Infiltration in unsaturated soils – An analytical approach

Ghotbi, Abdoul R.; Omidvar, Mehdi; Barari, Amin

doi:10.1016/j.compgeo.2011.05.007

Cited by 34 publications

(30 citation statements)

References 32 publications

(36 reference statements)

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“…Note that there are two convergence-control parameters c 1 and c 2 in the homotopy-series (18) and (19), respectively. Assume that c 1 and c 2 are properly chosen so that the homotopy-series (18) and (19) converge at q = 1. Then according to Eq.…”

Section: Mathematical Formulasmentioning

confidence: 99%

Analytic approximations of Von Kármán plate under arbitrary uniform pressure—equations in integral form

Zhong

Liao

2017

Sci. China Phys. Mech. Astron.

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In this paper, the homotopy analysis method (HAM) is successfully applied to solve the Von Kármán's plate equations in the integral form for a circular plate with the clamped boundary under an arbitrary uniform external pressure. Two HAM-based approaches are proposed. One is for a given external load Q, the other for a given central deflection. Both of them are valid for an arbitrary uniform external pressure by means of choosing a proper value of the so-called convergence-control parameters c 1 and c 2 in the frame of the HAM. Besides, it is found that iteration can greatly accelerate the convergence of solution series. In addition, we prove that the interpolation iterative method [1, 2] is a special case of the HAM-based 1st-order iteration approach for a given external load Q when c 1 = −θ and c 2 = −1, where θ denotes the interpolation parameter of the interpolation iterative method. Therefore, like Zheng and Zhou [3], one can similarly prove that the HAM-based approaches are valid for an arbitrary uniform external pressure, at least in some special cases such as c 1 = −θ and c 2 = −1. Furthermore, it is found that the HAM-based iteration approaches converge much faster than the interpolation iterative method [1,2]. All of these illustrate the validity and potential of the HAM for the famous Von Kármán's plate equations, and show the superiority of the HAM over perturbation methods.

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Section: Mathematical Formulasmentioning

confidence: 99%

Analytic approximations of Von Kármán plate under arbitrary uniform pressure—equations in integral form

Zhong

Liao

2017

Sci. China Phys. Mech. Astron.

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“…They established a relationship between Green-Ampt model and the algebraic solution to analyze physical features of the soil parameters. Ghotbi et al (2011) applied homotopy analysis method (HAM) to solve the equation analytically and showed that the method is superior over traditional perturbation techniques in the sense that it was not dependent on the assumption of a small parameter as the initial step. Nasseri et al (2012) presented three major cases for the governing PDE solved by traveling wave solution (TWS) method using general and modified forms of tanh functions.…”

Section: Introductionmentioning

confidence: 99%

New Analytical Solutions to 2-D Water Infiltration and Imbibition into Unsaturated Soils for Various Boundary and Initial Conditions

Sanayei

Rakhshandehroo

Talebbeydokhti

2016

Iran J Sci Technol Trans Civ Eng

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Fluid infiltration and imbibition into unsaturated soil are of vital significance from many perspectives. Mathematically, such transient flows are described by Richards' equation, a nonlinear parabolic partial differential equation with limited analytical solutions in the literature. However, the choice of exponential model for water content and hydraulic conductivity linearizes the nonlinear Richards' equation, making it possible to obtain an analytical solution via classical approaches. In this study, separation of variables and Fourier series expansion techniques are used to derive new analytical solutions to 2-D vertical and horizontal infiltration and imbibition into unsaturated soils for nonsymmetrical boundary and nonuniform initial conditions. A total of 11 cases are considered, where high water content is imposed on the top, side, or bottom edges of the sample and water is infiltrated (from the top and/or side boundaries) and imbibed (from the bottom boundary) into the sample. Residual water content and/or noflow boundary condition are assumed on other edges of the sample. Initial conditions include 7 cases of constant residual water content, 2 cases of sinusoidal, and 2 cases of exponential water content functions over the sample. Presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed-form solution. Two-dimensional and 3-D plots of water content are presented for the transient as well as steady-state conditions. To illustrate the use of the derived equations, water content values from numerical solutions are compared to those from analytical solutions for four cases, showing a maximum error of\2 %. The presented analytical solutions may be used as a benchmark for verification and accuracy assessment of numerical approaches where nonsymmetrical boundary and/or nonuniform initial conditions exist.

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“…Especially, there is a convergence-control parameter c 0 in the series solutions, which provides us a convenient way to guarantee the convergence of series solutions gained by the HAM. It is these merits that distinguish the HAM from other analytic approaches, and thus enable the HAM to be successfully applied to many complicated problems with high nonlinearity [39][40][41][42][43][44][45][46][47]. Note that the HAM was successfully applied to give an analytic approximation with much longer expiry for the optimal exercise boundary of an American put option than perturbation approximations [48].…”

Section: Introductionmentioning

confidence: 99%

On the homotopy analysis method for backward/forward-backward stochastic differential equations

Zhong

Liao

2017

Numer Algor

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In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forwardbackward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6 dimensional case, within less than 1% CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering and finance.

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Infiltration in unsaturated soils – An analytical approach

Cited by 34 publications

References 32 publications

Analytic approximations of Von Kármán plate under arbitrary uniform pressure—equations in integral form

Analytic approximations of Von Kármán plate under arbitrary uniform pressure—equations in integral form

New Analytical Solutions to 2-D Water Infiltration and Imbibition into Unsaturated Soils for Various Boundary and Initial Conditions

On the homotopy analysis method for backward/forward-backward stochastic differential equations

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