AF: Small: Collaborative Research: Scalable, high-order mesh-free algorithms applied to bulk-surface biomechanical problems

对物质过程进行建模，如药物分子与受体部位的相互作用，必须捕捉相互作用，通常是通过偏微分方程组(PDE)，既在整体上，也在表面上。这些偏微分方程组必须进行数值计算，因为它们经常在体积和表面之间具有非线性耦合，其几何形状往往随着时间的变化而变化。本项目开发并实现了基于径向基函数(RBFS)的可伸缩、高阶、无网格算法，用于三维演化几何形状中复杂的多尺度体积-表面生物力学建模。在此项资助下开发的算法和软件将直接使研究人员能够探索凝血过程中脂质膜形态和血小板生理的科学问题。这些算法将在生物学、材料科学和许多工业问题中广泛适用于其他耦合的体积-表面过程。这笔拨款将有助于支持博伊西州立大学新的计算科学与工程(CSE)博士项目的研究组合，并将支持该项目的首批研究生之一。作为该项目的一部分，PI将继续建立在他们从代表不足的群体中招收计算数学研究生的成功记录的基础上，通过与博伊西州立大学的LSAMP计划合作，这项建议的一个具体重点是为两个生物力学和生理问题开发RBF算法，这两个问题处于当前数值技术可以处理的前沿，并具有一般整体表面问题的共同特征：脂质双层的形态和血小板聚集和凝血。这些问题将推动以下数值离散和算法方面的发展对于径向基函数法：1)表面偏微分方程组高阶解的无尺度核；2)几何复杂区域内稳定的、可伸缩的无网格格式；3)具有最佳计算复杂性的高阶无网格几何建模技术；4)具有可变空间的散乱节点自动生成的SIMD友好算法；5)低成本的自动模板选择算法和自适应节点精化；6)体表面隐式离散的预处理策略；7)一致的、保持精度的无网格技术，用于可视化基于径向基函数的高阶方法的解。这些发展将通过开源软件包公开。

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.