کاربرد روش توابع پایه-شعاعی چندربعی برای حل معادله هلمهولتز به‌منظور آنالیز امواج لرزه‌ای در مخازن سدهای صلب

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه مهدسی عمزان، دانشگاه قم

2 عضو هیات علمی دانشگاه قم

3 استاد دانشکده فنی دانشگاه تهران، گروه عمران

چکیده

هزینه بالای ساخت شبکه، نیاز به حل اساسی وابسته به شرایط مسئله، تکینگی، شبیه‌سازی کل میدان، نیاز به شبکه منظم از خطوط متقاطع و ... از برجسته‌ترین نقاط ضعف روش‌های عددی باشبکه پرکاربرد در حل مسائل مکانیک محیط‌های پیوسته می‌باشد. در این پژوهش، با هدف رفع برخی از این نواقص، روش بدون شبکه پایه-شعاعی چندربعی برای آنالیز دوبعدی امواج لرزهای در مخازن سدهای صلب توسعه داده شد. به این منظور، معادله هلمهولتز و شرایط مرزی مختلط حاکم بر مسئله با استفاده از تابع چندربعی در حوزه فرکانس بازتولید و روند حل آن ارائه گردید. نتایج نشان داد که استفاده از فرم اصلی و مختلط این تابع به ترتیب برای فرکانس‌های کمتر و بیشتر از فرکانس طبیعی مخزن، زمان محاسبات را بهینه می‌کند. همچنین برای تعیین مهمترین عامل در دقت و همگرایی روش مذکور یعنی پارامتر شکل بهینه، ابتدا ناکارآمدی برخی از روش‌های پرکاربرد پیشین به اثبات رسید سپس یک الگوریتم جدید و پرسرعت معرفی گردید. نتایج این پژوهش نشان داد که پارامتر شکل بهینه برحسب فرکانس‌های مختلف بارگذاری قابل فرمول‌بندی است. چنین ویژگی کاربرد روش چندربعی در این مسئله خاص را نسبت به سایر روش‌ها آسان‌تر و هزینه‌های محاسباتی آن را کمتر می‌کند. دقت بالای روش حاضر طی دو مثال مختلف به ترتیب با و بدون در نظر گرفتن اثر جذب رسوبات کف مخزن در مقایسه با حل‌های دقیق نشان داده شد که خطای ناچیز آن به دلیل تعریف یک تابع تخمین پیوسته دقیق در کل دامنه مسئله و نیز استفاده از یک الگوریتم کارا برای پیدا کردن پارامتر شکل بهینه می‌باشد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Application of Multiquadric Radial Basis Function method for Helmholtz equation in seismic wave analysis for reservoir of rigid dams

نویسندگان [English]

  • Reza Babaee 1
  • Ehsan Jabbari 2
  • Morteza Eskandari-Ghadi 3
1 University of Qom
2 University of Qom
3 University of Tehran
چکیده [English]

 The high costs of mesh generation in mesh-dependent solution, weakness in capturing 
singularities, the need of modeling all over the domain, the need of problem dependent fundamental 
solutions, etc. are some of weaknesses in the common numerical mesh-dependent methods for solving 
continuum mechanics boundary value problems. In this study, aiming for eliminating some of these 
shortcomings, one of the well-known Radial Basis Functions (RBF) methods, Multiquadric (MQ), is 
developed for dynamic analysis of 2D reservoirs of rigid dams in frequency-domain. To this end, the 
Helmholtz equation and the governing complex boundary conditions are reproduced using MQ function 
in the frequency domain. The results show that with the use of real and complex forms of the MQ 
function, the computational time will be respectively optimized for frequencies smaller and larger than 
the natural frequency of the reservoir. Also, to determine the most important factors affecting both the 
accuracy and convergence of MQ method, first the inefficiency of some of the previously introduced 
methods is proved, and then a new high-speed algorithm is presented. It is shown that the optimal 
shape parameter for MQ method can be formulated in terms of the frequencies of seismic records. 
This advantage simplifies the application of MQ method in this particular problem and reduces the 
computational time, considerably. The high accuracy of the present method is shown in two different 
examples, where the effects of sediment absorption may either be considered or not. The high accuracy 
compared to the exact solutions achieved in this paper is due to a continuous estimation function defined 
all over the domain and also due to the simple algorithm used for finding the optimal shape parameter.

کلیدواژه‌ها [English]

