Hierarchical model reduction techniques for incompressible fluid dynamics and fluid-structure interaction problems

不可压缩流体动力学和流固耦合问题的分层模型简化技术

• 批准号: 1419060
• 负责人: Alessandro Veneziani
• 金额: $ 24.84万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419060&HistoricalAwards=false
• 关键词: Hierarchical; model; reduction; techniques; incompressible

Networks perfused by fluids are found in several engineering applications, ranging from hydrogeology, oil distribution, and internal combustion engines to hemodynamics. A quantitative analysis of these problems is of utmost interest for understanding fluid dynamics in the network, for predicting effects of local changes on the network (for instance, the effects of a surgical operation over the fluid dynamics in the arterial tree), and for optimizing flow distribution. Mathematical description and numerical approximation of these problems are challenging when coupling the accurate description of local dynamics with the large scale of the network. This proposal investigates a novel numerical method to undertake the quantitative analysis of fluid dynamics in complex networks called HiMod (Hierarchical Model Reduction). The primary (but not exclusive) application is the physiopathology of the arterial system, including in the mathematical model up to almost 2000 segments of the network. Several specific properties of this method need to be investigated for its development and engineering. The research provides a graduate student the opportunity of working on advanced mathematical and numerical techniques - including theoretical as well as practical aspects - in a truly interdisciplinary framework with frequent contacts with engineers and doctors expected to be the end users of these methodologies.Network of pipes are often modeled by assembling simplified equations describing each segment, like the well known Euler equations. Originally proposed for blood flow (incompressible fluid in compliant pipes) they have been extensively used in gas dynamics - for instance - in internal combustion engines (compressible fluid in rigid pipes). These equations are the result of several approximations to reduce the fully 3D mathematical model to a 1D set of hyperbolic equations. Unfortunately, this model reduction prevents proper capture the local features of the network that affects the global dynamics. The HiMod approach moves from a different perspective. We couple different numerical approximation techniques along the mainstream and the transversal directions. We use a finite element approximation for the mainstream and a spectral or modal approximation transversally. The number of modes can be locally and adaptively tuned to get the best possible trade-off between accuracy and computational efficiency. The rationale is that a relatively small number of modes is enough to guarantee good accuracy for the transversal dynamics, leading to a system of 1D problems (called a "psychologically 1D" model). Moving from preliminary promising studies for advection-diffusion problems, in this proposal we aim at developing the method for the 3D incompressible Navier-Stokes equations and fluid-structure interaction problems. Inf-sup stability and accuracy of the HiMod discretization as well as its role as preconditioner of the full problem will be investigated, together with adaptive techniques for the appropriate (automatic) selection of the transversal modes.

由流体灌流的网络在几个工程应用中都有发现，从水文地质学、石油分配、内燃机到血液动力学。对这些问题的定量分析对于了解网络中的流体动力学、预测局部变化对网络的影响(例如，外科手术对动脉树中流体动力学的影响)以及优化流量分配是非常有意义的。当将局部动力学的精确描述与网络的大规模相结合时，这些问题的数学描述和数值逼近是具有挑战性的。该方案研究了一种新的数值方法来进行复杂网络中流体动力学的定量分析，该方法被称为HiMod(分层模型降阶)。主要(但非排他性的)应用是动脉系统的生理病理，包括在数学模型中多达2000个节段的网络。为了其发展和工程应用，需要对该方法的几个特性进行研究。这项研究为研究生提供了在真正跨学科的框架内研究高级数学和数值技术的机会-包括理论和实际方面-与工程师和医生频繁接触，这些工程师和医生预计是这些方法的最终用户。管道网络通常通过组合描述每一段的简化方程来建模，如众所周知的欧拉方程。最初被提出用于血液流动(柔顺管道中的不可压缩流体)，它们已被广泛用于气体动力学-例如-在内燃机(刚性管道中的可压缩流体)中。这些方程是将全3D数学模型简化为一维双曲方程组的几次近似的结果。不幸的是，这种模型简化妨碍了正确地捕捉影响全局动态的网络的局部特征。HiMod方法从不同的角度出发。我们沿着主流方向和横向方向耦合了不同的数值逼近技术。我们对主流采用有限元近似，横向使用谱或模式近似。模式的数量可以局部地自适应地调整，以在精度和计算效率之间获得可能的最佳折衷。其基本原理是，相对较少的模式数量足以保证横向动力学的良好准确性，从而导致一维问题的系统(从心理上称为一维模型)。在这个方案中，我们从对流-扩散问题的初步研究进展到三维不可压缩的N-S方程和流固耦合问题的方法。将研究HiMod离散化的稳定性和精度，以及它作为整个问题的预条件的作用，以及适当(自动)选择横模的自适应技术。