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Lab 5:  Constrained Optimization

engineering工程作业代写 Start from scratch to build a new HEEDS MDO model for the problem described in Exercise 5 of the HEEDS Training Guide (below).

Goals

The primary goals of this exercise are to:

• learn how to solve constrained optimization problems,
• understand the effects that constraints have on an optimization solution, and
• understand the differences between active and inactive constraints.

Tasks  engineering工程作业代写

Start from scratch to build a new HEEDS MDO model for the problem described in Exercise 5 of the HEEDS Training Guide (below).

Turn In

Your assignment should be typed. Use a 12 pt font and 1 inch margins. On the first page,include the pledge in the form below:

Pledge: I have neither given nor received any unauthorized assistance on this lab assignment.

Signature: __________________________________  Print Name: ________________________

1.Answer all of the bulleted questions in the Post‐Processing section of the Exercise 5 tutorial. Copy and paste the relevant figures (plots and tables) into your report to justify your answers.

You must provide justification for all of your answers. Often, this justification will be in the form of relevant figures that you have used to generate your conclusions.

2.Answer the following questions:

a.How can we tell if a constraint is active?

b.If your optimized design is not as good as you hoped and you have active constraints, how could you possibly improve your design?

c. Why are responses normalized in the Performance function?

d.Why is the Performance function defined so that it always increases?

e.Based on the definition of the Performance function, is it possible for an infeasible design to be ranked higher than a feasible design?

3.Extra credit (not required): Perform the optimization described in the third Additional Experiment at the end of the tutorial. Provide a half‐page description of the results.  Exercise 5: Constrained Optimization

Exercise 5: Constrained Optimization  engineering工程作业代写

In this exercise, we will create a new HEEDS MDO study that demonstrates the use of constraints.

Imagine that a really mean groundskeeper planted a large tree in the middle of the fairway of your favorite golf hole. Now, your shot must go around or over the tree, but not through it. At the Red Cedar Hills golf course, this situation exists at Hole #2.

Let’s also imagine that the grass far away from the green is very high, so the ball must land (first impact) close to the green. Otherwise, it may get stuck in the tall grass and not roll to the green.

We call these restrictions constraints. When we place restrictions on our designs, we limit the acceptable regions of the design space.

In order to more easily satisfy the above two constraints, we will add two additional variables in this optimization study: Tee Back Spin and Tee Side Spin.

Our new optimization statement can now be written as:

Objective: Minimize final Ball Distance to the Hole (achieve a hole-in-one)  engineering工程作业代写

Constraints: Ball Distance To Tree ≥ 10 yards

First Impact Distance To Green ≤ 10 yards

Variables: 100 ≤   Tee Ball Speed (mph)   ≤ 150

20≤  Tee Launch Angle (degrees) ≤ 35

-30≤  Tee Target Angle (degrees)   ≤ -12

0  ≤ Tee Back Spin (rpm)  ≤ 6000

1000  ≤ Tee Side Spin (rpm)  ≤ 2000

Model: SwingSim: Hole 2

Because you are now more familiar with the HEEDS MDO user interface and the process of setting up a design study, the instructions below will have less detail than in previous exercises. If you get stuck, refer to the detailed instructions for similar steps that were performed in a previous exercise.

HEEDS MDO Introduction Training Exercises

Step 1: Creating a new HEEDS MDO project

We will build a new project from scratch. To prepare for this, we need to close the old project, start a new one and save the new blank project to the working directory.

1. Close your previous project so that we can build a new one from scratch.
2. From within the File menu, select New.
3. Save the new project to the working directory for Exercise 5. Name the project: E5_Constrained_Opt.

To save the project, return to the File menu and select Save As. In the Save Project As dialog,navigate to the folder:

…\HEEDS_MDO_Introduction_Projects\E5_Constrained_Opt\E5_Working_Directory

Enter E5_Constrained_Opt as the project name, and click Save.

The file will be saved with a .heeds extension.  engineering工程作业代写

Step 2: Defining the Analysis  

Next, we define the process automation steps for simulating the golf shot in the Process tab.

In preparation for this step, we need to know the batch command for SwingSim and the input/output files

associated with our baseline model:

Batch execution command: SwingGui.exe input_filename -swing output_filename

Input file name: Hole2_Tee_Shot_in.txt

Output file name:

Hole2_Tee_Shot_out.txt

Here are the general steps to define the analysis. If you are unsure how to perform any of these steps, refer to

Step 2 of Exercise 2 for more details.

1. Click the Process tab.
2. Rename the process SwingSim_Hole2_Tee_Shot.  engineering工程作业代写
3. Rename the analysis Hole2_Tee_Shot.
4. In the Execution sub-tab, specify the Execution command by browsing to the location of the executable file and selecting it.

5. In the Command options field, enter: Hole2_Tee_Shot_in.txt -swing Hole2_Tee_Shot_out.txt
6. In the Files sub-tab, add the input file Hole2_Tee_Shot_in.txt located in the Working Directory for Exercise 5.

