My research interests center on two disciplines: computational mathematics for material science and data-enabled industrial mathematics.
Data- Enabled Industrial Mathematics
One of my significant accomplishments at ERAU has been the establishment of a new research path in the subject of Data- Enabled Industrial Mathematics (DEIM). Industrial mathematics is a relatively new subject that focuses on converting technological, organizational, and data-enabled challenges offered by business, industry, or government into mathematical problems. I have set up a number of externally funded academia and industry partnerships.
I was successful in obtaining a National Science Foundation grant for the establishment of a Research Experiences for Undergraduates (REU) site at ERAU. The REU site allows undergraduate students to participate in real-world research projects in data-enabled industrial mathematics. The goals of this one-of-a-kind research experience are to educate students for challenges in data-driven industrial mathematics and to provide them with the abilities needed to navigate successfully in today's data-rich environment. The ultimate goal is to simulate a real-world industrial research experience, with an emphasis on student development and encouraging students to interact with content at a level that goes beyond the usual classroom. Students collaborate on challenges presented to them directly by business, government, and industry.
Computational mathematics for material scienceMy long-term research interest in computational mathematics for material science is focused on adaptive algorithms for numerical simulations of dynamic energy redistribution in advanced and novel materials such as spatio-temporal composites and micro-structured, multifunctional, and hierarchical metamaterials. My research in this field can be separated into the three following major categories:
Numerical methods for hyperbolic PDEs:
- Development of higher-order finite-volume algorithms for wave propagation in linear and nonlinear media, including thermo-elastic, electromagnetic, and shallow water waves.
- Higher-order finite-volume algorithms for dispersive waves in materials with microstructure, periodic and functionally graded media.
- Font tracking algorithms for shock waves and phase-transition fronts.
- Analysis and implementation of constitutive models for wave propagation in heterogeneous and anisotropic material with microstructure at different time and space scales.
- Numerical analysis of dynamic response under impact loading of advanced and novel materials such as micro-structured, bio-inspired, hierarchical, and dynamic materials.
- Numerical simulation of wave propagation in acoustic metamaterials (phononic crystals) with focus on negative index materials and acoustic diode.
- Dynamic energy redistribution by wave handling in complex material structures.