پایان نامه بکارگیری روش های عددی بدون شبکه در مدلسازی امواج غیرخطی سطح آب ناشی از باد

فهرست:

فصل اول: مقدمه

1-1- کلیات.. 2

1-2- معرفی تحقیق حاضر. 2

فصل دوم: مروری بر پژوهش های پیشین

2-1- مقدمه. 10

2-2- پیشینه ی تحقیقات انجام شده بر روی موج.. 11

2-2-1- مدل های اوّلیه ی امواج غیرخطی.. 11

2-2-2- مدل های جدید امواج غیرخطی.. 13

2-2-3- روش های عددی بدون شبکه در مدلسازی امواج غیرخطی.. 15

2-3- پیشینه ی تحقیقات انجام شده بر روی روش عددی مورد استفاده. 16

2-3-1- روش عددی دیفرانسل کوادرچر (DQ) 16

2-3-2- توابع پایه ی شعاعی (RBF) 20

2-3-2-1- انواع توابع پایه ی شعاعی.. 20

2-3-2-2- کاربرد توابع پایه ی شعاعی در درونیابی.. 21

2-3-2-3- کاربرد توابع پایه ی شعاعی در حل معادلات دیفرانسیل.. 22

2-3-2-4- روش عددی RBF-DQ.. 23

2-3-2-5- تابع شعاعی MQ.. 24

عنوان                                                                                                                      صفحه

2-3-3- عوامل موثر بر دقت و خطای مدل.. 25

2-3-3-1- چگالی گره ها 26

2-3-3-2- پارامتر شکل.. 26

2-3-3-2-1- تاثیر پارامتر شکل بر خطا 26

2-3-3-2-2- پارامتر شکل بهینه. 29

2-3-3-3- پدیده ی رانچ.. 32

2-3-3-4- دقت محاسبات، خطای گرد کردن و عدد وضعیت... 33

2-4- جمع بندی و نتیجه گیری.. 33

فصل سوم: تئوری تحقیق

3-1- مقدمه. 36

3-2- تئوری های موج.. 36

3-2-1- تئوری موج خطی.. 36

3-2-2- تئوری موج غیرخطی.. 39

3-2-2-1- دسته بندی تئوریهای اولیهی امواج غیرخطی.. 39

3-2-2-1-1- تئوری استوکس.... 39

3-2-2-1-2- تئوری Cnoidal 41

3-2-2-1-3- تئوری Boussinesq. 42

3-2-2- شبیه سازی عددی انتشار موج غیرخطی.. 42

3-2-2-1- هندسه ی مسئله و تعریف مخزن عددی.. 42

3-2-2-2- معادله ی حاکمه و شرایط مرزی.. 44

3-2-2-2-1- تئوری موج ساز. 44

3-2-2-2-2- تابع صعودی.. 46

3-2-2-3- روش مرکب اویلری و لاگرانژی (MEL) 48

عنوان                                                                                                                      صفحه

3-2-2-4- ناحیه ی استهلاک یا ساحل مصنوعی.. 49

3-2-2-5- بکارگیری روش RBF-DQ برای تخمین مشتقات مکانی.. 50

3-2-2-5-1- انتخاب تابع پایه. 50

3-2-2-5-2- تخمین مشتق های مکانی با روش RBF-DQ.. 51

3-2-2-5-3- روش RBF-DQ محلی.. 52

3-2-2-5-4- چگونگی اعمال شرایط مرزی.. 53

3-2-2-5-6- انتخاب پارامتر شکل مناسب... 53

3-2-2-6- انتگرال گیری بر روی زمان.. 54

3-2-2-7- تابع یکنواختکننده. 56

فصل چهارم: نتایج و بحث روی آزمایش های عددی

4-1- مقدمه. 58

4-2- مثال های عددی.. 59

4-2-1- مثال عددی اول: معادله ی برگرز. 59

4-2-1-1- بررسی عوامل موثر بر افزایش دقت روش... 60

4-2-1-1-1- بررسی تاثیر فاصله ی گرهها بر مدل.. 61

4-2-1-1-2- بررسی تاثیر پارامتر شکل بر مدل.. 61

4-2-1-1-3- بررسی تاثیر پارامتر شکل و فاصله ی گره ها بصورت همزمان.. 64

4-2-1-1-4- دقت محاسبات.. 65

4-2-1-1-5- پدیدهی رانچ.. 66

4-2-1-2- مقایسه ی روش های RBF-DQ و DQ.. 67

4-2-1-3- حل مسئله با استفاده از مقدار پارامتر شکل بهینه. 68

4-2-2- مثال عددی دوم: معادله ی هلمهلتز. 69

4-2-2-1- بررسی عوامل موثر بر افزایش دقت روش... 70

عنوان                                                                                                                      صفحه

