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VISUAL DESIGN STEERING TO AID DECISION-MAKING IN OPTIMAL DESIGN

A. SamantGPTS, Schenectady, NY

P. ShahTIBCO Software Inc., Palo Alto, CA

E.H. WinerNew York State Center for Engineering Design and Industrial Innovation (NYSCEDII)

Department of Mechanical and Aerospace EngineeringUniversity at Buffalo

AbstractThis paper describes the development of a capability to thoroughly examine multiple design variable changes and their impact on an optimal design problem. This work is built under the paradigm of Visual Design Steering, specifically, upon the previous work of Graph Morphing. A designer has the ability to study design variables over their bounds and see the impact on the corresponding objective function and constraints in a single, graphical, interactive representation created in real-time. Two major components have been added to the original implementation of Graph Morphing: 1) Robust constraint handling and 2) Increased interactivity with the visual representations. In addition, a detailed benchmark study on the effects Graph Morphing has the solution of optimal design problems was conducted. The description of the new capabilities is discussed in detail, as is the benchmark study that was performed.

IntroductionWith the intelligent use of new and emerging technologies many design processes are becoming more detailed and accurate while reducing time and the associated resources required. However, it is very important to make sure that human traits such as thinking and creativity are still included when dealing with the increasing amounts of data being produced. Many design processes simply can’t be automated nor should they be. Humans are still the innovators and users of technology so, attempts must be made that will incorporate human characteristics into areas that up to now, only computers worked on.

Optimal design is the process of finding a value of a specified parameter, named the objective function1. This value must meet certain requirements called constraints, in order to be acceptable and feasible. A formally stated optimization problem has numerous parameters. The objective function is the cost-function that is being either minimized or maximized. The

design variables are parameters, which can be changed. Design variable values are the main driving force for either moving toward a maximum or minimum value of an objective function. Constraints may exist which are limitations on the design space. Constraints can be of numerous forms, including equality, inequality, or side constraints. A typical optimization problem has the mathematical form given in Equation 1.

Minimize: F(X)Subject to: gj(X) ≤0 j = 1,m

hk(X) = 0 k =1, lXil ≤ Xi ≤ Xiu i = 1,n

(1)

Where, X = {X1 X2 X3 ….Xn}T is the design variable

vector, gj(X) are the inequality constraints, and hk(X) are the equality constraints.

Optimal design solution methods essentially provide a means for investigating different values of design variables that best meet the requirements of the design problem (objective and constraints). As the number of design variables and constraints increase, so does the time needed to find the optimal point. These types of problems are the fundamental building blocks for developing a design process for a complex system better known as Multidisciplinary Design Optimization (MDO).

MDO attempts to deal with complex problems and designs in an all-at-once approach, rather than in a sequential manner. For example, consider the design of an aircraft, where for simplicity it is assumed that the design depends on three disciplines: aerodynamics, structures, and control systems (in reality there are many more areas that would be represented).

MDO methods strive to take advantage of the non-hierarchical nature of the coupled system rather than force an over-the-wall approach. Figure 1 demonstrates a simplified design process for a non-hierarchical

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization4-6 September 2002, Atlanta, Georgia

AIAA 2002-5623

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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representation in what is called a Multiple Discipline Feasible (MDF) approach2. In this representation, each discipline solves its own subsystem analysis while maintaining the coupling requirements of the other disciplines involved. To be successful, the disciplines must communicate and coordinate with each other throughout the design process.

Figure 1 – Sample MDO Design Process for a simplified Aircraft.

In Figure 1 we can see that optimization is a part of the iterative cycle. Solving an optimization problem where there is a tremendous amount of data and variables to work with might require hundreds or even thousands of iterations to attain numerical convergence, and this may have to occur each cycle of the overall MDO process. As might be expected, this high level of iteration is costly both in terms of time and computational resources.

Visual Design SteeringBy speeding up the solution of sub-system and system level optimal design problems, the overall time to complete an MDO design process can be significantly decreased. One question that arises here is how can one make more use of design information to make the process faster and more efficient? The aim here is to present a designer with insight into the complex relationships design variables have on all on aspects of an optimization problem so that intelligent decisions can be made to improve the solution efficiency, accuracy, and quality. Data generated by the design process must be presented to the designer in such a form that it will be very easy to grasp the intricacies of the problem and make quick, informed decisions. Creating a visual representation of the data is one such approach. A modified paradigm termed Visual Design

Steering (VDS)3,4 can be used to achieve this.

