auto des ing simple examples for v7r2

8/14/2019 Auto Des Ing Simple Examples for V7R2

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Part I.

Simple Example for AutoDesign

2009

Institute of Design Optimization

&

FunctionBay Inc


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Sample-A: Three-ball contact problem

In order to explain the basic function of AutoDesign, lets solve a simple design problem. The

design model consists of three balls. The yellow & the blue ball are fixed on ground. When the redball is thrown with an initial velocity, the red ball should be contacted with the yellow ball and go

through the blue ball as near as possible.

1. Model Definition

2. Design Parameter Definition

3. Analysis Response Definition

4. Design Study

5. Design Optimization

Next, the refined optimization is explained to find more accurate results. This design uses the

simulation results to solve the former design problem.

6. Refining the Design Formulation

Finally, in order to explain the characteristics of Meta-model based optimization. The final approach

employs another design formulation, which represents the contact phenomenon by minimizing AR3.

7. Another Design Optimization


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1.Model DefinitionThe contact between the red ball and the yellow ball is defined but is not done between the red

ball and blue ball, because the blue ball is just the target point. The design variables are the initialvelocity of red ball along x-direction and the contact stiffness coefficient in the contact force

between the red and the yellow balls. Now, for the red ball to pass the nearest way to the center

of the blue ball, what can you define as the design objective and constraints?

To do this, the design system is molded as follows:

Figure A-1-1. MBD Model of the ball contact design problem

The below is the procedure for defining the balls, joint and contact force shown in Figure A-1-1.

1. Make balls shown in Figure A-1-1 using Ellipsoid icon in the body module folder.

2. Fixed the Yellow ball and the Blue ball with ground using the Fixed joint in the joint

module folder.

3. Define the contact force between the red ball and the yellow ball using the " Sphere To

Sphere" contact in the contact module folder.


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2.Design Parameter DefinitionLets study the procedure for defining design parameters.

1. Define parametric values shown in Figure A-2-1.[refer to the parametric value in this

manual (RecurDyn Menus and Tools/Menu Bar Users Guide/Subentity

commands/Parametric Value)]

Figure A-2-1. Parametric value definition

2. Link the InitialVX with the x direction initial velocity of the red ball body shown in Figure

A-2-2.

Figure A-2-2. Link the InitialVX

3. Link the K with the stiffness coefficient of the contact force between the red ball and the


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yellow ball shown in Figure A-2-3.

Figure A-2-3. Link the stiffness K

4. Define the design parameters from parametric values using the Design Parameter

command in AutoDesign menu shown in Figures A-2-4 and A-2-5. First, you push Create button to define the design parameters as Figure A-2-4. In Figure A-2-5, you

should link the design parameters to the parametric values that were defined in Figure A-

2-1. The initial values are the current parametric values defined in Figure A-2-1. You

should define the lower and the upper bounds on the design variable. This represents that

the optimization process should change the design values within these bounds during

iterations. After you create the design parameters, you check the active design

parameters for this design problem.


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Figure A-2-4. Check DV check box

Figure A-2-5. Define DP1 and DP2


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3.Analysis Response DefinitionThe design goal is that the center of the red ball passes the nearest way from the center of the

blue ball, the target point. You need to define the performance indexes for solving theoptimization problem. In AutoDesign, performance indexes are objectives and constraints in

design optimization, which are composed of analysis responses. Then, in order to define the

performance index, analysis responses are defined first. The procedure of defining analysis

responses is explained as:

Define Expressions shown in Figure A-3-1. Each expression is defined shown in the Figures A-

3-2, A-3-3, and A-3-4, sequentially.

Figure A-3-1. Expression List


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Figure A-3-2. Detailed Definition of Expression Ex1



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Register expressions for analysis response shown in Figure A-3-5. Other figures are

representing the dialogue of each analysis response. The detailed information for each

analysis response is shown in Figure A-3-6. Also, their physical relations are shown in Figure

A-3-7.

Figure A-3-4. Analysis Response List


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Figure A-3-5. The detailed information for three analysis responses

Figure A-3-6. Three analysis responses in the model

AR1

AR2 & AR3


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4.Design StudyBefore you start to solve this optimization problem, it needs to know the relationship between

design variables and analysis responses or the correlation between analysis responses. To get thatkind of information, you need the effect analysis from design of experiments. AutoDesign provides

such functions as effect analysis, correlation analysis and even design variable screening in Design

Study menu. Design Study is composed of six sub-menus listed in Table A-4-1.

Table A-4-1. Description of sub-menu in Design Study

Design Variables Select DOE method and define the level for variable

Performance Index Show the ARs checked in Analysis Response menu

Simulation Control Define the solving option of RecurDyn solver

Effect Analysis Perform the effect analysis from the analysis results

Screening Variables Screening procedure from the effect analysis results

Correlation Analysis Perform the correlation analysis from analysis results

Basic Procedure for Design Study

In order to design study such as effect analysis, screening variables and correlation analysis, you

select the DOE method and define the level for each variable and perform the simulation of

RecurDyn. First, these procedures are explained as:

1. In the sub-menu of design variables, select Boses Orthogonal Design inDOE methods,

and set the level of the study as 5. Then, one defines the required runs as 25. This

method is a strength-II orthogonal array design. For more information, one may refer to

the theoretical manual of AutoDesign.