4-2-2-1-1- بررسی تاثیر پارامتر شکل و تعداد گره ها بصورت همزمان.. 70

4-2-2-1-2- پدیدهی رانچ.. 71

4-2-2-2- حل مسئله با استفاده از مقدار پارامتر شکل بهینه. 72

4-3- شبیه سازی انتشار موج در مخزن عددی.. 73

4-3-1- انتشار موج خطی.. 73

4-3-1-1- بررسی تاثیر همزمان تعداد گره ها و پارامتر شکل.. 75

4-3-1-1-1- تاثیر پارامتر شکل و تعداد گره ها در راستای افقی.. 78

4-3-1-1-2- تاثیر پارامتر شکل و تعداد گرهها در راستای عمق.. 80

4-3-1-1-3- بررسی تاثیر همزمان تعداد گره ها در دامنه ی تاثیر
 و پارامتر شکل.. 83

4-3-1-2- حل مسئله با استفاده از پارامتر شکل مناسب و مقایسه ی
نتایج با نتایج روش تحلیلی.. 85

4-3-1-3- تاثیر طول ناحیهی استهلاک... 88

4-3-1-4- مقایسه ی نتایج با نتایج  روش عددی RBF. 88

4-3-2- شبیه سازی انتشار موج غیرخطی در مخزن عددی.. 89

4-3-2-1- بررسی تاثیر همزمان تعداد گرهها و پارامتر شکل.. 91

4-3-2-1-1- تاثیر پارامتر شکل و تعداد گرهها در راستای افقی.. 91

4-3-2-1-2- تاثیر پارامتر شکل و تعداد گره ها در راستای عمق.. 94

4-3-2-1-3- بررسی تاثیر همزمان تعداد گره ها در دامنه ی
تاثیر و پارامتر شکل.. 96

4-3-2-2- حل مسئله با استفاده از پارامتر شکل مناسب و مقایسه ی
نتایج با نتایج روش تحلیلی.. 99

4-3-2-3- مقایسه ی نتایج با نتایج روش عددی RBF. 102

4-4- انتشار موج ایجاد شده توسط موج ساز در مخزن آزمایشگاهی.. 102

عنوان                                                                                                                      صفحه

4-4-1- بررسی عوامل موثر بر غیرخطی شدن موج.. 105

فصل پنجم: نتیجه گیری و پیشنهادات

5-1- مقدمه. 109

5-2- جمع بندی و نتیجه گیری.. 109

5-3- پیشنهادات.. 110

مراجع.. 111

منبع:

AIRY, G. B. 1845. Tides and waves. Encyclopaedia Metropolitana, 241-396.

BELLMAN, R., CASTI, J. 1971. Differential quadrature and long term integration. J. of mathematical analysis applications, 34, 235-238.

BELLMAN, R., KASHEF, B. G. & CASTI, J. 1972. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. of Comput. Physics, 10, 40-52.

BELLMAN, R., KASHEF, B. G., LEE, E. S. & VASUDEVAN, R. 1975a. Solving hard problems by easy methods: differential and integral quadrature. Comp. & Math. with Appl, 1, 133-143.

BELLMAN, R., KASHEF, B. G., LEE, E. S. & VASUDEVAN, R. 1975b. Differential quadreture and splines. Comp. & Math. with Appl, 1, 371-376.

BELLMAN, R., KASHEF, B. G. & VASUDEVAN, R. 1974. The inverse problem of estimating heart parameters from cardiograms math. Biosci., 19, 221-230.

BELLMAN, R. & ROTH, R. S. 1986. Methods in approximation: techniques for mathematical modelling, Dordrecht, Holland, D Redial Publishing Company.

BERT, C. W., JANG, S. K. & STRIZ, A. G. 1988. Two new approximate methods for analyzing free vibration of structural components. AIAA J., 26, 612-618.

