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Dynamics-based analytical solutions to singular integrals for elastodynamics by time domain boundary element method. (English) Zbl 1480.74157

Summary: The singularities in 2-D time domain boundary element (TD-BEM) formulation for elastodynamics are divided into three categories: the wave front singularity, the space singularity and the dual singularity. A fully analytical procedure for dealing with the three singularities is proposed by adopting the concept of the finite part of an integral (Hadamard principle integral). In order to reduce the computation time, the conventional numerical procedure is adopted for the non-singular integrals in 2-D TD-BEM formulation. Therefore, the algorithm including the fully analytical procedure for dealing with singular integrals and the numerical procedure for dealing with non-singular integrals is implemented in this study. Two examples, 1-D rod and 2-D cavity representing the problems for the finite domain and the infinite domain respectively, are chosen to verify the effectiveness of the proposed algorithm. It shows that the results obtained from the proposed algorithm agree with the analytical solutions with good accuracy, indicating that the proposed algorithm is applicable for elastodynamics in both finite and infinite domains.

MSC:

74J10 Bulk waves in solid mechanics
65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
45E05 Integral equations with kernels of Cauchy type
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[1] Carrer, J. A.M.; Telles, J. C.F., A boundary element formulation to solve transient dynamic elastoplastic problems, Comput. Struct, 45, 707-713 (1992) · Zbl 0772.73089
[2] Polyzos, D.; Tsepoura, K. G.; Tsinopoulos, S. V., A boundary element method for solving 2-D and 3-D static gradient elastic problems: part I: integral formulation, Comput. Method Appl. Mech. Eng., 192, 2845-2873 (2003) · Zbl 1054.74740
[3] Tsepoura, K. G.; Tsinopoulos, S. V.; Polyzos, D., A boundary element method for solving 2-D and 3-D static gradient elastic problems: part II: numerical implementation, Comput. Method Appl. Mech. Eng., 192, 2875-2907 (2003) · Zbl 1054.74742
[4] Abreu, A. I.; Carrer, J. A.M.; Mansur, W. J., Scalar wave propagation in 2D: a BEM formulation based on the operational quadrature method, Eng. Anal. Bound. Elem., 27, 101-105 (2003) · Zbl 1018.74534
[5] Gaul, L.; Schanz, M., A comparative study of three boundary element approaches to calculate the transient response of viscoelastic solids with unbounded domains, Comput. Method Appl. Mech. Eng., 179, 111-123 (1999) · Zbl 0974.74074
[6] Lubich, C., Convolution quadrature and discretized operational calculus I, Numer. Math, 52, 129-145 (1988) · Zbl 0637.65016
[7] Lubich, C., Convolution quadrature and discretized operational calculus II, Numer. Math, 52, 413-425 (1988) · Zbl 0643.65094
[8] Schanz, M.; Antes, H., Application of ‘operational quadrature methods’ in time domain boundary element methods, Meccanica, 32, 179-186 (1997) · Zbl 0913.73075
[9] Kattis, S. E.; Polyzos, D.; Beskos, D., Vibration isolation by a row of piles using a 3-D frequency domain BEM, Int. J. Numer. Method Eng., 46, 713-728 (1999) · Zbl 1073.74549
[10] de Lacerda, L. A.; Wrobel, L. C.; Mansur, W. J., A boundary integral formulation for two-dimensional acoustic radiation in a subsonic uniform flow, J. Acoust. Soc. Am, 100, 98-107 (1996)
[11] Manolis, G. D., A comparative study on three boundary element method approaches to problems in elastodynamics, Int. J. Numer. Method Eng., 19, 73-91 (1983) · Zbl 0497.73085
[12] Carrer, J. A.M.; Pereira, W. L.A.; Mansur, W. J., Two-dimensional elastodynamics by the time-domain boundary element method: Lagrange interpolation strategy in time integration, Eng. Anal. Bound. Elem., 36, 1164-1172 (2012) · Zbl 1351.74098
