کاربرد تابع پایه-شعاعی چندربعی در حل مسائل تراوش با الگوریتمی جدید برای بهینه‌سازی پارامتر شکل

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه مهندسی عمران، دانشگاه قم

2 گروه مهدسی عمزان، دانشگاه قم

3 عضو هیات علمی دانشگاه قم

چکیده

دقت روش بدون شبکه چندربعی کاملا به انتخاب پارامتر شکل آن وابسته است. هدف از پژوهش حاضر، پیشنهاد یک الگوریتم نوین برای تعیین پارامتر شکل بهینه است، به‌طوری‌که برخی از مشکلات پیشین شامل؛ وابسته بودن به تعداد نقاط محاسباتی و یک حل دقیق از مسئله، هزینه بالا و دقت پایین محاسبات، تجربی بودن، همگرا شدن روش‌های بهینه‌سازی کلاسیک به نقاط بهینه محلی و دیگر موارد را برطرف نماید. به این منظور ضمن معرفی یک تابع هدف جدید، از الگوریتم ژنتیک استفاده شد و برای سرعت بخشیدن به روند حل آن، حد پایین پارامتر شکل؛ کمینه شعاع نقاط محاسباتی با شرط عدم تکینگی ماتریس ضرایب و حد بالای آن؛ بیشینه شعاع پیشنهاد گردید. الگوریتم مذکور از چهار مرحله تشکیل می‌شود: 1 )تولید پارامتر شکل توسط الگوریتم ژنتیک در بازه پیشنهادشده، 2 )تشکیل تابع چندربعی با تعداد پایینی از نقاط محاسباتی، 3 )تشکیل تابع چندربعی با تعداد بالایی از نقاط محاسباتی و 4 )کمینه‌سازی اختلاف جواب توابع به‌دست آمده از دو مرحله قبل. در الگوریتم فرا ابتکاری حاضر، توزیع‌های یکنواخت و غیریکنواخت منظم از نقاط محاسباتی با موفقیت به‌کار رفت و نشان داده شد که با آن می‌توان به پارامتر شکلی بهینه و ثابت و نیز مستقل از تعداد نقاط محاسباتی برای هندسه‌های دلخواه دست یافت. برای صحت‌سنجی نیز مسائلی همگن، ناهمگن و ناهمسان از پدیده تراوش حل شد و از فن تجزیه دامنه برای کاهش حجم محاسبات و شبیه‌سازی راحت‌تر میدان‌های ناهمگن و پیچیده استفاده گردید. مقایسه نتایج با سایر حل‌های دقیق و عددی، توانایی و دقت بالای الگوریتم پیشنهادی را نشان می‌دهد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Application of MQ-RBF method for solving seepage problems with a new algorithm for optimization of the shape parameter

نویسندگان [English]

  • Masoumeh Koushki 1
  • Reza Babaee 2
  • Ehsan Jabbari 3
1 University of Qom
2 University of Qom
3 University of Qom
چکیده [English]

The accuracy of the meshless method, Multiquadric, depends completely on the choice of its optimal shape parameter. The purpose of this research is proposing a new algorithm for determining the optimal shape parameter. It resolves some of the previous difficulties, such as depending on the number of computational nodes or an exact solution of the problem, high cost and low accuracy of calculations, being experimental, convergence of classical optimization methods to local optimal points and so on. For this purpose, in addition to introducing a new objective function, Genetic Algorithm(GA) was used and for speeding up the process of its solution, lower bound and upper bound of the shape parameter are suggested as minimum (when the coefficient matrix is not singular) and maximum radius of computational nodes, respectively. The algorithm consists of four steps: 1) producing initial shape parameters by GA in the proposed range, 2) introducing the MQ function with a few numbers of computational points, 3) introducing the MQ function with a large number of computational points, and 4) minimizing the difference between solutions of two functions obtained from the two preceding steps. In the meta-heuristic algorithm, uniform and non-uniform regular distributions of computational nodes have been successfully applied and it was shown that with this approach, an optimal constant shape parameter independent of the number of computational points could be obtained for arbitrary geometries. For verification, examples of homogeneous, inhomogeneous and anisotropic types of the seepage phenomena were solved so that domain decomposition technique was used for inhomogeneous problems and complex geometries. A comparison of results with other exact and numerical solutions showed the high ability and accuracy of the proposed algorithm.

