A Bayesian Inversion Approach to Parameter Estimation in Fluid-Structure Interaction Problem

利用贝叶斯反演估计固液耦合系统中的参数

尹海安

11749024

硕士

应用数学

张振

2019-05-15

2019-07-10

哈尔滨工业大学

深圳

The parameter estimation of the Fluid-Structure Interaction (FSI) is a mathematical inverse problem that acquires great significance in both theory and practical applications. It has a wide range of applications in industry and engineering such as geoscience, mechanical design, and biomedicine. Since the fluid-solid coupling problem exhibits strong nonlinearity, and the parameter estimation problem contains substantial uncertainty, it is challenging to design an algorithm that is both efficient and effective. This paper focuses on how to model the blood vessel with the FSI system and estimate the underlying parameters. A systematic analysis and research on the related algorithms and theories are carried out. It is worth mentioning that although the model is built for the simulation of blood vessels, the methodology used in this paper can also be applied to other FSI systems.This paper consists of two parts. The first is to find the forward numerical solution of the FSI problem, and the second is solving the inverse problem based on the Kalman filter and its variants. The detailed work of this paper is listed as follows:(1) Firstly, we gave an introduction to the finite element method (FEM) and a brief but complete explanation of its core ideas. This paper mainly uses P1 or P2 elements to solve the finite numerical problem. The FEM package is implemented in Matlab by ourselves. Two simple examples are presented to validate the correctness of our FEM package, which can meet the need for numerical simulation.(2) Secondly, considering that the critical model explored in this paper can be regarded as a coupled system of fluid (blood) and structure (vessel), we investigate the elasticity and fluid PDEs independently by the finite element method described above. The research on the elastic body provides support for the movement operation of the mesh and provides an excellent basic model to validate the Kalman filter.(3) After the investigation of these two underlying models, we set up to solve the coupling system. There are some technical problems in how to deal with the movement of the fluid and solid region together. In order to solve this problem, we introduce the ALE framework and propose a scheme which obeys geometric conservation law within under the framework. Moreover, an example is proposed to verify its effectiveness.(4) Based on the above, we have carried out a series of simulations and analyses on the vascular model. It is worth mentioning that the inlet boundary conditions we used come from the real velocity data of blood flow, and our experimental results are satisfactory;(5) The second part is to look at the problem from the perspective of system science. Thus we introduce the state space model firstly. Based on which, we introduce the idea and details of the Kalman filter, the Unscented Kalman filter, and the Cubature Kalman filter. We designed a series of experiments to illustrate the power of these filters;(6) We perform experiments on the estimation of the parameters in fluid and solid dynamics. Finally, we use them to estimate the parameters of the vascular model designed in the first part.In the work, the following methods and examples are worth noting:(1) In the ALE section, we propose an example where the grid velocity field is random. This is a well-designed example which shows the excellent numerical properties of ALE, and it also explains that grid speed is independent of the numerical model. The traditional idea is reasonable but limited by the choice of grid (which is either static or consistent with the motion of material points). To break through the limit, we need to sacrifice some convenience. That is, using a more complicated scheme and performing more calculations;(2) The ALE scheme in this paper is a monolithic scheme while the interface evolution and mesh movement are performed explicitly, which makes our system always a linear system during the computation, which reduces computation cost significantly;(3) The perturbed linear elastic system in the section of the Kalman filter is also a well-designed example. It is to illustrate that the modeling error itself can also be regarded as a random variable to participate the information analysis.The contribution of this paper is mainly reflected in the following aspects:(1) Firstly, this paper presents a scientific and comprehensive evolutionary methodology for research. It exhibits a complete process including modeling, analysis, and simulation for a practical problem. During the process, every part is extended rationally;(2) Secondly, this paper gives several well-designed examples with innovative consideration and satisfactory performance, which can provide a reference for other researchers;(3) Finally, in the study of inverse problem in parameter estimation, this paper focus on the discrete filtering system. However, the continuous situation shares some similar characteristics with much more theoretic content in it. Although we do not explore that system in this paper, it is definitely one right choice for research.