The principle of Visual Design Steering is as follows: Before, after, or during an analysis or optimization process, a designer is presented with visual representations of the most recent data. The visual representations provide understanding about the problem otherwise not available. These decisions can then be implemented immediately and the results examined. This is effectively the designer steering the solution process with the assistance of visual tools. These problems have become so complex that human interaction with the data is very limited without a paradigm such as VDS that offers methodologies to incorporate human traits into a complex design process.

Graph MorphingAn example to illustrate the point just made can be shown through the growth of optimization problems. Typical optimization problems may have hundreds of design variables and constraints. In many cases there may be thousand to millions of design variables and constraints. Without formalized concepts and methods, effective human interaction with these problems would be a very difficult task to say the least.

In the case of an n-dimensional optimization problem, the VDS paradigm can be applied through a concept called Graph Morphing3,4,5. Typical optimization problems have numerous design variables. Traditional attempts to visualize these problems has been limited to 3-dimensional space. Graph Morphing allows a designer to use 3-dimensional graphics to effectively interact with n-dimensional problems. A designer assigns one design variable to each of the axes of a 2D or 3D coordinate system (depending on the level of visual representation required). The remaining variables are assigned values and a visual representation of a subset of the design space is produced. A designer can then vary any of these remaining variables between the limits prescribed by the optimization problem so as to create new representations. By doing this, the effect design variables (whether alone or in combination) have on all other aspects of the optimization problem can be explored and understood. This allows the designer to get a good “feel” for the design problem and thus, where potential solutions may exist.

To understand the concept of Graph Morphing, it is best to review an example. We shall use GmorphVR to visualize this example. Gmorph and GmorphVR are two software tools developed to produce a visual representation of the design space in an n-dimensional optimization problem. These software tools are based on Graph Morphing. Gmorph uses a package called DISLIN6 to produce two and three dimensional flat

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screen displays, whereas GmorphVR makes use of the Virtual Reality Markup Language (VRML)7,8 to produce three dimensional, interactive displays. The advantage in using GmorphVR is that the designer can navigate through the visual representations, and examine constraint surfaces and objective function contours as if handling the object in physical space. This interactive 3D behavior allows for more exploration of a visual representation which results in more information gained about the problem. In contrast, Gmorph generates 3D representations from a fixed viewpoint. To look at the design space from a different angle, a designer must go back a few steps in the process and change the viewpoint. In addition, since the 3D representations are truly graphical projections, features of the representations may appear distorted or lumped together. This can make it difficult to distinguish features and thus the behaviors they are exhibiting.

Consider an optimization example problem expressed in terms of the mathematical equation below.

.15615

;50550

;646

;15315

;15215

;15115

0573.105

4406.16857.023657.02

0107.146017.05

4604.022193.02

1093.11

0221.95

26789.0667.14968.02

049.4524754.0

2517.021401.01438.01

..

088.2126504.05

34421.14

3903.1

:

≤≤−≤≤−≤≤−≤≤−≤≤−≤≤−

=+

−+−=

=+−−−−=

≤+

−−−−=

≤+−++−−=

++−+−=

X

X

X

X

X

X

where

X

XXXh

XX

XXXh

X

XXXg

XX

XXXg

ts

XxXXf

Minimize

(2)

This problem has six design variables namely X1, X2, X3, X4, X5 and X6. A GmorphVR representation of this problem is shown in Figure 2. In this representation, design variables X4, X5, X6, were placed on the coordinate axes, while the remaining variables were given constant values. The objective function contours are shown in varying shades going from red to white denoting the direction they are increasing or decreasing in value. This particular representation contains a single

objective function contour. A designer has the option of how many contours to be drawn, if any. Inequality constraint surfaces are drawn in green and equality constraint surfaces are in blue. A designer may choose different variables for the coordinate axes or concentrate on some smaller part of the existing design variable bounds and review the design space. Also, values of the design variables not assigned to the coordinate axes can be altered to produce new visual representations.

