Numerical Results

Numerical results are for a = 60, b = 40, a finite difference mesh with equal step of 10 in the x and v directions, Q(x, v) = 1 in D so that the load vector is Y = у = 1, and foundation stiffness К with parameters ai = 1, «2 = 7, n = 12, a> = l/v/б, v = – V4 = (1, 2), V2 = —V5 = (2, 1), and v3 = —v6 = (2, 2). The resulting stochastic algebraic equation depends on n = 12 independent N(0, 1) variables, the entries of Z, and has dimension d = 15, so that X and Y are vectors in R15.

We use for the optimal Galerkin solution nr = 5 rings and partition of 5n(1) as follows. The sphere 52(1) has been divided in 4 equal parts. The image of each of this parts in 53(1) has been divided in two equal parts. The division in two equal parts has been continued to partition the sphere 5k(1), k = 4,…, 12. The resulting number of partitions of 5i2(1) is 4096. Two partitions of have been used to construct sub-optimal Galerkin solutions, the partition {Aq} defining the optimal Galerkin solution with zq e Aq and a coarser partition. The corresponding solutions are referred to as sub-optimal 1 and sub-optimal 2 Galerkin solutions, respectively. The points zq for the sub­optimal 2 Galerkin are the points of intersections of spheres of radii (ru—1 + ru)/2, u = 1,… ,nr, with the coordinates of Rn. Each of these spheres intersects the coordinates of Rn at 2 n points, so that there are m = 2 nnr = 70 points zq and subsets Aq. The probability of the sets Aq is P(Aq) = 1/(2nnr) for all q’s.

Let Xms and Xmsn denote the plate displacement at the mid span node of the finite difference mesh and at a node neighboring it, respectively. Figure 1 shows estimates of the probability density function of Xms obtained by Monte Carlo simulation and optimal/sub-optimal Galerkin solutions. The estimate obtained by the sub-optimal 2 Galerkin solution exhibits relatively large fluctuations. The estimates of the density of Xms by the other Galerkin solutions are similar to that obtained by direct Monte Carlo. The density of Xms by the sub-optimal 1 Galerkin solution and Monte Carlo simulation nearly coincide. The Monte Carlo estimates have been calculated from 100,000 independent samples of Z and the Galerkin estimates have been calculated from Equations (11) and (12). Figure 2 shows estimates of the joint probability density of (Xms, Xm, sn). The estimates of this density by optimal and sub-optimal 1 Galerkin solutions and by Monte Carlo are similar. The sub­optimal 2 Galerkin solution provides a less satisfactory approximation. As in the previous figure, the

Monte Carlo estimates are based on 100,000 independent samples of Z and the Galerkin solutions have been calculated from Equations (11) and (12). The differences between the mean displacements by Galerkin solutions and Monte Carlo simulation are less then 0.0958%, 0.1267%, and 1.1056% for optimal, sub-optimal 1, and sub-optimal 2 Galerkin solutions, respectively. These differences and the plots in Figures 1 and 2 show that the optimal Galerkin solution is closer to the Monte Carlo result in the average, consistent with the fact that this solution is unbiased, but the sub-optimal 1 Galerkin solution provides a superior approximation for the probability law of the displacement field.

2. Conclusions

Galerkin solutions have been presented for a class of stochastic algebraic equations, that is, linear algebraic equations with random coefficients. The construction of the Galerkin solution is based on partitions of the sample space associated with the random parameters in the definition of a stochastic algebraic equation. It was shown that there is a Galerkin solution for an arbitrary stochastic algebraic equation that is optimal in the mean square sense. Moreover, the optimal Galerkin solution is equal to the conditional expectation of the exact solution of a stochastic differential equation taken with respect to a a – field coarser than the a-field relative to which this solution is measurable.

Galerkin solutions that are not optimal are referred to as sub-optimal. Algorithms are presented for the construction of both optimal and sub-optimal Galerkin solutions.

