A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

Gaussian closure method is commonly used in the analysis of nonlinear stochastic systems. However, Gaussian closure may lead to unacceptable errors when system response is very much different from being Gaussian, and accuracy of the method decreases as the nonlinearity of the system increases. The need for better accuracy in strongly non-linear problems has caused the development of non-Gaussian closure schemes. In this paper, we develop a new copula-based Gaussian mixture closure method for randomly excited nonlinear systems. Our method relies on the assumption of marginal PDF of response in terms of finite Gaussian mixture model, and the derivation of joint PDF with aid of dependence modeling of Gaussian copula. By substituting the non-Gaussian PDF representation into moment equations of nonlinear system, we further develop an optimization-based closure scheme for the solution of the unknown parameters in joint PDF. In this way, PDF and thus, moments of response of highly nonlinear system can be described in a more flexible and robust way. Effectiveness of the new closure method is demonstrated by a nonlinear and a Duffing oscillator that are subjected to Gaussian white noise. The results are compared with the Gaussian closure and exact solution. It has been shown that Gaussian closure is a special case of the new closure method, and accuracy of Gaussian closure is the lower bound of that of the new closure method.