# HS-5000: Stochastic Method Comparison: Stochastic Study of the Arm Model

Learn how to perform Stochastic studies and a Reliability-Based Design Optimization (RBDO) using the same fitting function.

Before you begin, complete HS-4000: Optimization Method Comparison: Arm Model Shape Optimization or import the HS-4000.hstx archive file, available in <hst.zip>/HS-5000/.

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.

1. In the Explorer, right-click and select Add from the context menu.
2. For Definition from, select an approach.
3. Select Stochastic, then Setup and click OK.
2. Go to the Stochastic 1 > Definition > Define Input Variables step.
4. 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 Bound-Lower 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.
1. Go to the Optimization 3 (GRSM, 6 IV, Exact Solver) > Evaluate step.
2. Click the Iteration History tab.
3. Select the optimal parameter values for length_1, length_2, length_3, length_4, and length_5 parameter, then right-click and select Copy from the context menu.
4. Go to Stochastic 1 > Definition > Define Input Variables step.
5. Select the Nominal fields for length_1, length_2, length_3, length_4, and length_5, then right-click and select Paste transpose from the context menu.
6. For height, change the Nominal value to 1.0.
5. 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.
1. For all active input variables, click (...) in the Nominal field.
2. In the pop-up window, Value field, enter 0.05 and click +/-.
3. Click OK.
The Lower and Upper bounds for all active input variables should replicate the image below.
6. 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 Bound-Lower Bound)/4)². For instance, the variance for the input variable length_1 is σ²=((-0.45+0.55)/4)²=6.25e-4.

7. Go to the Stochastic 1 > Definition > Define Output Responses step.
8. 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.
9. Define specifications.
1. Go to the Stochastic 1 > Specifications step.
2. In the work area, set the Mode to Hammersley.
Hammersley sampling belongs to the category of quasi-Monte Carlo methods. This method uses a quasi-random number generator, based on the Hammersley points, to uniformly sample a unit hypercube.
3. Click Apply.
1. Go to the Stochastic > Evaluate step.
11. Study the method convergence.
1. Return to the Specifications step. In the Settings tab, change the Number of Runs to 300 and then 1000.
2. Go to the Evaluate step and click Evaluate Tasks.

## Post-Process Stochastic Results

1. Go to the Stochastic 1 > Post- Processing step.
2. Review the statistics of the input variables and the output responses around the nominal design.
1. Click the Integrity tab.
2. 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.
3. Review histogram of the Stochastic results.
1. Click the Distribution tab.
2. 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 sub-range 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.
4. 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:
5. Estimate the probability to failure for the output response (probability for an output response to violate a user selected bound).
1. Click the Reliability tab.
3. 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.
The results show the convergence of the Hammersley method in function of the number of runs. The convergence is enhanced by increasing the number of runs. As the results obtained with 300 or 1000 runs are quite similar, you can assume that in this case 300 runs are enough to get accurate results.
4. Study the effects of bounds on the reliability by entering different values in the Bound Value columns.
5. Optional: Check the reliability on Max_Stress.
6. 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 Reliability-Based Optimization (RBDO)

In this step, you will be searching for 95% reliability on the Optimization constraint (max_disp < 1.5 mm).

You will use fitting functions as opposed to the exact solver to evaluate the output responses. Among the approximations, you will use the RBF that is created with MELS DOE. As a result, you will use the SORA method. You will continue using the six important input variables that all follow a normal distribution. You will start the Optimization from the previously obtained optimum solution with GRSM.
1. In the Explorer, right-click and select Add from the context menu.
2. In the Add dialog, select Optimization.
3. For Definition from, select Setup and click OK.
2. Go to the Optimization 7 > Definition > Define Input Variables step.
4. Copy the parameter values at the optimal design.
1. Go to the Optimization 3 (GRSM, 6 IV, Exact Solver) > Evaluate step.
2. Click the Iteration History tab.
3. Locate the optimal design and copy the length_1, length_2, length_3, length_4, and length_5 parameter values.
4. Go to Optimization 7 > Definition > Define Input Variables step.
5. Select the Nominal fields for length_1, length_2, length_3, length_4, and length_5, then right-click and select Paste transpose from the context menu.
6. For height, change the Nominal value to 1.0.
5. Modify distribution.
1. Click the Distributions tab.
2. In the Distribution Role column, select Design with Random for all active input variables.
3. In column 2, enter 6.25e-04 for all active input variables.
6. Go to the Optimization 7 > Definition > Define Output Responses step.
7. Click the Objectives/Constraints - Goals tab.
8. Apply an objective on the Volume output response.
2. In the Apply On column, select Volume.
3. In the Type column, select Minimize.
9. Apply a Constraint on the Max_Disp output response.
2. In the Apply On column, select Max_Disp.
3. In the Type column, select More.
4. In column 1, select Probabilistic Constraint.
5. In column 2, enter <= 1.5000000,CDF=95.000000.
10. Modify evaluation source.
1. Click the Define Output Responses tab.
2. In the Evaluate From column, select Fit - RBF (fit_4) for all output responses.
11. Define specifications.
1. Go to the Optimization 7 > Specifications step.
2. In the work area, set the Mode to Sequential Optimization and Reliability Assessment (SORA).
3. Click Apply.
1. Go to the Optimization 7 > Evaluate step.
13. 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.

1. Copy Stochastic approach.
1. In the Explorer, right-click on the Stochastic 1 (Hammersley) and select Copy from the context menu.
2. In the Copy dialog, Label field, enter Validation SORA and click OK.
2. Copy and paste the optimum solution from the SORA Optimization into the Nominal column of the input variables for the Validation SORA Stochastic study.
1. Go to the Optimization 7 > Evaluate step.
2. Click the Iteration History tab.
3. Copy the length_1, length_2, length_3, length_4, and length_5 optimal parameter values.
4. Go to the Validation SORA > Definition > Define Input Variables step.
5. Select the Nominal fields for length_1, length_2, length_3, length_4, and length_5, then right-click and select Paste transpose from the context menu.
3. 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.
1. For all active input variables, in the Nominal field, click (...).
2. In the pop-up window, Value field, enter 0.05 and click +/-.
3. Click OK to accept the changes and close the pop-up window.
4. Define specifications.
1. Go to the Validation SORA > Specifications step.
2. Verify that the Mode is set to Hammersley.
3. In the Settings tab, verify that the Number of Runs is 1000.
4. Click Apply.