# American Institute of Mathematical Sciences

2015, 5(4): 351-357. doi: 10.3934/naco.2015.5.351

## Solving the seepage problems with free surface by mathematical programming method

 1 College of Science, Dalian Nationalities University, Dalian 116600, China, China

Received  January 2015 Revised  October 2015 Published  October 2015

The nonsmooth equations model for seepage problems is proposed based on the basic principles of the seepage dynamic system and the finite element discrete method. The mathematical programming method is therefore applied. The free surface of seepage is plotted through interpolation with pressure intensity on the nodes. The numerical results show the new method is simple and rapid in convergence rate.
Citation: Jinzhi Wang, Yuduo Zhang. Solving the seepage problems with free surface by mathematical programming method. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 351-357. doi: 10.3934/naco.2015.5.351
##### References:
 [1] K. J. Bathe, Finite element free surface seepage analysis without mesh iteration, Int. J. Numer and Analytical Methods in Geomechanics, 3 (1979), 13-22. [2] W. J. Chen and Z. L. Wang, Finite element method of invariable mesh Gauss point for transient seepage problem with free surface, Journal of dalian university of technology, 31 (1991), 537-543. [3] C. S. Desai and G. C. Li, A residual flow procedure and application for free surface in porous media, Advances in Water Resources, 6 (1983), 27-35. [4] J. S. Pang and L. Q. Qi, Non-smooth equations: motivation and algorithms, SIAM. J. OPTIM., 3 (1993), 443-465. doi: 10.1137/0803021. [5] H. Peng et al, Imaginary element for numerical analysis of seepage with free surface, China Rural Water and Hydropower, 3 (1997), 26-27. [6] L. Q. Qi, Convergence analysis of some algorithms for solving non-smooth equation, Math Oper Res., 18 (1993), 227-224. doi: 10.1287/moor.18.1.227. [7] J. Z. Wang and W. J. Chen, Mixed fixed-Point FE method for seepage problems with free surfaces, Journal of Dalian University of Technology, 47 (2007), 793-797. [8] Y. T. Zhang, P. Chen and L. Wang, Initial flow method for seepage analysis with free surface, Chinese journal of Hydraulic, 8 (1988), 18-26. [9] H. Zheng et al., A new formulation of Signorini's type for seepage problems with free surface, International Journal for Numerical methods in engineering, online, 2005

show all references

##### References:
 [1] K. J. Bathe, Finite element free surface seepage analysis without mesh iteration, Int. J. Numer and Analytical Methods in Geomechanics, 3 (1979), 13-22. [2] W. J. Chen and Z. L. Wang, Finite element method of invariable mesh Gauss point for transient seepage problem with free surface, Journal of dalian university of technology, 31 (1991), 537-543. [3] C. S. Desai and G. C. Li, A residual flow procedure and application for free surface in porous media, Advances in Water Resources, 6 (1983), 27-35. [4] J. S. Pang and L. Q. Qi, Non-smooth equations: motivation and algorithms, SIAM. J. OPTIM., 3 (1993), 443-465. doi: 10.1137/0803021. [5] H. Peng et al, Imaginary element for numerical analysis of seepage with free surface, China Rural Water and Hydropower, 3 (1997), 26-27. [6] L. Q. Qi, Convergence analysis of some algorithms for solving non-smooth equation, Math Oper Res., 18 (1993), 227-224. doi: 10.1287/moor.18.1.227. [7] J. Z. Wang and W. J. Chen, Mixed fixed-Point FE method for seepage problems with free surfaces, Journal of Dalian University of Technology, 47 (2007), 793-797. [8] Y. T. Zhang, P. Chen and L. Wang, Initial flow method for seepage analysis with free surface, Chinese journal of Hydraulic, 8 (1988), 18-26. [9] H. Zheng et al., A new formulation of Signorini's type for seepage problems with free surface, International Journal for Numerical methods in engineering, online, 2005
 [1] B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463 [2] A. M. Bagirov, Moumita Ghosh, Dean Webb. A derivative-free method for linearly constrained nonsmooth optimization. Journal of Industrial and Management Optimization, 2006, 2 (3) : 319-338. doi: 10.3934/jimo.2006.2.319 [3] Jueyou Li, Guoquan Li, Zhiyou Wu, Changzhi Wu, Xiangyu Wang, Jae-Myung Lee, Kwang-Hyo Jung. Incremental gradient-free method for nonsmooth distributed optimization. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1841-1857. doi: 10.3934/jimo.2017021 [4] Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial and Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247 [5] Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial and Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235 [6] Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008 [7] Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure and Applied Analysis, 2002, 1 (3) : 379-415. doi: 10.3934/cpaa.2002.1.379 [8] Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025 [9] Daniel Coutand, Steve Shkoller. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 429-449. doi: 10.3934/dcdss.2010.3.429 [10] Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315 [11] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 [12] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [13] Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial and Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003 [14] Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015 [15] Bouthaina Abdelhedi. Existence of periodic solutions of a system of damped wave equations in thin domains. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 767-800. doi: 10.3934/dcds.2008.20.767 [16] Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592 [17] Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial and Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171 [18] T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201 [19] Xiaojiao Tong, Felix F. Wu, Yongping Zhang, Zheng Yan, Yixin Ni. A semismooth Newton method for solving optimal power flow. Journal of Industrial and Management Optimization, 2007, 3 (3) : 553-567. doi: 10.3934/jimo.2007.3.553 [20] Zhi-Feng Pang, Yu-Fei Yang. Semismooth Newton method for minimization of the LLT model. Inverse Problems and Imaging, 2009, 3 (4) : 677-691. doi: 10.3934/ipi.2009.3.677

Impact Factor: