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A Neumann expansion approach to flow through heterogeneous formations (1991)

Zeitoun, D. G. ; Braester, C.

Springer

Stochastic environmental research and risk assessment 5 (1991), S. 207-226

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ISSN: 1436-3259

Keywords: Stochastic groundwater flow ; Neumann expansion ; stochastic partial differential equation

Source: Springer Online Journal Archives 1860-2000

Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences

Notes: Abstract A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF01544058

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2

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A least-squares procedure for the solution of transport problems (1990)

Bensabat, J. ; Zeitoun, D. G.

Chichester : Wiley-Blackwell

International Journal for Numerical Methods in Fluids 10 (1990), S. 623-636

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ISSN: 0271-2091

Keywords: Least-squares method ; Optimal control problem ; Conjugate gradient ; Advection-diffusion equation ; Perturbation series ; Engineering ; Engineering General

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics

Notes: In this paper a least-squares formulation associated with a conjugate gradient algorithm is proposed for the solution of transport problems. In this procedure the advection-diffusion equation is first discretized in time using an implicit scheme. At each time step the resulting partial differential equation is replaced by an optimal control problem. This minimization problem involves the minimization of a functional defined via a state equation. This functional is chosen in order to force the numerical solution of the advection-diffusion equation to be equal to the hyperbolic advective part of this equation. The effectiveness of the method is shown through a one-dimensional example involving advective and diffusive transport. No oscillation and high accuracy have been obtained for the entire range of Peclet numbers with a Courant number well in excess of unity.

Additional Material: 4 Ill.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1002/fld.1650100603

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3

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A weighted least squares method for first-order hyperbolic systems (1995)

Zeitoun, D. G. ; Laible, J. P. ; Pinder, G. F.

Chichester : Wiley-Blackwell

International Journal for Numerical Methods in Fluids 20 (1995), S. 191-212

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ISSN: 0271-2091

Keywords: Weighted least squares ; Hyperbolic system of equations ; Shallow water equations ; Newton-Raphson method ; Engineering ; Engineering General

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics

Notes: The paper presents a generalization of the classical L2-norm weighted least squares method for the numerical solution of a first-order hyperbolic system. This alternative least squares method consists of the minimization of the weighted sum of the L2 residuals for each equation of the system. The order of accuracy of global conservation of each equation of the system is shown to be inversely proportional to the weight associated with the equation. The optimal relative weights between the equations are then determined in order to satisfy global conservation of the energy of the physical system.As an application of the algorithm, the shallow water equations on an irregular domain are first discretized in time and then solved using Laplace modification and the proposed least squares method.

Additional Material: 3 Ill.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1002/fld.1650200302

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4

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Derivation of material variability from settlement measurements (1993)

Zeitoun, D. G. ; Uzan, J.

New York, NY [u.a.] : Wiley-Blackwell

International Journal for Numerical and Analytical Methods in Geomechanics 17 (1993), S. 303-322

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ISSN: 0363-9061

Keywords: Engineering ; Engineering General

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Architecture, Civil Engineering, Surveying , Geosciences

Notes: In pavement evaluation using non-destructive testing (NDT), a large amount of deflection bowls are analysed in terms of the elastic moduli of the layers. The results are used to evaluate the material variability, which could serve in an overlay design procedure based on the concept of reliability. The model currently used for interpreting deflection bowls is based on the random variable theory which neglects the spatial distribution of the elastic modulus of the material. Since the subgrade and pavement materials have a spatial distribution, the analysis of NDT could lead to an underestimate of the material variability. The random field theory, which is more adequate than the random variable theory, is presented and used to correct the NDT analysis.The pavement is modelled as a multilayer random elastic medium with, for each layer, a constant Poisson's ratio and a random shear modulus characterized by three statistical moments: average value, standard deviation and autocorrelation function. The stochastic integral formulation presented in the previous publications is generalized here for a multilayer system. The multilayer system is analysed with the random field theory and the covariance matrix of the deflection bowl is obtained and used to generate deflection bowls corresponding to the properties of the random field. These bowls are then interpreted with the current procedure and elastic modulus variabilities are computed. It is found that the current procedure for interpreting deflection bowls underestimates the variability of the subgrade, by a factor of 0.4-1.0. It is interesting to note that the average moduli of the Boussinesq layer and of the two layers are not affected by the type of theory used. The variability of the upper layer in the two-layer system is also unaffected for a small variation range.

Additional Material: 7 Ill.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1002/nag.1610170503

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An iterative penalty method for the least squares solution of boundary value problems (1997)

Zeitoun, D. G. ; Laible, J. P. ; Pinder, G. F.

New York, NY [u.a.] : Wiley-Blackwell

Numerical Methods for Partial Differential Equations 13 (1997), S. 257-281

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ISSN: 0749-159X

Keywords: boundary value problems ; collocation least squares method ; augmented Lagrangian method ; Uzawa's algorithm ; preconditioned conjugate gradient ; Mathematics and Statistics

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Mathematics

Notes: This article is concerned with iterative techniques for linear systems of equations arising from a least squares formulation of boundary value problems. In its classical form, the solution of the least squares method is obtained by solving the traditional normal equation. However, for nonsmooth boundary conditions or in the case of refinement at a selected set of interior points, the matrix associated with the normal equation tends to be ill-conditioned. In this case, the least squares method may be formulated as a Powell multiplier method and the equations solved iteratively. Therein we use and compare two different iterative algorithms. The first algorithm is the preconditioned conjugate gradient method applied to the normal equation, while the second is a new algorithm based on the Powell method and formulated on the stabilized dual problem. The two algorithms are first compared on a one-dimensional problem with poorly conditioned matrices. Results show that, for such problems, the new algorithm gives more accurate results. The new algorithm is then applied to a two-dimensional steady state diffusion problem and a boundary layer problem. A comparison between the least squares method of Bramble and Schatz and the new algorithm demonstrates the ability of the new method to give highly accurate results on the boundary, or at a set of given interior collocation points without the deterioration of the condition number of the matrix. Conditions for convergence of the proposed algorithm are discussed. © 1997 John Wiley & Sons, Inc.

Additional Material: 15 Ill.

Type of Medium: Electronic Resource

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