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Articles Published in International Journal for Numerical Methods in Engineering

  1. Liu, W.K., Belytschko, T., and Mani, A.. "Random field finite elements" International Journal for Numerical Methods in Engineering. vol. 23. 1986. pp. 1831--1845.


  2. Xiu, D. and Karniadakis, G.E.. "Supersensitivity due to uncertain boundary conditions" International Journal for Numerical Methods in Engineering. vol. 61. NOV 28 2004. pp. 2114--2138.

    We study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi-polynomial-chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar `stochastic supersensitivity' for the mean location of the transition layer. Subsequently, we employ up to fourth-order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a `truncated' Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied. Copyright (C) 2004 John Wiley Sons, Ltd.


  3. Huang, S.P., Quek, S.T., and Phoon, K.K.. "Convergence study of the truncated Karhunen-Loeve expansion for simulation of stochastic processes" International Journal for Numerical Methods in Engineering. 52 (9). 2001. pp. 1029--1043.

    A random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen-Loeve (K-L) series expansion is based on the eigen-decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non-stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K-L expansion are investigated by comparing the second-order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen-solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K-L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K-L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K-L expansion method is that it can be easily generalized to simulate non-stationary processes with little additional effort.