Figure 2 - Typical GmorphVR Representation of and optimization problem

The above problem only had six design variables two equality and two inequality constraints. In reality, much larger size problems are more commonly used. It is with these problems that the benefits of Graph Morphing become apparent. Using GmorphVR, a designer can understand the behavior of the objectiveand constraint functions in a problem. Many intricacies of the problem behavior that otherwise would have gone unnoticed, may now be observed. The designer can gain more and more information about the behavior of the constraint equations and the objective function, relative to changes in the values of design variables. This knowledge can be applied to many aspects of the overall design process, for example, to improve the efficiency and accuracy of a formal optimization solution technique. This knowledge may even provide the designer insight as to the optimization technique that should be used based on the characteristics of the problem Or, as another example, this knowledge may be used to eliminate redundant constraints or design variables that are found to have little impact on the optimization problem. There are many areas in which this newly acquired insight can be applied. In this paper, one will be focused on: Using it to improve a

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formal optimization solution technique. This will be done by using various tools in a newly developed version of GmorphVR to select an initial point from which to start the formal solution technique.

Constraint HandlingDespite being a powerful tool in helping a designer in making decisions, GmorphVR has certain shortcomings. The current set-up of GmorphVR needs improvements, specifically with respect to the manner in which constraint information is presented.

The generation of a graphical representation of a design space using GmorphVR depends upon various factors including: 1) Which variables are chosen to be represented on the coordinate axes 2) What limits are set for design variables represented on the coordinate axes 3) The number of contours of the objective function to be displayed and 4) The values of the design variables not represented on the coordinate axes.

During the solution process, a designer would typically vary some or all of these factors. By varying these factors, new visual representations are created. A large number of different visual representations may be created as a designer explores the behaviors inherent in the problem. Although the main goal is to provide insight about these problems, it must be done in a fast, efficient manner. If possible, keeping the number of visual representations as low as possible can help to achieve this. Thus, the maximum amount of relevant information about the optimal design space must be displayed in each visual representation. A visual representation contains objective function contours and constraint surfaces. The objective function contours are displayed in varying shades of red, and are quite straightforward to interpret. This leaves constraint surfaces. The critical question becomes “What information about the constraint surfaces would a designer desire?”

An optimization problem may have many equality as well as inequality constraints. GmorphVR uses blue to identify equality constraints and green for inequality constraints. However, there is no effective way to identify inequality constraint g1 from inequality constraint g2. Thus, if a problem has two inequality constraints, the user sees two green surfaces, but can not immediately identify which surface corresponds to which constraint. If a designer has to deal with many inequality and equality constraints in a problem, and is looking at two or more different visual representations at the same time, it is difficult to keep a record of all the constraints, their locations, and values in a typical GmorphVR representation. Further, it may sometimes occur that some constraints are outside the design

variable bounds and therefore are not drawn in the graphical representation.

Consider a case in which design variable X1 is designated to a coordinate axis, and has some upper and lower bounds defined by the problem. Let these values be 0 for the lower bound and 10 for the upper bound. Also, assume that the problem has two inequality constraints, g1 and g2, that appear in a GmorphVR representation. Then, the designer creates a second representation concentrated on a smaller part of the design space by making the representation display the area between 5 to 10 for X1. What might occur is that constraint g2 moves out of the current GmorphVR representation and the designer sees only one green surface. Thus, it is necessary to provide the designer with the information that g2 is out of the existing visual representation view and that g1 is inside. Further, it is important to provide information as to whether g2 is satisfied or violated.

This can be done, by displaying a constraint summary table. This table lists all the constraints involved in the current problem. Through some visual means, the designer is informed which constraints are within the design variable bounds and which are not. This table may also contain other useful information about constraints that are outside the design variable bounds, including information such as which constraints are satisfied or violated.

The next task is to be able to identify each constraint surface. Imagine, for example, that there are two green inequality constraint surfaces (g1 and g2) in a GmorphVR representation. The designer should be able to identify g1 from g2 with relative ease. This is accomplished by adding additional interactivity with the representation. A designer can click on any surface in a GmorphVR representation and is instantly informed of the identity (and value if appropriate) of that surface. This is for either objective function contours or constraint curves.

