نوع مقاله : مقاله پژوهشی
نویسندگان
1 دانشگاه تهران
2 دانشگاه تربیت مدرس
چکیده
روشهای عددی یکی از ابزارهای مورد استفاده برای تحلیل پاسخ لرزه‌ای است، که در این میان روش اجزای مرزی جایگاه ویژه‌ای دارد. در این پژوهش، مطالعه جامعی بر چگونگی تاثیر ابعاد المان و طول گام زمانی بر پایداری و دقت نتایج روش اجزای مرزی حوزه زمان صورت پذیرفته است. به این منظور دو پارامتر β و L/λ که به‌طور گسترده در ادبیات فنی برای ارزیابی پایداری روشهای عددی شناخته شده‌اند مورد استفاده قرار گرفته‌اند. سه محیط همگن، شبه‌همگن و ناهمگن با تحلیل مجموعا 280 مدل عددی مورد مطالعه قرار گرفته و مشخص شده است که پایداری و دقت نتایج در تحلیل محیط ناهمگن وابستگی اساسی به دو پارامتر مذکور داشته و نتایج قابل قبول تنها هنگامی حاصل می‌شود که موج در هر گام زمانی فاصله‌ای در حدود یک چهارم تا نصف طول المان را طی کرده (β=0.24-0.40) و حداقل یک و نیم گره به ازای طول موج کمینه تعریف شود (λ/L >1.0). همچنین مشخص شد در تحلیل محیط ناهمگن، ضریب β مربوط به محیطی که سرعت کمتری دارد، تعیین‌کننده پایداری و دقت نتایج تحلیل اجزای مرزی حوزه زمان خواهد بود.
کلیدواژه‌ها
عنوان مقاله [English]
Evaluation of the Stability of Time Domain Boundary Element Method in Seismic Analysis of Heterogeneous Environments
نویسندگان [English]
- Shahram Maghami 1
- Abdollah Sohrabi-Bidar 1
- Niloufar Babaadam 2
چکیده [English]
Numerical approaches are one of the best tools for seismic response analysis. In between, the Boundary Element Method (BEM) has attracted special attention. In this paper, a comprehensive study has been performed to characterize the dependence of stability and accuracy of the time domain BEM on the chosen time step duration and effective length of the elements. To this end, the two parameters β and λ/L, widely known and used in the literature for the investigation of numerical stability and accuracy, have been employed. Three different environments as homogeneous, pseudo-homogeneous and non- homogeneous have been analyzed through total number of 280 numerical models. It is found that the stability and accuracy of the used algorithm is considerably influenced by the mentioned parameters, in a way that stable and accurate results will be achieved merely when the wave travels one-fourth to less than half the element size during each time step (0.24<β<0.4) and also when at least one and a half node is defined per the shortest wave-length (λ/L>1.5). It also became clear that in the modeling of non-homogeneous environments, the β value for the environment with the lowest wave velocity specifies the range of acceptable results.
کلیدواژه‌ها [English]
- Stability
- Numerical Intermittent Instability
- Boundary Elements
- Time Domain
- Time Step Duration
- Element Size
- β Parameter
- Copley, L. G., “Integral Equation Method for Radiation from Vibrating Bodies,” Journal of the Acoustical Society of America, Vol. 41, No. 4p1, pp. 807, 1967.
- Kamalian, M., Gatmiri, B., Sohrabi-Bidar, A., and Khalaj, A., “Amplification Pattern of 2D Semi-Sine Shaped Valleys Subjected to Vertically Propagating Incident Waves”, Communications in Numerical Methods in Engineering, Vol. 23, pp. 871-887, 2007.
- Kamalian, M., Jafari, M. K., Sohrabi-Bidar, A., and Razmkhah, A., “Seismic Response of 2-D Semi-Sine Shaped Hills to Vertically Propagating Incident Waves: Amplification Patterns and Engineering Applications”, Earthquake Spectra, Vol. 24, No. 2, pp. 405-430, 2008.