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Figure A-4-1. Define DOE method for design study

2. Confirm the Performance index that is checked in Analysis Responses.

Figure A-4-2. The selected PI list for design study

3. In the sub-menu of simulation control, define analysis setting shown in Figures A-4-4 and

A-4-5. This setting is the same as that in RD analysis. If you increase the accuracy of

effect analysis and optimization results, it is recommended that the plot multiplier factor

should be 1.0 and increase the number of steps. After setting the options, push the

simulation button. Then, Recurdyn is analyzed for the given number of trials.


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Figure A-4-3. Simulation control page

Figure A-4-4. General analysis setting


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Figure A-4-5. Integrator parameter setting

4. After all analyses are completed, one can select the effect analysis, correlation analysis and

screening design variables.

5. Now, select the effect analysis menu. Effect analysis gives the relation between one

performance index and all design variables.

Effect Analysis

Figure A-4-6 shows the effect analysis menu. Lets study the effect analysis procedure.

Select the performance index in PI row. Then check the design variables to see the effect

analysis for the selected PI. Then, push DRAW button. Figure A-4-7 shows the effect analysis

between PI_1 and design variables. This shows that DV1 is more nonlinear than DV2 in the

distance between red ball and blue ball.


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Figure A-4-6. Sub-menu for effect analysis

Figure A-4-7. Effect analysis result for the first PI


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Similarly, you can see the effect analysis for PI_2, which is shown in Figure A-4-8. For the 4th

and 5th levels of DV1, the contact forces are zero. This represents that two balls are not

contacted for those cases. It is noted that this discontinuity makes the accuracy of meta-

model to be worse.

Figure A-4-8. Effect analysis result for the second PI

Finally, you can see the effect analysis for PI_3, which is shown in Figure A-4-9. This

represents the distance between red and yellow balls. Unlike Figure A-4-8, this shows a

continuous result even though two balls didnt contact for 4th and 5th levels of DV1. This

represents that PI_3 is suitable to define the contact constraint in the design optimization.


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Figure A-4-9. Effect analysis result for the third PI

The explanation of effect analysis is completed. However, you have a question for the

minimization or maximization combinations shown in Figure A-4-10.

Figure A-4-10. Selection of minimization or maximization combination


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Suppose that you select the third PI. Then, you see the effect analysis result shown in Figure A-4-

9. Then, Figure 5-10 shows the design variable combination for minimizing PT_3 and maximizing

PI_3. If you need the confirmation for minimum or maximum set, then select one of them and

push simulation button in Figure A-4-10 menu.

Screening Variables

Figure A-4-11 shows the menu for screening variables. First, you can see the scatter points. This

represents the design variables. In this problem, there are only two design variables. Thus, variable

screening is not required but we study only the screening variable method.

1. First, select the first performance index, PI_1. Figure A-4-11 shows that two design

variables have severely different effectiveness. Now, you need to know which variable is

effective for PI_1.

2. Define the cutoff values as 1.0 and push Draw button. You can see Figure A-4-12. Then,

push Screening DV button. Figure A-4-13 shows the screening result. It shows that design

variable DV1 (or DP1) is more effective than DV2.

Figure A-4-11. Sub-menu for screening variables


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Figure A-4-12. Defining the cutoff value for screening variables

Figure A-4-13. Screened result for the first performance index


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3. Next, change the performance index AR3. Then, define the cutoff value as 12. Perform the

similar procedure in step 2. Then you can see the result shown in Figure A-4-14. In the

figure, Current represents the screening results for PI_3. Total represents the union of

screening results for PI_1 and PI_3. If you push update button, only active designvariables (marked on) are remained in New Model or Current Model.

Figure A-4-14. Screened result for the third performance index

Correlation Analysis

Figure A-4-15 shows the menu for correlation analysis. This shows the relation between two

selected ARs from the analysis results. If you see the relation between the first PI and the third PI,

check Horizontal Axis as PI_1 and Vertical Axis as PI_3 and push Draw button. Then, you can see

the correlation result shown in Figure A-4-16. Figure A-4-16 shows that they have no trend or

slightly reverse trend.


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Figure A-4-15. Sub-menu for correlation analysis

Figure A-4-16. Correlation result between PI_1 and PI_3


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5.Design OptimizationLets remind the following design problem:

Find the initial velocity of red ball along x-direction and the contact stiffness between red andyellow balls for red ball to hit the blue ball after red ball hit yellow ball.

Next process is for defining the design option and executing the optimization analysis. The first

step is to define the design variables shown in the Figure A-5-1. This can start using the Design

Optimization command in the Auto Design menu.

1. In Design Variable menu, the selected DPs are listed. In this menu, DP can be design

variable or constant during optimization process. If you define a DP as constant, you

should define its constant value.


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Figure A-5-1. Definition of design variables

2. The next process is to define the performance indexes in Figure A-5-2, which is named as

performance index of the dialogue of Figure A-5-1. Performance Index is a design

optimization formulation part. Figure A-5-3 shows the mathematical definition for design

optimization. Lets discuss the optimization formulation in Figure A-5-2.

In the first performance index, choose AR1 and define it as objective. Also, select the

design goal as minimization and define its weighting coefficient as 1.0.

In the second performance index, choose AR2 and define it as objective. Unlike AR1, the

design goal is defined as maximization and its weight coefficient is defined as 1.0.


