Solution

Home Screen Process Flowchart

1. Select the pct2 box.                                                                 

A window Stage pct2 shall open.

Define Pre-Processor

1. Select the Setup tab.
2. For Command specify the LS-DYNA executable ls-dyna (This name can be different on your computer). On Windows, the command has to be specified using the absolute path.
3. For Input File browse the parameterized file pct2.k. Parameters in LS-DYNA input files can be definied using *PARAMETER or the LS-OPT parameter format <<>>.
4. For efficient usage of the computing power from the machine, choice on handling number of concurrent jobs can be made suitably in this section. (E.g., if the machine has 2CPUs, and to run each job on a single CPU: Units per job = 1, Global limit = 2).

The parameters located in the selected Input File can be visualized in the adjoining tab Parameters.

Design Parameters.

1. Select the Parameters tab.
2. The Name of the various design parameters are listed here.
3. The created parameters are read in from the file parameters.k.
4. Select OK to proceed.

• parameters.k is shown below for reference.

*PARAMETER
rCB,650.0
rSIGY,528.0
rC,110.0
rK,30.
rRSAT,500.0
rSB,166.0
rH,1.0
rC1,0.01
rC2,0.35

Extract Histories

• The various histories for the optimization problem can be added by the following steps:

Add First History

1. Select the Histories tab.
2. Select the suitable history type from the various option available from the list under Add new.
3. For the first history definition select the option D3PLOT, which represents an interface to LS-DYNA d3plot files.

Define yy-stress on element 1

1. For Name enter s2.
2. For element ID enter 1.
3. For Results Type select the radio button on Stress.
   
4. Component of the stress required is yy_stress.
5. Select the OK button to add the history.

•  The New history tab closes and returns to the main page of the Histories tab in the Stage pct2 window.

Add Second History

• To add an additional history select from the option D3PLOT  from the Add new options.

Define M-surface yy strain element 1

1. For Name enter e2.
2. For element ID enter 1.
3. For Results Type select the radio button on Strain.
   
4. Component of the stress required is M_surf_yy_strain.
5. Select the OK button to add the history.

•  The New history tab closes and returns to the main page of the Histories tab in the Stage pct2 window.

Add Third History

• To add an additional history select from the option Crossplot  from the Add new options.

Define Crossplot stress vs strain

1. For Name enter s2_vs_e2.
2. For z(t) enter e2.
3. For F(t) enter s2.
4. Select the OK button to add the history.

•  The New history tab closes and returns to the main page of the Histories tab in the Stage pct2 window.

Histories Review

1. The defined histories can be reviewed in the tab of the Histories under the History definitions. Necessary changes can be made by selecting the choice.
2. To delete a History definition click on the cross button (denoted as X).
3. Additional histories can be added from the choice available under the Add new list as earlier.  
4. Select the OK button to proceed.

Additional Stages

Since the other stages are very similar, the Clone option may be used to create them from the already defined stage:

1. Right click on the Satge pct2 box on the main GUI.
2. Select the option Clone.
3. Add the name of the New Stage : pct4

• A new stage has been created in the main GUI with the same definitions than the basis stage (see figure below).
• The New Stage can be edited using the similar step as shown in the previous tabs, see below.
• As per the problem in hand 3 additional stages need to be added viz. pct6, pct8 and pct10.

Changes on Additional Stages

The following changes are made on the new stages

 

Home Screen Process Flowchart

1. Select the Setup box.                                                                 

• A window Problem global setup shall open.

Define the Parameters.

1. Select the Parameter Setup tab.
2. Change the Type of parameters to Continuous.
3. Enter the various ranges of each  input parameter viz. Starting, Minimum, Maximum as given in the figure.

Stage Matrix

1. Select the Stage Matrix tab.

• You find a matrix of parameters vs. stages, and symbols that visualize if parameters are found in input files.

Sampling Matrix

1. Select the Sampling Matrix tab.

• If you don't want to use a parameter as variable for a respective sampling, the variables can be switched off here. The parameters are treated as constants then.                          

2. Select the OK button to proceed.

Home Screen Process Flowchart

1. Select the Optimization box.                                                                 

A window Optimization shall open.

Add Objective

1. Select the Objectives tab.
2. From Composites select Residual2, Residual4, Residual6, Residual8 and Residual10 as the objective.

Define Objective

• For Weight leave the values to default 1.

Optimization Algorithm.

1. Select the Algorithms tab. Here, the algorithm used for the optimization on the metamodel can be selected.

• Make sure that the radio button for Adaptive Simulated Annealing (ASA) is selected and Switch to LFOP is checked. These are the default settings.     
  

