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 a n adaptive polynomial chaos expansion is investigated. Adaptivity in t his context refers to retaining, through an iterative procedure, only those terms in a representation of the solution process that are signi ficant to the numerical evaluation of the solution. The technique can be applied to the calculation of statistics of extremes for nongaussia n processes. The only assumption involved is that these processes be t he response of a nonlinear oscillator excited by a general stochastic process. The proposed technique is an extension of a technique develop ed by the second author for the solution of general nonlinear random v ibration problems. Accordingly, the response process is represented us ing its Karhunen-Loeve expansion. This expansion allows for the optima l 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 orthog onal set in the space of second-order random variables. The time depen dent coefficients in this expansion are then computed by using a Galer kin 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 approximat ion can be assessed by comparing the contribution of successive coeffi cients. A significant contribution of this paper is the development an d implimentation of adaptive schemes for the polynomial chaos expansio n. These schemes permit the inclusion of only those terms in the expan sion that have a significant contribution. (C) 1997 Elsevier Science L td.