求解任意波数的三维Helmholtz方程.pdf

262015.51(2)
Computer Engineering and Applications计算机工程与应用
求解任意波数的三维 Helmholtz方程
毛崎波
MAO Qibo
南昌航空大学飞行器工程学院,南昌330063
School of Aircraft Engineering, Nanchang Hangkong University Nanchang 330063, China
MAO Qibo Solution for three dimensional Helmholtz equations under arbitrary wave numbers. Computer Engi-
neering and Applications, 2015, 51(2): 26-29
Abstract: The Adomian Decomposition Method( ADM is employed in this paper to solve three dimensional Helmholtz
equations under abi ray ave numbers Based on te ADM, the three dimensional Helmholtz differential equation becomes
a recursive algebraic equation. Furthermore, the boundary conditions become simple algebraic equations which are suit
able for symbolic computation. By using boundary conditions, the closed-form series solution can be easily obtained. The
main advantages of ADM are computational simplicity and do not involve any linearization or discretization. Finally, two
numerical examples are presented to check the reliability of the proposed method for solving the three dimensional Helm
holtz equations with different wave numbers. The numerical results on three dimensional problems with known anal ytic
solutions demonstrate that the ADM is quite accurate and readily implemented. Furthermore, the good convergence and
the excellent numerical stability of the solution based on the ADM can also be found for high wave numbers. It means that
the ADM is quite efficient and is practically well suited for solving three dimensional Helmholtz equations at differen
Wave numbers
Key words: three dimensional Helmholtz equations; Adomian decomposition method; wave numbers
摘要:提出通过 Adomian分解法求解任意波数的三维 Helmholtz方程。通过 Adomian分解法可以把三维 Helmholtz
微分方程转换成递归代数公式,并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以
很容易得到方程的解析解表达式。 Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散
化。最后通过数値计算以验证 Adomian分解法求解任意波数下三维 Helmholtz方程的有效性。数值计算结果表明:
Adomian分解法的计算结果非常接近精确解,并且该方法在大波数情况下还具有良好的收敛性
关键词:三维 Helmholtz方程; Adomian分解法;波数
文献标志码:A中图分类号:0242.2doi:10.3778/1S.1002-8331.1405-0047
算工作量増加,并且效率低下。因此有必要寻找适合任
Helmholtz方程在力学、声学及电磁学等领域有着意波数ド Helmholtz方程的计算方法。
泛的应用背景,因此有众多学者对该方程的求解进行
20世纪80年代,美国数学物理学家 Adomian提出
了广泛的研究。注意到 Helmholtz方程在高波数时它的了一种新型的数学方法一一 Adomian分解法用以求解
解旱现高振荡的特性,传统方法(如无网格法、微分容高阶微分方程。与经典的数值方法如有限元方法相比
积解法阳、有限元法、有限差分8等)求解 Helmholtz Adomian修正分解法的主要优点在于不需要进行离散
方程时,计算精度会随着波数的增加而急遽降低。如果化从而不用修正误差,并且计算快速。近来,本文把
要保证计算精度,就需要増加单元网格的数量,导致计 Adomian分解法应用于求解振动力学问题2,并成功
基金项日:国家自然科学基金(No.51265037.No.11464031):江西省高等学校科技落地项目(No.KJLD12075);江西省教育厅科技
项目(No.G13524)。
作者简介:毛崎波(1975-),男博上,教授、研究领域为结构振动与噪声控制。E-mal:qiao@nchu.ecdu.cn
收稿日期:2014-05-07修回日期:2014-07-25文章编号:1002-8331(2015)02-0026-04
CNKI?络优先出版:2014-08-25,htp:/w.cnki. net/kcms/doi/10.3778/.5.1002-831.1405-0047.htm