Research Applying Mathematics to Current Scientific Problems
Bioelectrostatics
I have a long running research program, with Prof. Jay Bardhan of Northeastern, to improve models of bioelectrostatics, as well as accelerate the computations. Using the BIBEE model approximation, we have produced rigorous upper and lower bounds on the solvation energy of a complex. This work has also been used to look at protein-ligand binding energies. We have recently incorporated the effects from both charge hydration asymmetry and a static surface potential, as well as nonlocal dielectrics. This work has scaled well on very large machines, including a 512 GPU cluster.
Crustal Dynamics, Magma Dynamics
I am the co-creator of PyLith, a flexible, efficient simulator for crustal deformation in 2D and 3D, across scales ranging from meters to hundreds of kilometers and milliseconds to hundreds of years. PyLith can simulate both dynamic and quasi-static tectonics problems. Using a finite element discretization, it can handle many types of cells, including lower dimensional cells used in cohesive element formulations. I am also beginning to work in magma dynamics, two-phase flow in a deforming solid matrix. I produced initial results for a simple shear problem generating porosity bands, which shows that nonlinear FAS multigrid can outperform Newton-Krylov methods.
Fracture Mechanics, Wave Mechanics
I have provided design and implementation guidance for codes using PETSc, and in particular the unstructured mesh component. I collaborated with Blaise Bourdin to produce a scalable code to simulate fracture mechanics. It is capable of predicting the creation of new cracks, their path, and their interactions, in two and three space dimensions. I co-designed the PetClaw extension to PyClaw with David Ketcheson and Aron Ahmadia. This code demonstrated excellent scalability on the entire Shaheen BG/Q at KAUST (65536 cores). This work was motivated by the first of a series of workshops held by David Ketcheson, which I co-organized.