- Sohrabi-Bidar, A., Kamalian, M., Jafari, M. K., “Seismic Response of 3D Gaussian Shaped Valleys to Vertically Propagating Incident Waves”, Geophysical Journal International, 183, pp. 1429-1442. 2010.
- Sohrabi-Bidar, A., and Kamalian, M., “Effects of Three-Dimensionality on Seismic Response of Gaussian-Shaped Hills for Simple Incident Pulses”, Soil Dynamics and Earthquake Engineering, Vol. 52, pp. 1-12, 2013.
- Peirce, A., and Siebrits, E. “Stability Analysis and Design of Time-Stepping Schemes for General Elastodynamic Boundary Element Models”, International Journal of Numerical. Methods in Engineering, Vol. 40, No. 2, pp. 319-342, 1997.
- Panji, M., and Mojtabazadeh-Hasanlouei, S., “Surface Motion of Alluvial Valleys Subjected to Obliquely Incident Plane SH-Wave Propagation”, Journal of Earthquake Engeneering, Vol. 26, No. 12, pp.1-26, 2021.
- Mojtabazadeh-Hasanlouei, S., Panji, M., and Kamalian, M., “On Subsurface Multiple Inclusions Model Under Transient SH-Wave Propagation, Waves in Random and Complex Media”, Waves in Random and Complex Media, Vol. 22, No. 4, pp.1937-1976, 2020.
- Nohegoo-Shahvari, A., Kamalina, M., and Panji, M., “A Hybrid Time-Domain Half-Plane FE/BE Approach for SH-Wave Scattering of Alluvial Sites”, Engineering Analysis with Boundary Elements, Vol. 105, pp. 194-206, 2019.
- Yu, G., Mansur, W. J., Carrer, J. A. M., and Gong, L., “A Linear θ Method Applied to 2D Time Domain BEM”, Communications in Numerical Methods in Engineering, Vol. 14, No. 12, pp. 1171–1179, 1998.
- Araújo, F. C., Mansur, W. J., and Nishikava, L. K., “Linear Θ Time-Marching Algorithm in 3D BEM Formulation for Elastodynamics,” Engineering Analysis with Boundary Elements, Vol. 23, No. 10, pp. 825-833, 1999.
- Kobayashi, S., “Fundamentals of Boundary Integral Equation Methods in Elastodynamics”, Topics in Boundary Element Research, pp. 1-54, 1985.
- Soares, D., and Mansur, W. J., “An Efficient Stabilized Boundary Element Formulation for 2D Time-Domain Acoustics and Elastodynamics”, Computational Mechanics, Vol. 40, No. 2, pp. 355-365, 2007.
- Manolis, G. D., and Dineva, P. S., “Elastic Waves in Continuous and Discontinuous Geological Media by Boundary Integral Equation Methods: A Review”, Soil Dynamics and Earthquake Engineering, Vol. 70, pp. 11-29, 2015.
- Marrero, M., and Dominguez, J., “Numerical Behavior of Time Domain BEM for Three-Dimensional Transient Elastodynamic Problems”, Engineering Analysis with Boundary Elements, Vol. 27, No. 1, pp. 39-48, 2003.
- Carrer, J. A. M., and Mansur, W. J., “Time-Dependent Fundamental Solution Generated by A Not Impulsive Source in the Boundary Element Method Analysis of the 2D Scalar Wave Equation”, Communications in Numerical Methods in Engineering, Vol. 18, No. 4, pp. 277-285, 2002.
- Carrer, J. A. M., and Mansur, W. J., “Time Discontinuous Linear Traction Approximation in Time-Domain BEM: 2-D Elastodynamics”, International Journal for Numerical Methods in Engineering, 2000.
- Dominguez, J., Boundary Elements in Dynamics, Wit Press, 1993.
- Kuhlemeyer, R. L., and Lysmer, J., “Finite Element Method Accuracy for Wave Propagation Problems”, Journal of the Soil Mechanics and Foundations Division, Vol. 99, No. 5, pp. 421-427, 1973.
