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

与心脏瓣膜相关的问题是导致心脏骤停和心力衰竭的原因，这可能对一个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会给患者带来严重的痛苦，并对他们的生活产生负面影响，可能出现严重的并发症。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故障瓣膜。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设计和选择的必要性，计算方法正成为一种有价值的工具。人体瓣膜内血液流动的性质使得从数学和计算的角度来看建模问题相当具有挑战性，因为多种物理现象相互作用。具体来说，主要的挑战是阀瓣所经历的大结构位移，同时保持对流固界面水动力的准确描述。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有助于未来的阀门设计优化，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，拟议的项目将对科学和工程的许多其他应用有用，并且对初级研究人员的培训，教育和职业生涯也有有益的影响，这些研究领域是一个重要的，令人兴奋的，数学，计算和社会影响的研究领域。该项目将在为期三年的项目中每年资助2名研究生。这个项目是关于开发、分析和实现流固耦合（FSI）问题的有限元方法（fem）和物质点法（MPMs）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场问题是一种强大的计算模拟工具。例如，将一维模型与多维模型相结合以降低计算成本，或将fem与有限体积方法相结合以利用这两种方法的算法和数学特点的优势。在同样的思想下，如果在一个物理模型的动力系统中出现不同的变形情况，FEM与MPM的耦合是一种很有前途的组合。事实上，有限元法在小变形时达到最佳精度，而MPM混合欧拉-拉格朗日公式在大变形时更有利。事实上，只有极少数作者（包括pi）研究了FEM-MPM耦合，而FSI框架与MPM方法的耦合还有待探索。材料点方法学的使用将避免困扰许多现有FSI方法的网格纠缠问题。为了设计理想的耦合方法，需要进行初步的工作。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固体之间的耦合问题。同时，将研究用混合FSI-MPM方法离散的固体力学性能，并利用圆柱撞击刚性壁的Taylor杆试验来检验该方法的精度。然后，从准备工作中获得的知识将用于实现生物瓣膜的FSI-MPM耦合方法，用MPM建模瓣膜小叶，并在FEM-FSI框架中描述血管和血流。由于离散化的非线性和线性系统可能是大型且高度耦合的，因此也将选择和研究适当的求解器和预调节器。最后，FSI-MPM耦合方法也将应用于支架动脉的模拟，使用MPM来描述支架。通过这种方式，可以避免支架的复杂网格划分过程，同时捕获其动力学行为。在提议的研究中开发的计算技术将适用并被证明是广泛应用的宝贵工具，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，水坝溃坝和机翼设计，仅举几例。我们所有的发现都将在FEMuS中实现，FEMuS是一个用c++语言编写的开源库，可在线免费下载。我们的努力将有助于标准化新的计算技术，目前只能在研究软件。尽管如此，来自世界各地的研究人员都有可能获得我们的发现，并加入我们的努力，这大大加快了标准化程序。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.