BERT, C. W., JANG, S. K. & STRIZ, A. G. 1989. Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature Comput. Mech., 5, 217-226.

BIESEL, F. & SUQUET, F. 1954. Laboratory wave generating apparatus. English Transl. by St. Anthony Falls Hydraulic Laboratory, Report 39.

BOUSSINESQ, J. 1872. Th´eorie des ondes et des remous qui se propagent le long d’uncanal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Math. Pures Appl, 17, 55-108.

BROEZE, J., DAALEN, E. F. G. V. & ZANDBERGEN, P. J. 1993. A three-dimensional panel method for nonlinear free surface waves on vector computers. Comput. Mech., 13, 12-28.

BUHMANN, M. D. 1990. Multivariate cardinal interpolation with radial-basis functions. Constructive Approximation, 6, 225-255.

BUHMANN, M. D. 2003. Radial basis functions: theory and implementations, Cambridge, Cambridge University Press.

BUHMANN, M. D. & DYN, N. 1993. Spectral convergence of multiquadric interpolation. Edinburgh Math. Soc., 36, 319-333.

BUHR, H. & SVENDSEN. 1974. Laboratory generation of waves of constant form, Proc. 14th Conf. on Coastal Eng., ASCE.

CAO, Y. S., SCHULTZ, W. W. & BECK, R. F. 1991. Three-dimensional desingularized boundary integral methods for potential problems. Int. J. Numer. Methods Fluids, 12, 785-803.

CARLSON, R. E. & FOLEY, T. A. 1991. The parameter R2 in multiquadric interpolation. Comput. and Math. with Applications, 21, 29-42.

CHAN, R. K., STREET, R. L. 1970. A numerical model for water waves. Tech. Rep. Dep. Civil Engrg.,135.

CHEN, C. N. 2000. Differential quadrature element analysis using extended differential quadrature. Comput. and Math. with Applications, 39, 65-79.

CHEN, C. N. 2004. DQEM and DQFDM irregular elements for analyses of 2-D heat conduction in orthotropic media. Applied Math. Modelling, 28, 617-638.

CHENG, A. H. D. 2012. 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, 220-239.

CIVAN, F. 1989. Differential cubature for multidimensional problems Proc of the 20th Annual Pittsburgh Conference on modeling and simulation, Vogt WG and Mickle MH (eds), Instrument Society of American, North Carolina, 1843-1847.

CIVAN, F. & SLIEPCEVICH, C. M. 1983a. Application of differential quadrature to transport processes. J. Math. Anal. Appl., 93, 206-221.

CIVAN, F. & SLIEPCEVICH, C. M. 1983b. Solution of the Poisson equation by differential quadrature. Int. J. Numer. Meth. Engng., 19, 711-724.

CIVAN, F. & SLIEPCEVICH, C. M. 1984a. On the solution of the Thomas-Fermi equation by differential quadrature. J. Comput. Phys., 56, 343-348.

CIVAN, F. & SLIEPCEVICH, C. M. 1984b. Differential quadrature for multi-dimensional problems. J. Math. Anal. Appl., 101, 423-443.

COOKER, M. J., PEREGRINE, D. H. & SKOVGGARD, O. 1990. The interaction between a solitary wave and a submerged semicircular cylinder. Fluid Mech., 215, 1-22.

DALRYMPLE, R. A. & ROGERS, B. D. 2006. Numerical modeling of water waves with the SPH method. Coastal Engineering, 53, 141-147.

DEAN, R. G. 1974. Evaluation and development of water wave theories for engineering application. Technical Report, 4, 133, 534p.

DEAN R. G., Dalrymple R. A. 1991. Water wave mechanics for engineers and scientists. World Scientific. 353p.

DEAN, R. G. 1965. Stream function representation of nonlinear ocean waves. J. Geophys. Res. 70, 4561-4572.

DING, H., SHU, C. & TANG, D. B. 2005. Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments. Int. J. Numer. Meth. Engng., 63, 1513-1529.

DOMMERMUTH, D. G. & YUE, D. K. P. 1987. Numerical simulations of nonlinear axisymmetric flows with a free surface. Fluid Mech., 178, 195-219.

DRISCOLL, T. A. & FORNBERG, B. 2002. Interpolation in the limit of increasingly flat radial basis functions. Comput. and Math. with Applications, 43, 413-422.

