In first-order methods, a linear approximation of the limit state is used to obtain the failure probability of Eq. 1) as depicted by the dashed line in Fig. This results in a multiplicative effect on the computational effort (i.e., the number of performance analyzes required for optimization is multiplied by the number of analyzes required for reliability evaluation.) For real design problems with significant computational effort for a single performance analysis, simple direct or reverse FORM RBDO may be insoluble. As in the previous chapter, η denotes all the input random variables of the system in uncorrelated standard normal space.
In this formulation, the limit state gη is a function of the input variables and auxiliary versions û of the intermediate variables uη(η). Similarly, the coefficients b0, bx and bu are derived from a first-order approximation of the compatibility constraint (between two disciplinary analyses), uη(η)-û. This is due to the fixed point iteration in the evaluation of the limit state and the gradient.
However, if the system limit state is differentiable near the MPP and the multidisciplinary system is feasible at the MPP, there is a solution for the FORM formulation. In this situation, a further modification of the multi-constraint FORM method would be necessary for the MPP search. However, as an alternative to fully integrated analysis, one can opt for the simultaneous analysis and design approach, as applied in the context of the direct FORM reliability analysis algorithms presented in Chapter II.
The gradient for the probabilistic constraint using inverse FORM (Equation 4)) is simply the gradient of the limit state with respect to the design variable vector, d. Sequential RBDO using inverse FORM is the basis of the SORA (sequential optimization and reliability analysis) method proposed by Du and Chen (2003). The difference is that the MPP search uses the SAND reliability analysis as in Eq. 14) instead of the fully integrated multidisciplinary analysis.
The Rackwitz-Fiessler (R-F) Newton step given in Eq. 6) improves the efficiency of the fully integrated methods described previously. The RF Newton step is based on a specific solution to the Karush-Kuhn Tucker (KKT) conditions for a quadratic program approximation of the MPP search optimization problem (i.e. Equation 3) with a first-order limit state approximation). The base point for this evaluation is the mean value optimum (i.e. the deterministic solution for the optimization when the random variable is fixed at its mean value). To establish consistency, three different assumptions were used to determine whether the algorithm is consistently capable of.
Thus, by maximizing P_regen_fraction, the infeasibility (in terms of meeting design constraints) of the system is minimized. In order to solve Equation 4), the RBDO-MDO methods from the previous chapter can be used. Each of the 12 RBDO-MDO methods from the previous chapter was applied to the UAV planning problem to solve Eqs.
The sensitivity factor is found by simply normalizing the derivative of the limit state with respect to the variable at MPP.
Sensitivity Analysis
At the higher design levels, more of the system is considered with less detail. Various systems design models (eg, System Engineering Vee, waterfall and spiral models) depict this common theme of iterative feedback between design levels (Buede, 2000). In this case, the weight distribution of the RLV system provides input for inertial loads required for the structural design of individual components (Cerro et al, 2002; Mahadevan and Smith, 2003).
The goal of liquid hydrogen tank design is therefore to reduce tank weight while meeting fuel volume and structural integrity requirements. Once the component design is complete, a more accurate estimate of the tank weight is available from Eq. Note that the function estimates for each phase of RBDO (i.e., the optimization phase and the probabilistic analysis phase) include the estimates required for the finite-difference gradient approximations.
Instead, a component-level optimization can be used for structural reservoir size analysis. This method maintains the autonomy between system optimization and component design, but samples the component models in order to characterize the error of the system analysis model. Furthermore, the sensitivity of system design constraints to the uncertainty associated with this error determines the effect of disciplinary model error on a multidisciplinary system and provides a useful metric for model selection.
This methodology is then applied to the RLV geometry and tank design problem of the previous chapter with a discussion of results. Collectively, the sources of uncertainties in the model predictions can be described as model error, or simply the difference between the model prediction and actual performance of the system. The previous chapter covered the integration of the reusable launch vehicle system geometry design and a component tank design.
The tank design provides a better estimate of tank weight based on structural integrity requirements and material properties. The original weight module, W, is divided into the conceptual calculation for the tank weight, Tconcept and the balance of the calculations in the module denoted W-. The conceptual design is thus the initial optimization of the RLV geometry based on the conceptual weight model and aerodynamic analysis.
The two-level integrated RBDO method in Chapter V appears promising for designs where (1) close interaction between design levels is both possible and desired, (2) the design process can "afford" the extra computational effort, and (3) detailed design is necessary to reduce otherwise unacceptable uncertainty associated with the conceptual design. Forsberg and Mooz, “The Relationship of Systems Engineering to the Project Management Cycle,” Engineering Management Journal 4(3), p.
