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

由流体灌注的网络在几个工程应用中被发现，从水文地质学、石油分配和内燃机到血液动力学。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，外科手术对动脉树中的流体动力学的影响）以及优化流量分布是最感兴趣的。这些问题的数学描述和数值近似是具有挑战性的耦合时，精确的局部动态与大规模的网络的描述。该提案研究了一种新的数值方法进行定量分析的流体动力学在复杂的网络称为HiMod（分层模型简化）。主要（但不是唯一的）应用是动脉系统的生理病理学，包括在数学模型中多达近2000个网络段。这种方法的几个具体性质需要研究其开发和工程。 该研究为研究生提供了在真正的跨学科框架内研究先进的数学和数值技术的机会-包括理论和实践方面-与工程师和医生经常接触，预计这些方法的最终用户。管网通常通过组装描述每个部分的简化方程来建模，如众所周知的欧拉方程。最初提出用于血液流动（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3D数学模型简化为1D双曲方程组的几个近似的结果。不幸的是，这种模型简化阻止了正确捕获影响全局动态的网络局部特征。HiMod方法从不同的角度出发。我们耦合不同的数值逼近技术沿着主流和横向方向。我们使用有限元近似的主流和频谱或模态近似横向。模式的数量可以局部和自适应地调整，以获得准确性和计算效率之间的最佳可能的权衡。基本原理是，相对少量的模式就足以保证横向动力学的良好精度，从而导致一维问题的系统（称为“心理一维”模型）。从对流扩散问题的初步有前途的研究，在这个建议中，我们的目标是发展的方法，三维不可压缩Navier-Stokes方程和流体-结构相互作用问题。的HiMod离散化，以及它的作用作为预条件的完整的问题的信息-超级稳定性和准确性将被调查，连同自适应技术的适当的（自动）选择的横向模式。

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.