DUCHON, J. 1976. Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Revue Francaise D Automatique Informatique Recherche Operationnelle, 10, 5-12.

ELLEX & ARUMUGAM. 1984. An experimental study of waves generated by an oscillating wedge, J. Hyd. Res., 22.

FASSHAUER, G. E. 1997. Solving partial differential equations by collocation with radial basis functions. Surface fitting and multiresolution methods. Proc. of the 3rd international conference on curves and surfaces.

FLICK & GUZA. 1980. Paddle Generated Waves in Laboratory Channels. J. Waterways and Harbors Eng. ASCE, 106, l.

FLORYAN, J. M. & RASMUSSEN 1989. Numerical methods for viscous flows with moving boundaries. Appl. Mech. Rev, 42, 323-341.

FOCHESATO, C. & DIAS, F. 2006 A fast method for nonlinear three-dimensional free-surface waves. Proc. R. Soc. Lond. A, 2715-2735.

FONTANET, P. 1961. Theorie de la generation de la houle cylindrique par un batteur plan. La Houille Blanche, 16, 1, 3-31.

FORNBERG, B., ZUEV, J. 2007. The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Computers and Mathematics with Applications, 54, 379-398.

FRANKE, C., SCHABACK, R. 1998. Solving partial differential equations by collocation using radial basis functions. Applied Math. and Comput., 73-82.

FRANKE, R. 1982. Scattered data interpolation—tests of some methods. Math. of Comput., 38, 181-200.

GERSTNER, F. 1809. Theorie der Wellen, Abh. d. k. bohm. Ges. d. Wiss. Ann. der Physik, 32, 412-442.

GILBERT, G., THOMPSON, D. M. & BREWER, A. J. 1971. Design curves for regular and random wave generators. J. Hyd. Res. 9, 2, 163-196.

GOLBERG M. A. & CHEN C. S. 1994. The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations. Bound. Elem. Commun., 5, 57-61.

GOLBERG M. A. & CHEN C. S. 1997. Discrete Projection Methods for Integral Equations. Southampton: Comput. Mechanics Publications.

GOLBERG, M. A., CHEN, C. S. & KARUR, S. 1996. Improved multiquadric approximation for partial differential equations. Engineering Analysis with Boundary Elements, 18, 9-17.

GOLBERG, M. A., CHEN, H. & BOWMAN, H. 1999. Some recent results and proposals for the use of radial basis functions in the BEM. Engineering Analysis with Boundary Elements, 23, 285-296.

GRILLI, S. T., GUYENNE, P. & DIAS, F. 2001. A fully nonlinear model for threedimensional overturning waves over arbitrary bottom. Intl. J. Numer. Meth. Fluids, 35.

GRILLI, S. T., SKOURUP, J. & SVENDSEN, I. A. 1989. An efficient boundary element method for nonlinear water waves. Eng. Anal. Boundary Elem., 6, 97-107.

HARDY, R. L. 1971. Multiquadric equations of topography and other irregular surfaces. Geophy. Res., 76, 1905-1915.

HASHEMI, M. R., ABEDINI, M. J. & MALEKZADEH, P. 2006. Numerical modeling of long waves in shallow water using incremental differential quadrature method. Ocean Engineering, 33, 1749-1764.

HASHEMI, M. R. & HATAM, F. 2011. Unsteady seepage analysis using local radial basis function-based differential quadrature method. Appl. Math. Model, 35, 4934-4950.

HAVELOCK, T. H. 1929. Forced surface-wave on water. Philosophical Magazine viii, 569-576.

HOLTHUIJSEN, L. H. 2007. Waves in oceanic and coastal waters. Cambridge University Press, 387p.

HON, Y. C., CHEUNG, K. F., MAO, X. Z. & KANSA, E. J. 1999. A Multiquadric Solution for the shallow water Equations. ASCE J. of Hydrodlic Engineering, 125, 524-533.

HON, Y. C. & KANSA, E. J. 2000. Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput. Math. Appl., 39, 123-137.

HON, Y. C., LU, M. W., XUE, W. M. & ZHU, Y. M. 1997. Multiquadric method for the numerical solution of a biphasic model. appl. Math. Comput., 88, 153-175.

HON, Y. C. & MAO, X. Z. 1997. A multiquadric interpolation method for solving initial value problems. Sci. Comput., 12, 51-55.

