Reliability Index based strategy for the Probability Damage approach in Fail-Safe Optimization
The β-PDFSO is a new fail-safe optimization strategy to include available information on the probability of occurrence of different accidental scenarios as well as uncertainty in parameters affecting structural responses. The optimization approach avoids obtaining oversized designs, as the value of the objective function is reduced compared to the fail-safe RBDO. Due to the lack of knowledge of which damaged configuration will occur, a new random reliability index
is defined for each limit-state of the damaged structure. This new random variable is constructed through the
in each damaged configuration. The method guarantees
in the limit-states of the intact configuration and
in limit-states affecting the damaged structure.
An optimization problem is defined using mathematical expressions to simulate the objective function
and the structural responses
.
These structural responses are defined by a set of equations, which depend on the design variables
,
the random variables
and a vector of coefficients
,
as shown in Eq.1, where a response
is defined per load case
and limit-state
.
By varying the vector
the structural response
is modified, being possible to obtain a different value for the intact and damaged configurations: the vector
is taken as the unit vector to represent the structural response in the intact model (
), while values between 0 and 1 are used to simulate the response in the damaged configurations (
). By adopting this approach, the structural responses defined in Eq. 1 increase for values of
lower than 1, being possible to simulate a loss of structural capacity in the damaged configurations. In this problem,
was taken as the number of damaged configurations.
The formulation of the optimization problem is presented in Eq. 2. The objective is to minimize the objective function
, defined as the sum of the design variables
.
The random variables are defined as normally distributed with mean
and standard deviation
.
The probability of occurrence
of each damaged configuration is shown in Table 1.
The vector of coefficients defined in Eq. 1,
,
is set for each load case
,
limit-state
and each model
,
expressed as
. Only some values of these coefficients are summarized in the arrays presented in Eq. 3. The full length matrices are available in the matlab file main.m.
The optimization problem was solved for a target reliability index and a target probability of failure of
, which correspond to a confidence level,
.
Edit the defineInputParameters function, where the following inputs are defined:
| Problem Parameters | Description |
|---|---|
| d0 | Initial value of design varaibles |
| nDV | Number of design variables |
| nLC | Number of load cases |
| nDcon | Number of limit-states |
| nDamages | Number of damaged configurations |
| pDamages | Probability of occurrence of each damaged configuration |
| pfPDFSO | Target probability of failure in the β-PDFSO |
| mu_xi | mean of the random variable i |
| sig_xi | standard deviation of the random variable i |
| beta_min | target reliability index for RA |
| resp | Structural Response (cell) |
| respMax | Maximum allowable response (cell) |
| GFun | Limit state (cell) |
| Cfd0 | coefficients for the intact model (cell) |
| CfDamages | coefficients for the damaged configurations (cell) |
| dGFun_dresp | derivative of GFun with respect to resp |
| dresp_dxi | derivative of resp with respect to xi |
| Optimization Parameters | Description |
|---|---|
| lb | Lower bounds of design variables |
| ub | Upper bounds of design variables |
| TolCon | Constraint tolerance |
| TolFun | Objective function tolerance |
| DiffMinChange | Minimum finite difference step |
| DiffMaxChange | Maximum finite difference step |
The program generates a CFD plot of each probabilistic constraint and the file opt_results_it.txt.