In addition, there exists a need for information about constraints that may not be visible on a representation but are still either satisfied or violated. In a GmorphVR representation, constraint values dictate their position in the representation. Some of the constraints may be outside of the design variable bounds being displayed. In this case, a designer must know if these constraints are satisfied or violated. For example, suppose an optimization problem is being investigated with two inequality constraints and six design variables. The designer chooses three variables for the coordinate axis system and assigns values to the remaining variables. These values cause one of the inequality constraints to

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be drawn above the upper bound for the variable that has been assigned to what would normally be the z-axis of a 3D coordinate axis system as shown in figure 3.

Figure 3 – Possible locations of constraint curves that must be handled in GmorphVR representations.

If the area below this constraint curve (the area shown by the representation) is infeasible, then the entire space shown by the visual representation is infeasible and there is no reason for a designer to look any further at that combination of design variables. However, if that region is feasible for the constraint curve, then the designer should continue to investigate to gain insight about the problem. Basically, a curve not being drawn in a representation means that it can’t be active inside the representation. It also means that the design space being shown could be feasible or infeasible. So, this is directly computed for each representation if any constraints are not drawn (but still present) in a GmorphVR representation. This data is then presented with the representation so a designer immediately knows if the design space being viewed is feasible or infeasible.

GmorphVR 2.0Incorporating all these features under the Graph Morphing concept was accomplished in the next major revision of the code, namely GmorphVR 2.0. This version is web based and works in any web browser that supports javascript and frames. The code has been created as a server side system, thus eliminating the need for special client side libraries or routines. A screenshot of the software is shown in Figure 4.

Figure 4 – Screenshot of GmorphVR 2.0 software

The control frame on the right hand side of Figure 4 displays the current values of the design variables that are not represented on the coordinate axes of the Graph Morphing representation as well as values that determine how the objective function is displayed. The designer sets these values for a particular representation. The main frame in the upper left contains the Graph Morphing representation of the optimization problem setup. The frame on the bottom right hand side is the color code frame. The designer uses this color code to understand the constraint information, which is tabulated in the constraint information frame which is located in the bottom center of the figure. For example, blue may be used to show that a constraint is within the design variable bounds and thus visible, whereas green may represent that the constraint is outside the design variable bounds but still remains satisfied. The constraint information frame displays all the constraint information in tabular form. Each constraint listed in this frame has two columns beside it. The first column tells the designer if the constraint is within the design variable bounds. This information is given by color. For example, if a constraint is listed with a blue box in the first column next to it, then this constraint is inside the design variable bounds and is displayed. In this case, the second column is left blank. However, if a constraint is listed with a black box in the first column next to it, then this constraint is outside the design variable bounds and is therefore not displayed in the current representation. When this situation occurs the designer must look to the second column. This column will be color coded to denote whether the area displayed in the representation is feasible or infeasible for this constraint. If the second column is red then this constraint is violated and the representation has no feasible points inside of it. The frame at the bottom left hand side is called the constraint indicator frame. The constraint indicator frame displays the constraint

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number when the user clicks the mouse on a particular constraint surface in the 3D design space.

The software is completely platform independent. The only requirement is a VRML enabled web-browser. This is most easily accomplished by installing a VRML plug-in. A comprehensive list is available at the Wed3D Repository (http://www.web3d.org/vrml/vrml.htm).

Benchmarking of GmorphVR 2.0GmorphVR was thoroughly tested on a number of optimal design problems. Before these results are presented, additional background on a problem ranking and reduction method must be given. When the concept of Graph Morphing was first implemented into software, it was quickly learned that it was just as easy for a designer to explore poor regions of the design space as easily as exploring good regions. For example, even though multiple regions of a design space may be feasible, one may yield far better results for objective function values than the another. If a designer does not thoroughly explore the design space with multiple representations, good regions may not be found at all. This necessitated the development of a formalized method for not only creating Graph Morphing representations, but also for reducing problem complexity for visual purposes5. This ranking method guides a designer through a step by step process that ends with the selection of an initial point located in the region of the design space close to the global optimum. The full optimization problem is then solved using traditional optimization methods starting from this initial point. This will decrease the time and cost associated with the formal solution procedure compared to starting it from a randomly generated initial point.

To summarize, the ranking method is composed of these steps:1. Rank and reduce constraints based on their behavior

over the design space.2. Rank and reduce design variables based on impact on

the objective function and constraints.3. Create representations of reduced optimal design

problem.4. Choose initial point from which to start formal

optimization algorithm.

