8452 X Ma,N.Zabaras foumal of Computational Physics 227(2008)8448-8471 where.the equality is interpreted in the l (o)sense and there is a one-to-one con spondence between r(( (平平》=（Ψ (19) 20 W( (21) X@=∑aΨ) (22) ()ndmntee P+1- (23) ygmciCEeeppammatctkaimpmmcp威nm Remark 1.The truncated GPCE expansion is characterized by the stochastic dimension and the order of the expansion.The checks are made a priori in order to determine the optimal order of the GPCE. 4.Stochastic finite element method formulation In this section.we will present the complete stochastic finite element formulation for this problem 4.1.Non-polynomial function evaluations of stochastic spectral expansion First.let us determine the spectral expansion of the product of the formcab-).where a andbare given by a0=工aΨ，(.b(8=∑bΨ，( (24 We want to find the coefficients cy of the expression (25) -含20 (26)where, the equality is interpreted in the L2ðXÞ sense and there is a one-to-one correspondence between Cnðni1 ðxÞ; ... ; nin ðxÞÞ and WjðnÞ. Since each type of polynomial in the Askey-series forms a complete basis for L2ðXÞ, we can expect the GPCE to converge to any L2 random process in the mean-square sense. The orthogonality relation of the Wiener–Askey polynomial chaos takes the form hWiWji¼hW2 i idij; ð19Þ where dij is the Kronecker delta and h; i denotes the ensemble average, which is the inner product in the Hilbert space of the variables n, hfðnÞgðnÞi ¼ Z fðnÞgðnÞWðnÞdn: ð20Þ Here, WðnÞ is the weighting function corresponding to the Wiener–Askey polynomial chaos basis Wj [9]. Note that, some types of orthogonal polynomials from the Askey scheme have weighting functions the same as the probability function of certain types of random distributions. For example, the weighting function of the p-dimensional Hermite polynomial is just the probability density function of multivariate standard normal distribution, i.e., WðnÞ ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffi ð2pÞ p p e1 2nTn : ð21Þ In practice, we then choose the type of independent variables n in the polynomials fWjðnÞg according to the type of random distribution and truncate the expansion at finite term P, i.e., XðxÞ ¼ XP j¼0 a^jWjðnÞ: ð22Þ The total number of expansion terms is (P + 1) and is determined by the dimension (M) of random vector n and the highest order (n) of the polynomials fWjg: P þ 1 ¼ ðn þ MÞ! n!M! : ð23Þ We choose a suitable order of the GPCE to capture strong non-linear dependence of the solution process on the input uncer￾tainty (uncertainty quantification or uncertainty propagation process). Remark 1. The truncated GPCE expansion is characterized by the stochastic dimension and the order of the expansion. The stochastic dimension is determined by the number of terms M in the truncated K–L expansion of the input random processes. Since the accuracy of the truncated GPCE depends on the order of the expansion, we require techniques to determine the optimal truncation order. We use the weak-Cauchy convergence criterion for this purpose [13]. Let the guess for optimal order be q. Then we construct an order m GPCE, where m ¼ q þ 1; q þ 2. In the criterion, we require that the L2 norm of the difference in the two approximations be negligible. Note the convergence should hold point-wise and these checks are made a priori in order to determine the optimal order of the GPCE. 4. Stochastic finite element method formulation In this section, we will present the complete stochastic finite element formulation for this problem. 4.1. Non-polynomial function evaluations of stochastic spectral expansion First, let us determine the spectral expansion of the product of the form c ¼ ab ¼ PP i¼0ciWiðnÞ, where a and b are given by aðnÞ ¼ XP i¼0 aiWiðnÞ; bðnÞ ¼ XP i¼0 biWiðnÞ: ð24Þ We want to find the coefficients ck of the expression cðnÞ ¼ XP k¼0 ckWkðnÞ ¼ XP i¼0 XP j¼0 aibjWiðnÞWjðnÞ: ð25Þ Following the method introduced in [42], we perform a Galerkin projection onto the polynomial orthogonal basis and use the orthogonality of the basis discussed in the previous section. Then the expression of the coefficients can be found as ck ¼ XP i¼0 XP j¼0 hWiWjWki hW2 k i aibj; ð26Þ 8452 X. Ma, N. Zabaras / Journal of Computational Physics 227 (2008) 8448–8471