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Figure A-5-2. Definition of performance indexes

Figure A-5-3. Design optimization formulation

3. Select the DOE & meta-modeling method shown in the Figure A-5-4. This type design

problem is somewhat noisy. Thus, Radial Basis Function Model (Multi-Quadratics) is

selected. Also, ISCD-2 is selected as DOE, which is an efficient DOE method, which


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requires only 5 points for two design variables.

Figure A-5-4. Selection of the DOE & meta-modeling method

4. Define the option of optimization control and execute analysis shown in the Figure A-5-5.

The analysis setting is the same that of Design Study. Finally, if you push the optimization

button, you can see the summary of the design optimization formulation shown in Figure

6-6. Then, check your formulation. If you see some mistakes, then push Cancel button and

correct the mistakes. Otherwise, push Simulate button. Then, AutoDesign runs until

convergence criteria are satisfied or maximum iteration is reached. During optimization

process, you can see the analysis results in Simulation History menu.


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Figure A-5-5. Control option definition for optimization and analysis

Figure A-5-6. Summary of design optimization formulation


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If AutoDesign is completed, then you can see the convergence results in Result Sheet. Figure

A-5-7 shows the optimization results. The final value of AR1 is 4.7106 (mm). Figure A-5-8

shows the trajectory of red ball for SAO15.

Figure A-5-7. Convergence history

Figure A-5-8. Animation of the final design


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6.Refining the Design OptimizationIn chapter 5, AutoDesign gives a good optimization result that the distance error between red and

blue balls is nearly 4.7(mm). Now, we try re-optimization to find more accurate solution. As youcan see, AR2 is maximized to guarantee the contact between red and yellow balls. Thus, we

modify the weight between AR1 and AR2. Put AR1 to twice weight than AR2, this represents that

MIN AR1 is more refined than MAX AR2 during optimization process. Also, We add one

inequality constraint as AR1 =< 4.0, this represents that the new design should be better than

the current optimal design. Figure A-6-1 shows the refined formulation.

Figure A-6-1. Modification of Optimization Formulation

As this new design problem uses the same design variables and ARs, we can use the simulation

results used to solve the design problem in Chapter 5. Thus, in DOE Meta Modeling Method, we

uncheck Select DOE Method and check Get From Simulation History shown in Figure A-6-2.

Then, the simulation history window will be shown. Figure A-6-3 shows the simulation history

results. You define the importing results by checking in import box. Then, push Confirmbutton.

Then, you can see the windows shown in Figure A-6-2, again. Select the RBF(Multi-Quadtatics) as

meta-model method.


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Figure A-6-2. DOE selection for Meta-Modeling

Figure A-6-3. Get From Simulation History


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In the optimization control window, all the convergence tolerances are used the same values in

Chapter 5. Then, pushOptimizationbutton. The summary of optimization formulation is shown in

Figure A-6-3. You can see the changed information in the performance index and meta-modelparts.

Figure A-6-4. Summary of re-optimization formulation


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Figure A-6-5 shows the re-optimization results. The refined value of AR1 is0.8976 (mm).

Figure A-6-5. Convergence history


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7.Another Design OptimizationWhen the effect analysis was performed, the contact force (AR2) was not continuous but the

distance between red and yellow balls (AR3) is continuous. In order to help ones understanding,lets consider the effect analysis results again. Figure A-7-1 shows the effect analysis results for

contact force. When the initial velocity (DV1) is higher, two balls (Red and Yellow balls) do not

make contact. The distance between red and yellow balls (AR3), however, gives + values for

those cases, which is shown in Figure A-7-2. This represents that AR3 is more suitable for meta-

models than AR2.

Figure A-7-1 Effect analysis result for AR2

Figure A-7-2 Effect analysis result for AR3

Now, we can formulate the design problem as:

Minimize AR1 & AR3,


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where this formulation is shown in Figure A-7-3.

Figure A-7-3 another formulation of the three ball contact problem

Figure A-7-4 DOE method and meta-modeling method


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Figure A-7-4 shows the initial DOE method and meta-modeling method that are the recommended

options for design optimization. ISCD-2 and RBF(Multi-quadratics) are selected, which are the

default methods. Also, for polynomial type, Auto is the default option. The convergence

tolerances for design optimization use the default values.

Figure A-7-5 the convergence tolerances

Finally check the analysis setting option before push optimization button in Figure A-7-5. The

value of plot multiplier step factor is set to1and the number of step should be highenough to

represent the dynamic behavior of system numerically. In this study, we uses step=1000.

Figure A-7-6 shows the summary of this new design formulation. Except replacing the contact

force with the distance, all other information is equal to that of the design optimization in section6. Figure A-7-7 shows the convergence history. AR1 is converged to 0.9346 (mm). This is nearly

equal to that of the refined design optimization described in section 6. The design approach,

however, is easier than the former one.


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Figure A-7-6 summary of another design formulation

Figure A-7-7 optimization results for another design formulation


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Sample-B: Catapult system design

Figure B-1-1 shows a catapult with a curved arm, which throws a ball using a target. Certain

aspects of the catapult can be changed to aim the catapult. These are the angle of the front linkat start position and the mount point of the main spring to the front link. As the engineer, your

goal is to control these parameters such that the ball will not only arrive at the mouth of the

target, but will also do so with the correct angle of attack which will allow the ball to go inside.