  • Multiquadric Radial Basis Function (MQ-RBF)
  • Concrete gravity dam
  • Shape parameter
  • Frequency domain
  • Hydrodynamic pressure
[1] H.M. Westergaard, Water pressures on dams during earthquakes, Trans. ASCE, 95 (1933) 418-433.
[2] A.K. Chopra, Earthquake behavior of reservoirdam systems, Journal of the Engineering Mechanics Division, 94(6) (1968) 1475-1500.
[3] J. Humar, M. Roufaiel, Finite element analysis of reservoir vibration, Journal of Engineering Mechanics, 109(1) (1983) 215-230.
[4] S.K. Sharan, Finite element modeling of infinite reservoirs, Journal of engineering mechanics, 111(12) (1985) 1457-1469.
[5] B.-F. Chen, Nonlinear hydrodynamic effects on concrete dam, Engineering structures, 18(3) (1996) 201-212.
[6] B.-F. Chen, Y.-S. Yuan, Nonlinear hydrodynamic pressures on rigid arch dams during earthquakes, reviewing, J. Hydraulic Engineering,  (1999).
[7] M. Millan, Y. Young, J. Prevost, The effects of reservoir geometry on the seismic response of gravity dams, Earthquake engineering & structural dynamics, 36(11) (2007) 1441-1459.
[8] N. Bouaanani, P. Paultre, A new boundary condition for energy radiation in covered reservoirs using BEM, Engineering analysis with boundary elements, 29(9) (2005) 903-911.
[9] S. Li, Diagonalization procedure for scaled boundary finite element method in modeling semi‐infinite reservoir with uniform cross‐section, International journal for numerical methods in engineering, 80(5) (2009) 596-608.
[10] G. Lin, Y. Wang, Z. Hu, An efficient approach for frequency‐domain and time‐domain hydrodynamic analysis of dam–reservoir systems, Earthquake Engineering & Structural Dynamics, 41(13) (2012) 1725-1749.
[11]R. Babaee, N. Khaji, M.T. Ahmadi, Development of decoupled equations methods for calculating hydrodynamic pressures on concrete gravity dams, Modares Civil Engineering journal, 15(4) (2014) 41-52.
[12] R. Babaee, Application of decoupled equations methods for solving dam-resorvior interaction problem, MSc thesis, Tarbiyat Moderes university, Tehran, Iran, 2013.
[13] E.J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & mathematics with applications, 19(8-9) (1990) 147-161.
[14] J. Li, A.H.-D. Cheng, C.-S. Chen, A comparison of efficiency and error convergence of multiquadric collocation method and finite element method, Engineering Analysis with Boundary Elements, 27(3) (2003) 251-257.
[15] S.A. Sarra, Integrated multiquadric radial basis function approximation methods, Computers & Mathematics with Applications, 51(8) (2006) 1283-1296.
[16] E.J. Kansa, J. Geiser, Numerical solution to timedependent 4D inviscid Burgers’ equations, Engineering Analysis with Boundary Elements, 37(3) (2013) 637-645.
[17] Z. Wu, S. Zhang, Conservative multiquadric quasiinterpolation method for Hamiltonian wave equations, Engineering Analysis with Boundary Elements, 37(78) (2013) 1052-1058.
[18] C. Bustamante, H. Power, Y. Sua, W. Florez, A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems, Applied Mathematical Modelling, 37(6) (2013) 4538-4547.
[19] G. Lin, Y. Wang, Z. Hu, An efficient approach for frequency‐domain and time‐domain hydrodynamic analysis of dam–reservoir systems, Earthquake Engineering & Structural Dynamics, 41(13) (2012) .9471-5271
[20] S. Patel, A. Rastogi, Meshfree multiquadric solution for real field large heterogeneous aquifer system, Water Resources Management, 31(9) (2017) 2869-2884.
[21] N. Li, H. Su, D. Gui, X. Feng, Multiquadric RBF-FD method for the convection-dominated diffusion problems base on Shishkin nodes, International Journal of Heat and Mass Transfer, 118 (2018) 734-745.
[22] R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, Journal of geophysical research, 76(8) (1971) 1905-1915.
[23] R. Franke, A critical comparison of some methods for interpolation of scattered data, NAVAL POSTGRADUATE SCHOOL MONTEREY CA, 1979.
[24] G.E. Fasshauer, Newton iteration with multiquadrics for the solution of nonlinear PDEs, Computers & Mathematics with Applications, 43(3-5) (2002) 423-438.
[25] E.J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates, Computers & Mathematics with applications, 19(8-9) (1990) 127-145.
[26] E.J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & mathematics with applications, 19(8-9) (1990) 147-161.
[27] S.A. Sarra, D. Sturgill, A random variable shape parameter strategy for radial basis function approximation methods, Engineering Analysis with Boundary Elements, 33(11) (2009) 1239-1245.
[28] A. Golbabai, H. Rabiei, Hybrid shape parameter strategy for the RBF approximation of vibrating systems, International Journal of Computer Mathematics, 89(17) (2012) 2410-2427.
[29] A. Golbabai, E. Mohebianfar, H. Rabiei, On the new variable shape parameter strategies for radial basis functions, Computational and Applied Mathematics, .407-196 )5102( )2(43
[30] J. Biazar, M. Hosami, Selection of an interval for variable shape parameter in approximation by radial basis functions, Advances in Numerical Analysis, 2016 (2016).
[31] W. Chen, Y. Hong, J. Lin, The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method, Computers & Mathematics with Applications, 75(8) (2018) 2942-2954.
[32] H.R. Azarboni, M. Keyanpour, M. Yaghouti, LeaveTwo-Out Cross Validation to optimal shape parameter in radial basis functions, Engineering Analysis with Boundary Elements, 100 (2019) 204-210.
[33] A. Fallah, E. Jabbari, R. Babaee, Development of the Kansa method for solving seepage problems using a new algorithm for the shape parameter optimization, Computers & Mathematics with Applications, 77(3) (2019) 815-829.
[34] M. Kooshki, R. Babaee, E. Jabbari, Aplication of the RBF multiquadric method for solving seepage problems using a new algorithm for optimization of the shape parameter, Amirkabir Civil Engineering Journal (accepted for publication) doi: 10.22060/ CEEJ.2019.15155.5840.
[35] E. Jabbari, A.R Fallah, Investigation of meshless methods in solving seepage equation, 12th Iranian Hydraulic Conference,Tehran, Iran, 2013.
[36] A.K. Chopra, Hydrodynamic pressures on dams during earthquakes, Journal of the Engineering Mechanics Division, 93(6) (1967) 205-224.
[37] N. Bouaanani, P. Paultre, J. Proulx, A closed-form formulation for earthquake-induced hydrodynamic pressure on gravity dams, Journal of Sound and Vibration, 261(3) (2003) 573-582.