流固耦合模型的参数反演问题是一类兼具极高的实际应用价值和理论意义的数学问题。它在地质科学、机械设计、生物医学等许多行业中都有着广泛的作用。同时，由于流固耦合问题具有很强的非线性性，而参数反演问题又包含很强的不确定性，如何得到一个高效且有效的算法，是十分具有挑战性的。本文围绕如何利用流固耦合模型对血管建模并反演其参数这一问题展开，对一些相关的算法和理论进行了一个比较系统的分析和研究。值得一提的是，虽然模型是基于对血管的模拟建立的，但本文使用的方法也同样可以应用于其他的流固耦合系统。本文从行文逻辑上讲主要分为两大部分，首先是流固耦合正问题的数值求解探索，然后是基于卡尔曼滤波器及其变种进行的反问题的分析。本文的具体工作如下:(1) 首先，我们对有限元方法进行了一个简单的介绍，对其核心思路进行了一个相对完备的阐述。本文主要使用P1/P2元进行有限元数值求解，并附上了两个简单的算例说明其具有良好的数值性质，可以满足本文的计算需求；(2) 其次，考虑到本文探究的核心模型可以看作是一个流体（血液）与弹性体（血管）的耦合模型，因此，我们利用前述的有限元方法对弹性体问题和流体问题分别进行了一定的分析和实验。除了对模型的认知以外，这里对于弹性体的研究，既为移动网格提供了支持，也为后续验证卡尔曼滤波器效果提供了一个良好的基础模型；(3) 在对这两个基础问题有了一定认识以后，我们开始考虑求解其耦合问题。然而除了如何耦合流体场与固体场以外，还有一个技术难题是如何去处理流体区域的移动问题。为了解决这个问题，我们引入了ALE框架并对在此基础上提出了一个几何守恒格式，同时提出了一个纯流体算例对其有效性进行了验证；(4) 在上述基础上，我们对血管模型进行了一系列简单的模拟和分析，值得一提的是，我们用的入口边界条件是真实的血液流速数据，我们的实验结果性状良好；(5) 第二部分则是站在在系统科学的视角看待问题，于是我们首先对状态空间模型进行了一定的说明，介绍说明了卡尔曼滤波器、无损卡尔曼滤波和容积卡尔曼滤波器的设计逻辑，同时我们设计了一系列实验说明这些滤波器的效果；(6) 在此基础上，我们再次对流体固体的一些参数进行单独的反演实验，最后，我们利用它们对第一部分设计的血管模型的参数进行了反演。 在这些工作中，以下的一些方法和算例是值得注意的：(1) 在介绍ALE的部分中，我们提出了一个在网格速度场随机的情况下计算纯流体问题的算例，这个算例是一个说明ALE优异数值性质的良好算例，同时它也说明了，网格速度与算例本身是相互独立的，传统的思路是符合逻辑但是存在局限性的，当然，不可否认的是，要打破这些局限性，我们需要牺牲一些便利性，即使用更复杂的格式和更多的计算量；(2) 我们使用的ALE算法是一种整体算法，然而我们的界面演化和网格移动是显式独立进行的，这使得我们的系统在计算过程中始终是线性系统，在很大程度上减少了计算量；(3) 我们在卡尔曼滤波算法验证时设计的带扰动的线性弹性系统也是一个设计良好的算例，然而值得注意的是，它本身的阐述作用大于其蕴含的物理意义，它是为了说明模型本身的建模误差也可以视为一种随机变量参与到信息分析中。总的来说，本文的价值主要体现在如下几个方面：(1) 首先，本文展示了一个科学全面循序渐进的研究方法，对一个实际问题，给了从建模到分析到模拟的一个完备的过程，同时在此过程中，每一步都给了较为充分而细致的分析和探究；(2) 其次，本文给出了几个有创新意义且性质良好的算例，能够为后续研究者提供参考；(3) 最后，在反问题研究中，本文的工作主要都是在离散滤波系统的内进行的。然而它们与连续滤波系统有着许多相似的特性，同时其内涵更丰富。虽然本文并未对该系统进行进一步探究，但它相对于离散滤波问题更具研究价值，是一个良好的课题选择。

FSI ALE Kalman Filter

流固耦合 ALE 卡尔曼滤波

英语

联合培养

南方科技大学

Yin HA. A Bayesian Inversion Approach to Parameter Estimation in Fluid-Structure Interaction Problem[D]. 深圳. 哈尔滨工业大学,2019.