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研究生： 賴聲泓
Sheng-Hong Lai
論文名稱：
Parallel Computation of Acoustic Eigenvalue Problems Using a Polynomial Jacobi-Davidson Method
指導教授： 黃楓南
Feng-Nan Hwang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 98
語文別： 英文
論文頁數： 51
中文關鍵詞： 平行計算 、多項式型 、特徵值 、有限元素法
外文關鍵詞： finite element method, acoustic, eigenvalue, precondition, additive Schwarz, Jacobi-Davidson method
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    • 在我們日常生活周遭隨處可遇到有關於聲音的問題。例如開車時、搭公車時或搭飛機時。這些問題可以經由數學模型來描述聲音在一些吸振材質加入之下的振動情況，再經由有限元素法離散之後可以得到一個多項式型的特徵值問題。在部份應用中，我們有興趣找出一些低頻率的特徵值，但這些特徵值通常都落在頻譜的內部。並且當網格切的非常細的時後，特徵值問題的維度會變成相當地大。所以我們需要一個可平行化的工具來解這種大型稀疏的多項式特徵值問題。Jacobi-Davidson 演算法提供了一個快速並且有效率的方式來解出這類問題的內部特徵值。我們在 additive Schwarz 的架構下平行化實作 acobi-Davidson 演算法，並且用來解由聲音的問題所產生的多項式型的特徵值問題。經由數值實驗結果，我們列出了一些 additive Schwarz preconditioned Jacobi-Davidson 的平行效能。在解 correction equation 時，經由 Krylov-Schwarz 演算法的幫助之下，Jacobi-Davidson 的效率有著顯著的進步。

    The acoustic problems usually happens around us in our daily life when we drive a car, take a bus or take a plane. From the problems of acoustic vibrations with damping, a polynomial eigenvalue problem is obtained by applying the Galerkin finite element method. For particular applications, we are interested in finding some selected low frequency eigenvalues which are located within the interior of the spectrum. The size of the resulting eigenproblem is typically large especially for very fined mesh case so that the parallel polynomial eigensolver is need to deal with such problem. The Jacobi-Davidson method provides a fast and efficient manner for solving the interior eigenvalues for the large sparse polynomial eigenvalue problems. We proposed an Jacobi-Davidson method based on an additive Schwarz framework in parallel implementation and used it to solve the polynomial eigenvalue problem arising from the acoustic. And we showed some parallel performance of the additive Schwarz preconditioned Jacobi-Davidson method by numerical experiments. With help of Krylov-Schwarz algorithm for the correction equation, the efficiency of JD algorithm is greatly improved.

    List of Figures . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Polynomial acoustic eigenvalue problems . . . . . . . . . . . . 4 2.1 Mathematical acoustic model . . . . . . . . . . . . . . . . 4 2.1.1 The continuity equation and Euler’s equation . . . . . 4 2.1.2 Acoustic wave equation . . . . . . . . . . . . . . . . 6 2.1.3 Boundary conditions and time-separation assumption . . 7 2.2 Galerkin finite element discretization . . . . . . . . . . . 9 2.3 Some acoustic polynomial eigenvalue problems . . . . . . . 10 2.4 A special case with semi-analytical solution available . . 12 3 Parallel solution algorithms for polynomial eigenvalue problem 17 3.1 Additive Schwarz preconditioned Jacobi-Davidson algorithm 17 3.1.1 Preconditioned correction equation . . . . . . . . . . 20 3.1.2 Restricted additive Schwarz . . . . . . . . . . . . . 21 4 Numerical experiment of ASPJD . . . . . . . . . . . . . . . . 22 4.1 A benchmark problem and setup for numerical experiments . 22 4.2 Parallel code validation . . . . . . . . . . . . . . . . . 24 4.3 ASPJD tunning . . . . . . . . . . . . . . . . . . . . . . 28 4.4 Parallel scalability analysis . . . . . . . . . . . . . . 31 4.5 Comments on other possible correction equation solvers . . 33 4.6 Linearized approaches . . . . . . . . . . . . . . . . . . 33 5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . 37 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A. Bilinear basis and the local stiffess and mass matrices . . 41

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