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

建模材料过程，如药物分子与受体位点对接的相互作用，必须捕获相互作用，通常通过偏微分方程（PDE）系统，在本体和表面上。这些偏微分方程必须进行数值计算，因为它们通常具有体与表面之间的非线性耦合，其几何形状通常随时间演变。 该项目开发并实现了可扩展的，高阶的，无网格算法-基于径向基函数（RBFs）-用于三维演变几何中复杂的多尺度体表面生物力学建模。 根据这项资助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血小板的脂质膜形态和生理学的科学问题。 该算法将具有广泛的适用性，以其他耦合体表面过程-在生物学，材料科学和许多工业问题。该补助金将有助于加强博伊西州立大学新的计算科学与工程（CSE）博士课程的研究组合，并将支持该计划的第一批研究生之一。 作为该项目的一部分，PI将继续与博伊西州立大学的LSAMP项目合作，在其成功招募计算数学研究生的记录基础上继续发展。该提案的一个具体重点是为两个生物力学和生理学问题开发RBF算法，这两个问题处于当前数值技术可以处理的最前沿，并且具有与现有技术相同的特征。一般体积表面问题：脂质双层的形态以及血小板聚集和凝固。 这些问题将推动径向基函数数值离散和算法的发展：1）表面偏微分方程高阶解的无标度核; 2）几何复杂区域中稳定的、可扩展的无网格平流格式，无需任何调整参数; 3）具有最佳计算复杂度的高阶无网格几何建模技术; 4）可变空间离散节点的SIMD友好自动生成算法5）低成本自动模板选择算法6）体表面隐式离散化的预处理策略6）离散化的网格优化算法7）网格优化算法8）网格优化算法10）网格优化算法11）网格优化算法12）网格优化算法13）网格优化算法14）网格优化算法15）网格优化算法16）网格优化算法17）网格优化算法18）网格优化算法19 7）一致的，保持精度的无网格技术，用于可视化基于RBF的高阶方法的解决方案。 这些发展将通过一个开放源码软件包向公众公布。

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.