[13] Dominguez, J., Boundary Elements in Dynamics (1993), Computational Mechanics Publications and Elsevier Applied Science: Computational Mechanics Publications and Elsevier Applied Science Southampton and London · Zbl 0790.73003
[14] Dominguez, J.; Gallego, R., Time domain boundary element method for dynamic stress intensity factor computations, Int. J. Numer. Method Eng., 33, 635-647 (1992) · Zbl 0825.73906
[15] Manolis, G. D.; Beskos, D., Boundary Element Methods in Elastodynamics (1988), Unwin Hyman: Unwin Hyman London
[16] Mansur, W. J., A time-stepping technique to solve wave propagation problems using the boundary element method (1983), University of Southampton: University of Southampton Southampton, Ph.D. thesis
[17] Telles, J. C.F.; Carrer, J. A.M.; Mansur, W. J., Transient dynamic elastoplastic analysis by the time-domain BEM formulation, Eng. Anal. Bound. Elem., 23, 479-486 (1999) · Zbl 0957.74074
[18] Muñoz-Reja, M. M.; Buroni, F. C.; Sáez, A., 3D explicit-BEM fracture analysis for materials with anisotropic multifield coupling, Appl. Math. Model, 40, 2897-2912 (2016) · Zbl 1452.74011
[19] Khambampati, A. K.; Kim, K. Y.; Lee, Y. G., Boundary element method to estimate the time-varying interfacial boundary in horizontal immiscible liquids flow using electrical resistance tomography, Appl. Math. Model, 40, 1052-1068 (2016) · Zbl 1446.76033
[20] Useche, J.; Harnish, C., A boundary element method formulation for modal analysis of doubly curved thick shallow shells, Appl. Math. Model, 40, 3591-3600 (2016) · Zbl 1459.74121
[21] Manolis, G. D., A comparative study on three boundary element method approaches to problems in elastodynamics, Int. J. Numer. Method Eng., 19, 73-91 (1983) · Zbl 0497.73085
[22] Niwa, Y.; Fukui, T.; Kato, S., An application of the integral equation method to two-dimensional elastodynamics, Appl. Mech. Lett., 28, 281-290 (1980)
[23] Mansur, W. J.; Brebbia, C. A., Numerical implementation of the boundary element method for two dimensional transient scalar wave propagation problems, Appl. Math. Model, 6, 299-306 (1982) · Zbl 0488.65057
[24] Mansur, W. J.; Brebbia, C. A., Formulation of the boundary element method for transient problems governed by the scalar wave equation, Appl. Math. Model, 6, 307-311 (1982) · Zbl 0488.65058
[25] Carrer, J. A.M.; Mansur, W. J., Stress and velocity in 2D transient elastodynamic analysis by the boundary element method, Eng. Anal. Bound. Elem., 23, 233-245 (1999) · Zbl 0955.74072
[26] Hadamard, J., Lectures On Cauchy’s Problem in Linear Partial Differential Equations (2003), Courier Dover Publications: Courier Dover Publications Dover · JFM 49.0725.04
[27] Ren, Y. T., Boundary element methods for wave propagation problems in isotropic or anisotropic media and its application in engineering (1995), Tsinghua University: Tsinghua University China, Ph.D. thesis in Chinese
[28] Lei, W. D.; Ji, D. F.; Li, H. J., On an analytical method to solve singular integrals both in space and time for 2-D elastodynamics by TD-BEM, Appl. Math. Model, 39, 6307-6318 (2015) · Zbl 1443.74073
[29] Guiggiani, M.; Gigante, A., A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, J. Appl. Mech., 57, 906-915 (1990) · Zbl 0735.73084
[30] Teng, B.; Gou, Y.; Ning, D. Z., A higher order BEM for wave-current action on structures—direct computation of free-term coefficient and CPV integrals, Chin. Ocean. Eng., 20, 395-410 (2006)
[31] Eringen, A. C.; E. S, Suhubi, Elastodynamics, Linear Theory, 2 (1975), Academic Press: Academic Press New York
[32] Chou, P. C.; Koenig, H. A., A unified approach to cylindrical and spherical elastic waves by method of characteristics, J. Appl. Mech., 33, 159-167 (1996)
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