The model supplied with this tutorial will have all of the geometry and joint modeling complete,

but are not ready for optimization. In this tutorial sample, you will learn how to prepare this

model for design optimization.

Figure B-1-1. Catapult system

Figure B-1-1. Catapult System


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1. Loading the Model and Viewing the Animation

1-1. To load the base model and view the animation:1. On your Desktop, double-click the RecurDyn tool.

RecurDyn starts and the New Model window appears.

2. Click Cancel to exit the New Model dialog box. You will use an existing model.3. In the toolbar, click the Open tool and select Sample_B.rdyn from the same directory

where this tutorial is located.

The catapult appears in the modeling window.

4. Click Play.The trajectory of the ball should appear as shown in Figure B-1-2. The contact between

the ball and the target has been disabled so that it doesnt interfere with the design

optimization results. Also, an inplane joint was added to remove out-of-plane movement

of the ball in the z-direction, as a small amount of this was inevitable due to the

tessellation of the catapult arms track surface. The main focus of the model is on the

balls movement in the x- and y-directions.

Figure B-1-2. Animation for the current design


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1.2. The Design VariablesThe design variables will be the factors of the model which you can control. Figure B-2-1 is a

diagram showing these factors on the model.

Figure B-2-1. Two design variables

Front link angle is defined as the angle of the front link from horizontal, measured at the rear

pivot point. As the front link angle is varied, the rear pivot point remains stationary while the

front pivot point moves to accommodate the angle change.

Spring mount height is the distance between the spring mount point and the front pivot, and is

expressed as a fraction of the entire link length (the distance between the front and rear pivots).

2.2-1. Exercising the ModelYou will now explore how changing the design variables affects the balls trajectory. In this model,

the design variables will be linked to parametric values, so for now you will actually vary the

parametric values, instead.

1-2. To exercise the model:


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1. In the Database window, double-click on one of the items under ParametricValues shownin Figure B-2-2.

Figure B-2-2. Parametric Value list

2. Click or double-click on the value next to PV_frontLinkAngle and change it to -40.3. Click OK, noting how this change affects the catapults physical configuration.4. Click Analysis.5. Click Simulate.6. When the simulation stops, click Play and view the results.7. Repeat the above steps, using different combinations of values for PV_frontLinkAngle and

PV_springMountHeight, within the following ranges:

-40 PV_frontLinkAngle -10 0.4 PV_springMountHeight 0.6If you do not wish to run through many simulations, Figures B-2-3 ~ B-2-6 are several

animation results representing the ends of the ranges for the two design variables.


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Figure B-2-3. Analysis result that front link angle = -10 degree and spring mount height = 0.4




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8. When you are finished exploring the model, return the parametric values to their original

values before continuing with the tutorial (PV_frontLinkAngle = -25, PV_springMountHeight

= 0.5).

3.2-2. Defining the Design Variables1-3. To create a design parameter:

1.

From the AutoDesign menu, click Design Parameter. This will bring up the DesignParameter List dialog shown below.


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2. Click Create to create a new design parameter.3. In the Direct Relation dialog that appears, change the name from DP1 to

DP_frontLinkAngle.

4. Press Pv to bring up the Parametric Value Listdialog. Select the PV_frontLinkAngle parametric

value by clicking on its name. When selected, it

should be highlighted in blue, as shown at right.

5. Click OK to choose this as the design parameter.

6. Back in the Direct Relation dialog, define upper and lowerbounds (-40, -10). Enter a description (Front link angle)

in the Description field. When completed, the dialog

should appear as shown at right.

7. Press OK to return to the Design Parameter List dialog.

8. Create design parameter for spring mount height, similarly, using the following settings: Name: DP_springMountHeight Parametric Value: PV_springMountHeight Lower Bound: 0.4 Upper Bound: 0.6


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Description: Spring mount height9. Return to the Design Parameter List dialog, and check the checkboxes under the DV

column for both of the design parameter you just created. This activates both as Design

Variables, which will be used in the Design Study and Design Optimizations to follow.

When completed, the Design Parameter List dialog should appear as shown below. Note:

To go back and edit a design parameter, click on the button under the Prop. column.

10. Press OK to close the Design Parameter List dialog.


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4.3. Defining the Performance IndexesThe performance indexes will tell you how well or poorly the model is able to perform its goal. In

this case, these are the error of the balls angle of attack and how close it gets to the target. To

obtain good optimization results, these goals are formulated as follows.

The position and velocity of the ball is measured with respect to the targets reference frame. To

evaluate the angular error of the ball, the balls velocity in the y -direction is measured as it

crosses the target plane. A small y-directional velocity indicates a small angular error. To

evaluate the positional error of the ball, the distance between the ball and the target is measured

in the y-direction as the ball crosses the target plane, as shown in Figure B-3-1.

Figure B-3-1. Performance indexes for design optimization


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In addition, a third metric is used, which is simply the

measurement of the closest the ball ever comes to the

target.

1-4.

This formulation of the performance indexes may not be the most intuitive one, but it provides results whi

ch are the least sensitive to numerical noise.

1-5.To create an analysis response:1. From the AutoDesign menu, click Analysis Response. This will bring up the Design Parameter

List dialog shown below.

2. Click Create to create a new analysis response.


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3. In the Analysis Response - Basic dialog that appears, change

the name from AR1 to AR_yVelocityError_t.

