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Seepage flow problems by variational inequalities: Theory and approximation
Author(s) -
Kikuchi Noboru
Publication year - 1977
Publication title -
international journal for numerical and analytical methods in geomechanics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.419
H-Index - 91
eISSN - 1096-9853
pISSN - 0363-9061
DOI - 10.1002/nag.1610010305
Subject(s) - variational inequality , mathematics , finite element method , quadratic equation , boundary value problem , a priori and a posteriori , mathematical analysis , variational principle , domain (mathematical analysis) , boundary (topology) , convergence (economics) , unilateral contact , mathematical optimization , geometry , philosophy , physics , epistemology , thermodynamics , economics , economic growth
The theory of variational inequalities enables us to formulate and solve free boundary problems in fixed domains, while most other methods assume the position of the unknown domain in solving the problem. Here the problem of seepage flow through a rectangular dam with a free boundary is formulated as a vertical inequality following the ideas of Baiocchi. In order to demonstrate the essential ideas of extending the domain of the solution of problems with free boundaries, the problem of the deflection, of a string on a rigid support is first examined. Next, variational inequalities are derived which are associated with several cases of seepage problems. An approximation theory, including a priori error estimates, is developed using finite element methods, and an associated numerical scheme is given. It is shown that for linear and quadratic finite element methods, the rates of convergence are 0( h ) and 0( h 1.25‐δ ), 0 < δ < 0.25, respectively, if the permeability is constant.