Geometric methods for fluid-structure interactions

流固耦合的几何方法

• 批准号: RGPIN-2018-05751
• 负责人: Putkaradze, Vakhtang
• 金额: $ 1.46万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677169
• 关键词: Geometric; methods; fluid; structure; interactions

When a grass or tree leaf bends in the wind, the motion deforms the internal structure of narrow channels and moves the fluid inside. A geologist looking for oil probes the Earth with sounds waves, which move the porous media and fluid inside it in a complex manner. Blood pulsing through our arteries and veins deforms the elastic walls with every heartbeat. Interactions of fluids and structures are everywhere, and many more examples from everyday life can be easily found. Describing interaction of fluids and structures is always challenging. This project will create a unified framework for considering interactions between fluids and structures, using the approach of geometric mechanics. ******The focus of the project will be on the cases when fluids are flowing inside the elastic materials, such as narrow tubes and porous material media. The unifying theme of the project is the use of general geometric ideas, such as the symmetry of space, and methods of analytical mechanics (variational procedure), yielding the derivation and analysis of equations from the first principles. The methods of geometric mechanics allow to treat a wide variety of problems. The first set of problems concerns the mechanics of tubes conveying fluid, a problem which is relevant for engineering (e.g. chemical), and biomedical applications (blood flow). The methods developed in this project will also lead to the development of variational computational methods which guarantee conservation of linear and angular momenta and absence of artificial sources and sinks of mass, forces and torques due to discretization. The second set of problems concerns the dynamics of flexible porous media, such as sponge filled with water, and sheets and rods made of such media. The theory developed in this project will allow to analyze the motion of fluid-filled sheets and rods, and in particular, compute internal dissipation under the motion, which is difficult to compute without geometric methods. Finally, we apply Hamel's theory of mechanics to our problems. This method, based on choosing the most convenient velocities for the problem at hand, will further simplify the analysis and allow to find the most convenient velocity coordinates for the complex problems considered here. ***We are also incorporating geometric elasticity models for describing the growth of glioma in the brain, with the focus on its mechanical effects, given that the volume of the brain is constrained and the composition the brain matter around glioma changes. Understanding the effects of mechanics and additional stresses on glioma growth may eventually contribute to treatment recommendations. In addition, since the methods we develop here are general, and are based on fundamental principles of symmetry, they can be applied to a wide variety of practical problems coming from different physics, forming the background for studies beyond the scope of the 5-year project described here.

当草或树叶在风中弯曲时，这种运动使狭窄通道的内部结构变形，并将液体移动到里面。寻找石油的地质学家用声波探测地球，声波以复杂的方式移动其中的多孔介质和流体。每次心跳时，流经我们动脉和静脉的血液都会使弹性血管壁变形。流体和结构的相互作用无处不在，从日常生活中可以很容易地找到更多的例子。描述流体与结构的相互作用一直是一个挑战。这个项目将创建一个统一的框架来考虑流体和结构之间的相互作用，使用几何力学的方法。******该项目的重点将放在流体在弹性材料内部流动的情况下，例如窄管和多孔材料介质。该项目的统一主题是使用一般的几何思想，如空间的对称性，以及分析力学的方法（变分过程），从第一原理中推导和分析方程。几何力学的方法允许处理各种各样的问题。第一组问题涉及管道输送流体的力学，这是一个与工程（如化学）和生物医学应用（血液流动）相关的问题。在这个项目中开发的方法也将导致变分计算方法的发展，这些方法保证了线性和角动量的守恒，并且由于离散化而没有人为的质量、力和扭矩的源和汇。第二组问题涉及柔性多孔介质的动力学，例如充满水的海绵，以及由这种介质制成的片和棒。在这个项目中发展的理论将允许分析充满流体的板和棒的运动，特别是计算运动下的内部耗散，这是没有几何方法难以计算的。最后，我们把哈默尔的力学理论应用到我们的问题中。这种方法基于为手头的问题选择最方便的速度，将进一步简化分析，并允许为这里所考虑的复杂问题找到最方便的速度坐标。考虑到大脑的体积受到限制，胶质瘤周围的大脑物质成分发生变化，我们还将几何弹性模型用于描述大脑中胶质瘤的生长，重点关注其机械效应。了解神经胶质瘤生长的机制和额外压力的影响可能最终有助于推荐治疗方法。此外，由于我们在这里开发的方法是通用的，并且基于对称的基本原理，它们可以应用于来自不同物理学的各种各样的实际问题，形成了超出这里描述的5年项目范围的研究背景。