4. Press EL to bring up the Expression List dialog. Select the

Ex_yVelocityError_t expression by clicking on its name.

When selected, it should be highlighted in blue, as shown at

right.

5. Click OK to choose this as

the expression to use.

6. Back in the AnalysisResponse - Basic dialog, for the Treatment, select End Value from the dropdown list. Enter a

description (Vertical velocity error of ball w.r.t. target.) in the Description field. When

completed, the dialog should appear as shown at right.

7. Press OK.


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Figure B-3-2. Analysis response list

10. Press OK to close the Analysis Response List dialog.

11. Save the model.


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5.4. Design Optimization1-6.

4-1. Objectives and Constraints

When performing optimizations, you can define objectives and constraints that will guide the

optimization to the desired solution. Objectives are used when you want to minimize or

maximize performance indexes. Our model has three performance indexes, of which one will be

minimized using an objective. For other models, it is possible to have multiple objectives, and

have different weights assigned to each one to specify how important it is relative to the other

objectives. Constraints, on the other hand, are used when a specific requirement must be met in

order for a solution to be considered successful.

For this problem, we may specify that the y-velocity error is less than 5(mm/sec), or that the y-

position error must be less than 5(mm). From the viewpoint of ideal solution, these values should

be equal to0.0. However, the numerical analysis can not give those accurate results. Thus, the

values of these limitations fully depend on the accuracy of numerical solvers.

1-7.4-2. AutoDesigns Design Optimization ProcessWhen AutoDesign performs an optimization, it first

performs a design of experiment (DOE). During

this process, it samples the design space at

several points, evaluating the performance indexes

given various combinations of the design variables.

These points are indicated by the blue squares in

the figure at right.

Next, AutoDesign fits an analytical surface to the

points, called a meta-model. Using this meta-

model, AutoDesign then determines the best point

in the design space to search for the optimal

solution. Then, evaluate the exact analysis for the

selected optimum point. If this new design can not

satisfy the convergence criterion, then, the

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

Design Variable 1

D

es

i

n

Var

ia

ble

2

Design Space


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analysis results and design point are added to the original DOE tables and re-construct the meta-

model. Then, the optimizer solves the optimization problem made from new-meta model. This

process is repeated until all convergence criteria are satisfied. The red squares, shown in the

figure, are the optimum points selected by optimizer. We call this optimization process asSequential Approximate Optimization with Meta-Models (SAOM). For more information on the

sampling algorithm, please refer to the AutoDesign Theoretical Manual.

6.4-3. Running a Design OptimizationWe will now run an optimization in which the objective is to minimize the position error, and

constraints are set on the y-velocity and y-position error.

To run a design optimization:

1. From the AutoDesign menu, select Design Optimization.The Design Variable dialog should appear as shown below, by default.


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2. Click on the Performance Index tab.3. Change the objective function according to the table below.

Performance Index Definition Goal Weight/Limit Value

AR_yVelocityError_t Constraint EQ 0

AR_yPositionError_t Constraint EQ 0

AR_positionError Objective MIN 1

After making the changes, the dialog should appear as shown below.

Here, we are defining that the target values for the errors are 0, and at the same time we

want to minimize the position error.

4. Click on the DOE Meta Modeling Methods tab.


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11. Change the settings so that they appear as shown at below. As explained in section 4-1,we will try to satisfy ABS(AR1) and ABS(AR2) .LE. 5. To do this, we define them as

equality constraints and set their convergence tolerances to 5.0. This limitation fully

depends on the resolution of dynamic analysis results.

12. Check the analysis setting by clickingAnalysis Setting button. In order to reduce

the numerical error, we increase the

number of time steps shown in right. If you

increase the resolution of optimization

solution, then increase the number of steps.

In chapter 6, we will show more accurate

design by only increasing the value.


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13. Click the Optimization button to run the optimization with the settings you just made.Then, you can see the summary of the optimization formulation shown at below.

Then, click the simulate button. The optimization will be progressed.

13. To view the result of the design optimization after optimization is completed, Click the

Result Sheet tab.

The Performance indexes for optimization iterations are listed at the top of the dialog. To

see the final result of the optimization shown at below, scroll down to the last iteration.

As shown below, the final vertical velocity error is 2.12885 (mm/sec), and the final vertical

position error is 0.47 (mm). Also, the final position error is 0.6235 (mm). The optimization

took 8 iterations to converge to these results. For the consecutive iterations between SAO7

and SAO8, All ARs have equal values. Thus, AutoDesign does not call dynamic analysis forSAO8 because it is the same result as SAO7. When you see the summary file, AutoDesign

explains this phenomenon. As the initial DOE requires 5 runs, SAO8 becomes RUN_13

because it is 5+8. For this case, you can use the analysis results of SAO7 for the final

design.


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7.5. Animating the Optimized ModelIn order to animate the optimized model, we will first have to update the model with the

optimized design variables.

1-8. To update the model with the optimized design variables:1. From the AutoDesign menu, select Simulation History.

Using the scrollbar on the right side, find the last simulation.

2. Check the box under Update for this simulation. You may have to use the horizontal scrollbar to see this checkbox.

3. At the bottom of the dialog, click the radio button next to Current Model.4. Click the Update button.5. Click OK.


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The models parametric values have

now been updated, as shown at right

(this window can be brought up by

selecting Parametric Value from theSubentity menu).

