Modeling material processes, like interaction of a drug molecule docking with a receptor site, must capture interactions, often by systems of partial differential equations (PDEs), both in the bulk and on the surface. These PDEs must be calculated numerically, since they often have non-linear couplings between bulk and surface whose geometries often evolve in time. This project develops and implements scalable, high-order, meshfree algorithms -- based on radial basis functions (RBFs) -- for complex multi-scale bulk-surface biomechanical modeling in three-dimensional evolving geometries. The algorithms and software developed under this grant will directly enable researchers to explore scientific questions in lipid membrane morphology and physiology of platelets in the clotting process. The algorithms will have broad applicability to other coupled bulk-surface processes -- in biology, material science, and many industrial problems. The grant will help bolster the research portfolio of the new Computational Science and Engineering (CSE) PhD program at Boise State University, and will support one of the first graduate students in this program. The PIs will continue to build upon their successful track record of recruiting graduate students in computational mathematics from underrepresented groups as part of this project by working with the LSAMP program at Boise State University.
A specific focus of this proposal is on developing RBF algorithms for two biomechanical and physiological problems that are at the forefront of what current numerical techniques can handle and that have features common to general bulk-surface problems:  morphology of the lipid bilayer and platelet aggregation and coagulation. These problems will drive the development of the following advances in numerical discretizations and algorithms for RBFs: 1) Scale-free kernels for high-order solutions of surface PDEs; 2) Stable, scalable meshfree schemes for advection in geometrically complex domains without any tuning parameters; 3) High-order meshfree geometric modeling techniques with optimal computational complexity; 4) SIMD-friendly algorithms for automatic scattered node generation with variable spatial; 5) Algorithms for low-cost automatic stencil selection for upwinding and adaptive node refinement; 6) Preconditioning strategies for implicit discretizations of bulk-surface; 7) Consistent, accuracy preserving meshfree techniques for visualizing solutions from RBF-based high-order methods. The developments will be made publicly available through an open-source software package.
