Available online at www.sciencedirect.com
Procedia Engineering 31 (2012) 1019 – 1023
International Conference on Advances in Computational Modeling and Simulation
Probabilistic Collocation Method for Yield Approach Index of Heterogeneous Slope Linfang Shena* a
Engineering institute , Beijing University, Beijingˈ100871, China
Abstract A stochastic uncertainty quantification approach has been introduced into heterogeneous slope stability evaluation with the aid of probabilistic collocation method. This method combines Karhunen-Loeve expansion and polynomial chaos expansion to estimate the stochastic properties by running the deterministic geomechanics simulation models independently. With this approach, yield approach index of a heterogeneous slope is computed. The results show that in the toe and middle region FAI have the largest value indicating more instability, and the standard deviation field demonstrates that uncertainty has the largest value in the toe.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Open access under CC BY-NC-ND license. Keywords: Karhunen-Loeve expansion, polynomial chaos expansion, probabilistic collocation, heterogeneity
1. Introduction Geological materials often have high degree of heterogeneity, which results the uncertainty of the numerical modeling. In dealing with this problem, stochastic field descriptions of the uncertainty input parameters are employed. Monte Carlo method is the most commonly used and easily implanted approach. With this method, a large number of realizations of the random input field are created. Deterministic equations are solved, and then the output field distribution can be obtained directly with statistical calculation. But it is time-consuming for that it needs a great number of realizations to make sure the accuracy. In some other methods, such as moment equations method and polynomial chaos expansion method, coupled statistical differential equations are generated which is difficult to be solved. * Corresponding author. Tel.:+86-010-62612240;. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.1136
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Linfang Shen / Procedia Engineering 31 (2012) 1019 – 1023
In this study, we present a new method, probabilistic collocation method (PCM), with combination of Karhunen-Loeve and polynomial chaos expansion. In this method, realizations are greatly reduced compared with Monte Carlo, and coupled equations are not confronted. We introduce this new method into geomechanical problem to confirm its validity and efficiency. 2. Mathematical formulation 2.1. Karhunen-Loeve expansion Let Y (x,T ) be a random input space function, where x D and T 4 (a probability space). One may write Y (x, T ) Y (x) Y c(x, T ) , where Y (x) the mean is and Y c(x,T ) is the fluctuation. The spatial property of the stochastic field can be described by the covariance CY (x, y) Y c(x, T )Y c(y, T ) ! . The covariance may be decomposed as:
CY (x, y )
f
Oi fi (x) fi (y ) ¦ i 1
(1)
Where Oi and f i (x) are eigenvalues and deterministic eigenfunctions, respectively, and can be solved from the following Fredholm equation:
³D CY (x, y) f (x)dx
(2)
Of ( y)
Then the stochastic field Y (x, T ) can be expressed as: f
(3)
Y (x) ¦ Oi fi (x)[i (T )
Y (x,T )
i 1
Where [ i (T ) are orthogonal Gaussian random variables with zero mean and unit variance. Take a truncated KL expansion in the computation: N
Y (x, T ) Y (x) ¦ Oi f i (x)[ i (T )
(4)
i 1
For a two-dimensional stochastic field, we assume that the covariance function CY (x, y) V Y2 exp( x1 y1 / K1 x 2 y 2 / K 2 ) in for example domain D
^( x1, x2 ) : 0 d x1 d L1;0 d x2 d L2 ` , V
CY (x, y) is separable, a rectangular
the eigenvalues and eigenfunctions can be obtained by 2 Y
and K are the variance and the correlation length of the combining those in each dimension, where field, respectively. The eigenvalues and eigenfunctions can be expressed as: 4K1K 2V Y2 (5) Oi (K12 Z12, j 1)(K 22 Z 22,i 1)
f i ( x1 , x2 )
f1, j ( x1 ) f 2,i ( x2 )
(6)
and
f k ,i ( x k ) where
K k Z k ,i cos(Z k ,i x k ) sin(Z k ,i x k ) (K k2 Z k2,i 1) Lk 2 K k
,
k
1,2
i 1,2,3!
(7)
Z ,i are the positive roots of the characteristic equation: (K 2Z 2 1) sin(ZL)
2KZ cos(ZL)
(8)
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Linfang Shen / Procedia Engineering 31 (2012) 1019 – 1023
2.2. Polynomial Chaos Expansion Since the input random fields are represented by Karhunen-Loeve expansion, dependent random fields of output spatial structure can be found with unknown covariance. In such case, polynomial chaos expansion may be used to describe the dependent random fields. With the polynomial chaos expansion, each random field can be expressed as:
y(x, T )
f
f
i1
a 0 (x) ¦ a i1 (x)*1 ([ i1 (T )) ¦¦ a i1i2 (x)*2 ([ i1 (T ), [ i2 (T )) ! i1 1
(9)
i1 1 i2 1
It can be truncated by finite terms, and can be written simply as P
y (x,T )
¦c
j
(x)< j ([ )
(10)
j 1
where c j (x) are the coefficients vectors, and < j ([ ) are Hermite Polynomials. 2.3 Probabilistic collocation method Consider a stochastic differential equation (11) Ly(x, t ,T ) f (x, t ) where y(x, t , T ) is the unknown random function of space x and time t , and f (x, t ) is the source term. L involves differentiations in space and can be nonlinear. Represent y(x, t ) by a set of truncated polynomial chaos expansion yÖ (x, t , T ) . P
c i (x, t )
