An algorithm to develop a stochastic model of the response using adaptive and sparse Polynomial Chaos Expansion (PCE) is proposed in this paper. This new algorithm is highly advantageous compared to the regular PCE in terms of the number of samples required and the computational cost to build a stochastic model. As a sampling strategy, the A-optimal design that minimizes the average of the variance of the PCE coefficients, and hence, increases the accuracy of the stochastic model is used. A non-intrusive stochastic collocation approach that uses linear regression is used to obtain the expansion coefficients, and the contribution of the individual expansion terms to the total variance of the response are used to form the sparse PCE. Also, a weighted sampling plan is utilized for the new sparse PCE to find a high-fidelity stochastic model which is then followed by the convergence analysis of the Kullback-Leibler (KL) divergence of the stochastic model's probability density function (PDF), not just the mean and standard deviation. The proposed algorithm has been implemented for two analytical problems and a fracture mechanics problem involving many random input variables to investigate its validity. The comparison of the results obtained with this algorithm to regular full PCE, q-norm truncation strategy for orthogonal polynomials, and Latin Hypercube Sampling (LHS) simulations are also provided. The results obtained from the application problems showed that highly accurate PCE models are obtainable with very few numbers of function evaluations and low computational cost than regular full PCE and q-norm truncation strategy with LHS samples. The absolute percentage error in mean and standard deviation of the converged PCE model was less than 0.2% and verified the superiority of this algorithm.
展开▼
