Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 126398, 9 pages
http://dx.doi.org/10.1155/2014/126398
Research Article

Numerical Simulation of Soil Water Movement under Subsurface Irrigation

Department of Applied Mathematics, Xi’an University of Technology, Xi’an 710048, China

Received 30 April 2014; Revised 14 July 2014; Accepted 14 July 2014; Published 3 August 2014

Academic Editor: Kim M. Liew

Copyright © 2014 Xinqiang Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. B. Lucyt, “A numerical approach to the testing of the fission hypothesis,” The Astronomical Journal, vol. 82, pp. 1013–1024, 1977. View at Google Scholar
  2. R. A. Gingold and J. J. Monaghan, “Smoothed particle hydrodynamics: theory and application to non-spherical stars,” Monthly Notices of the Royal Astronomical Society, vol. 181, pp. 375–389, 1977. View at Google Scholar
  3. B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element method: diffuse approximation and diffuse elements,” Computational Mechanics, vol. 10, no. 5, pp. 307–318, 1992. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229–256, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. R. Schaback, “Improved error bounds for scattered data interpolation by radial basis functions,” Mathematics of Computation, vol. 68, no. 225, pp. 201–216, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. N. Sukumar, B. Moran, and T. Belytschko, “The natural element method in solid mechanics,” International Journal for Numerical Methods in Engineering, vol. 43, no. 5, pp. 839–887, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. Z. Wu, “About the convergence of interpolation with radial basis function,” Annals of Mathematics, vol. 14A, pp. 480–486, 1993. View at Google Scholar
  8. H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree,” Advances in Computational Mathematics, vol. 4, no. 1, pp. 389–396, 1995. View at Publisher · View at Google Scholar · View at Scopus
  9. H. Wendland, “Meshless Galerkin methods using radial basis functions,” Mathematics of Computation, vol. 68, no. 228, pp. 1521–1531, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. S. N. Atluri and S. Shen, The Meshless Local Petrov—Galerkin (MLPG) Method, Tech Science Press, Encino, Calif, USA, 2002.
  11. G. R. Liu, Meshfree Methods: Moving beyond the Finite Element Method, CRC Press, Boca Raton, Fla, USA, 2002. View at MathSciNet
  12. W. K. Liu, Y. Chen, R. A. Uras, and C. T. Chang, “Generalized multiple scale reproducing kernel particle methods,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 91–157, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. C. A. Duarte and J. T. Oden, “An {$h$}-{$p$} adaptive method using clouds,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 237–262, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. I. Babuška and J. M. Melenk, “The partition of unity method,” International Journal for Numerical Methods in Engineering, vol. 40, no. 4, pp. 727–758, 1997. View at Google Scholar · View at MathSciNet · View at Scopus
  15. E. Oñate, S. Idelsohn, O. C. Zienkiewicz, and R. L. Taylor, “A finite point method in computational mechanics: applications to convective transport and fluid flow,” International Journal for Numerical Methods in Engineering, vol. 39, no. 22, pp. 3839–3866, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. N. R. Aluru, “A point collocation method based on reproducing kernel approximations,” International Journal for Numerical Methods in Engineering, vol. 47, no. 6, pp. 1083–1121, 2000. View at Publisher · View at Google Scholar · View at Scopus
  17. T. J. Liszka, C. Duarte, and W. W. Tworzydlo, “Hp-meshless cloud method,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 263–288, 1996. View at Publisher · View at Google Scholar · View at Scopus
  18. T. Zhu, J. Zhang, and S. N. Atluri, “A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach,” Computational Mechanics, vol. 21, no. 3, pp. 223–235, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. X. Zhang and Y. Liu, Meshless Methods, Tsinghua University Press, Beijing, China, 2004.
