HS5000: Stochastic Method Comparison: Stochastic Study of the Arm Model
Learn how to perform Stochastic studies and a ReliabilityBased Design Optimization (RBDO) using the same fitting function.
You will run a Stochastic study around the nominal point.
In the Stochastic study, you will be using a Hammersley distribution successively with 100, 300, and 1000 runs in order to compare the convergence of statistical results. You defined all six input variables as random variables following a normal distribution.
Run Stochastic
In this step, you will check the robustness of the optimal solution found with GRSM.

Add a Stochastic.
 In the Explorer, rightclick and select Add from the context menu.
 In the Add dialog, select Stochastic and click OK.
 Go to the Stochastic 1 > Definition > Define Input Variables step.

In the Active column, clear the radius_1,
radius_2 and radius_3
checkboxes.

Copy the parameter values at the optimal design.
For Stochastic studies, you must provide data about the standard variation σ (or variance σ²) of parameters in order to take into account uncertainties. By default, σ² is computed in HyperStudy using the range rule σ² = ((Upper BoundLower Bound)/4)² which is a function of the input variable's bounds. If you do not have reliable data about the standard deviation, the default σ² can be modified by changing the upper and lower bounds of the parameters.
 Go to the Optimization 3 (GRSM, 6 IV, Exact Solver) > Evaluate step.
 Click the Iteration History tab.

Select the optimal parameter values for length_1, length_2, length_3,
length_4, and length_5 parameter, then rightclick and select
Copy from the context menu.
 Go to Stochastic 1 > Definition > Define Input Variables step.
 Select the Nominal fields for length_1, length_2, length_3, length_4, and length_5, then rightclick and select Paste transpose from the context menu.
 For height, change the Nominal value to 1.0.

Change the lower and upper bounds for every active input variable to match the
dimensions' tolerances.
For this tutorial, assume that the tolerances are within 0.05 mm.
 For all active input variables, click (...) in the Nominal field.

In the popup window, Value field, enter 0.05
and click +/.
 Click OK.
The Lower and Upper bounds for all active input variables should replicate the image below. 
Click the Distributions tab, and verify that
Distribution is set to Normal Variance for all active
input variables.
Look at the columns 1 and 2. Column 1 displays the nominal parameter values, and column 2 displays the variance σ² which is computed using the standard deviation σ of a parameter around its mean.
Variance is computed as follows: σ² = ((Upper BoundLower Bound)/4)². For instance, the variance for the input variable length_1 is σ²=((0.45+0.55)/4)²=6.25e4.
 Go to the Stochastic 1 > Definition > Define Output Responses step.

In the work area, Evaluate From column, select Fit  RBF
(fit_4) for all output responses.
Even though Max_stress was not used in the Optimization, you will use it in the Stochastic study to check the reliability.

Define specifications.
 Go to the Stochastic 1 > Specifications step.

In the work area, set the Mode to
Hammersley.
Hammersley sampling belongs to the category of quasiMonte Carlo methods. This method uses a quasirandom number generator, based on the Hammersley points, to uniformly sample a unit hypercube.
 Click Apply.

Evaluate tasks.
 Go to the Stochastic > Evaluate step.
 Click Evaluate Tasks.

Study the method convergence.
 Return to the Specifications step. In the Settings tab, change the Number of Runs to 300 and then 1000.
 Go to the Evaluate step and click Evaluate Tasks.
PostProcess Stochastic Results
 Go to the Stochastic 1 > Post Processing step.

Review the statistics of the input variables and the output responses around
the nominal design.
 Click the Integrity tab.

Using the Channel selector, select the
Health category to get a summarized view of
statistics and spot eventual, missing, or bad values.
The Range column can be useful to understand the spread of values in the data from the minimum to the maximum.

Review histogram of the Stochastic results.
 Click the Distribution tab.
 Using the Channel selector, select Length 1.
The chart shows three pieces of information about the distribution of values for the selected input variable. The Histogram uses the left axis, and represents the frequency of runs yielding a subrange of response values. The Probability density uses the right axis, and indicates the relative likelihood of the input variable to take a particular value. A higher value indicates that the values are more probable to occur. The Cumulative distribution is another curve that uses the right axis. It is equal to the integral of the Probability density. The value of the Cumulative distribution indicates what percentage of the data falls below the value’s threshold. Note that the initial value of the Cumulative distribution will always equal 0, and the final value of the CDF will always be 1.0. This is because all of the data will reside between the upper and lower bounds. 
Review the Probability density and the Cumulative distribution of Max_Disp.
Compare the distributions obtained with the different number of runs for
Hammersley (100, 300, and 1000).
Note: You can see the pattern of the distribution changes quite a bit from 100 to 300 runs, but very little from 300 to 1000 runs.Note:

Estimate the probability to failure for the output response (probability for an
output response to violate a user selected bound).
 Click the Reliability tab.
 Click Add Reliability.

