寬頻元素法應用於解非結構性元素Helmholtz方程式

Helmholtz Solver in unstructured mesh with Spectral Element Method

洪立昕(Li-Hsin Hung)

指導教授 ： 顏瑞和

國立臺灣大學/工學院/機械工程學研究所/碩士(2005年)

https://doi.org/10.6342/NTU.2005.00386

摘要

寬頻元素法應用於解非結構性元素Helmholtz方程式

摘要

本文主旨在使用寬頻元素法發展二維的三角形元素，三維的四面體元素組成計算模型的Helmholtz方程式泛用型解程式，程式可解Helmholtz方程式在二維和三維任意幾何外型。在本文中由展開基底的建立到數值的運算，還有演算法的原理均有所研究，完整描述實作程式的理論基礎。程式經過連續邊界條件問題驗證準確性，也實際驗證複雜幾何形狀問題，確定可達到寬頻收斂特性。目前對於移動邊界的研究很多，考慮重新計算元素分布的效率和元素形狀的性質，非結構性元素是較適合採用的元素外型。本研究提供一個有效率的解題核心，作為未來發展非結構性元素流場模擬所需的基礎。

關鍵字

寬頻元素 ； 有限元素 ； 非結構性元素 ； Helmholtz方程式

並列摘要

Helmholtz Solver in unstructured mesh with
Spectral Element Method

Abstract
Due to fast convergence, small diffusion and dispersion errors, higher order numerical methods, such as the spectral element methods have been shown computationally more efficient than the conventional lower order methods in a full Navier Stokes simulations. Traditionally, the element for the spectral element method is in a structured quadrilateral region. The extension from one dimension to higher dimensions is relatively straight forward. In order to broaden the application of spectral element methods to more complex geometries, the use of unstructured elements in the triangular region for two dimensions, and tetrahedral region in three dimensions is to be investigated in this thesis.
The objective of this study is to develop an efficient solver to solve a Helmholtz equation in 2D or 3D unstructured meshed domain. A computer code has been developed. The details of algorithms are addressed in this thesis.
The code is validated by various examples with complex geometries. Different problems with Dirichlet and Neumann boundary conditions are also tested. .

Key Words: spectral element, hp method, finite element, unstructured element, Helmholtz equation

並列關鍵字

spectral element ； hp method ； finite element ； unstructured element ； Helmholtz equation

參考文獻

Oxford University Press, New York, 1999

4. D. Gottlieb and S.A. Orszag. Numerical Analysis of Spectral Methods:Theory and Applications. SIAM-CMBS,1997

6. S.A. Orszag and G. S. Patterson, Numerical simulation of the three dimensional homogeneous isotropic turbulence, Phy. Rev. Letter., 28-76, (1972)

7. A. T. Patera, A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comp. Physics., 54-68, (1984)

8. Josep Sarrate and Antonio Huerta, Efficient Unstructured quadrilateral mesh Generation, Int. J. Numer. Meth. Engng 2000; 49:1327-1350.