These steps utilize the concepts of Design of Experiments (DOE) as well as some of the mathematics from Concurrent Subspace Optimization (CSSO) to reduce the original optimal design problem. It is important to note that the problem is reduced in complexity for only visualization purposes. Once the initial point is chosen, the original problem is solved and compared against a solution from a random initial point using the same optimization algorithm.

Test Case 1An example is now presented to demonstrate the use and feasibility of the ranking method with the newest version of the Graph Morphing software, GmorphVR 2.0, which was previously discussed. This continuous optimization problem9 contains five design variables, five inequality constraints, and five side constraints and is presented in Equation 3. In this case, the entire ranking procedure was applied in an attempt to simplify the problem. For visual purposes, neither the problem design variables nor constraints were able to be reduced. Thus, the full problem was used for creating the Graph Morphing representations.

Min.F = x1x2x3x4 + 2x5x4 - x1

3

s.t.g1 = 4x1

0.5 - x22 x3- 3x5 ≤ 0

g2 = 6 + 1/8 (1/ x12) - (x4 x2 ) - 2 x5≤ 0

g3 = 18 + 4 x32 x2

2 - 1.8 x5- x32 x4

2 ≤ 0g4 = 16 + 3.5 x3 x1

2 - 5x5- x32≤ 0

g5 = -5 + 4 x32 x4 + 0.5 x5- x4 x2

2≤ 0

where

(3)

1≤ x1 ≤ 160 ≤ x2 ≤ 160 ≤ x3 ≤ 160 ≤ x4 ≤ 160 ≤ x5 ≤ 16

The next step of the ranking method provides a designer with a methodology for deciding which and how many visual representations need to be created. This is done by selecting an appropriate orthogonal array that adequately fits the size of the optimization problem. This array provides different design variable combinations to explore. By creating the visual representations to view these combinations, a designer is provided a mechanism with which to gain understanding of the critical relationships that exist between the design variables and all other aspects of the optimization problem.

A Graph Morphing representation was subsequently created with design variables x1, x3 and x5 on coordinate axes. The settings for variables x2 and x3 were set according to values prescribed by the orthogonal array. In this case, it was not necessary to examine all the combinations of points set forth in the orthogonal array, as the relationships among design variables and their effect on the constraints and the objective function became evident by implementing only about half of the experiments. Figures 5 and 6 are representations with design variables x2 and x3 set to zero. The constraints in these figures have been marked for visual clarity.

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Figure 5 – GmorphVR 2.0 plot of test case 1 with x2= 0 and x4=0 (view 1).

Figure 6 – GmorphVR 2.0 plot of test case 1 with x2= 0 and x4=0 (view 2).

In these figures, the objective function appears to have a linear relationship with the design variables x1, x3 and x5. The objective function value decreases as the color moves from red to white (left to right in the figures). The feasible region lies at the upper top right of the plot, where constraints g3, g4 and g5 intersect each other. A more detailed investigation of this region shows that

there are several points which could lead the objective function to a minimum value, The most appealing is [16.0, 0.0, 0.0, 0.0, 10.0]T which yields an objective function value of –4096. The investigation of other design experiments led to the generation of additional representations such as the one in Figure 7 where x4 is set equal to 8.0.

Figure 7 – GmorphVR 2.0 plot of test case 1 with x2= 0 and x4=8.0 (constraints only).

Using representations such as those in figures 5, 6, and 7 a feel for how the constraints are behaving and the associated objective function values obtained can be attained. For example, in Figure 7, even though much of the space is feasible the objective function values are significantly higher than areas in figures 5 and 6. The only change is that x4 has been increased from 0.0 to 8.0. This directly led to the conclusion that x4 should be kept at its lower bound. Once the ranking method had been completed for this problem the initial point of [16.0, 0.0, 0.0, 0.0, 10.0]T was chosen.

A formal optimization was then conducted with the commercial optimization program DOT (Design Optimization Tools)10. The problem was solved using the Modified Method of Feasible Directions, Sequential Linear Programming, and Sequential Quadratic Programming. Table 1 gives the initial points that were used while applying these traditional techniques.