6. In the Database window, under Contacts, right-click onSphereToSurface_ballToTarget and uncheck the Inactive

option.

7. Click Analysis.8. Uncheck the checkbox next to Output File Name, if it is checked.9. Click Simulate.10. When the simulation is done, click Play.

The ball should successfully make it into the target, as shown below. The results will be

easier to see if the model is displayed in Wireframe with Silhouettes mode.


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Before continuing with the tutorial, the model needs to be reset for the next optimizations.

1-9. To reset the model for the next optimizations:1. Inactivate the contact between the ball and target.2. Change the parametric values back to the initial values:

PV_frontLinkAngle = -25 PV_springMountHeight = 0.5

3. Save the model.

8.


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9.6. Refining the design optimizationAs we explained in Chapter 4-3, the accuracy of optimization results fully depend on that of

analysis results. In order to show their relations, lets reconsider the design problem described in

chapter 4-3. Now, lets increase the number of steps twice as follows:

Then, all information for the design optimization formulation uses the same values except time

steps and convergence tolerance. The convergence tolerance for equality constraints are reduced

by 2.0. In chapter 4, we use the value as 5.0.


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Sample-C: Robust Design Optimization

Robust Design Optimization differs from the deterministic design optimization you just performed

in Sample-A and Sample-B, in that it takes into account the variability of the components whichmake up the system being optimized. For example, if temperature fluctuation or manufacturing

conditions caused variability in the front link angle, you could measure this variability and input

the standard deviation into the robust design optimization. The optimization would then give

results which would tell you the variability of the system performance, and therefore aid in the

design of a system robust to the variation of its individual components.

RecurDyn AutoDesign allows you to define the sigma level to which you want to optimize. That is,

you can define the percent feasible, or the fraction of the produced systems which will satisfy

the quality constraints. A common standard is to design to 6-sigma, which means that99.9999998% of the produced systems will satisfy the quality constraints. For more information

on robust design optimization and design for 6-sigma (DFSS), refer to the RecurDyn Theoretical

and Guideline Manual.


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1. Loading the Model and Viewing the Animation

1-10. To load the base model and view the animation:1. On your Desktop, double-click the RecurDyn tool.

RecurDyn starts and the New Model window appears.

2. Click Cancel to exit the New Model dialog box. You will use an existing model.

3. In the toolbar, click the Open tool and select Sample_C1.rdyn from the same directory

where this tutorial is located. The MTT2D appears in the modeling window.

4. Click the model on the screen to enter MTT2D module. Then, Model name is changed from

Model1 into MTT2D@Model1 on the left upper part in screen.

5. Click the analysis button.

6. Click the play button.

The sheet runs through the upper and lower baffles and reaches to the second tray.

Figure C-1-2. Animation for the current design


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10.2. The Design VariablesThe design variables will be the factors of the model which you can control. Figure C-2-1 and

Figure C-2-2 are the window and the diagram showing these factors on the model.

Figure C-2-1. Three design variables for random constants

In the above window, the thickness, youngs modulus and curl radius are selected as DV1 ~ DV3.

These variables are un-controllable factors from the viewpoint of mechanism designers. Thus, we

consider them as random constants.

Figure C-2-2. Two design variables with tolerances


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In Figure C-2-2, the vertical locations of two guides are selected as design variables. In MTT2D,

the guide positions can not be defined by using parametric values. Thus, they are not design

variables directly. To overcome this situation, we introduce the motion that can include the

parametric values. Two motions use the following expressions, respectively.

PV_Yupper*STEP(TIME, 0, 0, 0.01, 1)

PV_Ylower*STEP(TIME, 0, 0, 0.01, 1)

11.2-1. Defining the Design Variables1-11. To create a design parameter:

11. From the AutoDesign menu, click Design Parameter. This will bring up the DesignParameter List dialog shown below.

12. Click Create to create a new design parameter.13. In the Direct Relation dialog that appears, change the name from DP1 to

DP_SheetCurlFactor.


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14. Press Pv to bring up the Parametric Value Listdialog. Select the PV_SheetCurlFactor

parametric value by clicking on its name.

When selected, it should be highlighted inblue, as shown at right.

15. Click OK to choose this as the designparameter.

16. Back in the Direct Relation dialog, define upper andlower bounds (-50, 50). Enter a description (Paper

Curl Factor) in the Description field. When

completed, the dialog should appear as shown at

right.

17. Press OK to return to the Design Parameter Listdialog.

18. Create design parameter for spring mount height, similarly, using the following settings:Name Parametric Value Lower

bound

Upper

Bound

Description

DP_Modulus PV_E 5200 7200 Paper_Modulus_E

DP_Thickness PV_Thickness 0.1 0.3 Paper_Thickness

DP_UpperPos PV_Yupper -0.1 0.1 Upper baffle Y loc

DP_LowerPos PV_Ylower -0.1 0.1 Lower baffle Y loc


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19. Return to the Design Parameter List dialog, and check the checkboxes under the DVcolumn for both of the design parameter you just created. This activates both as Design

Variables, which will be used in the Design Study and Design Optimizations to follow.

When completed, the Design Parameter List dialog should appear as shown below. Note:

To go back and edit a design parameter, click on the button under the Prop. column.

20.Press OK to close the Design Parameter List dialog.

