A Fast Newton-Raphson Method in Stochastic Linearization

Owing to its accessible implementation and rapidity, the equivalent linearization has become a common probabilistic approach for the analysis of large-dimension nonlinear structures, as encountered in earthquake and wind engineering. It consists in replacing the nonlinear system by an equivalent linear one, by tuning the parameters of the equivalent system, in order to minimize some discrepancy error. Consequently classical analysis tools such as the spectral analysis may be reconditioned to approximate the solution of structures with slight to moderate nonlinearities. The tuning of the equivalent parameters requires the solution of a set of nonlinear algebraic equations involving integrals. It is typically performed with the fixed-point algorithm, which is known to behave poorly in terms of convergence. We therefore advocate for the use and implementation of a Newton-Raphson approach, which behaves much better, even in its dishonest formulation. Unfortunately, this latter option requires the costly construction of a Jacobian matrix. In the approach described in this paper, this issue is answered by introducing a series expansion method that provides a fast and accurate estimation of the residual function (whose solution provides the equivalent parameters) and a fast and approximate estimation of the Jacobian matrix. An illustration demonstrate the good accuracy obtained with the proposed method.