# Cantilever Beam (Reliability Based Optimization)

In this page is shown how to perform the reliability based optimization of the Cantilever Beam defined in Cantilever Beam. In this example two design variables are introduced for representing the beam width and the beam height. The objective of the optimization analysis is to design a cantilever beam with an associated failure probability (the probability to exceed the maximum allowed displacement) of 1e-3.

## Prepare Input

### Design Variable

Two design variable are created in the GUI that defined the design parameter of the Cantilever Beam. The beam width is allowed to vary between 0.01 and 0.2 [m] while the beam height is allowed to vary between 0.02 and 0.4 [m]. The following screenshots show the definition of the design variables in the GUI.

Prior to the reaching the stage displayed in the following image, A new parameter that defines the target failure probability needs to be defined, as shown in the sub section below.

### Parameter

A new parameter that defines the target failure probability needs to be defined as follow:

## Prepare Evaluator

The RBO analysis requires the definition of an objective function or constraint involving the failure probability.

### Objective Function

In this example the objective function is defined as the square difference between the target failure probability and the failure probability associated to the component or system. The objective function is defined as:

The script required by the objective function is shown in the following figure:

```for n=1:length(Tinput),
Toutput(n).fobj=(Tinput(n).targetPf -Tinput(n).pf)^2;
end
```

Please note that in order to have access to the design variable variables it is necessary to add them as a inputs for objective function.

## Optimization Problem

The optimization is defined simply by creating a new optimization problem (right click on the folder optimization problem in the model section) and adding the above define objective function and the probabilistic model defined in the section Cantilever Beam (Reliability Analysis).

Please note that the output shows in the optimization model are the output of the probabilistic model (i.e. the output of the performance function, vg, and the output of the model, w) plus the output of the objective function (fobj).

## Analysis

The analysis of the optimization problem starts clicking on the green bottom on the right top corner of the editor.

### Reliability Based Optimization

In the first page of the wizard it is necessary to define the name of the analysis.

The the type of analysis Reliability based optimization is selected. The type of optimization algorithm is also selected from the same page. In this example the Simplex method is used:

The the details of RBO analysis are selected in the next wizard page. Here, the user has to define the mapping between the design variables used during the optimization loop an the variables (inputs) of the probabilistic model used to estimated the failure probability. The following mapping is defined:

• The current value of the design variable Xdvb is used to replace the value of the parameter b.
• The current value of the design variable Xdvh is used to replace the mean value of the random variable h.
Then, the type of RBO analysis must be selected. In this examples we select the direct approach.

The simulation method used to evaluate the probabilistic model and estimate the failure probability is selected. In this example the Line Sampling method is adopted.

Finally the termination criteria and the advanced setting of the optimization algorithm have to be defined:

### Results

The results of the analysis are shown in the folder: Analysis -> Optimization Analsys-> RBOanalysis. This folder contains the output object EngineOutputs that contains an Optimum object. This object can open in the editor as shown in the following:

The following results are obtained;

• h: 2.642e-01
• b: 1.258e-01
• Objective function value: 1.16e-10