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A meshless method namely, discrete least square method (DLSM), is presented in the paper for the solution of free surface seepage problem. In this method computational domain is discredited by some nodes and then the set of simultaneous equations are built using moving least square (MLS) shape functions and least square technique. The proposed method does not need any background mesh therefore it is a truly meshless method. Several numerical two dimensional examples of Poisson partial differential equations (PDEs) are presented to illustrate the performance of the present DLSM. And finally a free surface seepage problem in a porous media is solved and results are presented.

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Evaluating the rate and maximum height of capillary rise is of prime interest in unsaturated soil mechanics. Antecedent solutions

to this problem have dwelled mostly on determining the maximum capillary rise height, overlooking moisture and suction changes

in the capillary region. A comprehensive improved solution for the capillary rise of water in soils is presented. Salient features of

the formulation including consideration of initial soil suction (if any) prior to capillary rise, and determination of water content

variation in the capillary region are elaborately discussed. Results reveal that suction head variation within the capillary region

is non-linear, where the curvature decreases as water rises to higher elevations. The solution is verified and compared with

existing solutions, by means of two sets of experimental data available in the literature. The comparison suggests that the

improved formulation is more accurate and versatile than previous solutions for capillary rise.

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Upstream blankets, drains and cutoff walls are considered as effective measures to reduce seepage, uplift pressure and exit gradient under the foundation of hydraulic structures. To investigate the effectiveness of these measures, individually or in accordance with others, a large number of experiments were carried out on a laboratory model. To extend the investigation for unlimited arrangements, the physical conditions of all experiments were simulated with a mathematical model. Having compared the data obtained from experiments with those provided from the mathematical model, a good correlation was found between the two sets of data indicating that the mathematical model could be used as a useful tool for calculating the effects of various measures on designing hydraulic structures. Based on this correlation a large number of different inclined angles of cutoff walls, lengths of upstream blankets, and various positions of drains within the mathematical model were simulated. It was found that regardless of their length, the blankets reduce seepage, uplift pressure and exit gradient. However, vertical cutoff walls are the most effective. Moreover, it was found that the best positions of a cutoff wall to reduce seepage flow and uplift force are at the downstream and upstream end, respectively. Also, having simulated the effects of drains, it was found that the maximum reduction in uplift force takes place when the drain is positioned at a distance of 1/3 times the dam width at the downstream of the upstream end. Finally, it was indicated that the maximum reduction in exit gradient occurs when a drain is placed at a distance of 2/3 times of the dam width from upstream end or at the downstream end.

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In this paper using the eigenvalues and eigenvectors symmetric block diagonal matrices with infinite dimension and numerical method of finite difference a closed form solution for exact solving of Laplace equation is presented. Moreover, the method of this paper has applications in different states of boundary conditions like Newman, Dirichlet and other mixed boundary conditions. Moreover, with the method of this paper, a mathematical model for the exact solution of the Poisson equation is derived. Since these equations have many applications in engineering problems, in each part examples like water seepage problem through the soil and torsion of prismatic bars are presented. Finally the method is provided for torsion problem of prismatic bars with non-circular and non-rectangular cross sections by using of conformal mapping.

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This paper presents numerical and theoretical studies on the stability of shallow shield tunnel face found in cohesive-frictional soil. The minimum limit support pressure was determined by superposition method; it was calculated by multiplying soil cohesion, surcharge load, and soil weight by their corresponding coefficients. The varying characteristics of these coefficients with soil friction angle and tunnel cover-to-diameter ratio were obtained by wedge model and numerical simulation. The face stability of shallow shield tunnel with seepage was studied by deformation and seepage coupled numerical simulation; the constitutive model used in the analysis was elastic-perfectly plastic Mohr–Coulomb model. The failure mode of tunnel face was shown related to water level. By considering the effect of seepage on failure mode, the wedge model was modified to calculate the limit support pressure under seepage condition. The water head around the tunnel face was fitted by an exponential function, and then an analytical solution to the limit support pressure under seepage condition was deduced. The variations in the limit support pressure on strength parameters of soil and water lever compare well with the numerical results. The modified wedge model was employed to analyze the tunnel face stability of Qianjiang cross-river shield tunnel. The influence of tide on the limit support pressure was obtained, and the calculated limit support pressure by the modified wedge model is consistent with the numerical result.