کلیدواژه‌ها [English]

  • Shape parameter
  • Multiquadric Method
  • Radial basis function
  • genetic algorithm
  • Seepage
[1] R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, Journal of geophysical research,79(8)(1971)1905-1915.
[2] R.L. Hardy, Theory and applications of the multiquadricbiharmonic method 20 years of discovery 1968–1988, Computers & Mathematics with Applications, 19(8-9) (1990) 163-208.
[3] Y.-C. Hon, K.F. Cheung, X.-Z. Mao, E.J. Kansa, Multiquadric solution for shallow water equations, Journal of Hydraulic Engineering, 125(5) (1999) 524-533.
[4] S. Wong, Y. Hon, T. Li, A meshless multilayer model for a coastal system by radial basis functions, Computers & Mathematics with Applications, 43(3-5) (2002) 585-605.
[5] R. Hardy, Hardy's multiquadric-biharmonic method for gravity field predictions II, Computers & Mathematics with Applications, 41(7-8) (2001) 1043-1048.
[6] D. Young, S. Jane, C. Lin, C. Chiu, K. Chen, Solutions of 2D and 3D Stokes laws using multiquadrics method, Engineering analysis with boundary elements, 28(10) (2004) 1233-1243.
[7] F. Gao, C. Chi, Numerical solution of nonlinear Burgers’ equation using high accuracy multi-quadric quasiinterpolation, Applied Mathematics and Computation, 229 (2014) 414-421.
[8] W. Bao, Y. Song, Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations, Numerical Methods for Partial Differential Equations, 30(1) (2014) 95-119.
[9] Z. Wu, S. Zhang, Conservative multiquadric quasiinterpolation method for Hamiltonian wave equations, Engineering Analysis with Boundary Elements, 37(7-8) (2013) 1052-1058.
[10] S. Patel, A. Rastogi, Meshfree multiquadric solution for real field large heterogeneous aquifer system, Water Resources Management, 31(9) (2017) 2869-2884.
[11] N. Li, H. Su, D. Gui, X. Feng, Multiquadric RBFFD method for the convection-dominated diffusion problems base on Shishkin nodes, International Journal of Heat and Mass Transfer, 118 (2018) 734-745.
[12] 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.
[13] E. Kansa, R. Carlson, Improved accuracy of multiquadric interpolation using variable shape parameters, Computers & Mathematics with Applications, 24(12) (1992) 99-120.
[14] S.A. Sarra, Integrated multiquadric radial basis function approximation methods, Computers & Mathematics with Applications, 51(8) (2006) 1283-1296.
[15] C.-S. Huang, C.-F. Lee, A.-D. Cheng, Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method, Engineering Analysis with Boundary Elements, 31(7) (2007) 614-623.
[16] A.-D. Cheng, Multiquadric and its shape parameter—a numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation, Engineering analysis with boundary elements, 36(2) (2012) 220-239.
[17] C.-S. Huang, H.-D. Yen, A.-D. Cheng, On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs, Engineering analysis with boundary elements, 34(9) (2010) 802-809.
[18] S. Xiang, K.-m. Wang, Y.-t. Ai, Y.-d. Sha, H. Shi, Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation, Applied Mathematical Modelling, 36(5) (2012) 1931-1938.
[19] M. Esmaeilbeigi, M. Hosseini, A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa’s method, Applied Mathematics and Computation, 249 (2014) 419-428.
[20] S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Advances in Computational Mathematics, 11(2-3) (1999) 193-210.
[21] M. Uddin, On the selection of a good value of shape parameter in solving time-dependent partial differential equations using RBF approximation method, Applied Mathematical Modelling, 38(1) (2014) 135-144.
[22] J. Biazar, M. Hosami, Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions, Advances in Numerical Analysis, (2016).
[23] 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.
[24] H.R. Azarboni, M. Keyanpour, M. Yaghouti, Leave-TwoOut Cross Validation to optimal shape parameter in radial basis functions, Engineering Analysis with Boundary Elements, 100 (2019) 204-210.
[25] 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.
[26] E. Kansa, A strictly conservative spatial approximation scheme for the governing engineering and physics equations over irregular regions and inhomogeneously scattered nodes, Computers & Mathematics with Applications, 24(5-6) (1992) 169-190.
[27] E. Jabbari, R. Ghiassi, Three-dimensional steady state seepage, a finite volume approach, WIT Transactions on Ecology and the Environment, 52 (2002).
[28] P.-O. Persson, G. Strang, A simple mesh generator in MATLAB, SIAM review, 46(2) (2004) 329-345.
[29] D.P. Hardin, T. Michaels, E.B. Saff, A Comparison of Popular Point Configurations on $mathbb {S}^ 2$, arXiv preprint arXiv:1607.04590,  (2016).
[30] M. Caliari, S. De Marchi, M. Vianello, Padua2D: Lagrange interpolation at Padua points on bivariate domains, ACM Trans. Math. Software,  (2008) 35-33.