Collaborative Research: Hybrid Fluid-Structure Interaction Material Point Method with applications to Large Deformation Problems in Hemodynamics

人体内与心脏瓣膜相关的问题是心脏骤停和心力衰竭的原因，这可能会对人的健康造成毁灭性的后果，甚至导致死亡。虽然不一定是致命的，但与腿部静脉瓣膜相关的病理可能会给受影响的人带来严重的痛苦，并对他们的生活产生负面影响，可能会出现重大并发症。对于瓣膜相关疾病的治疗，目前最常见的做法是用假体装置替换发生故障的瓣膜。不幸的是，人工瓣膜存在长期耐用性和植入后并发症的问题。鉴于改进现有人工瓣膜的设计和选择的必要性，计算方法正在成为一个有价值的工具。由于多种物理现象相互作用，人体瓣膜内血液流动的性质使得建模问题从数学和计算的角度来看具有相当大的挑战性。具体地说，主要的挑战是阀门叶片经历了巨大的结构位移，同时保持了对流固界面处流体动力的准确描述。这个项目的重点是开发新的流体-结构相互作用方法，特别是在大变形的情况下。该项目提供的重要见解将使未来的阀门设计优化成为可能，同时避免昂贵的经验设计迭代。除了对社会的明显潜在影响外，拟议的项目还将对科学和工程中的许多其他应用有用，并对初级研究人员在一个重要的、令人兴奋的、在数学、计算和社会影响方面的研究领域的培训、教育和职业生涯产生有益的影响。这个项目将在三年的项目中每年资助2名研究生。这个项目是关于开发、分析和实施流体-结构相互作用(FSI)问题中有限元方法(FEMS)和物质点方法(MPM)耦合的新计算技术。使用不同的离散化技术来研究多尺度和多物理问题是计算模拟的有力工具。例如，将一维模型与多维模型耦合以降低计算成本，或将有限体积法与有限体积法耦合以发挥这两种方法的算法和数学特性的优势。基于同样的思想，如果在一个物理模型的动力学区域内出现不同的变形区域，有限元和MPM的耦合代表着一种很有前途的组合。事实上，有限元在小变形时达到了最好的精度，而MPM混合欧拉-拉格朗日公式在大变形时是有利的。事实上，有限元-MPM耦合只有很少的作者研究过，包括PI，而FSI框架与MPM方法的耦合还有待探索。物质点方法的使用将避免困扰许多现有FSI方法的网状纠缠问题。要设计所需的耦合方法，需要进行前期工作。首先，将使用FSI文献中的基准问题来解决浸入有限元流体中的MPM实体之间的耦合问题。同时，将研究用混合FSI-MPM方法离散的固体的力学性质，并用圆柱撞击刚性壁的泰勒杆试验来考察该方法的精度。然后，将利用从准备工作中获得的知识来实现生物瓣膜的FSI-MPM耦合方法，利用MPM对瓣叶进行建模，并在有限元-FSI框架中描述血管和血液流动。还将选择和研究适当的求解器和预条件，因为离散化的非线性和线性系统可能是大的和高度耦合的。最后，FSI-MPM耦合方法也将被应用于支架动脉的模拟，支架的描述使用MPM。通过这种方式，可以避免复杂的支架啮合过程，同时捕捉其动态行为。在拟议的研究中开发的计算技术将适用于广泛的应用，并被证明是无价的工具，例如人体阀门流体和结构动力学、航空航天和土木工程问题、溃坝和翼型设计等等。我们所有的发现都将在FEMuS中实现，这是一个用C语言编写的开源库，可以在线免费下载。我们的努力有望促进目前仅在研究软件中可用的新型计算技术的标准化。然而，来自世界各地的研究人员可能会访问我们的发现并加入我们的努力，并大幅加快标准化程序。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Heart valve associated issues in the human organism are the cause of cardiac arrest and heart failure, which may have devastating consequences on a person's health and even lead to death. While not necessarily fatal, pathologies associated with leg vein valves can nevertheless cause severe distress to the people affected and have a negative impact on their life with possibly major complications. For the treatment of valve associated diseases, the most common practice nowadays is the replacement of the malfunctioning valve with a prosthetic device. Unfortunately, prosthetic valves have issues with long term durability and post-implantation complications. Given the necessity of improving the design and selection of existing prosthetic valves, computational methodologies are becoming a valuable tool. The nature of blood flow inside a human valve renders the modeling problem considerably challenging from the mathematical and computational standpoints, as multiple physical phenomena mutually interact. Specifically, the major challenges are the large structural displacements experienced by the valve leaflets, while preserving accurate description of the hydrodynamic force at the fluid-solid interface. The focus of this project is on developing new fluid-structure interaction methodologies with specific interest in the case of large deformations. The important insight provided in this project will enable future valve design optimization while avoiding costly empirical design iterations. In addition to the obvious potential impact on society, the proposed project will be useful to many other applications in science and engineering, and also have beneficial impact on the training, education, and careers of junior researchers in an important, exciting, and mathematically, computationally, and societally impactful area of research. This project will support 2 graduate students per year for each year of the three year project.This project is about the development, analysis, and implementation of novel computational techniques for the coupling of finite element methods (FEMs) to material point methods (MPMs) in fluid-structure interaction (FSI) problems. The use of different discretization techniques for the study of multiscale and multiphysics problems is a powerful tool for computational simulations. For instance, one-dimensional models are coupled with multi-dimensional models for computational cost reduction, or FEMs are coupled with finite volume methods to exploit the advantages of the algorithmic and mathematical features of these two methods. With the same idea, the coupling of FEM with MPM represents a promising combination, if different deformation regimes occur within the dynamical regime of a physical model. As a matter of fact, the FEM reaches its best accuracy for small deformations whereas the MPM mixed Eulerian-Lagrangian formulation becomes beneficial when large deformations occur. FEM-MPM coupling has, in fact, been studied only by very few authors, including the PIs, and the coupling of an FSI framework with an MPM approach is yet to be explored. The use of the material point methodology would avoid the mesh entanglement issues that plague many existing FSI methods. To design the desired coupling approach, preliminary work is needed. First, the coupling between an MPM solid body immersed in an FEM fluid will be addressed, using benchmark problems from the FSI literature. At the same time, the mechanical properties of a solid body discretized with the mixed FSI-MPM approach will be studied and the accuracy of the method will be investigated using the Taylor bar test in which a cylinder impacts a rigid wall. Then, the knowledge gained from the preparatory work will be used to realize an FSI-MPM coupling methodology for biological valves, with the valve leaflets modeled with the MPM and the blood vessel and blood flow described in an FEM-FSI framework. Appropriate solvers and preconditioners will also be selected and studied because the discretized nonlinear and linear systems will likely be large and highly coupled. Lastly, the FSI-MPM coupling approach will also be applied for the simulations of stented arteries, with the stent described using the MPM. In this way, complex meshing procedure for the stent can be avoided, while capturing its dynamical behavior. The computational techniques developed within the proposed research will be applicable and prove to be invaluable tools for a broad spectrum of applications such as human valve fluid and structural dynamics, aerospace and civil engineering problems, dam breaking, and airfoil design, to name a few. All our findings will be implemented in FEMuS, an open source library written in C++ language, freely downloadable online. Our effort will hopefully contribute to the standardization of novel computational techniques that are currently available only in research software. Nevertheless, researchers from all over the world can potentially access our findings and join us in this effort, with a substantial speed up in the standardization procedure.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.