Define the reliability.
The Bound Values are those from the Optimization problem definition (Max_Disp<=1.5).
 Set Response to Max_Disp (r_1).
 Set Bound Type to <=.
 For Bound Value, enter 1.50.
 Study the effects of bounds on the reliability by entering different values in the Bound Value columns.
 Optional:
Check the reliability on Max_Stress.

Click the Reliability Plot tab to see the relationship
between the desired threshold and the reliability of the system.
The values reported in the Reliability column of the Reliability tab can be observed on these reliability curves. On the Max_Disp reliability curve, only 11.4% of the designs have a value above 1.5, which means 88.6% are below 1.5. A reliability of 95% can be reached by changing the threshold for Max_Disp to 1.49763.
Run ReliabilityBased Optimization (RBDO)
In this step, you will be searching for 95% reliability on the Optimization constraint (max_disp < 1.5 mm).

Add an Optimization.
 In the Explorer, rightclick and select Add from the context menu.
 In the Add dialog, select Optimization and click OK.
 Go to the Optimization 7 > Definition > Define Input Variables step.
 In the Active column, clear the radius_1, radius_2, and radius_3 checkboxes.

Copy the parameter values at the optimal design.
 Go to the Optimization 3 (GRSM, 6 IV, Exact Solver) > Evaluate step.
 Click the Iteration History tab.
 Locate the optimal design and copy the length_1, length_2, length_3, length_4, and length_5 parameter values.
 Go to Optimization 7 > Definition > Define Input Variables step.
 Select the Nominal fields for length_1, length_2, length_3, length_4, and length_5, then rightclick and select Paste transpose from the context menu.
 For height, change the Nominal value to 1.0.

Modify distribution.
 Click the Distributions tab.

In the Distribution Role column, select Design with
Random for all active input variables.

In column 2, enter 6.25e04 for all active input
variables.
 Go to the Optimization 7 > Definition > Define Output Responses step.
 Click the Objectives/Constraints  Goals tab.

Apply an objective on the Volume output response.
 Click Add Goal.
 In the Apply On column, select Volume.
 In the Type column, select Minimize.

Apply a Constraint on the Max_Disp output response.
 Click Add Goal.
 In the Apply On column, select Max_Disp.
 In the Type column, select More.
 In column 1, select Probabilistic Constraint.
 In column 2, enter <= 1.5000000,CDF=95.000000.

Modify evaluation source.
 Click the Define Output Responses tab.
 In the Evaluate From column, select Fit  RBF (fit_4) for all output responses.

Define specifications.
 Go to the Optimization 7 > Specifications step.
 In the work area, set the Mode to Sequential Optimization and Reliability Assessment (SORA).
 Click Apply.

Evaluate tasks.
 Go to the Optimization 7 > Evaluate step.
 Click Evaluate Tasks.

Click the Iteration History tab to review the results of
the Optimization in a table.
A deterministic optimum is found first. As you can see in iteration 1, the displacement is at the constraint bound. This design is not 95% reliable as required in this probabilistic Optimization study. The SORA will work to make sure the constraint satisfies the probabilistic requirement. As seen in iteration 2, the design that meets 95% reliability is the one that has a max displacement shifted away from the bound of 1.5. Corresponding to this improvement, there is an increase in the objective value (volume).
Run Validation Stochastic
As SORA uses the approximation method for the reliability calculation, it is recommended that you make a validation Stochastic study after the Optimization.

Copy Stochastic approach.
 In the Explorer, rightclick on the Stochastic 1 (Hammersley) and select Copy from the context menu.
 In the Copy dialog, Label field, enter Validation SORA and click OK.

Copy and paste the optimum solution from the SORA Optimization into the Nominal
column of the input variables for the Validation SORA Stochastic study.
 Go to the Optimization 7 > Evaluate step.
 Click the Iteration History tab.
 Copy the length_1, length_2, length_3, length_4, and length_5 optimal parameter values.
 Go to the Validation SORA > Definition > Define Input Variables step.
 Select the Nominal fields for length_1, length_2, length_3, length_4, and length_5, then rightclick and select Paste transpose from the context menu.

Change the lower and upper bounds for every active input variable to match the
dimensions’ tolerances.
For this tutorial, assume that the tolerances are within 0.05 mm.
 For all active input variables, in the Nominal field, click (...).
 In the popup window, Value field, enter 0.05 and click +/.
 Click OK to accept the changes and close the popup window.

Define specifications.
 Go to the Validation SORA > Specifications step.
 Verify that the Mode is set to Hammersley.
 In the Settings tab, verify that the Number of Runs is 1000.
 Click Apply.

Evaluate tasks.
 Go to the Validation SORA > Evaluate step.
 Click Evaluate Tasks.
 Go to the Validation SORA > PostProcessing step.

Add a reliability on Max_Disp.
 Click the Reliability tab.
 Click Add Reliability.

Define the reliability.
 Set Response to Max_Disp.
 Set Bound Type to <=.
 For Bound Value, enter 1.50.
The result confirms that the optimum solution found with SORA is reliable at 95%.