Nonlinear Geometric PDEs: Modeling, Analysis and Approximation

Understanding slender structures is one of the great unresolved challenges of modern science and technology. Such structures permeate biological systems (flowers, leaves, tissue, and active matter), materials science (programmable materials), robotics (deployable devices), and biomedical engineering (soft robotics). These problems are typically lower-dimensional and susceptible to geometric effects such as metric constraints, length and area constraints, and curvature. Corresponding models are thus governed by geometric partial differential equations (PDEs), which, much like nature, are nonlinear. Fabrication of slender materials is time-consuming, expensive, and often erratic, which makes the development of predictive computational tools of paramount importance in engineering and science. The numerical treatment of nonlinear geometric PDEs must cope with the dynamic deformation of geometries, the presence of strong nonlinearities, and the development of self-penetrating structures and topological changes. Central to this proposal is the essential role of liquid crystals (LCs) as key constituents in the fabrication and actuation of slender structures. Models of nematic LC films are used to describe morphogenesis (shape formation) and active matter. Prestrained plates and LC networks are used to comprehend the shapes of flowers and leaves as well as to design and actuate programmable materials. Moreover, approximating local and nonlocal geometric problems, governed by fully nonlinear PDEs and singular integro-differential operators, constitutes a formidable yet distinct computational challenge. This research program combines reduced-order modeling (using differential geometry), structure-preserving algorithms, and efficient computation, and is supplemented by analysis (asymptotics and G-convergence). This project consists of three intertwined thrusts involving modeling, analysis, and approximation of several nonlinear geometric PDEs and nonlocal equations. The research is suitable for student training in exciting, mathematically and computationally challenging, and practically relevant areas of contemporary research, and is conducted together with former students, postdocs, and collaborators, who visit regularly. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.