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Figure 19.3 The Turning Bands Method. (Mantoglu, A. and Wilson, J. L., 1982, The Turning Bands Method for Simulation of Random Fields Using Line Generation by a Spectral Method, Water Resources Research, Vol. 18, No. 5, pp. 1379 1394, (1982) American Geophysical Union, Reproduced by permission of American Geophysical Union.)
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two-dimensional case, 4 16 lines are adequate, more lines yielding ner results. In the three-dimensional case, they cite the nding of Journel and Huijbregts (1978) that accurate results are obtained with 15 lines connecting the opposite midpoints of the edges of a regular icosahedron (or, equivalently, the nodes of a regular dodecahedron) enveloping the region. To generate the random process along the turning bands lines, it is necessary to develop a radial spectral density function f ( ) corresponding to the covariance function C(r). Mantoglu and Wilson give the following pair for a simple exponential correlation C(r) = 2 exp( br) f ( ) = and for a double exponential C(r) = 2 exp( b2 r 2 ) f ( ) = /b exp 2b 2b
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/b b[1 + ( /b)2 ]3/2
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(19.6)
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(19.7)
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They further show that the spectral density function of the one-dimensional process, S1 ( ), is related to the radial spectral density function by S1 ( ) = 2 f ( ) 2 (19.8)
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The one-dimensional process along line i is generated by summing M harmonics
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[S1 ( k ) ]1/2 cos( k + k )
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(19.9)
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in which k are independent random angles uniformly distributed between 0 and 2 , k = (k 1/2) , and k = k + . The range of frequencies is [ , + ], and is /M. The term is a small random frequency added to prevent periodicities. Mantoglu and Wilson give examples showing that, for a correlation distance b 1 , adequate accuracy results from using = 40b and M = 50 to 100. It should be noted that the points N at which the random eld is simulated are not necessarily the centroids of simple elements. They could be Gauss integration points for more complicated elements. Therefore, this approach could be used in conjunction with sophisticated elements incorporating higher order expansions or coupled behavior. The turning bands method has found its greatest application in the analysis of water resources and ground water hydrology. Gui et al. (2000) used the turning bands technique in successive Monte Carlo simulations of a homogeneous embankment. They assumed that the hydraulic conductivity was logNormally distributed about a constant mean value and that the spatial correlation was anisotropic. Therefore, the random eld is de ned by the mean lnK , the standard deviation lnK , and the spatial covariance function 1/2 vy 2 vx 2 2 CP (v) = P exp + (19.10) Ix Iy where P is the function ln(K), Ix and Iy are the correlation lengths of P in the x and 2 y directions, P is the variance of P , v is the offset vector with components vx and vy . For each realization of the random eld, 50 simulations were run. The ow regime was evaluated at 50, 100, and 500 days, and the factor of safety and reliability index were computed. In this case, the other material properties were held constant. They concluded that the reliability index was very sensitive to the uncertainty in K. They also found that the factor of safety was adequately modeled as Normally or logNormally distributed when lnK 0.5, but was not well represented by either distribution when lnK > 0.5.
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19.2.2 Local Average Subdivision
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The local average subdivision (LAS) method grew out of Fenton s doctoral work at Princeton University (Fenton 1990; Fenton and Vanmarcke 1991). The method works by successive division of the region over which the random eld is to be de ned. At each stage of the process the equal subdivisions must (1) have the correct variance according to local averaging theory, (2) be properly correlated with each other, and (3) average to the parent value. Fenton, Grif ths, and their colleagues (Fenton and Grif ths 1996; Fenton et al. 1996; Paice et al. 1996; Grif ths and Fenton 1997; Grif ths and Fenton 1998) have used this technique to study a variety of problems, primarily in steady-state seepage through porous media. Grif ths and Fenton (1998) studied the classic problem of a sheet-pile wall penetrating into a single layer of soil. The particular geometry they used consisted of 3.2 m of soil and sheet piling penetrating 1.6 m. They performed 2000 realizations for each two-dimensional case and 1000 for each three-dimensional case. The variation in the exit gradient was considerably higher than that of other quantities of interest. They found that, for all values of the coef cient of variation of the hydraulic conductivity, the maximum values of the mean and standard deviation of the exit gradient corresponded to a correlation distance of 2 4 m. In other words, the most severe impact of randomness in the
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