A Bayesian Inversion Approach To Parameter Estimation In Fluid-structure Interaction Problem

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.