New Variational Inequality Formulation for Seepage Problems With Free Surfaces

ZHENG Hong; LIU De-fu; LEE C. F.; THAM L. G.

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Article Navigation > Applied Mathematics and Mechanics > 2005 > 26(3): 363-371

ZHENG Hong, LIU De-fu, LEE C. F., THAM L. G.. New Variational Inequality Formulation for Seepage Problems With Free Surfaces[J]. Applied Mathematics and Mechanics, 2005, 26(3): 363-371.

Citation:

ZHENG Hong, LIU De-fu, LEE C. F., THAM L. G.. New Variational Inequality Formulation for Seepage Problems With Free Surfaces[J]. Applied Mathematics and Mechanics, 2005, 26(3): 363-371.

Citation:

ZHENG Hong, LIU De-fu, LEE C. F., THAM L. G.. New Variational Inequality Formulation for Seepage Problems With Free Surfaces[J]. Applied Mathematics and Mechanics, 2005, 26(3): 363-371.

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New Variational Inequality Formulation for Seepage Problems With Free Surfaces

1.
College of Civil & Hydroelectric Engineering, Three Gorges University, Yichang Hubei 443002, P. R. China;

Received Date: 2004-08-17
Rev Recd Date: 2004-11-27
Publish Date: 2005-03-15

Abstract

Abstract

A new variational inequality formulation for seepage problems with free surfaces was presented, in which a boundary condition of Signorini's type was prescribed over the potential seepage surfaces. This made the singularity of seepage points eliminated and the location of seepage points determined. Compared to other variational formulations, the proposed formulation owns better numerical stability.
- free boundary problem,
- seepage,
- variational inequality,
- finite element

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References(19)

References

[1]	Desai C S, Li G C. A residual flow procedure and application for free surface in porous media[J].Advances in Water Resources,1983,6(1):27—35. doi: 10.1016/0309-1708(83)90076-3
[2]	张有天，陈平，王镭. 有自由面渗流分析的初流量法[J].水利学报，1988，(8)：18—26.
[3]	Bathe K J, Khoshgoftaar M R. Finite element free surface seepage analysis without mesh iteration[J].International Journal for Numerical and Analytical Methods in Geomechanics,1979,3(1):13—22. doi: 10.1002/nag.1610030103
[4]	Alt H W. Numerical solution of steady state porous flow free boundary problems[J].Numerische Mathematik,1980,36(1):73—96. doi: 10.1007/BF01395990
[5]	Brezis H, Kinderlehrer D, Stampacchia G. Sur une nouvelle formulation due probleme de l'ecoulement a travers une digue[J].C R Acad Sci Paris,Ser A,1978,287：711—714.
[6]	Lacy S J, Prevost J H. Flow through porous media: a procedure for locating the free surface[J].International Journal for Numerical and Analytical Methods in Geomechanics,1987,11(6):585—601. doi: 10.1002/nag.1610110605
[7]	Borja R I, Kishnani S S. On the solution of elliptic free boundary problems via Newton's method[J].Computer Methods in Applied Mechanics and Engineering,1991,88(2):341—361. doi: 10.1016/0045-7825(91)90094-M
[8]	Oden J T, Kikuchi N. Recent advances: theory of variational inequalities with applications to problems of flow through porous media[J].International Journal of Engineering Science,1980,18(10):1173—1284. doi: 10.1016/0020-7225(80)90111-1
[9]	Westbrook D R. Analysis of inequality and residual flow procedures and an iterative scheme for free surface seepage[J].Internal Journal for Numerical Methods in Engineering,1985,21(10):1971—1802. doi: 10.1002/nme.1620211104
[10]	Chipot M.Variational Inequalities and Flow in Porous Media[M].New York:Springer-Verlag, 1984.
[11]	佘颖禾，孙鹰，郭小明. 具有自由边界的二维渗流问题[J]. 应用数学和力学，1996，17(6)：523—527.
[12]	Harr M E.Groundwater and Seepage[M].Columbus: McGraw-Hill, 1990.
[13]	Sperb R.Maximum Principles and Their Applications[M].Washington, DC: Academic Press,1981.
[14]	Troianiello G M.Elliptic Differential Equations and Obstacle Problems[M]. New York:Plenum Press,1987.
[15]	Glowinski R.Numerical Methods for Nonlinear Variational Problems[M].Berlin:Springer-Verlag,1984.
[16]	Kinderlehrer D, Stampacchia G.An Introduction to Variational Inequalities and Their Applications[M].New York: Academic Press,1980.
[17]	Quarteroni A, Valli A.Numerical Approximation of Partial Differential Equations[M].Berlin:Springer-Verlag,1997.
[18]	Aitchison J M,Poole M W. A numerical algorithm for the solution of Signorini problems[J].Journal of Computational & Applied Mathematics,1998,94(1):55—67.
[19]	郑铁生，李立，许庆余. 一类椭圆型变分不等式离散问题的迭代算法[J].应用数学和力学，1995，16(4):329—335.