- Park, D., and Hashash, Y. M. A., “Soil Damping Formulation in Nonlinear Time Domain Site Response Analysis”, Journal of Earthquake Engineering, Vol. 8, No. 2, pp. 249-274, 2004.
- Bao, H., Bielak, J., Ghattas, O., Kallivokas, L. F., O’Hallaron, D. R., Shewchuk, J. R., and Xu, J., “Earthquake Ground Motion Modeling on Parallel Computers”, Proceedings of ACM/IEEE Conference on Supercomputing, pp. 1-19, 1996.
- Komatitsch, D., and Vilotte, J. P., “The Spectral Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geological Structures”, Bulletin of the Seismological Society of America, Vol. 88, No. 2, pp. 386-392, 1998.
- Marburg, S., "Discretization Requirements: How Many Elements Per Wavelength Are Necessary?", Computational Acoustics of Noise Propagation in Fluids-Finite and Boundary Element Methods, Springer, Berlin, Heidelberg, 2008. 309-332.
- Marburg, S., “Six Boundary Elements Per Wavelength: Is That Enough?”, Journal of Computational Acoustics, Vol. 10, No. 01, pp. 25-51, 2002.
- Xu, J., Bielak, J., Ghattas, O., and Wang, J., “Three-Dimensional Seismic Ground Motion Modeling in Inelastic Basins”, Physics of the Earth and Planetary Interiors, Vol. 137, No. 1-4, pp. 81-95, 2003.
- Bouchon, M., and Sánchez-Sesma, F. J., “Boundary Integral Equations and Boundary Elements Methods in Elastodynamics”, Advances in Geophysics, Vol. 48, No. 06, pp. 157-189, 2007.
- Chaillat, S., Bonnet, M., and Semblat, J, F., “Multi-Level Fast Multipole Multi-Region Method for 3D Seismic Response of Alluvial Basins”, Eighth World Congress on Computational Mechanics, 2008.
- Dineva, P. S., Manolis, G. D., and Rangelov, T. V., “Site Effects Due to Wave Path Inhomogeneity by BEM”, Engineering Analysis with Boundary Elements, Vol. 32, No. 12, pp. 1025-1036, 2008.
- Kamalian, M., Jafari, M. K., Sohrabi-Bidar, A., Razmkhah, A. and Gatmiri, B., “Time-Domain Two-Dimensional Site Response Analysis of Non-Homogeneous Topographic Structures by a Hybrid FE / BE Method”, Soil Dynamic and Earthquake Engineering, Vol. 26, pp. 753-765, 2006.
- Kamalian, M. and Sohrabi-Bidar, A., “Dynamic analysis of non-homogeneous 2D topographic features in time domain using the boundary element method”, Esteghlal, Vol. 24, No. 2, pp. 51-68, 2006.
- Kamalian, M., Jafari, M.K. and Sohrabi-Bidar, A., “Seismic Behavior of 2D Semi-sin Hills Subjected Vertical IncidentWwave”, Esteghlal, Vol. 26, No.1, pp. 109-130, 2006.
- Kamalian, M., Jafari, M. K., Sohrabi-Bidar, A. and Razmkhah, A., “Amplification Pattern of Vertical Incident Waves by 2D Trapezoidal Hills”, Modares Technical and Engineering Journal, Vol. 29, pp. 11-30, 2007.
- Dravinski, M., and Mossessian, T. K., “Scattering of Plane Harmonic P, SV, and Rayleigh Waves By Dipping Layers of Arbitrary Shape”, Bulletin of the Seismological Society of America, Vol. 77, No. 1, pp. 212–235, 1987.
- Mossessian, T. K., and Dravinski, M., “Application of A Hybrid Method for Scattering of P, SV, and Rayleigh Waves By Near-Surface Irregularities”, Bulletin of the Seismological Society of America, Vol. 77, No. 5, pp. 1784-1803, 1987.
- Qom regional water company., “Report of Geoelectrical studies of Qom basin”, 2010
)