II. BACKGROUND
II.5 Reliability Based Design Optimization
One final reliability method to be studied in the context of M-E pavement design is the Advanced Mean Value Method. This method is similar to the FORM method, but the AMV method makes one simplifying assumption. The AMV method assumes that when the limit state function approaches zero, that point represents the most probable point. Therefore, the limiting function can be forced to zero by changing the β value. This method has an advantage over the second moment method in accuracy because it, like FORM, uses computation in the rotationally symmetric standard uncorrelated normal space. However, while AMV in general is not as accurate as FORM due to the imprecise calculation of the MPP, it only needs to evaluate the gradients of the limit state function once. Because the u-space gradients, evaluated at the origin in u-space, are used to approximate the α vector, system reliability analysis can be performed.
𝑃1 = min𝑑{𝑐0(𝑑)|𝑝𝑘(𝑑,𝑥) ≤ 𝑝̂𝑘} (II.37)
The objective function is a cost function (co) of stochastic variables (d) and the constraint requires evaluation of the probability of failure with respect to both the stochastic and deterministic variables (x) for each k constraint function.
The P2 problem seeks to minimize the failure probability of the component (k) with the largest probability of failure, termed the critical component. This formulation minimizes the maximum probability of failure, but does not guarantee that the probability of failure meet a required threshold. Again, the probability of failure is with respect to both stochastic and deterministic variables.
𝑃2 = min𝑑{max𝑘𝑝𝑘(𝑑,𝑥)} (II.38) The third formulation (P3) differs slightly from the P1 problem in that the cost function is written with respect to the probability of failure of the components.
𝑃3 = min𝑑{𝑐0(𝑑) +∑𝐾𝑘=1𝑐𝑘(𝑑)𝑝𝑘(𝑑,𝑥)|𝑝𝑘(𝑑,𝑥) ≤ 𝑝̂𝑘} (II.39)
These three formulations are easily re-written for system optimization problems.
II.5.2 RBDO Problem Formulations
Various formulations of RBDO methods have been developed and applied to numerous engineering applications (49). Solution techniques utilizing First Order Reliability Methods have been shown effective for both component and system RBDO problems.
RBDO using Efficient Global reliability Analysis (EGRA) provides another practical design process that has been shown to be accurate and efficient.
Two popular formulations for the FORM based optimization method are common: single loop direct FORM (also known as the reliability index approach) and inverse FORM (also known as performance measure approach). The single loop direct FORM based model can be described mathematically as:
min𝒅,𝒙𝑓(𝒅) (II.40)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:𝑔(𝒅,𝒙) = 0 (II.41)
𝒖∗
‖𝒖∗‖=−‖∇∇ug(𝐝,𝐱)
ug(𝐝,𝐱)‖ (II.42)
‖𝒖∗‖= 𝛽𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 (II.43)
Where the objective function, Eq. II.40, is minimized with respect to design parameters (d) and random variable input parameters (x). Equations II.41 and II.42 are the constraints required to satisfy the Karush-Kuhn-Tucker (KKT) conditions, and Eq. II.43 is the
reliability constraint required by FORM. The reliability index (βrequired) is defined as the norm of the vector of input parameters transformed to the equivalent standard normal space (u). The vector of variables in the transformed space is related to the gradient of the limit state equation (g) and its norm.
The single loop inverse FORM model can be described as:
min𝒅,𝒙𝑓(𝒅) (II.44)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:𝑔(𝒅,𝒙) ≥0 (II.45)
𝒖∗
‖𝒖∗‖=−‖∇∇ug(𝐝,𝐱)
ug(𝐝,𝐱)‖ (II.46)
‖𝒖∗‖= 𝛽𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 (II.47)
Here, the objective function is similar to the direct FORM method, Eq.s II.46 and II.47 satisfy the KKT conditions, and Eq. II.45 satisfies the inverse FORM optimality condition. The inverse FORM method has numerous advantages, as described by McDonald and Mahadevan in (49).
FORM methods are efficient with respect to required function evaluations, but approximations in these methods can cause them to fail to find the MPP. Methods such as the direct FORM method can result in inaccuracies if the approximation to the shape of the limit state is poor. The EGRA process is an alternative to these FORM based methods that improves accuracy and maintains efficiency, primarily through the use of surrogate
modeling. EGRA evaluates the function at a small number of samples, constructs a surrogate model for the function, and solves an auxiliary optimization problem finding the point of maximum expected feasibility. This point of feasibility is determined through an expected feasibility function which searches for potential training points near the limit state, the area where accuracy is most important. The process iterates by selecting this point as a new training point and re-training the surrogate repeatedly until the expected feasibility converges. The final surrogate model can then be used to make predictions of reliability for the true function.
The EGRA RBDO method can be formulated as a nested, single-loop, or sequential optimization problem. The nested loop formulation is the most computationally expensive process as each iteration requires full reliability analyses and no information from these analyses are shared in later iterations. A single-loop method improves the efficiency of the nested method through use of a surrogate model that evaluates the reliability across a domain rather than at individual candidate points. The potential for model error is introduced with the inclusion of the surrogate, but results are easily verified after convergence of the EGRA analysis. Sequential formulation improves on the single-loop process by intermittently improving the accuracy of the surrogate model to incorporate verification into the iterative process. (50)