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Table 1 – Starting points for formal solution procedures(* - Indicates initial point found using Graph Morphing)

Point # x1 x2 x3 x4 x5

1 1.0 0.0 0.0 0.0 0.0

2 16 16 16 16 16

3 8.5 8 8 8 8

4* 16 0.0 0.0 0.0 10.0

Initial points 1 and 2 are the design variables lower and upper bounds respectively. Point 3 represents the averages of each design variable between its lower and upper bounds. The problem was then solved using each of the indicated methods from each of the initial design points. The optimization parameters (convergence criteria, move limits, etc.) were held constant for the solution runs for the various methods. Thus, the only difference in runs with the same method was the initial point from which the solution procedure began. The total function evaluations required to solve the problem from the various methods are shown in Figure 7.

0

50

100

150

200

250

MMFD SLP SQP

Solution Methods

Fu

nct

ion

Eva

luat

ion

s

Figure 7 – Total function evaluations required for each solution method to complete from each of the 4 initial points

In this figure, there are four bars over each solution method. Each bar represents a solution run using this method from one of the specified initial points. The bars are ordered from left to right with the leftmost bar representing initial point 1.

This figure shows that initial point 1 required approximately 200 more function evaluations than the Graph Morphing point to reach a solution using the MMFD solution strategy; this is a savings of approximately 90%. Where point 3 required 17 more function evaluations than the Graph Morphing point. This is a savings of approximately 45%. The MMFD strategy was only able to find a solution with the first, and the Graph Morphing initial point, hence only three

bars are plotted for MMFD. Point 2 for MMFD is not plotted because it was not able to converge to a solution. For the SLP solution method Point 1 required 17, Point 2 required 78 and Point 3 required 47 more function evaluations than the Graph Morphing point to identify a solution. This is an overall savings of approximately 65%. For the SQP solution strategy, Point 1 required about same number of function evaluations as the Graph Morphing point, where Point 2 & Point 3 required 14 more function evaluations than the Graph Morphing point to reach a solution. This is an overall savings of approximately 42%.

A different aspect to consider is the final objective function value obtained. Figure 8 displays these results for each of the solution runs.

-4500

-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

MMFD SLP SQP

Solution Methods

Ob

j. F

un

c. V

alu

e

Figure 8 – Final Objective function value found for each solution method from each of the 4 initial points

The best solution point obtained was the design variable vector [16.0, 0.0, 0.0, 0.0, 10.00]T. This gives an objective function value of –4096 with constraints g3

and g5 active. This is the global optimum point as identified by the problem reference. All the methods converged to the global optimum when started from the Graph Morphing point (Point 4). Figure 8 shows that none of the methods were able to identify the global optimum when started from Point 2. In addition, for Point 1, MMFD was not able locate the global optimum. However, each method was able to locate the global optimum when started from the Graph Morphing initial point. All remaining objective function values for their respective strategies were within 0.8% of one another.

This test case offers some interesting results. We can see that the Graph Morphing initial point and final global optimum point are actually the same. This is due to the small size of problem. A designer should not expect to obtain such results with all optimization problems. However, this example is useful for explaining the solution process used for benchmarking Graph Morphing.

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Results and DiscussionIn all, 20 test problems were analyzed. This resulted in more than 80 comparisons. The problems ranged in size from 4 design variables and 1 constraint to 40 design variables and 15 constraints, and all are nonlinear in nature. The CASCADE11 system, developed at the University at Buffalo, was used to generate many of the test problems. CASCADE produces simulated optimization and MDO systems for testing purposes. All test problems were used for minimizing the objective function and had constraints in the form of inequalities. All design variables have a lower bound of 0.0 and an upper bound of 99.9. Several of the test problems were taken from more real world applications such as the design of a speed reducer12 and design of a flywheel13.

For all test problems, the entire ranking method was applied successfully in order to simplify them. The GmorphVR tool was then used to choose an initial point from which to start a formal optimization algorithm. Two or three other initial points were also used for comparison. These three other initial points are the lower bounds, the upper bounds, and the average of the upper and lower bounds for all design variables in each test case. The problems were then solved for these different initial starting points using various solution algorithms of the optimization software package DOT as was done for the previous test case presented.

