Reliability-Based Design Optimization (RBDO)
Reliability-Based Design Optimization (RBDO): Theoretical Foundations
What is RBDO?
Professor, what is RBDO?
RBDO (Reliability-Based Design Optimization) is optimization that considers the variability of design variables (e.g., manufacturing errors). Deterministic optimization optimizes using "standard values," whereas RBDO optimizes such that "the probability of satisfying constraints, even including variability, is above a specified value."
$\beta_t$ is the target reliability index. With $\beta = 3$, the failure probability $\approx 10^{-3}$.
Summary
The "6σ" in reliability design was trademarked by Motorola in 1986
"Six Sigma" is a statistical quality control concept. In 1986, Motorola engineer Bill Smith (known as the "Father of Six Sigma") proposed it internally as a framework to reduce manufacturing defect rates to 3.4 ppm and established it in a proprietary manner. In Reliability-Based Design Optimization (RBDO), this Six Sigma criterion is mathematically modeled as a probabilistic constraint, enabling robust design that accounts for uncertainties (material variability, load fluctuations). GE adopted Six Sigma company-wide in 1995 under CEO Jack Welch, and publicly stated that applying RBDO to engine parts reduced warranty costs by $1 billion annually.
Computational Methods for Reliability-Based Design Optimization (RBDO)
RBDO Calculation
1. FORM/SORM — Efficiently calculates the reliability index $\beta$.
2. Surrogate Model — Replaces FEM. Speeds up Monte Carlo.
3. OptiSlang + FEM — Probabilistic wrapper + FEM.
Summary
FORM is the standard first-order reliability analysis method for over 40 years
The First Order Reliability Method (FORM) was proposed by Hasofer and Lind in the Journal of Engineering Mechanics in 1974. It is a reliability assessment method based on the design point (most probable failure point). The algorithm transforms variables to standard normal space and finds the minimum distance (reliability index β) to the failure surface (limit state surface). With low computational cost, it has been the standard engineering design method for over 40 years. Its combination with Monte Carlo methods (Importance Sampling) to compensate for FORM's approximation accuracy limits is widely used in seismic design and nuclear structure evaluation.
Reliability-Based Design Optimization (RBDO) in Practice
RBDO in Practice
Automotive crash safety (ensuring safety under manufacturing variability), aerospace structural reliability.
Practical Checklist
B787 quantified fastener hole failure probability using RBDO
In the design of CFRP fuselage panels for the Boeing 787, RBDO using probabilistic FEM was adopted to evaluate the probability of fatigue crack initiation. Material property statistical variability (fiber strength, interlaminar shear strength) was sampled via Monte Carlo methods, and the design process optimized the allowable range for fastener pitch and tightening torque against a target failure probability (10⁻⁷ per flight hour), as introduced in the Boeing Technical Journal (2009).
Reliability-Based Design Optimization (RBDO): Software & Solver Comparison
RBDO Tools
OptiSlang is a specialized probabilistic design tool acquired by Ansys
optiSlang (by German company Dynardo, founded 2001) is a probabilistic design optimization platform integrating sensitivity analysis, robust design, RBDO, and coefficient of variation analysis. It has been adopted for Volkswagen's crash safety robust design and ZF's gearbox reliability design. Ansys acquired it in 2019 and integrated it as Ansys optiSLang. Its seamless ability to build FEA ⇔ probabilistic optimization loops from within the Ansys Workbench environment is a key differentiator from competitors like NESSUS (by SwRI, for space/aerospace).
Advanced Technologies
Advanced RBDO
The difference between FORM and SORM lies in the presence of curvature correction
FORM (First Order Reliability Method) linearly approximates the failure surface at the design point, so error can be large for highly nonlinear failure surfaces with significant curvature. SORM (Second Order Reliability Method), proposed by Hohenbichler & Rackwitz (1983), is a curvature-corrected method that corrects the probability using principal curvatures κi at the design point. For cases with high curvature (e.g., buckling reliability of thin-walled structures), the estimated reliability can differ by more than 10 times between FORM and SORM. Guidelines for nuclear and spacecraft design mandate the use of SORM or Monte Carlo.
Reliability-Based Design Optimization (RBDO): Common Issues & Debugging
RBDO Troubles
Monte Carlo method requires 100 million runs to find P(f)=10⁻⁶
When using Monte Carlo methods for reliability analysis, the number of samples needed to ensure estimation accuracy for failure probability Pf is on the order of the inverse of Pf (e.g., ~10⁸ samples for Pf=10⁻⁶). For a system where one FEA evaluation takes 10 seconds, 10⁸ samples would take 32 years, making it impractical. Importance Sampling or Line Sampling (LS) can reduce sample counts by 100-1000 times, but their efficiency depends on prior estimation accuracy of the failure region, so combination with FORM is the standard practical workflow.
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