Application of Nelder-Mead simplex method for unconfined seepage problems

In unconfined seepage problems, the phreatic line resulted from mesh deforming methods is rarely a smooth and continuous curve. The main problem is at the meeting point of the phreatic line with the down stream face of the dam where the phreatic line must be tangent to the seepage face according to the fluid continuity principle. In this paper a mesh deforming finite element method based on Nelder-Mead simplex optimization is presented to solve this problem. The phreatic line is approximated by a 4th degree polynomial and Nelder-Mead simplex method is used to calculate the polynomial’s coefficients minimizing an error function which is introduced based on the conditions on the phreatic line. Tangentiality of the phreatic line to the seepage face is introduced in the solution by a constraint in optimization procedure. The results of the presented method are verified by the results of the nonlinear finite element and other mesh deforming methods.