Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration

This paper presents a new module towards the development of efficient computational stochastic mechanics. Specifically, the possibility of an adaptive polynomial chaos expansion is investigated. Adaptivity in this context refers to retaining, through an iterative procedure, only those terms in a representation of the solution process that are significant to the numerical evaluation of the solution. The technique can be applied to the calculation of statistics of extremes for nongaussian processes. The only assumption involved is that these processes be the response of a nonlinear oscillator excited by a general stochastic process. The proposed technique is an extension of a technique developed by the second author for the solution of general nonlinear random vibration problems. Accordingly, the response process is represented using its Karhunen-Loeve expansion. This expansion allows for the optimal encapsulation of the information contained in the stochastic process into a set of discrete random variables. The response process is then expanded using the polynomial chaos basis, which is a complete orthogonal set in the space of second-order random variables. The time dependent coefficients in this expansion are then computed by using a Galerkin projection scheme which minimizes the approximation error involved in using a finite-dimensional subspace. These coefficients completely characterize the solution process, and the accuracy of the approximation can be assessed by comparing the contribution of successive coefficients. A significant contribution of this paper is the development and implimentation of adaptive schemes for the polynomial chaos expansion. These schemes permit the inclusion of only those terms in the expansion that have a significant contribution.