12.3. Defining the Performance IndexesLets consider the paper distributing system shown in Figure C-3-1. The goal of the mechanism is

to design the baffler y-positions for the paper to pass through the target position nevertheless the

material property (Youngs modulus, thickness and curl radius etc.) variations of the paper. In this

problem, the material property variations are called asnoise factorsand the baffler positions aredone as design variables. If the design variables have +/- tolerances, we call them as random

design variables. In AutoDesign, the noise factors are called asrandom constants.


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Figure C-3-1. Performance indexes for design optimization

1-12. To create an analysis response:1. From the AutoDesign menu, click Analysis Response. This will bring up the Design Parameter

List dialog shown below.

2. Click Create to create a new analysis response.

3. In the Analysis Response - Basic dialog that


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appears, change the name from AR1 to AR_Ysensor1.

4. Press EL to bring up the Expression List dialog. Select the ExSensorexpression by clicking

on its name. When selected, it should be highlighted in blue, as shown at right.

5. Click OK to choose this as the expression to use.

6. Back in the Analysis Response -

Basic dialog, for the Treatment,

select End Valuefrom the dropdown list. Enter a description (Y where x is 894mm.) in the

Description field. When completed, the dialog should appear as shown at right.

7. Press OK.

To go back and edit an analysis response, click on the button under the Prop. column

8. Create two more analysis responses using the following values:

Name: Error Sum

Expression Name: ExSensorError_Square

Treatment: End Value

Description: Error Square


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The treatment parameter is used to control how you extract a single numerical value from a

curve which varies over time. For example, setting the treatment to End Value will assign the

value of the curve at the end of simulation, while Min Value will assign the lowest value that the

curve reaches during the simulation.

9. Return to the Analysis Response List window, and check the checkbox under the PI column

corresponding to the analysis responses you just created. This activates them as Performance

Indexes, which will be used in the Design Study and Design Optimizations to follow.

When completed, the Analysis Response List window should appear as shown in Figure B-3-2.

Figure C-3-2. Analysis response list

10. Press OK to close the Analysis Response List dialog.

11. Save the model.

4. Running a Robust Design Optimization

We will now run an optimization in which the objective is to minimize the variance of position error,


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and constraints are set on the y-position error.

The random design variables and random constants are listed in Table C-4-1.

Table C-4-1. List of random design variables and constants

Current Values Deviations Remark

Paper curl radius 0 -/+ 30 Random Constant

Paper Youngs modulus 6200 -/+ 10% Random Constant

Paper Thickness 2.0 -/+ 0.05 Random Constant

Upper baffle position -0.31 -/+ 0.05 Random Design Variable

Lower baffle position -0.22 -/+ 0.05 Random Design Variable

The robust design optimization problem is defined as:

Minimize Variance

Subject to

The paper position at x=894 (mm) = Target position

and

-1.0 =< Upper baffle position -/+ deviation =< 1.0

-1.0 =< Lower baffle position -/+ deviation =< 1.0.

The value of variance is affected from the tolerance of design variables and the deviations of noise

factors. Thus, the robust design optimization problem is to find the design variables to minimize

the variation of position errors, which is a typical example of robust design optimization.

To run a robust design optimization:

1. From the AutoDesign menu, select DFSS/Robust Design Optimization.The Design Variable dialog should appear as shown below. You should define the red-box


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parts.

Unlike the design optimization, three selections are newly shown. They areStatistical Info,

Dev. Typeand Dev. Value. The detail descriptions of them are explained in Guideline

manual. In the Statistical info, you can define which variables are random or

deterministic. If the variable has tolerance or deviation, then it is random. Otherwise, it

is adeterministicvariable. In theDev. Type, you can define that the deviation of variable

is an absolute magnitude or the ratio of design value.SDdenotes the absolute magnitude.

COVdoes the relative ratio. Paper properties are defined as random constantbecause

they are only noise factors.

2. Click on the Performance Index tab.

3. Change the objective function according to the table below.

AR Definition Goal Weight/Limit

Value

Robust Index Alpha Weight

AR_Ysensor1 Constraint EQ 12.3 NA NA

AR_Ysensor1 Objective MIN 1 1 0

It is noted that AR2 is not used in Sample_C1.

After making the changes, the dialog should appear as shown below. The grey part

represents the deactivated values.

In DFSS/Robust design optimization, the design objective is internally represented as


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PI = Weight*(AR*Alpah_Weight+Sigma*Robust_Index)

The value ofWeightrepresents the relative importance of the selected AR in the multiple

objectives. Alpha_Weight and Robust_Index are the flags of two responses. These valuesshould be0or1.0represents that the corresponding response is neglected.

If Alpha_Weght=1 and Robust_Index=0, then minimize Weight*AR.

If Alpha_Weght=0 and Robust_Index=0, then minimize Weight*Sigma.

If Alpha_Weght=0 and Robust_Index=1, then minimize Weight*(AR+Sigma).

If both values are0, then make no design formulation. It is a logical error.

4. Click on the DOE Meta Modeling Methods tab.

5. Click the checkbox next to SelectDOE Method.

6. Select Discrete Latin Hypercube

Design, which is the recommended

method when the random constants

are included. The number of

sampling points is recommended

more than 2*k+k*(k-1)/2, where

k is the number of design variablesand constants. For this problem,

k=5. Thus, the number of sampling

points should be more than

20(=2*5+5*4/2).