Optimal and sub-optimal Galerkin solutions have been used to calculate statistics for the displace­ment of the simply supported plate sitting on a random elastic foundation. The elastic foundation is modeled by a homogeneous translation random field with uniform marginal distribution. The accur­acy of the Galerkin solutions depends on the partition of the sample space used in their definition, and was evaluated by Monte Carlo simulation.


Babuska, I. M., Temptone, R., and Zouraris, E. (2004). Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM Journal of Numerical Analysis, 42(2), 800-825.

Deb, M. K., Babuska, I. M., and Oden, J. T. (2001). Solution of stochastic partial differential equa­tions using Galerkin finite element techniques. Computational Methods in Applied Mechanics and Engineering, 190, 6359-6372.

Field, Jr., R. V. and Grigoriu, M. (2004). On the accuracy of the polynomial chaos approximation.

Probabilistic Engineering Mechanics, 19(1-2), 65-80.

Ghanem, R. G. and Spanos, P. D. (1991). Stochastic Finite Elements: A Spectral Approach. Springer – Verlag, New York.

Grigoriu, M. (1995). Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, andMATLAB Solutions. Prentice Hall, Englewoods Cliffs, NJ.

Grigoriu, M. (2002). Stochastic Calculus. Applications in Science and Engineering. Birkhauser, Boston.

Resnick, S. I. (1998). A Probability Path. Birkhauser, Boston.

H. P. Hong

Department of Civil and Environmental Engineering,
University of Western Ontario, N5A 6B9 Canada


A new sampling technique referred to as the hypercube point concentration sampling technique is proposed. This sampling technique is based on the concepts of the Latin hypercube sampling technique and the point concentration method. In the proposed technique, first, the probability density function of the random variables is replaced by a sufficiently large number of probability concentrations with magnitudes and locations determined from the moments of the random variables. In other words, the probability density function is replaced by the probability mass function determined based on the point estimate method. The probability mass function is then used with the Latin hypercube sampling technique to obtain samples. For evaluating statistics of a complicated performance function of an engineering system, the proposed technique could be more efficient than the Latin hypercube sampling technique since for a given simulation cycle the required number of evaluations of the performance function in the former is less than that in the latter. The proposed sampling technique is illustrated through numerical examples.


Many random variables are involved in an engineering system. Direct or simple Monte Carlo method and efficient reliability methods such as the first-order and second-order reliability methods (Madsen et al. 1986) can be employed to carry out probabilistic assessment of the system. The reliability methods are almost exclusively used to estimate the probability of failure while the Monte Carlo method are employed to calculate the statistics of the responses or the probability of failure of the system. Although the direct Monte Carlo method is simple to use, however it can be computationally intensive. To reduce the number of simulation cycles, more efficient simulation methods can be used (Iman and Conover 1980, Rubinstein 1981).

In this study, a new sampling technique is proposed. This new sampling technique is based on the Latin hypercube sampling (LHS) technique and the point estimate method (Rosenblueth 1975, 1981). In the LHS technique (Iman and Conover 1980), the domain of the random variables is partitioned into many mutually exclusive and collectively exhaustive hypercubes. The hypercubes that do not have common domain in subspaces are selected randomly; and a point within each of the selected hypercubes is chosen randomly for the analysis. For the proposed technique, the marginal probability density functions of the random variables are replaced by probability mass functions with a sufficiently large number of concentrations. The locations and their associated probability concentrations are obtained based on Rosenblueth’s point estimate method (Rosenlueth 1975, 1981). The probability mass functions are then used with the Latin hypercube sampling technique to generate values of the random variables for the simulation analysis. The advantage of the new technique is that it requires less number of evaluations of the performance function of the system than the LHS technique. This is because several generated samples are likely to fall into a same concentration point. The proposed technique is described in detail in the following, and is illustrated by numerical examples.


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 527-536.

© 2006 Springer. Printed in the Netherlands.