A Time Domain Dual Reciprocity Precise Integration Method for Scalar Wave Propagation Problems

doi:10.19596/j.cnki.1001-246x.8003

Abstract

Abstract: To avoid computation of multi-frequencies pressure in solution of frequency method, a dual reciprocity method (DRM) is applied to convert domain integral, which is related to derivatives of pressure over time, in boundary integral equation into boundary integral. Firstly, field points are collocated at all boundary nodes. With boundary condition, boundary quantities at boundary nodes can be represented linearly by quantities at domain nodes, which are applied for radial basis function interpolation and are arbitrarily distributed in DRM. Secondly, field points are further collocated at domain nodes. A system of ordinary differential equations (ODEs) of the second order is obtained. Thirdly, variation rate of pressure was introduced as an unknown quantity to reduce order of resulted ODEs. Finally, a precise integration method is adopted to solved the first ordered ODEs. Numerical examples demonstrated validity and stability of the method.

Key words: scalar wave propagation, boundary integral equation, dual reciprocity method, precise integration method

CLC Number:

P631.4

ZHOU Fenglin, WANG Weijia, LIAO Haiyang, LI Guang. A Time Domain Dual Reciprocity Precise Integration Method for Scalar Wave Propagation Problems[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(1): 26-36.

References

[1] GIBSON W C. The method of moments in electromagnetics[M]. New York:Chapman & Hall/CRC Press, 2007.
[2] LIU Y J. Fast multipole boundary element method:Theory and applications in engineering[M]. Cambridge:Cambridge University Press, 2011.
[3] WU H, LIU Y, JIANG W. 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.
[4] WU H, YE W, JIANG W. A collocation BEM for 3D acoustic problems based on a non-singular Burton-Miller formulation with linear continuous elements[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 332:191-216.
[5] COOX L, ATAK O, VANDEPITTE D, DESMET W. An isogeometric indirect boundary element method for solving acoustic problems in open-boundary domains[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 316:186-208.
[6] 王现辉, 乔慧, 张小明, 等. 基于等几何分析的边界元法求解Helmholtz问题[J]. 计算物理, 2017, 34(1):61-66.
[7] 王现辉, 郑兴帅, 乔慧,等. 二维Helmholtz边界超奇异积分方程解析研究[J]. 计算物理, 2017, 34(6):666-672.
[8] ERGIN A A, SHANKER B, MICHIELSSEN E. Fast analysis of transient acoustic wave scattering from rigid bodies using the multilevel plane wave time domain algorithm[J]. The Journal of the Acoustical Society of America, 2000, 107(3):1168-1178.
[9] CHEN W, LI J, FU Z. Singular boundary method using time-dependent fundamental solution for scalar wave equations[J]. Computational Mechanics, 2016, 58(5):717-730.
[10] 王道平, 张量, 沈晶, 等. 基于紧致差分格式的高效时域有限差分算法[J]. 计算物理, 2014, 31(1):91-95.
[11] WU H, YE W, JIANG W. Isogeometric finite element analysis of interior acoustic problems[J]. Applied Acoustics, 2015, 100:63-73.
[12] 付梅艳, 陈再高, 王玥, 等. 基于三角形单元的时域有限积分方法研究[J]. 微波学报, 2010, (s1):83-87.
[13] GIMPERLEIN H, STARK D. On a preconditioner for time domain boundary element methods[J]. Engineering Analysis with Boundary Elements, 2018, 96(11):109-114.
[14] ABREU A I, CARRER J A M, MANSUR W J. A step by step scheme in BEM formulation for scalar wave equation[J]. Engineering Analysis with Boundary Elements, 2003, 27(2):101-105.
[15] CARRER J A M, MANSUR W J. Time-domain BEM analysis for the 2D scalar wave equation:initial conditions contributions to space and time derivatives[J]. International Journal of Numerical Methods in Engineering. 1996, 39:2169-2188.
[16] MERAL G, TEZER-Sezgin M. DRBEM solution of exterior nonlinear wave problem using FDM and LSM time integrations[J]. Engineering Analysis with Boundary Elements, 2010, 34:547-580.
[17] ZHOU F L, ZHANG J M, SHENG X M, LI G Y. A dual reciprocity boundary face method for 3D non-homogeneous elasticity problems[J]. Engineering Analysis with Boundary Elements, 2012, 36(9):1301-1310.
[18] TANAKA M, MATSUMOTO T, TAKAKUWA S. Dual reciprocity BEM for time-stepping approach to the transient heat conduction problem in nonlinear materials[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195(37):4953-4961.
[19] 高强, 谭述君, 钟万勰. 精细积分方法综述[J]. 中国科学:技术科学, 2016, 46(12):1207-1218.
[20] 钟万勰. 结构动力方程的精细时程积分法[J]. 大连理工大学学报, 1994, 34:131-136.
[21] 钟万勰. 暂态历程的精细计算方法[J]. 计算结构力学及其应用, 1995, 12:1-6.
[22] ZHONG W X, ZHU J P, ZHONG X X. On a new time integration method for solving time dependent partial differential equations[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 130:163-178.
[23] FASSHAUER G E, ZHANG J G. On choosing "optimal" shape parameters for RBF approximation[J]. Numerical Algorithms, 2007, 45:345-368.
[24] 谭述君. 精细积分方法的改进及其在动力学与控制中的应用[D]. 大连:大连理工大学,2009.