  20. 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, vol. 19, no. 8-9, pp. 127–145, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. G. E. Fasshauer, “Solving differential equations with radial basis functions: multilevel methods and smoothing,” Advances in Computational Mathematics, vol. 11, no. 2-3, pp. 139–159, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. Y. C. Hon, K. F. Cheung, X. Z. Mao, and E. J. Kansa, “Multiquadric solution for shallow water equations,” Journal of Hydraulic Engineering, vol. 125, no. 5, pp. 524–533, 1999. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Rippa, “An algorithm for selecting a good value for the parameter c in radial basis function interpolation,” Advances in Computational Mathematics, vol. 11, no. 2-3, pp. 193–210, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. M. D. Buhmann, “A new class of radial basis functions with compact support,” Mathematics of Computation, vol. 70, no. 233, pp. 307–318, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. L. W. Zhang, Y. J. Deng, and K. M. Liew, “An improved element-free Galerkin method for numerical modeling of the biological population problems,” Engineering Analysis with Boundary Elements, vol. 40, pp. 181–188, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  26. R. J. Cheng, L. W. Zhang, and K. M. Liew, “Modeling of biological population problems using the element-free kp-Ritz method,” Applied Mathematics and Computation, vol. 227, pp. 274–290, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  27. P. F. Guo, L. W. Zhang, and K. M. Liew, “Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method,” Applied Mathematics and Computation, vol. 240, pp. 91–101, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  28. J. W. Yan, L. W. Zhang, K. M. Liew, and L. H. He, “A higher-order gradient theory for modeling of the vibration behavior of single-wall carbon nanocones,” Applied Mathematical Modelling, vol. 38, no. 11-12, pp. 2946–2960, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Z. X. Lei, L. W. Zhang, K. M. Liew, and J. L. Yu, “analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the element-free kp-Ritz method,” Composite Structure, vol. 113, pp. 328–338, 2014. View at Publisher · View at Google Scholar
  30. X. F. Pan, X. Zhang, and M. W. Lu, “Meshless Galerkin least-squares method,” Computational Mechanics, vol. 35, no. 3, pp. 182–189, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  31. Y. Gu and L. Zhang, “Coupling of the meshfree and finite element methods for determination of the crack tip fields,” Engineering Fracture Mechanics, vol. 75, pp. 986–1004, 2008. View at Publisher · View at Google Scholar
  32. Y. Duan and Y. Tan, “On condition number of meshless collocation method using radial basis functions,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 141–147, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. H. Hu, Z. Li, and A. H. Cheng, “Radial basis collocation methods for elliptic boundary value problems,” Computers and Mathematics with Applications, vol. 50, no. 1-2, pp. 289–320, 2005. View at Publisher · View at Google Scholar · View at Scopus
  34. L. Su, X. Qin, B. Miao, and Q. Wang, “Radial basis function collocation method with difference for nonlinear convection-dominated diffusion equations,” in Proceedings of the 6th International Conference on Natural Computation (ICNC '10), pp. 3203–3207, Yantai, China, August 2010. View at Publisher · View at Google Scholar · View at Scopus
  35. W. Guo, B. Li, Z. Ji, Y. Jiang, and F. Yan, “Two-dimensional numerical simulation of soil water infiltration under bed-irrigating sowing,” Transactions of the Chinese Society of Agricultural Engineering, vol. 17, no. 2, pp. 24–27, 2001. View at Google Scholar · View at Scopus
  36. S. P. Neuman, “Saturated-unsaturated seepage by finite elements,” Journal of the Hydraulics Division, vol. 99, no. 12, pp. 2233–2250, 1973. View at Google Scholar · View at Scopus
  37. H. Zhang and S. Chen, “Numerical simulation of infiltration in unsaturated soil,” Rock and Soil Mechanics, vol. 24, no. 5, pp. 715–718, 2003. View at Google Scholar
  38. X. Lu and D. H. Wu, “Numerical simulation of infiltration in unsaturated soil,” China Water Transport, vol. 4, no. 4, pp. 136–137, 2006. View at Google Scholar
  39. H. R. Li and Z. D. Luo, “Semi-discrete finite volume element simulation for two-dimensional unsaturated soil water flow problem,” Mathematica Numerica Sinica, vol. 33, no. 1, pp. 57–68, 2011. View at Google Scholar · View at MathSciNet
  40. D. Zhou and H. Wang, “Application of RBF in simulation of groundwater flow,” Journal of Liaoning Normal University, vol. 31, no. 4, pp. 390–392, 2008. View at Google Scholar