7. Add the output file Hole2_Tee_Shot_out.txt located in the Working Directory for Exercise 5.

Note that the input and output files were generated for us ahead of time. Normally, we would have to create these ourselves by building a baseline model (to create the input file) and running the simulation of the baseline design (to create the output files).

Exercise 5: Constrained Optimization  engineering工程作业代写

The Process tab should now look similar to this:

8. Link the SwingSim screen shots to this analysis so that we can use them during post-processing.Referring to the picture above, click the Visualization sub-tab near the bottom of the screen.Add the Image file: swing.png

9. Save your project.

Step 3: Defining the Project Variables and Responses

There are five variables to define in the Parameters tab. The definition of these variables gives rise to what we often refer to as our design space, or parameter space.For continuous variables, the properties that must be defined are the min and max values, the baseline and the resolution. While sometimes there are obvious limits or available data to help in the definition of these properties, often they are defined based on intuition or experience.

The results of a study sometimes suggest that a better solution could be found with a different set of variable definitions. For example, if one or more variable values tend toward the max or min value during an optimization study, it may be worthwhile to expand the range for those variables in the next run, if this is allowable for your problem. HEEDS MDO Introduction Training Exercises

For the current study, the variables have the following properties:

Min Variable Name Max Baseline Resolution

If you are unsure how to define these variables, refer to Step 3 of Exercise 2.

When completed, your variable definitions should look like this:

There are three output responses to define:  engineering工程作业代写

BallDistanceToHole

BallDistanceToTree

FirstImpactDistanceToGreen

Since the value corresponding to each of these responses exists in the analysis output file, the source for each of them will be Tag. In other words, we will tag a location in the output file so that HEEDS will know where to extract these values.

If you are unsure how to define these responses, refer to Step 3 of Exercise 2.

When completed, your response definitions should look like this:

Save your project.

Step 4: Tagging the Input and Output Files

You could think of the tagging process as connecting your analysis model (input and output files) to the design study statement (variables and responses).

More literally, tagging tells HEEDS where to change values of the variables in the input files, and where to extract values of the responses in the output files.

If you are unsure how to tag your variables or responses, refer to Step 4 of Exercise 2.

When completed, your tagged input file should look like this:

And your tagged output file should look like this:

Save your project.

Step 5: Defining the Optimization Study  engineering工程作业代写

Now that the process automation steps are fully defined, we need to define the optimization study. The study details are provided below. The budget for SHERPA will be 250 evaluations in this study.

If you are unsure how to complete the study definitions, refer to Step 2 of Exercise 3.

Note that constraints are defined in a similar manner as objectives, except that the constraint limit must also be specified.  engineering工程作业代写

Study name: Constrained_Optimization

Study type: Parameter Optimization

Optimization algorithm: SHERPA

Number of Evaluations: 250

Objective: minimize BallDistanceToHole

Constraints: BallDistanceToTree ≥ 10 yards

FirstImpactDistanceToGreen ≤ 10 yards

When completed, your study response definitions should look like this:

If you select any objective or constraint, you can then expand the properties pane on the far right by clicking the symbol.

In the properties pane, you can modify the Normalization factor and the weights used in the performance function. Note that for objectives the default Normalization factor is the baseline value of that response. The default Linear weight is 1 and default Quadratic weight is 0. For constraints, the default Normalization factor is the value of the upper or lower bound. If the bound for a constraint is zero, then you will need to modify the Normalization factor for that constraint. The default Linear weight is 0 and Quadratic weight 10,000.  engineering工程作业代写

For all responses, you can set an ERROR criterion in the properties pane. This allows you to automatically declare the design as an ERROR if the response value is outside the specified limits. This functionality is useful when you know that any response outside the supplied range is not a legitimate result. The default action here is not to define an ERROR criterion.

ERROR limits are quite different than constraint limits. A design that violates a constraint may still be a valid design, even though it is not necessarily a high performing one. Infeasible design points provide useful information to many types of search algorithms, so they should not be discarded. On the other hand, ERROR designs can severely mislead a search algorithm, so they should be ignored.

For the current study, leave all response properties at their default values and close the properties pane.

Save the project.

Step 6: Running the Project  engineering工程作业代写

Run the project.

As described in the previous exercise, monitor the intermediate results to verify that they are as expected.

Post-Processing

If you have been monitoring the results during execution, you have already noticed that there are a lot of infeasible designs in this study. Apparently it is not easy to achieve a hole-in-one while hitting a golf shot around a big tree, and while landing the ball very close to the green.

The SHERPA search algorithm is able to find very good designs on highly constrained problems, even when the baseline (starting) design is infeasible.

Let’s see what else we can learn from the study data. Recall that your results may be different than those in the screen shots below.

The Design Study Summary  engineering工程作业代写

Open HEEDS Post and answer the following questions based on the study Summary page.

