By the Numb3rs Spring 2022 - Grants & Awards

Grants & Awards

Anna Vainchtein: National Science Foundation Award

Congratulations to Anna Vainchtein, who was awarded a National Science Foundation Award for a project that investigates the dynamics of transition fronts and other nonlinear waves in spatially discrete systems. These waves play a major role in transporting energy in lattices and mechanical metamaterials, engineered structures that exploit the instabilities of their components to yield a desired collective response.

The project aims to advance the fundamental understanding of the energy transfer phenomena associated with wave propagation. This information is important for designing novel mechanical metamaterials with desired characteristics that enable applications in soft robotics, morphing surfaces, reconfigurable devices, and mechanical logic, among others. Due to the ubiquity of nonlinear transition waves in physical and biological settings, understanding their properties, as well as the conditions for their existence and stability, is relevant in fields such as mechanical engineering, materials science, condensed matter physics, and biophysics. The project will provide research training opportunities for doctoral students.

Juan Manfredi: Simons Collaboration Grant for Mathematicians Award

Congratulations to Professor Juan Manfredi who was a recipient of a Simons Collaboration Grant for Mathematicians Award starting on Sept. 1, 2022, for a period of five years.

“We study Asymptotic Mean Value Properties for regular and viscosity solutions,” Manfredi said, “to general nonlinear, possibly degenerate, elliptic equations and the convergence of the semi-discrete approximations they suggest.”

The aim of the Simons Collaborations in MPS program is to stimulate progress on fundamental scientific questions of major importance in mathematics, theoretical physics, and theoretical computer science.

Manfredi was part of a team that was formed by former University of Pittsburgh doctoral students Andras Domokos and Diego Ricciotti at California State University in Sacramento,  Bianca Stroffolini in Naples (Italy), Julio Rossi in Buenos Aires (Argentina), and Fernando Charro at Wayne State University. 

Catalin Trenchea: NSF Award for Collaborative Research: Time Accurate Fluid-Structure Interactions

Congratulations to Professor Catalin Trenchea, who received an NSF Award for Collaborative Research: Time Accurate Fluid-Structure Interactions.

In realistic problems describing fluid flow, sometimes the dynamics are not known, or the variables are changing rapidly. Hence, to accurately compute the solution, one might need to use small temporal discretization parameters. For example, in simulations of blood flow, the pressure rapidly increases and then decreases during the systole, which lasts 3/8 of the cardiac cycle, followed by slower and smaller changes in the pressure during diastole, lasting 5/8 of the cardiac cycle. To accurately capture the peak systolic flow, a small time step has to be used in that interval. However, that same time step might be unnecessarily small during diastole and could lead to longer computational times. Therefore, robust adaptive time-stepping is central to accurate and efficient long-term predictions of the solution. The adaptive time-stepping methods for partial differential equations describing flow problems are under-investigated and this project will make a major contribution in that field. The methods developed in this project will be used to model problems involving transport and fluid-elastic/poroelastic structure interaction, such as the transport of contaminants in hydrological systems where surface water percolates through rocks and sand, transport of nutrients and oxygen between capillaries and tissue, or spread of a disease across a border. This project will involve the training of graduate students.

The focus of this project is the development of adaptive time-stepping methods for two classes of coupled flow problems: the fluid-porous medium coupled problems and the fluid-structure interaction problems. A monolithic and a partitioned method will be developed for the fluid-porous medium problem described using the Stokes-Darcy system. Partitioned numerical methods will be developed for the fluid-structure interaction problems with both thin and thick structures. The proposed methods will be semi-discretized in time based on the refactorized Cauchy’s one-legged theta-like method, which is B-stable when used with a variable time step. Furthermore, when theta is 0.5, the method is also second-order accurate and conserves all linear and quadratic Hamiltonians. However, the application of this method to coupled problems, especially when partitioned methods are designed, has to be carefully performed to allow the use of black-box and legacy codes. The proposed methods will be mathematically and computationally analyzed. Various adaptive strategies will be considered. The performance of each method will be investigated with respect to the parameters in the problem. In both classes of multi-physics problems, the underlying equations will be coupled with a transport equation. The proposed techniques will also be applied to the transport problem, with particular attention to mass and energy conservation. Conservative properties of the transport problem will be investigated.

Dehua Wang: National Science Foundation Grant

Congratulations to Professor Dehua Wang, who received an NSF grant for his collaborative research on “Stability Analysis for Nonlinear Partial Differential Equations Across Multiscale Applications" for the period of Aug. 1, 2022, through July 31, 2025. This is a joint grant from NSF (USA) and EPSRC (UK) with a total amount over one million dollars. The other five co-PIs are from the University of Oxford, University of Wisconsin at Madison, Penn State University, and University of Texas at Austin.