Second-Order Reliability Methods
The SORM is an improvement over the FORM by accounting for the curved nature of the failure boundary given by $g(\vec{X}) = 0$ around the design point $\vec{x}^{*}$; thus, providing a better approximation of the probability of failure $P_{f}$.
Curve-Fitting Method
The CF method fits a hyper-paraboloid surface with a vertex at the design point $\vec{x}^{*}$ and the principal curvatures matching the principal curvatures of the failure boundary given by $g(\vec{X}) = 0$ at that point. The probabilities $P_{f}$ of failure are estimated using Hohenbichler and Rackwitz (1988) and Breitung (1984) approximations of the exact solution provided by Tvedt (1990). The calculated probabilities of failure $P_{f}$ are then used to estimate the generalized reliability indices $\beta$, which account for the curved nature of the failure boundary given by $g(\vec{X}) = 0$ around the design point $\vec{x}^{*}$.
Point-Fitting Method
The PF method fits a series of hyper-semiparaboloid surfaces with a vertex at the design point $\vec{x}^{*}$. The principal curvatures of each surface are estimated using fitting points found at the intersections of a hyper-cylinder with axis coinciding with the design point $\vec{u}^{*}$ and the failure boundary given by $g(\vec{U}) = 0$ in $U$-space. The PF method provides a better estimate of the probability of failure $P_{f}$ than the CF method since it provides a better approximation of highly non-linear failure boundaries given by $g(\vec{X}) = 0$ that are unsymmetrical about the design point $\vec{x}^{*}$.
A great description of both methods can be found in Der Kiureghian (2022).
API
Fortuna.solve
— Methodsolve(Problem::ReliabilityProblem, AnalysisMethod::SORM; FORMSolution::Union{Nothing, HLRFCache, iHLRFCache} = nothing, FORMConfig::FORM = FORM(), diff::Symbol = :automatic)
Function used to solve reliability problems using Second-Order Reliability Method (SORM).
If diff
is:
:automatic
, then the function will use automatic differentiation to compute gradients, jacobians, etc.:numeric
, then the function will use numeric differentiation to compute gradients, jacobians, etc.
Fortuna.SORM
— TypeSORM <: AbstractReliabililyAnalysisMethod
Type used to perform reliability analysis using Second-Order Reliability Method (SORM).
Submethod::Fortuna.SORMSubmethod
Fortuna.CF
— TypeCF <: SORMSubmethod
Type used to perform reliability analysis using Curve-Fitting (CF) method.
ϵ::Real
: Step size used to compute the Hessian at the design point in $U$-space
Fortuna.CFCache
— TypeCFCache
Type used to perform reliability analysis using Point-Fitting (PF) method.
FORMSolution::Union{HLRFCache, RFCache, iHLRFCache}
: Results of reliability analysis performed using First-Order Reliability Method (FORM)β₂::Vector{Union{Missing, Float64}}
: Generalized reliability indices $\beta$PoF₂::Vector{Union{Missing, Float64}}
: Probabilities of failure $P_{f}$κ::Vector{Float64}
: Principal curvatures $\kappa$
Fortuna.PF
— TypePF <: SORMSubmethod
Type used to perform reliability analysis using Point-Fitting (PF) method.
Fortuna.PFCache
— TypePFCache
Type used to perform reliability analysis using Point-Fitting (PF) method.
FORMSolution::iHLRFCache
: Results of reliability analysis performed using First-Order Reliability Method (FORM)β₂::Vector{Union{Missing, Float64}}
: Generalized reliability index $\beta$PoF₂::Vector{Union{Missing, Float64}}
: Probabilities of failure $P_{f}$FittingPoints⁻::Matrix{Float64}
: Fitting points on the negative side of the hyper-cylinderFittingPoints⁺::Matrix{Float64}
: Fitting points on the positive side of the hyper-cylinderκ₁::Matrix{Float64}
: Principal curvatures on the negative and positive sidesκ₂::Matrix{Float64}
: Principal curvatures of each hyper-semiparabola