Solving Eigenproblems with application in collapsible channel flows

Hao, Yujue (2013) Solving Eigenproblems with application in collapsible channel flows. PhD thesis, University of Glasgow.

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Abstract

Collapsible channel flows have been attracting the interest of many researchers, because of the physiological applications in the cardiovascular system, the respiratory system and urinary system. The linear stability analysis of the collapsible channel flows in the Fluid-Beam Model can be finalized as a large sparse asymmetric generalized eigenvalue problem, where the stiffness matrix is sparse, asymmetric and nonsingular, and the mass matrix is sparse, asymmetric and singular. The dimensions of the both matrices can reach about ten thousand or more, and the traditional QZ Algorithm is so expensive for this size of eigenvalue problem, due to its large requirement of computational resources and the quite long elapsed time. Unlike the traditional direct methods, the projection methods are much more efficient for solving some specified eigenpairs of the large scale eigenvalue problems, because normally a small subspace is made use of, and the original eigenvalue problem is projected to this small subspace. With this projection, the size of the eigenvalue problem is reduced significantly, and then the small dimensional eigenvalue problem can be easily and rapidly worked out by employing a traditional solver. Combined with a restarting strategy, this can be used to solve large dimensional eigenvalue problem much more rapidly and precisely. So far as we know, the Implicitly Restarted Arnoldi iteration(IRA) is considered as one of the most effective asymmetric eigenvalue solvers. In order to improve the efficiency of linear stability analysis in collapsible channel flows, an IRA method is employed to the linear stability analysis of collapsible channel flows in FBM. A Frontal Solver, which is an efficient solver of large sparse linear system, is also used to replace the process of shift-and-invert transformation. After applying these two efficient solvers, the new eigenvalue solver of collapsible channel flows---Arnoldi method with a Frontal Solver(AR-F), not only gets rid of the restriction of memory storage, but also reduces the computational time observably. Some validating and testing work have been done to variety of meshes. The AR-F can solve the eigenvalues with largest real parts very quickly, and can also solve the large scale eigenvalue problems, which cannot be solved by the QZ Algorithm, whose results have been proved to be correct with the unsteady simulations. Compared with the traditional QZ Algorithm, not only a great deal of elapsed time is saved, but also the increasing rate of the operation numbers is dropped to $O(n)$ from $O(n^3)$ of QZ Algorithm. With the powerful AR-F, the stability problems of refined meshes in collapsible channel flows are no long a barrier to the study. So AR-F is used to solve the eigenvalue problems from two refined meshes of the two different boundary conditions(pressure-driven system and flow-driven system), and the two neutral curves obtained are both revised and extended. This is the first time that IRA is made use of in the problem of fluid-structure interaction, and this is also a critical footstone to adopt a three dimensional model over FBM. Recently, the energy analysis and the energetics are the centre of research in collapsible channel flow. Because the linear stability analysis is much more accurate and faster than the unsteady simulation, the energy solutions from eigenpairs are also achieved in this thesis. The energy analysis with eigenpairs has its own advantages: the accuracy, the timing, the division, any mode and any point. In order to analyze the energy from eigenpairs much more clearly, the energy results with different initial solutions are presented first, then the energy solutions with eigenpairs are validated with those presented by Liu et al. in the pressure-driven system. By using the energy analysis with eigenpairs, much more energy results in flow-dirven system are obtained and analyzed.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Keywords: Collapsible channel flow, Linear stability analysis, Large asymmetric generalized eigenvalue problem, Implicitly Restarted Arnoldi iteration, Frontal solver, Energy analysis.
Subjects: Q Science > QA Mathematics
Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Funder's Name: UNSPECIFIED
Supervisor's Name: Luo, Prof. Xiaoyu
Date of Award: 2013
Depositing User: Dr YUJUE HAO
Unique ID: glathesis:2013-4589
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 20 Nov 2013 08:59
Last Modified: 03 Dec 2013 13:16
URI: http://theses.gla.ac.uk/id/eprint/4589

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