The RBDO problem presented in Equation2.27 can be solved directly applying a two level optimization approach, being the outer loop’s purpose to solve the optimization problem in terms of the design variables d and the inner loop’s purpose to solve the reliability analysis problem in terms of the random variables x. In this section the most popular two level RBDO methods are discussed, i.e. the Reliability Index Approach (RIA) and the Performance Measurement Approach (PMA). A flowchart of two level RBDO methods is presented in Figure 2.26.
Initial Design Structural
Analysis Reliability Analysis min G(d,u) subject to: kuk = βT min kuk subject to: G(d,u) = 0 Convergence? Optimum design Optimization algorithm Modified design yes no PMA RIA
Figure 2.26: Flowchart of the two-level methods.
2.4.2.1 Reliability Index Approach (RIA)
The Reliability Index Approach (RIA) was first introduced in Enevoldsen and Sorensen [66], where the classical RBDO problem is formulated in terms of the reliability index β as follows:
subject to:
gj(d, x) ≤ 0 (j = 1, ..., n) (2.28b) βi(d, x) ≥ βiT (i = 1, ..., m) (2.28c)
where βi and βiT are respectively the structural and target reliability indexes for the i-esime limit state. The RIA employs the First Order Reliability Method (FORM) to perform the reliability analysis and consequently obtain βi. Details of the FORM method were discussed at Section2.2.2.
After transforming the random variables x into uncorrelated and normalized variables u through u = T (x), the reliability index β can be obtained by solving the optimization problem presented in Equation 2.12. The larger the reliability index becomes, the safer the structural system is. The search scheme of the RIA consists on searching u∗ under zero or negative limit-state conditions (Equation 2.12b) relying on the linear approximations of the limit-state function ˜G at each design point uk. An illustration of the RIA search plan is presented in Figure2.27.
u1 u2 β4 β1 β3 β2 ˜ G1 ˜ G2 ˜ G3 ˜ G4 u1 u2 u3 u4= uM P P G(u) = 0
Figure 2.27: Illustration of the RIA search plan for a 2-variable case.
A flowchart of the RIA method where the nested optimization loops can be seen clearly is presented in Figure 2.28.
Initial Design Structural Analysis Reliability Analysis (FORM) Transform xk→ uk Evaluation of Gi(uk) and ∇uGi(uk) Obtention of βiand uk+1 Convergence of β? Transform uk+1 → xk+1 x∗ = xk+1 k=k+1 Convergence? Optimum design Optimization algorithm Modified design yes no yes no
Figure 2.28: Flowchart of the Reliability Index Approach (RIA) method.
2.4.2.2 Performance Measure Approach (PMA)
The Performance Measure Approach (PMA) was proposed in Tu et al [201] and in this method the probabilistic constraints are transformed to performance measures corresponding to the target reliability level βT. In the PMA the classical RBDO formulation is given by an inverse reliability analysis:
min F (d, x) (2.29a)
subject to:
gj(d, x) ≤ 0 (j = 1, ..., n) (2.29b) Gi(d, x) ≥ 0 (i = 1, ..., m) (2.29c)
where Gi represents the performance measure associated to the target reliability βiT, being evaluated through an inverse reliability analysis. The essential difference from the PMA respect to the RIA lies in solving the inner optimization loop corresponding to the RA. Although it also requires the transformation of the random variables x into uncorrelated and normalized variables u, the formulation of the problem is the inverse, imposing the target reliability index that the structure must reach (βiT) in each i limit-state function, as expressed in Equation2.30:
min Gi(d, u) (2.30a)
subject to:
kuki = βiT (i = 1, ..., m) (2.30b)
This change is done based on the principle that it is usually easier to solve an opti- mization problem with complex objective function and simple constraints than vice versa. The optimum point u∗ is the MPP correponding to a prescribed reliability βT. The search plan in the PMA consists of exploring the hyper-shere of radius βT aiming to find the point of Gi that is tangent to the hyper-sphere, which has proven to be far more robust than the search scheme of the RIA (Lee et al [128] or Aoues and Chateauneuf [9]). Figure 2.29 shows an example of the PMA search plan for a 2-variable case. u1 u2 βT uMPP Decrease of G Target reliability hyper-sphere (circle in 2D) G(u)
Figure 2.29: Illustration of the PMA search plan for a 2-variable case.
Instead of using general optimization algorithms, there are several ones that were devel- oped specifically for solving Equation2.30 such as the Advanced Mean Value (AMV),
the Conjugate Mean Value (CMV) and the Hybrid Mean Value (HMV). As stated in Youn et al [221], the AMV method is suitable for convex limit-state functions but shows instability and inefficiency in dealing with concave functions since the method updates the steepest direction α using just the information of the current MPP. On the other hand, the CMV method, which considers the information of the last three MPP to update α, improves the rate of convergence for concave functions although worsens when treating with convex ones. In this sense, the HMV method, which adaptively selects the AMV and CMV methods depending on the convexity of the limit-state function, was developed. These three methods are briefly described below:
Advanced Mean Value (AMV)
This method excels when the performance function Gi is convex, but if the function is concave the AMV tends to be slow and/or divergent.
1. The initial design point k = 0 corresponds to the mean value of the random variables in the u-space, uk = µ.
2. The value of the limit-state function G(uk) and its gradient with respect to the random variables ∇uG(uk) are obtained.
3. The steepest direction α is obtained as: αk = −
∇uG(uk)
k∇uG(uk)k
4. The next design point is obtained as: uk+1 = βT· αk. 5. Go back to step 2 until convergence of u.
Conjugate Mean Value (CMV)
This method excels when the performance function Gi is concave, but by contrast is slow and shows bad convergence when the function Gi is convex.
1. The initial design point k = 0 corresponds to the mean value of the random variables in the u-space, uk = µ.
2. The value of the limit-state function G(uk) and its gradient with respect to the random variables ∇uG(uk) are obtained.
3. The steepest direction α is obtained as: αk = −
∇uG(uk)
k∇uG(uk)k
4. For iterations k = 1, 2, 3 the next design point is obtained like in the AMV: uk+1 = βT·αk. However, when k ≥ 3 the next design point is obtained combining the search directions of the two previous iterations as: uk+1 = βT·
αk+αk−1+αk−2
kαk+αk−1αk−2k.
5. Go back to step 2 until convergence of u. Hybrid Mean Value (HMV)
The Hybrid Mean Value (HMV) first identifies if the performance function Gi is convex or concave and then employs the AMV (convex) or the CMV (concave). This algorithm can be described as follows:
1. Perform the AMV during three consecutive iterations.
2. Obtain the function type criteria ξ that identifies if the performance function Gi is concave or convex, and which is defined as: ξ = (αk+1− αk) · (αk− αi,k−1). 3. If ξ > 0 the performance function is convex and thus the AMV is used. Otherwise
if ξ < 0 the function is concave and the CMV is used. ξ > 0 → AM V
ξ < 0 → CM V
4. Go back to step 2 until convergence of u.
A flowchart of the PMA method where the nested optimization loops can be seen clearly is presented in Figure 2.30.