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（PDEs）系统，无论是在体上还是在表面上。这些偏微分方程必须进行数值计算，因为它们通常具有体积和表面之间的非线性耦合，其几何形状经常随时间而变化。该项目开发并实现了基于径向基函数（rbf）的可扩展、高阶、无网格算法，用于三维演化几何中复杂的多尺度体面生物力学建模。在这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血小板脂膜形态和生理学的科学问题。该算法将广泛适用于其他耦合体表面过程-在生物学，材料科学和许多工业问题。这笔拨款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计划合作，从代表性不足的群体中招募计算数学研究生，这是他们成功的记录。本提案的一个特别重点是开发RBF算法来解决两个生物力学和生理问题，这两个问题是当前数值技术可以处理的前沿问题，并且具有一般体表面问题的共同特征：脂质双分子层的形态和血小板聚集和凝固。这些问题将推动rbf数值离散化和算法的发展：1)表面偏微分方程高阶解的无标度核；2)稳定的、可扩展的无网格平流方案在几何复杂的区域，没有任何调整参数；3)计算复杂度最优的高阶无网格几何建模技术；4)基于simd的可变空间离散节点自动生成算法；5)低成本自动上卷模板选择和自适应节点细化算法；体面隐式离散化的预处理策略；7)基于rbf的高阶方法可视化解决方案的一致性，保持精度的无网格技术。这些开发将通过一个开源软件包向公众开放。

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.