano-materials and nano-structures. In particular, it allows for the identification of nano-materials which are likely to demonstrate collective materials properties such as ferro-magnetism and ferro-electricity, as well as materials which are likely to show electro-mechanical coupling at the nano-scale in the form of flexo-electricity. Prospective engineering applications of some of these novel nano-materials include a new paradigm of sensors, actuators and energy harvesting technologies.
In order to have predictive tools that incorporate the objective structures framework, I have been working towards integrating this framework within Density Functional Theory (DFT). The resulting suite of computational tools, called Objective Density Functional Theory (Objective DFT), allow nano-science to be done in a systematic and informed manner , ab inito. Additionally, the computational packages that I have been developing as part of Objective DFT, are leading to novel computational methods in nano-mechanics. Using these new and powerful tools, I have been investigating experimentally realizable micro-meter length scale atomistic structures, strain engineering of the electronic properties of existing and new nano-structures (with applications to electronic / optical meta-materials), and multi-physics coupling at the nano-scale (with applications to the design of sensors and actuators).
It appears very likely that Objective DFT will be instrumental in the discovery and characterization of completely new classes of functional nano-materials and nano-structures, in the near future.
2) Pushing the envelope of large scale first principles simulations of complex materials
Attractive as first principles calculations are, the solution of the equations of Density Functional Theory requires substantial computational resources while dealing with large and/or complex materials systems. In spite of active research in this area for several decades, first principles calculations (particularly, first principles molecular dynamics simulations) are still routinely limited to a few hundred atoms, unless simplifying assumptions about the nature of the system are made. With this in mind, I have been working towards overcoming some of these computational bottlenecks by means of designing and implementing, novel mathematical and computational strategies.
My collaborators and I have recently made great progress towards carrying out ab intio molecular dynamics simulations of non-insulating (i.e., metallic or semi-conducting) systems of unprecedented size, on large scale computational platforms. As a result of these developments, it is now becoming possible for example, to carry out reliable large scale simulations of Lithium ion battery materials, materials defects, and catalytic reactions. Such computational studies are expected to be instrumental in the design of next generation materials, as well as, energy storage/conversion devices. In particular, a vast number of problems of interest to engineers and mechanicians that have been outside the capabilities of existing first principles techniques, can now be successfully attacked.
3) Using symmetry principles in various areas of science and engineering
The principles of symmetry can provide key physical and mathematical insights into the behavior of physical systems across all length-scales, and varied physical theories. My collaborators and I have been investigating the usage of symmetry principles in the context of a variety of problems :
Symmetry mediated self assembly: The large degrees of symmetry associated with Objective Structures is likely to make them ideal candidates for synthesis via self-assembly. This observation serves to provide a qualitative explanation of the the prevalence of these structures as the building blocks of viruses, proteins and other biological materials. I have been making progress in using the tools of statistical mechanics for investigating the process of self assembly of objective structures with the hope that, new nano and biological structures can be eventually designed and synthesized following this line of thought.
Wave propagation in novel symmetric structures and composites: Conventionally, wave-propagation problems have been studied for periodic structures and composites. The objective structures framework however, enables one to envisage novel classes of structures and composites that could have a variety of engineering applications (including acoustic vibration control and damping). With the recent progress in additive manufacturing, it also seems likely that such novel structures could be fabricated and tested with relative ease. I have been analyzing the propagation of elastic waves (both conventional plane-waves and other waves with non-conventional symmetries) in such novel structures and composites, with the goal of establishing universal mathematical properties of such problems.
Design of computational solvers: While the importance of physical symmetry seems to have been well recognized and utilized at the theoretical or conceptual levels, examples of the general use of symmetry principles in computational problems appear to be much more limited. A large majority of existing examples are concerned with problems having periodic symmetries, for which the underlying mathematical tools and computational techniques are well known. In contrast, the usage of non-periodic symmetries within a computational context appears to have been carried out in a manner that is rather inadequate, or at best, ad hoc. I have been taking important steps in setting up a systematic and efficient computational framework for symmetry adaptation within the context of scalable solvers, with a particular focus on non-periodic symmetries. Due to the preponderance of physical problems associated with non-periodic symmetries across different length scales and varied physical theories, there are numerous potential benefits (such as gain in computational efficiency and scalability), of such an enterprise.