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

流体灌注网络在许多工程应用中都有发现，从水文地质学、石油分布、内燃机到血流动力学。对这些问题的定量分析对于理解网络中的流体动力学、预测局部变化对网络的影响（例如，外科手术对动脉树流体动力学的影响）以及优化血流分布至关重要。这些问题的数学描述和数值逼近在网络的大尺度下对局部动力学的精确描述具有挑战性。本文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，称为HiMod（分层模型简化）。主要的（但不是唯一的）应用是动脉系统的生理病理，包括在数学模型中多达2000段的网络。该方法的若干特性需要进一步研究，以便进行开发和工程化。该研究为研究生提供了在真正的跨学科框架内研究高级数学和数值技术（包括理论和实践方面）的机会，并与有望成为这些方法最终用户的工程师和医生经常接触。管道网络通常通过组合描述每个部分的简化方程来建模，例如众所周知的欧拉方程。最初提出的血液流动（柔性管道中的不可压缩流体），它们已广泛用于气体动力学-例如-内燃机（刚性管道中的可压缩流体）。这些方程是将全三维数学模型简化为一维双曲方程集的几个近似的结果。不幸的是，这种模型缩减阻止了正确捕获影响全局动态的网络的局部特征。HiMod方法从一个不同的角度出发。我们在主流方向和横向方向上结合了不同的数值近似技术。我们对主流采用有限元近似，横向采用谱或模态近似。模式的数量可以局部自适应调整，以获得精度和计算效率之间的最佳平衡。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，从而导致一维问题系统（称为“心理一维”模型）。从对平流扩散问题的初步研究开始，在本建议中，我们的目标是发展三维不可压缩Navier-Stokes方程和流固相互作用问题的方法。将研究HiMod离散化的稳定性和准确性，以及它作为整个问题的前置条件的作用，以及适当（自动）选择横向模式的自适应技术。

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.