Discrete optimization

Home Screen Process Flowchart

• The results can be viewed by selecting the view button on the task bar. A seperate window of LS-OPT Viewer opens up.

LS-OPT Viewer

1.  Choose History under the category Optimization.

• We can choose from the left side the entities we want to observe. 

Select Variables → thood

Fig. 1(a) shows:

• The optimization history of the variable thood.
• The development of the optimal variable value (red line)
• How the range of thood (set from 1 to 5 at the beginning) decreases after every iteration (blue lines).

• Observe the thood variable at every iterations takes discrete steps as assigned in the parameter setup.

• This variable is important (see Sensitivities) to reach the bounds of constraint and to minimize the objective and seems to be converged.

Select Variable → tbumper

Fig. 1(b)  shows:

• The optimization history of the variable tbumper.
• The development of the optimal variable value (red line)
• This variable is rather insignificant (see Sensitivites) and therefore differs between the iteration without affecting the objective.

As you can see in the figure that the value of HIC doesn't change a lot between iteration 5 and 6. The tolerance for termination of 1% is reached and this will make LS-OPT stop the optimization process after 6 iterations although 10 iterations were specified in the Run panel. 

Select Response → crash → HIC

Fig. 2(a) shows:

• The predicted result (black line) of the optimal HIC response for every iteration.
• The computed result (red points) of the HIC response for every iteration.
• The convergence trend of the HIC response.

Select Constraint → Intrusions

Fig. 2(b) shows:

• The predicted result (black line) of the optimal Intrusion constraint for every iteration.
• The computed result (red points) of the Intrusion constraint for every iteration.
• The constraint upper bound is reached (red line).

We can plot the response surface of the metamodel we've built. For example, we want to see the response surface of HIC with respect to the variables thood and tbumper, do it as following: 

New Plot

• To view a new plot select the plot botton on the task bar. A seperate window of LS-OPT Viewer opens up.

LS-OPT Viewer

1. Select under Metamodel the item Surface.
2. Slide to the last iteration.
3. Select the Setup tab.
4. Set the response HIC as the z-coordinate, the variables tbumper and thood as the x-coordinate and y-coordinate, respectively.
5. Pick Constraints to show the constraints.  The feasible region is drawn in green while the infeasible region is drawn in red.
6. Choose Predicted value. A purple cross will appear somewhere on the surface. You can move it by changing the values of the variables tbumper and thood.
7. Click Center on Opt. to locate the cross at the optimal point.
8. Select the Points tab.
9. Pick Predicted Optimum and Computed Optimum.

• We zoom up the plot and rotate it with mouse by pressing Ctrl at the meantime. The purple cube shows the predicted optimum value, while the green cube above denotes the computed optimum value, which is computed directly through the simulation model. Green indicates that the point is feasible.

 
 

Discrete variable - thood

• At every step of iteration the varable thood takes discrete steps to reach the optimum. This is necessary because of manufaturing  limitation.
• The discrete steps in the optimization process at every iteration is shown in the picture below.

New Plot

• To view a new plot select the plot botton on the task bar. A seperate window of LS-OPT Viewer opens up.

LS-OPT Viewer 

1. Select under Metamodel the item Accuracy.

RMS Error and R² Indicator

1. From the left side of the window select Responses → HIC.
2. RMS Error, SPRESS and R² (R-sq) can be found above the plot.
3. We can slide to observe the accuary of results at each iteration. In this case for Iteration 1.

•  At the last iteration (Iteration 6) we obtain a result with RMS Error = 8.23 (4.61%) and R² = 0.994.
• A small RMS error and a coefficient of determination (R²) ~1 indicates good fit (see table below).
• The SPRESS value is average (=24.5%) indicating a moderate prediction of the metamodel fit.

We can find the development of RMS Error and R² in History under Optimization.

New Plot

• To view a new plot select the plot botton on the task bar. A seperate window of LS-OPT Viewer opens up.

LS-OPT Viewer

1. Choose History under the category Optimization.
2. Select from the left of the window Response → HIC 
3. Select RMS Error. We see that the RMS error is large at the 1st iteration and progressivly reduces till the 3rd iteration.
4. Select R2 Error. The R² error reaches its maximum near 1 at the 4th and remains till the 6th iteration.