2. Select the OK button to proceed.

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. Select under Simulations the item Histories.

Plot Setup

Setup the plot for pct2

1. Select the Setup tab.
2. Select n for the iterations setup.
3. Plot co-ordinate selections :

x axis : time

y axis : s2

• The first plot in the image depicts a plot of the stress against time.

4. In the similar way plot the strain curve by changing the y axis entity to e2.

5. The loading cycle curve can be illustrated using the choice for the y axis as s2_vs_e2 and test2. We can notice the hysteresis for the cyclic loading case.

Multiple Plots

1. Select the Multi option to view all the simulation crossplots and test curves simultaneously.
2. Select n for the iterations setup and select the last iteration to display the optimal curves.
3. To view all the stress-strain crossplots viz. s2_vs_e2, s4_vs_e4, etc. select the respective boxes.
4. Similarly select all the test curves viz. test2, test4, etc.
5. To distinguish between the crossplots it's convinient to select the  Neutral option for the Color entities. 

Convergence

1. Select any of the history curves. In this case s2_vs_e2 plot has been selected.
2. From the file histories select the corresponding test curve, e.g. test2.
3. To observe the convergence over the iterations choice is made for all the iterations.
4. To distinguish between the iterations choice of Color entities is made on the Iterations.
5. Select the Options tab.
6. To show the optimal curves over the iterations the Only optimal option has been selected. 

We can plot the several surface plots of the metamodel we've built. For example, we want to see a surface plot of the objective function Residual2 with respect to the variables CB and SIGY, do it as following:

• Note : The surface plot for Residual2 shows a non-linear surface, although we used linear polynomials. The reason is that Curve Mapping is defined as a composite expression that is calculated using the metamodels of the components, and the expression is non-linear.
• The optimum values for the input variables can be visualized in the top-left hand corner, whereas the location of the optimum values on the surface plot can be seen with the purple cross.

• We rotate the plot with the mouse by pressing Ctrl at the meantime. We can change the surface plot for each iteration (at step 2.) and visualize the effect of domain reduction (as shown below). 

What is the approximation error of the result?

The approximation error indicators (predict the metamodel accuracy of the results) can be visualized in the LS-OPT Viewer.

New Plot

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

LS-OPT Viewer

1. Select under Metamodel the item Accuracy.
2. From Entity select the Objective Residual2.

→ Good approximation: low distance between black line and green points.

→Similar observations can be made by selecting the other objectives.

Optimization History

• Alternatively the accuracy of the optimal values at various iterations can be studied under a Optimization History plot.

New Plot

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

LS-OPT Viewer

1. Select under Optimization the item History.
2. From Setup tab, select the Objective Residual2.

→ Good approximation of optima: The computed values (red points) almost coincide with the predicted values (black line) for the optimum.

→Similar observations can be made by selecting the other objectives.

→For observing the multiobjective select the option.   

New Plot

• To view a new plot select the plot 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. 

Convergence of Input variables

Setup

1. Select the input variable to observe the convergene behaviour across the iterations.

Use the split plot from the task menu to compare the various Optimization Histories of the input variables. Interesting revelations can be made from the plots. (Shown in figure below)

Convergence of Objectives

1.  Select Objective → Multiobjective

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

Which variable appears to be the most important? 

The significance of a variable for a response can be illustrated with ANOVA (analysis of variance) or GSA/Sobol (global sensitivity analysis). In this case, only GSA is available, since ANOVA is only calculated for responses, not for composites.

New Plot

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

LS-OPT Viewer

   GSA/Sobol

1. Restart the LS-OPT Viewer and select under Metamodel the item Sensitivity
2. We can slide to observe the sensitivity of results at each iteration. In this case for Iteration 1.
3. The linear ANOVA measure is unavailable. Choice can be kept at GSA/Sobol.
4. Select  any of the Composite to observe the sensitivity measure. In this case select Residual2.
5. Select Sort for the variables.

→ For the composite Residual2 the variable CB is the most important or significant, followed by the variable SIGY.

→ Similar observation can be made for the other composites upon selection.

Multiple Responses Sensitivity

The viewer allows the user to select multiple responses and composites for sensitivity analysis to get the influence of the variables on e.g. the whole problem or a load case using the following steps:

6. Select the Multi option.
7. Select the desired entities, in this case all the composites are selected .

•  The plot describes the cumulative values on a single graph w.r.t. the variables. This is only availabe for GSA, since the values are normalized.