A new method for reliability analysis and reliability-based design optimization

Abstract

We present a novel method for reliability-based design optimization, which is based on the approximation of the safe region in the random space by a polytope-like region. This polytope is in its turn transformed into quite a simple region by using generalized spherical coordinates. The failure probability can then be easily estimated by considering simple quadrature rules. One of the advantages of the proposed approach is that by increasing the number of vertices, we can improve arbitrarily the accuracy of the failure probability estimation. The sensitivity analysis of the failure probability is also provided. We show that the proposed approach leads to an optimization problem, where the set of optimization variables includes all the original design variables and all the parameters that control the shape of the polytope. In addition, this problem can be solved by a single iteration scheme of optimization. We illustrate the performance of the new approach by solving several examples of truss topology optimization.

Appendix: Closed formula for the MPP

Since the random load is f = Fξ, and x ≥ ε, then c(x,ξ) = ξ^⊤F^⊤K(x)^− 1Fξ. By calling M(x) = F^⊤K(x)^− 1F, we have that the MPP can be found as the solution ξ^∗ to

$$\begin{array}{@{}rcl@{}} &\ &\underset{\boldsymbol{\xi}}\min \boldsymbol{\xi}^{\top} \boldsymbol{\xi} , \\ &&\text{s.t. } \boldsymbol{\xi}^{\top} \mathbf{M}(\mathbf{x}) \boldsymbol{\xi} \ge c_{0} . \end{array} $$

Note that the inequality constraint must be active at the solution (the MPP is in the failure surface), since the only possible interior solution is ξ^∗ = 0 which does not satisfy the constraint for a positive c₀. Hence, let λ be the nonnegative Lagrange multiplier of the constraint at the solution. The first order optimality condition is ξ^∗− λM(x)ξ^∗ = 0. The value λ = 0 is not a possible solution since it leads to ξ^∗ = 0. Hence λ > 0. Let μ = λ^− 1. Then, from the optimality condition we obtain M(x)ξ^∗ = μξ^∗. Hence ξ^∗ must be an eigenvector of M(x) of eigenvalue μ. Let v be a unit eigenvector of eigenvalue μ, i.e. M(x)v = μv with v^⊤v = 1. Then ξ^∗ = αv for certain value α. Since the constraint is active ξ^∗⊤M(x)ξ^∗ = α²v^⊤M(x)v = α²μ = c₀. Then α² = c₀/μ and ξ^∗ = αv = (c₀/μ)^1/2v. Since the objective function is ξ^∗⊤ξ^∗ = α² = c₀/μ, then μ must be the maximum eigenvalue of M(x).