HON, Y. C. & MAO, X. Z. 1998. An efficient numerical scheme for Burgers equation. Appl. Math. Comput., 95, 37-50.

HON, Y. C. & MAO, X. Z. 1999. A radial basis function method for solving options pricing model Financial Engineering, 8, 31-50.

HU, L. C. & HU, C. R. 1974. Identification of rate constenls by differential quadrature in partly measurable compartmental models math. Biosci., 21, 71-76.

HUANG, C. S., LEE, C. F. & CHENG, A. H. D. 2007. Error estimate, optimal shape factor,and high precision computation of multiquadric collocation method. Engineering Analysis with Boundary Elements, 31, 614-623.

HUANG, C. S., YEN, H. D. & CHENG, A. H. D. 2010. On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs. Engineering Analysis with Boundary Elements, 34, 802-809.

HYUN. 1976. Theory for hinged wavemakers of finite draft in water of constant depth, J. Hydronantics, 10, 1.

ISSACSON, M. 1982. Nonlinear wave effects on fixed and floating bodies. Fluid Mech., 120, 267-281.

JANG, S. K., BERT, C. W. & STRIZ, A. G. 1989. Application of differential quadrature to static analysis of structural components Int. J. Numer. Meth. Engng, 28, 561-577.

KAMPHUIS, J. W. 2000. Coastal and Estuary Processes. Hydraulic Modeling.

KANSA, E., J. 1990. Multiquadrics – A scattered data approximation scheme with applications to computational fluid-dynamics –1. Surface approximations and partial derivative estimates. Computers Mathematics Application, 19, 127-145.

KANSA, E. J. 1990. Multiquadrics – A scattered data approximation scheme with applications to computational fluid-dynamics –2. Solutions to parabolic, hyperbolic and elliptic partial-differential equations. Computers and Mathematics with Applications, 19, 147-161.

KANSA, E. J. 1999. Motivation for using radial basis functions to solve PDEs [Online]. Available: http://www.cityu.edu.hk/rbf-pde/files/overview-pdf.pdf [Accessed jan. 17 2011].

KASHEF, B. G. & BELLMAN, R. 1974. Solution of the partial differential equation of the Hodgkin-Huxley model using differential quadrature. Math. Biosci., 19, 1-18.

KASHIWAGI, M. 1996. Full-nonlinear simulations of hydrodynamic forces on a heaving two-dimensional body. J. Soc. Nav. Archit., 180, 373-381.

KOO, W. C., KIM, M. H. 2006. Numerical simulation of nonlinear wave and force generated by a wedge-shape wave maker. Ocean Engineering, 33, 983-1006.

KORTEWEG, D. J. & VRIES, G. D. 1895. On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag., 39, 422-443.

LARSSON, E. & FORNBERG, B. 2003. A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl., 46, 891-902.

LIU, Y., XUE, M. & YUE, D. K. P. 2001. Computation of fully-nonlinear threedimensional wave-wave and wave-body interactions — Part II: nonlinear waves and forces on a body. Fluid Mech., 438, 41-66.

LONGUET-HIGGINS, M. S. & COKELET, E. D. 1976. The deformation of steep surface waves on water I. A numerical method of computation. Proc. R. Soc. Lond. A, 350, 1-26.

LUH, L. T. 2010. The mystery of the shape parameter 2. arXiv:1002.2082v1.

LUH, L. T. 2010. The mystery of the shape parameter 3. arXiv:1004.0759v1.

LUH, L. T. 2010. The mystery of the shape parameter 4. arXiv:1004.0761v1.

LUH, L. T. 2010. The mytery of the shape parameter arXiv:1001.5087v1.

LUH, L. T. 2010. The shape parameter in the Gaussian function. arXiv:1006.2318v1.

LUH, L. T. 2008. The crucial constants in the exponential-type error estimates for Gaussian interpolation. Analysis in Theory and Applications, 24, 2, 183-94.

MA. 2004. Meshless local Petrov–Galerkin method for two-dimensional nonlinea water wave problems. J. of Comput. Physics, 205, 611-625.

MA, R., LI, G. 2006. Spectral analysis of Stokes waves. Ocean Eng., 29, 6, 593-604.

MADSEN, O. S. 1971. On the generation of long waves. J. of Geophysical Res. 76, 8672-8683.