7. Select Simultaneous Kriging/DACE methodfor the Meta Modeling Method, which is also the

default method only when Latin hypercube design is employed.

8. Select Auto for the Polynomial Function, which is the default option.

9. Click R to update the number of trials to 21, which includes the current design.

10. Click on the Optimization Control tab.


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11. Change the settings so

that they appear as

shown at the right.

For the convergence

tolerance, the default values are used. Unlike the design optimization module,

DFSS/Robust Design Control is newly shown. The detail information of them is explained in

Guideline Manual. As you can see, AutoDesign solves the robust design problem by using

the meta-models. Although the analysis responses are validated when the meta-model is

updated during optimization process, the standard deviation is estimated from meta-

models. Thus, the variance values of final design are not validated. In theValidation Type,

there are three types such asNone,ValidationandValidation & Re-Optimization. When

Validation is selected, AutoDesign performs the exact analyses for the sampled points

within the deviation ranges at the final design. Then, estimate the sample variance. In the

Variance Estimation Method, there are two types such as Taylor Series method and

Random Sampling method, which are the variance approximation method from meta-

models, which are explained inGuideline manual.

12. Check the analysis setting by clicking

Analysis Setting button. As explained inSample-A and Sample-B, it is noted that

the accuracy of analysis responses

depends on the number of STEP.


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13. Click the Optimization button to run the optimization with the settings you just made.

Then, you can see the summary of the optimization formulation shown at below. Then,

click the simulate button. The

optimization will be progressed.


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14. To view the result of the design optimization after optimization is completed, click the

Result Sheet tab.

The optimization process is converged at the 7 th iteration. The final design gives that AR1

is 12.299713 and itsapproximate Sigma is 0.31 and the sample Sigma is obtained as

0.0697. The error between the approximate Sigma and the sample Sigma is caused from

the accuracy of meta-models. When the sample Sigma is greater than the estimated ones,

you may re-optimize by usingGet from Simulation History.

15. Now, check the analysis results in Simulation History. When constructing the meta-

models, the values of design variables and constants are sampled within their bounds and

deviations. These results are marked as Initial Runs for Meta Model in the column of

description of simulation. The iterative runs of robust design optimization are marked as

RSAO#. During optimization process, DV1 ~ DV3 are constants. Finally, as the validation

type is selected asValidation, the final 8 analyses are marked as Variance Verification#


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in the description column. It is noted that DV1~DV3 are changed within their deviations

for the verification runs.

16. Savethe model. In order to study 6-Sigma design, save as the model as Sample_C2.


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5. Running a 6-Sigma Design Optimization

Load the model Sample_C2.rdyn. As you can see, 6-Sigma design uses the same design

variables and analysis responses as Sample_C1. Only the design formulation is different fromthe robust design. Figure C-5-1 shows the design variables, which is the same asSample_C1.

Figure C-5-1. The design variables for 6-Sigma Design Optimization

To run a robust design optimization:

1. Lets consider the 6-sigma design optimization formulation, shown in Figure C-5-2. The design

goal is to minimize the sum of error, which represents (Y_position - 12.3)**2. From the

viewpoint of 6-sigma design, the Y_position should satisfy the following inequality relations.

9.3 =< Y_position -/+ 6*Sigma =< 15.3

AutoDesign describes the above relation by using two inequality constraints.

9.3 =< Y_position - 6*Sigma

Y_position + 6*Sigma =< 15.3

The signs of - and + positioned before sigma are automatically defined for the inequality

constraint types such as GEor LE. Hence, you can define those two inequality constraints

shown in Figure C-5-2. The grey coloured parts are deactivated.

As explained inSample_C1. DFSS/Robust design module defines the design objective as


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UncheckSelect DOE methodand ChekGet From Simulation History. Then, the button will

be activated. Click the button.

4. Then, you can see the simulation history as below. Then, check the runs in the Importcolumn. In order to compare 6-sigma design optimization with the robust design result,

select only the results ofInitial Runs for Meta Model.Finally, click Ok button notImport

button. TheImportbutton is used to import the text file of simulation history.

5. Now, you will back to the window ofDOE Meta Modeling Methods. Then, select Simultaneous

Kriging/DACE method for meta-model and Auto for polynomial type.

6. Click theOptimization Controltab. We will the same convergence tolerances and the same

validation information. Thus, click theRobust Design Optimizationbutton. If all information

is the same as Figure C-5-3, then push theSimulatebutton.


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Figure C-5-3. Summary of 6-Sigma design optimization formulation

7. When the optimization process is converged, click the Result Sheet tab. Then, the

optimization results are shown in Figure C-5-4. The final design gives that AR1=12.335 and

the estimate Sigma and the sample Sigma are 1.30 and 0.0698, respectively. In the robust

index check, the relation ofAR1+/-6*0.0698satisfies the limit values of 9.3 and 15.3. Thus,

the relaxed values for robust indexes are obtained as 6. For more information of 6-sigma

design, refer to theguideline manual.


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Figure C-5-4. The convergence history of 6-Sigma design optimization

Now, compare the optimization results ofSample_C1 and Sample_C2. Both designs have

different design variables (DV4 & DV5) but give nearly equal values of the sample-Sigma. Thus,

you can select one of them.

auto des ing simple examples for v7r2

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