If the Summary page does not appear immediately on the screen, click on the Study in the tree.

• How many designs were evaluated? __________

The answer should be the number that was requested (your optimization budget). If this is not the case, then an error may have occurred or the study might have been terminated prematurely.

• How many of the designs were feasible? __________ Infeasible? __________
• What is the evaluation number of the best design found? __________
• What is the Distance to the Hole for the best design found? __________

The Objective History

Click on the Objective History plot and answer the following questions:

• For the best design found, did the shot go to the right, to the left, or over the tree? __________

Hint: right click on the best design and select Show model. You might have to zoom in to more easily select the best design. Alternatively, select the Hole2_Tee_Shot Image from the tree and view the trajectory image for the best design.

• Why does the blue line move upward for a short time early in the search?  engineering工程作业代写

_____________________________________

Hint: The blue line tracks the highest ranked design found so far. Design ranking depends on the

value of all objectives and constraints as they are represented in the Performance function.

The Parallel Data Plot  engineering工程作业代写

Click on the Parallel Data plot. Perform the following filtering process and answer the questions below.

• With so many infeasible designs, the plot looks messy and difficult to interpret. Filter out the infeasible designs by selecting the Plot ribbon and deselecting the Infeasible Designs in the Plot Data grouping. The result is:

• Click and drag to select multiple designs that are close to the hole. Refer to step 6.3 in Exercise 4 if you are unsure how to do this. Based on the characteristics of these best shots, to which design variables is the objective most sensitive? Least sensitive?
• There are vertical red lines along parts of the graph below “BallDistanceToTree” and above “FirstImpactDistanceToGreen.” What do these red lines represent?
• Highlight the Best Design. For this design, are the two constraints active or inactive?
• Try further honing the results by filtering out designs that are more than 3 yards from the hole.Based on the data, how many shot types (concepts) yield good solutions to this problem?  engineering工程作业代写

The Constraint Violations Plot

Click on the Constraint Violations plot. This plot displays the number of designs that did not satisfy each of the constraints. Recall that constraints limit the acceptable regions of the design space.Generally, those constraints that are violated most often during the study are the ones that are most difficult to satisfy. These constraints may have a significant influence on our ability to improve the objective or even to satisfy other constraints. It follows that we may be able to find better solutions by reducing the severity of some of these constraints, if it is acceptable to do so.

• In the current problem, which of the two constraints was most difficult to satisfy? __________
• How could we have increased the number of feasible solutions found in the study?

The 2D Bubble Plot

2D bubble plots are not created by default, but they can be useful in many problems. A 2D bubble plot is a 2D Relation Plot, with a third dimension represented by the size of the symbol (a bubble in this case).

Let’s create a 2D bubble plot:

1. Click on the 2D Relation Plot button in the General Plots grouping on the Home ribbon.
2. Select All Designs as the Design Set to be plotted.
3. In the X-Axis Parameter pull-down menu, select BallDistanceToTree (one of our constraints).
4. In the Y-Axis Parameter menu, select FirstImpactDistanceToGreen (our other constraint).
5. Click on the Finish button to create the plot.

The 2D Relation Plot creation wizard is shown in the figure below.

6.To complete the bubble plot, we can select any response or design variable to be represented as the size of the bubble. In the Series Data grouping in the Plot tab ribbon,select BallDistanceToHole from the Size pulldown menu, as shown in the figure to the right.  engineering工程作业代写

The constraint boundaries are represented as blue hashed lines (see the figure below).

• Are the two constraints active or inactive? ________________

Hint: Highlight the Best Design and determine how close it is to the constraint boundaries.

• For this problem, why do you think the best design found is not located at the constraint boundaries?

Hint: Are the constraints competing with the objective in this problem?

Additional Experiments

Here are a few suggestions for additional experiments:

1. Change the range of variables (primarily the TeeTargetAngle) to force your shot to the left, to the right, or over the tree. You might need to broaden the range of some of the other variables to find a good shot for some of these cases.  engineering工程作业代写

2. Change the values of the constraint limits to make the shot easier or harder.
3. Try a different optimization problem statement:

Objective: maximize Ball Distance To Tree

Constraints: Ball Distance to the Hole ≤ 3 yards

First Impact Distance To Green ≤ 10 yards

The above problem statement emphasizes the reliability of the design. By maximizing the Ball Distance To Tree, we are minimizing the chance that the ball will hit the tree if the shot is not executed exactly as prescribed. At the same time, we are still enforcing a minimum level of shot quality (less than 3 yards from the hole).

Discussion Questions  engineering工程作业代写

Develop your own answers to these questions, then read some possible responses on the following page.

1. How can we tell if a constraint is active?
2. If your optimized design is not as good as you hoped and you have active constraints, how could you possibly improve your design?

3. Why are responses normalized in the Performance function?
4. Why is the Performance function defined so that it always increases?
5. Based on the definition of the Performance function, is it possible for an infeasible design to be ranked higher than a feasible design?

 

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