MADSEN, O. S. 1970. Waves generated by a piston type wavemaker, Proc. 12th Conf. on Coastal Eng., ASCE.

MADYCH, W. R. 1992. Miscellaneous error-bounds for multiquadric and related interpolators. Comput. and Math. with Applications, 24, 121-138.

MADYCH, W. R. & NELSON, S. A. 1988. Multivariate interpolation and conditionally positive definite functions. Approximation Theory and its Applications, 4, 77-89.

MADYCH, W. R. & NELSON, S. A. 1990. Multivariate interpolation and conditionally positive definite functions 2. Math. of Comput., 54, 211-230.

MADYCH, W. R. & NELSON, S. A. 1992. Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. Approx. Theory, 70, 94-114.

MALEKZADEH , P. & KARAMI, G. 2003. Out-of Plane analysis of circular arches by DQM. Int. J. of Solids and Structures, 40, 6527-6545.

MALIK, M. & BERT, C. W. 1996. Implementing multiple boundary conditions in the DQ solution of higher-order PDE's: application to free vibration of plates. Int. J. Numer. Meth. Engng., 39, 1237-1258.

MCCOWAN, J. 1891. On the Solitary Wave. Phil. Mag., 5, 45-58.

MCCOWAN, J. 1894. On the Highest Wave of Permanent Type. Phil. Mag., 5, 351-357.

MEI, C. C. 1978. Numerical methods in water wave diffraction and radiation. Ann. Rev. Fluid Mech, 10, 393-416.

MINGLE, J. O. 1977. The method of differential quadrature for transient nonlinear diffusion J. Math. Anal. Appl., 60, 559-569.

MONAGHAN, J. J. 1992. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics, 30, 543-574.

MONAGHAN, J. J. 1994. Simulating free surface flows with SPH. J. of Comput. Physics, 110, 399-406.

MUNK, W. H. 1949. The Solitary Wave Theory and its Application to Surf Problems. Ann. New York Acad. of Science, 51, 376-424.

NAADIMUTHU, G., BELLMAN, R., WANG, K. M. & LEE, E. S. 1984. Differential quadrature and partial differential equations: some numerical results. J. Math. Anal. Appl., 98, 220-233.

OHYAMA, T. & NADAOKA, K. 1991. Development of a numerical wave tank for analysis of nonlinear and irregular wave field. Fluid Dyn. Res., 8, 231-251.

PEREGRINE, D. H. 2000. Coastal and Estuary Processes. Hydraulic Modeling.

PEREGRINE, D. H. 1967. Long waves on a beach. J. of Fluid Mechanics, 27, 4, 815-827

QUAN, J. R., CHANG, C. T. 1989. New insights in solving distributed system equations by the quadrature methods - I. Comput. Chem. Engrg., 13, 779-788.

QUAN, J. R., CHANG, C. T. 1989. New insights in solving distributed system equations by the quadrature methods - II. Comput. Chem. Engrg., 13, 1017-1024.

REEVE, D., CHADWICH, A. & FLEMING, C. 2004. Coastal Engineering. Taylor and Francis Group.

RIPPA, S. 1999. An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Comput. Math., 11, 193-210.

SCARDOVELLI, R., ZALESKI, S. 1999. Direct numerical simulation of free surface and interfacial flow. Ann. Rev. Fluid Mech., 31, 567-604.

SCARDOVELLI, R., ZALESKI, S. 1999. Direct numerical simulation of free surface and interfacial flow. Ann. Rev. Fluid Mech., 31, 567-604.

SCHABACK, R. 1999. A Unified Theory of Radial Basis Functions (Native Hilbert Spaces for Radial Basis Functions II).

SCHULER, M., 1936. Erzeugung von Oberflachenwellen durch schwingende Korper, Zeit. Angew. Math. Mech., 16, 2, 65-72.

SCHWARTZ, L. W., FENTON, J. D. 1982. Strongly nonlinear waves. Ann. Rev. Fluid Mech, 14, 39-60.

SCULLEN, D., TUCK, E. O. 1995. Free-surface Flow Computations for Submerged Cylinders. Ship Res., 39, 185-193.

SENTURK, U. 2011. Modeling nonlinear waves in a numerical wave tank with localized meshless RBF method. Computers & Fluids, 44, 221-228.

SHARAN, M., KANSA, E. J. & GUPTA, S. 1997. Applications of the multiqudric method for the solution of elliptic partial differential equations. appl. math. comput., 84, 275-302.