The average savings in function evaluations was approximately 52%. In addition, for each problem, every time a solution was started from a Graph Morphing initial point, it converged to the global optimum for the problem. Many of the comparison runs were not able to even move from the initial point, much less locate the global optimum.

There is some overhead with using the Graph Morphing concept. There is some time involved to perform the entire ranking method as well as create the visual representations. For all the test cases, this time was limited to 30 minutes in which to choose an initial point. This meant that some problems were not as thoroughly investigated as they might have been had additional time be allotted. However, it does show that significant improvement in computational time can be achieved with a modest amount of time spent on exploration with Graph Morphing.

The newly added constraint handling features in GmorphVR 2.0 made useful investigations in this short time frame possible. For example, many representations that may have been infeasible due to a constraint above or below the bounds of the visual representation were immediately identified. By simply looking for a specific

color in the constraint information frames it was known whether the current visual representation should have been explored any further. Also, the increased interactivity made it easy to obtain objective function values for the current combination of design variables. Thus, even if a current representation was feasible, it may have been a combination of variables that resulted in higher values for the objective function than other combinations. With a few clicks of the mouse, contour values were quickly obtained and compared to values obtained from other visual representations.

Summary and ConclusionsIn this paper, the new version of the 3D Graph Morphing software, GmorphVR 2.0, has been described. This new version has detailed handling of problem constraints as well as additional interactivity for a designer to obtain information about a design space with increased ease and effectiveness. This new software was then thoroughly tested on a number of optimal design problems inside the previously developed ranking method to assess its viability.

It was found that with minimal effort and time, using GmorphVR 2.0 in the described framework offered substantial improvement in computational efficiency. In addition, a designer may obtain valuable insight about the complex relationships that exist between the design variables and other aspects of the problem.

AcknowledgementsThe authors wish to acknowledge the generous support of this work under the National Science Foundation PFF Award DMII 9553210 and from the New York State Center for Engineering Design and Industrial Innovation (NYSCEDII). Further, the authors wish to thank Dr. Christina Bloebaum and Dr. Jarek Sobieski. It was initial interactions between and with these individuals which brought about the concept of Graph Morphing and all the subsequent innovations.

References[1] G. N. Vanderplaats, Numerical Optimization Techniques for Engineering Design with Applications, McGraw-Hill, Inc., New York, NY, 1984, pp. 1-21.

[2] R J. Balling and J. Sobieszczanski-Sobieski,“Optimization of Coupled Systems: A Critical Overview of Approaches”, AIAA Paper 94-4339, September, 1994.

[3] E.H. Winer and C.L. Bloebaum, “Development of Visual Design Steering as an Aid in Large Scale Multidisciplinary Design Optimization –Part 1: Method Development”, Journal of Structural and Multidisciplinary Optimization, to appear 2002.

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[4] E.H. Winer and C.L. Bloebaum, “Development of Visual Design Steering as an Aid in Large Scale Multidisciplinary Design Optimization –Part 2: Method Validation”, Journal of Structural and Multidisciplinary Optimization, to appear 2002.

[5] E.H. Winer and C.L. Bloebaum, “Visual Design Steering for Optimization Solution Improvement”, Journal of Structural and Multidisciplinary Optimization, vol. 22, no. 3, October 2001, pp. 219-229.

[6] H. Michels, DISLIN Manual, Max-Plank-Institut fur Aeronomie, Katlenburg-Lindau, 1997.

[7] A.L. Ames et.al., The VRML 1.0 sourcebook. John Wiley & Sons, Inc., 1996.

[8] A.L. Ames et.al., The VRML 2.0 sourcebook. John Wiley & Sons, Inc., 1997.

[9] G.V. Reklaits, A. Ravindran, and K.M. Ragsdell, Engineering Optimization Method and Application, A Wiley –Interscience Publication, John Wiley and sons, New York, NY, 1982, pp. 556-560.

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[11] K.F. Hulme and C.L. Bloebaum, “Development of CASCADE – A Multidisciplinary Design Test Simulator” Sixth AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA, September, 1996, pp. 438-447.

[12] S. Azarm, “Local Monotonicity in Optimal Design”, Ph. D. dissertation, University of Michigan.

[13] J. Golinski, “Optimal Synthesis Problems Solved by Means of Nonlinear Programming and Random Methods”, Journal of Mechanisms, vol. 5, pp. 287-309.