Keywords: response surface method, first order reliability method, sequential quadratic program solver, probability of failure, structural reliability, finite element method.

The estimation of the probability of failure by means of the Monte Carlo method and the finite element method is generally an expensive task. Several authors have focused attention on the response surface method (RSM) [1]. The objective of this method is to replace the limit state function (LSF) with a simple polynomial expression. The further application of an importance sampling simulation permits a fast estimation of the probability of failure, because the response surface (RS) is used instead of the LSF in the evaluation of the structural performance.

The sequential quadratic programming (SQP) and the RS methods are coupled in the paper in order to solve reliability problems in an accurate and efficient way. From a computational point of view, the disadvantage of the SQP method is that it works on the Hessian matrix of the associated Lagrangian.

Three update methods of the Hessian matrix of the LSF are compared. The first two are the well known Broyden-Fletcher-Goldfarb-Shanno (BFGS) [2] and the Symmetric Rank One (SR1) [3] formulas. The third one uses a rotation of the coordinate axes in order to approximate the second order derivatives by fitting a RS along the principal directions of the Hessian matrix. Two different initial conditions of the Hessian matrix are tested: the identity matrix and numerical approximation from samples of the LSF.

The proposed improvement of the RSM is tested with a set of five examples concerning applications in the structural field and problems with a large number of random variables. The examples show that the coupling of the SQP and RS methods is an interesting alternative to the classical RSM in terms of accuracy and efficiency.

In most of the examples, a significant reduction of the CPU time has been obtained. Within the SQP method, it has been observed that the initialization of the Hessian matrix as the identity matrix may lead to non-convergence to the solution.

In terms of efficiency, the Hessian update based on the spectral decomposition is highly advisable. The reduction of the CPU time is obtained by a fitting second order RS without cross-terms in the principal coordinate system. Then a good approximation of the Hessian matrix leads to faster convergence than using the BFGS and SR1 formulas.

1

C.G. Bucher, U. Bourgund, "A fast and efficient response surface approach for structural reliability problems", Structural Safety, 7, 57-66, 1990. doi:10.1016/0167-4730(90)90012-E

2

J. Nocedal, S.J. Wright, "Numerical optimization", 2nd ed. New York, Springer, 2006. doi:10.1007/b98874

3

M.J. Goldsmith, "Sequential quadratic programming methods based on indefinite hessian approximations", PhD Thesis, Stanford University, USA, 1999.

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