求解声辐射特性的Burton-Miller改进型积分方程
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摘要
提出了一种求解结构声辐射问题的Burton-Miller改进型边界积分方程,利用拉普拉斯方程的特性对传统边界积分方程及其法向偏导方程进行处理,转化其中与频率相关的高阶奇异积分项和柯西型积分项分别为弱奇异积分项和不含奇异性的积分项;进一步联立求解结构内外拉普拉斯问题下的边界积分方程,将与频率无关的高阶奇异积分项和柯西型积分项转化为弱奇异积分乘积的形式,以保证计算的精度.以脉动球源和横向振动球源为例,将所得结果与传统边界积分方程相比较,表明该方法不仅可以保证全波数范围内解的唯一性,且具有很高的计算精度.
A modified Burton-Miller improved boundary integral equation for structural acoustic radiation problem was proposed.The traditional boundary integral equations and its normal derivation were dealt with the properties of the Laplace equation,and the high-order singular integration and Cauchy-type integration in the equations associated with frequency were respectively transformed into weak singular integration and non-singular integration.Furthermore,by solving the boundary integral equation in the form of Laplace problem inside and outside the structure,the frequency-independent high-order singular integral terms and the Cauchy integral terms can be transformed into the product of weak-singular integrals to ensure the accuracy.Taking the pulsating sphere source and the lateral vibration sphere as examples,the results show that the method can not only guarantee the uniqueness of solutions in the range of full wave number,but also has high computational accuracy compared with the traditional boundary integral equation.
A modified Burton-Miller improved boundary integral equation for structural acoustic radiation problem was proposed.The traditional boundary integral equations and its normal derivation were dealt with the properties of the Laplace equation,and the high-order singular integration and Cauchy-type integration in the equations associated with frequency were respectively transformed into weak singular integration and non-singular integration.Furthermore,by solving the boundary integral equation in the form of Laplace problem inside and outside the structure,the frequency-independent high-order singular integral terms and the Cauchy integral terms can be transformed into the product of weak-singular integrals to ensure the accuracy.Taking the pulsating sphere source and the lateral vibration sphere as examples,the results show that the method can not only guarantee the uniqueness of solutions in the range of full wave number,but also has high computational accuracy compared with the traditional boundary integral equation.
引文
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[2]Burton A J,Miller G F.The application of the integral equation method to the numerical solution of some exterior boundary value problems[J].Proc R Soc London:Ser A,1971,323(1553):201-210.
[3]Marburg S.The Burton and Miller method:unlocking another mystery of its coupling parameter[J].J Comp Acoustic,2015,23(7):1550016.
[4]Mathews I C.Numerical techniques for three dimensional steady-state fluid-structure interaction[J].J Acoust Soc Am,1986,79(1):1317-1325.
[5]Yang S A.A numerical method for scattering from acoustically soft and hard thin bodies in two dimensions[J].J Sound Vib,2002,250(2):773-793.
[6]Harris P J,Chen K.On efficient preconditioners for iterative solution of a Galerkin boundary element equations for the three-dimensional exterior Helmholtz problem[J].Journal of Computational and Applied Mathematics,2003,156(3-4):303-318.
[7]Gray L J,Glaeser J M,Kaplan T.Direct evaluation of hypersingular Galerkin surface integrals[J].SIAM Journal on Scientific Computing,2004,25(2):1534-1556.
[8]Matsumoto T,Zheng C,Harada S,et al.Explicit evaluation of hypersingular boundary integral equation for 3-d Helmholtz equation discretized with constant triangular element[J].Engineering Analysis with Boundary Elements,2011,35(11):1225-1235.
[9]Wu H J,Liu Y J,Jiang W K.A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems[J].Engineering Analysis with Boundary Elements,2013,37(2):309-318.
[10]王雪仁,季振林.快速多极边界元法应用于预测消声器的声学性能[J].中国科学技术大学学报,2008,37(2):207-214.
[11]吴海军,蒋伟康.二维声学多层快速多极子边界元及其应用[J].声学学报,2012,37(1):55-61.
[2]Burton A J,Miller G F.The application of the integral equation method to the numerical solution of some exterior boundary value problems[J].Proc R Soc London:Ser A,1971,323(1553):201-210.
[3]Marburg S.The Burton and Miller method:unlocking another mystery of its coupling parameter[J].J Comp Acoustic,2015,23(7):1550016.
[4]Mathews I C.Numerical techniques for three dimensional steady-state fluid-structure interaction[J].J Acoust Soc Am,1986,79(1):1317-1325.
[5]Yang S A.A numerical method for scattering from acoustically soft and hard thin bodies in two dimensions[J].J Sound Vib,2002,250(2):773-793.
[6]Harris P J,Chen K.On efficient preconditioners for iterative solution of a Galerkin boundary element equations for the three-dimensional exterior Helmholtz problem[J].Journal of Computational and Applied Mathematics,2003,156(3-4):303-318.
[7]Gray L J,Glaeser J M,Kaplan T.Direct evaluation of hypersingular Galerkin surface integrals[J].SIAM Journal on Scientific Computing,2004,25(2):1534-1556.
[8]Matsumoto T,Zheng C,Harada S,et al.Explicit evaluation of hypersingular boundary integral equation for 3-d Helmholtz equation discretized with constant triangular element[J].Engineering Analysis with Boundary Elements,2011,35(11):1225-1235.
[9]Wu H J,Liu Y J,Jiang W K.A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems[J].Engineering Analysis with Boundary Elements,2013,37(2):309-318.
[10]王雪仁,季振林.快速多极边界元法应用于预测消声器的声学性能[J].中国科学技术大学学报,2008,37(2):207-214.
[11]吴海军,蒋伟康.二维声学多层快速多极子边界元及其应用[J].声学学报,2012,37(1):55-61.