Parallel Nonlinear Preconditioning Algorithms and Applications in Biomechanics

流固耦合问题的计算机模拟在科学和工程中有许多应用，如飞机、汽车和悬索桥的振动分析。最近，该技术已扩展到某些医疗问题的诊断和治疗计划，如先天性和获得性心血管疾病，以及医疗设备的设计和优化。在具有大量处理器核的超级计算机上求解流固耦合问题是具有挑战性的，因为数学模型由一个耦合的复杂非线性系统组成，而大多数用于解决此类问题的现有算法和软件都不能很好地扩展到几百个处理器核以上。首席研究员将开发高度可扩展的算法和软件，适用于大型超级计算机，并适用于不同的血流模型和动脉壁的材料参数。成熟的技术可用于解决许多类型的线性问题，但对于耦合的，高度非线性的多物理场问题，迫切需要鲁棒和可扩展的技术，特别是在大型并行计算机上实现。本项目的技术重点是一类非线性预条件牛顿方法，该方法结合了非线性消去技术和多层域分解的并行化。通过这项研究，主要研究者将解决非线性难题，建立一系列具有不同非线性水平的物理模型。将设计提供高度并行性的算法，以便有效地使用大规模并行计算机。目标应用是生物力学中的一系列流固相互作用问题。对于流体模型，将研究牛顿模型和非牛顿模型。对于实体模型，对于小变形将考虑线性弹性，对于大变形将考虑几何非线性弹性，对于材料非线性问题将考虑超弹性。为了在具有大量处理器内核的并行计算机上获得高可扩展性，将研究两级和多层版本的算法。这项研究将对需要解决非线性难题的计算科学和工程领域产生重大影响。该研究为对生物力学、并行计算和一般计算科学与工程应用感兴趣的研究生和本科生提供了丰富的机会。

Computer simulation of fluid-structure interaction problems has many applications in science and engineering, such as the vibration analysis of aircraft, automobiles, and suspension bridges. More recently, the technique has been extended to diagnosing and treatment planning of certain medical problems such as congenital and acquired cardiovascular diseases, and to the design and optimization of medical devices. Solving the fluid-structure interaction problems on supercomputers with a large number of processor cores is challenging because the mathematical model consists of a coupled complicated nonlinear system, and most existing algorithms and software for the solution of problems of this kind do not scale well beyond a few hundred processor cores. The principal investigator will develop highly scalable algorithms and software that are suitable for large scale supercomputers and applicable for different models of blood flows and material parameters for the arterial wall. Mature technologies are available for solving many types of linear problems, but for coupled, highly nonlinear multi-physics problems, robust and scalable techniques are badly needed, especially for implementation on large scale parallel computers. The technical focus of this project is a class of non-linearly preconditioned Newton methods that combines a nonlinear elimination technique with multilevel domain decomposition for parallelization. Through this research the principal investigator will solve non-linear difficult problems modeling a wide range of physical models with different levels of non-linearities. Algorithms that provide a high degree of parallelism will be designed so that large scale parallel computers can be used efficiently. The target application is a family of fluid-structure interaction problems in biomechanics. For the fluid model, both Newtonian and non-Newtonian models will be studied. For solid models, linear elasticity will be considered for small deformations, geometric nonlinear elasticity will be considered for large deformations, and hyper-elasticity will be considered for materially nonlinear problems. Two- and many-level versions of the algorithms will be investigated to obtain high scalability on parallel computers with a large number of processor cores. This research will have a great impact in areas of computational science and engineering where non-linear difficult problems need to be solved. The research is rich in opportunities for both graduate and undergraduate students interested in applications in biomechanics, parallel computing, and general computational science and engineering.