SHU, C. 2000. Differential quadrature and its application in engineering, Berlin, Springer.

SHU, C., CHEW, Y. T. & LIU, Y. 1997. An efficient approach for numerical simulation of flows in Czochralski crystal growth J. Cryst. Growth., 181, 427-436.

SHU, C., DING, H., CHEN, H. Q. & WANG, T. G. 2005. An upwind local RBF-DQ method for simulation of inviscid compressible flows. Computer Methods in Applied Mechanics and Engineering, 194, 2001-2017.

SHU, C., DING, H. & YEO, K., S. 2003. Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg., 192, 941-954.

SHU, C., DING, H. & YEO, K. S. 2004. Solution of partial differential equations by a global radial basis function–based differential quadrature method. Engineering Analysis with Boundary Elements, 28, 1217-1226.

SHU, C., DING, H. & YEO, K. S. 2005. Computation of incompressible Navier–Stokes equations by local RBF-based differential quadrature method. Computer Modeling in Engineering and Sciences, 7, 195-205.

SHU, C., KHOO, B. & YEO, K. S. 1994. Numerical solutions of incompressible Navier-Stokes equations by generalized differential quadrature. Finite Elem. Anal. Design, 18, 83-97.

SHU, C., RICHARDS, B. E. 1990. High resolution of natural convection in a square cavity by generalized differential quadrature Proc. of 3rd Conf. on Adv. in Numer. Methods in Eng: Theory and Appl, Swansea, Uk, 2, 978-985.

STEAD, S. 1984. Estimation of gradients from scattered data. Rocky Mount. j. Math, 14, 265-279.

STOKES, G. G. 1847. On the theory of oscillatory waves. Trans. Camb. Phil. Soc., 8, 441-455

TARTWATER, A. E. 1985. Parameter study of Hardy’s multiquadric method for scattered data interpolation. Technical Report UCRL-54670. 69p.

TANIZAWA, K. 2000. The state of the art on numerical wave tank. Proc. of the 4th Osaka colloquium on seakeeping performance of ships. 95-114.

TSAI, W., YUE, D. K. P. 1996. Computation of nonlinear free-surface flows. Ann. Review of Fluid Mech., 28, 249-278.

URSELL, F., DEAN, R. G., Yu, Y.S. 1960. Forced small amplitude waves: A comparison of theory and experiment. J. of Fluid Mechanics 7, 33-52.

WANG, J. G., LIU, G. R. 2002. On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Computer Methods in Applied Mechanics and Engineering, 191, 2611-2630.

WANG, Q. X. 2005. Unstructured MEL modeling of nonlinear unsteady ship waves. Comput. Phys., 210, 368-385.

WANG, S. 1974. Plunger-type wavemakers: Theory and experiment, J. of Hyd. Res., 12, 3, 357-388

WENDLAND, H. 2005. Scattered data approximation. Cambridge University Press.

WERTZ, J., KANSA, E. J. & LING, L. 2006. The role of the multiquadric shape parameters in solving elliptic partial differential equations. Computers and Mathematics with Applications, 51, 1335-1348.

WU, Y. L. & SHU, C. 2002. Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli. Comput. Mechanics, 29, 477-485.

WU, Y. L. & SHU, C. 2007. Integrated radial basis functions-based differential quadrature method and its performance. Int. J. Numer. Meth. Fluids, 53, 969-984.

WU, Z. M. & SCHABACK, R. 1993. Local error-estimates for radial basis function interpolation of scattered data. IMA J. of Numerical Analysis, 13, 13-27.

WU, Y. C. 1988. Plunger-type wavemaker theory theorie d'un generateur de houle a plongeur oscillant, J. of Hyd. Res., 26, 4,483-491.

XUE, M., XU, H. B., LIU, Y. & YUE, D. K. P. 2001. Computations of fullynonlinear three-dimensional wave-wave and wave-body interactions, Part I: three-dimensional steep waves. Fluid Mech., 438, 11-39.

YEUNG, R. W. 1982. Numerical methods in free-surface flows. Ann. Rev. Fluid Mech., 14, 395-442.

ZHANG, X., KHOO, B. & LOU, J. 2006. Wave propagation in a fully nonlinear numerical wave tank: a desingularized method. Ocean Eng, 33, 10-31.