Multidisciplinary Design and Optimization

A Practical Global Optimization Procedure Shenghua Zhang†, Xiaoming Yu† and Erwin Johnson‡ MSC.Software Corporation 2 MacArthur Place Santa Ana, CA 92707

† ‡

Member of the Technical Staff Project Manager, AIAA Associate Fellow

1 American Institute of Aeronautics and Astronautics

ABSTRACT Automated design techniques often assume that a given design task has a single best design that can be reached from any starting point. This assumption is contradicted by many industry optimization applications that show multiple local minima. This paper addresses this local minima behavior by presenting a global optimization strategy that can be characterized as a modified multi-start method. The method includes four major aspects: 1) the global design space is sampled 2) heuristic techniques are applied to enhance the chance of finding the global optimum 3) the efficient gradient-based design optimization tool contained in MSC.Nastran is used as the local search engine 4) the process is continued until a specified time budget is exhausted or an asymptotically correct condition is met. These aspects combine to produce a global optimization procedure that can be applied to practical design tasks. The paper provides details of the Global Optimization algorithm and how it has been integrated with the existing local optimization capability of MSC.Nastran. Two academic and two small industry problems are demonstrated to show the effectiveness and the efficiency of the procedure. Concluding comments summarize the results of the study and suggest further developments that could be pursued.

INTRODUCTION Designer/analysts now routinely exploit the power of computers and sophisticated automated design techniques to evaluate numerous design options. Often the user of these techniques is satisfied if they produce a design that shows improvement relative to what has been achieved by standard design techniques. However, there is always a question as to whether the design obtained using these methods is a true global optimum or if the procedure, instead, has led to an optimum that is only local in nature. The existence of local optimum is a well known phenomenon in some very practical applications. Dynamic response design tasks are known to be particularly prone to having many local minima. A particular example is in the design of an engine mount while considering variations in the rates of springs, dampers and the geometric locations of the mount. The load conditions are time and/or frequency dependent and the design objective is typically to minimize the vehicle response. Another example is in the design of structures that utilize composite materials. It is well known that selecting ply orientation angles as design variables can result in local minima. Because of the highly competitive market for faster and better product design, there is a strong motivation to seek the true optimal design and hence in tech-

niques that assist in the search for the global optimum. The major challenges in finding a global solution are: 1) there is no single method can solve all the GO problems since no global optimality criteria exist for general problems and 2) it is impossible to find a global optimum unless all the possibilities (infinite number) in the design space are exhausted. A realistic goal of a global optimization procedure then is to obtain an approximation to the global minimum while expending an acceptable amount of computer resources. Using the notation of Ref. 1, a global optimization problem is defined as: Problem P. Find a design variable vector x to minimize a cost (objective) function

f (x) for x S R n , where S is the constraint set defined as S

( x | g (x) j

0, j

1, l , g ( x ) j

0, j

l

1, m )

(1) *

A point x is called local minimum for the above *

problem if f (x )

f (x) for all x in a small feasi* * ble neighborhood of the point x . A point x G is defined as a global minimum for the above problem if

f ( x *G )

f ( x ) for all x in S.

Problem P defined in Eq.1 is a very general form. It may be classified into various types. For example, it is a constrained GO problem when the S set is both bounded by the design variables and the constraints while it is an unconstrained GO when the S set is only bounded by the limits on the design variables. In addition it is a convex GO problem when the S constraint set is convex. In this case, the local minimum is the global minima. Finally, when the S set assumes the finite number of discrete points, it is a combinatorial GO problem [1]. In general, no global optimality conditions exist for the classes of the GO problems specified here. The development of strategies for global optimization remains an area of active research. This paper does not attempt to find the best overall solution to this search, but to demonstrate a practical technique that can be of utility in many engineering applications. This led to the selection of a multi-start technique as a practical algorithm that did not make an excessive demand on computer resources. The multi-start approach combines a local search (LS) in a global search space. Specifically, each local optimization task is randomly started from a set of points distributed over the design space. The best local minimum found from each starting point is a


candidate for the global minimum. Although the method holds the potential to find the global minimum, it is not efficient since the same local minimum may be visited repeatedly. An ideal case would be that each local minimum was visited only once. Then, a problem having k local minima could be solved with only k local searches. The so-called clustering method is built on the above concept [1,2]. This method defines

As indicated in the Introduction, the method presented in this paper tries to strike a balance between two often conflicting factors: finding the global minimum and expending an acceptable amount of computer resources. The following four subsections detail the four major aspects of the method. .

Approximate global search space

*

a region of attraction, R x as the subset of points in S starting from which any LS will always arrive at the same local minimum, x*. Then, finding the global solution becomes a process of identifying the regions *

of attraction. This paper utilizes the R x concept although it is not explicitly computed. The multi-start method of this paper can solve both unconstrained and constrained global optimization problems. It has four major aspects: 1.

2.

3.

4.

The global design space is first approximated by a small but intelligent set of points that are derived using DOE methods, such as orthogonal arrays, fractional or full factorial design techniques. This important step enables the subsequent local searches to be performed in a well balanced but a reduced design space. For a problem having n design variables and two levels being taken for each design variable, the number of starting points is reduced from the magnitude of ~2n to the magnitude of n. Thus, a problem with hundreds of design variables can be solved efficiently. Heuristic techniques are applied to quickly find the global optimum. The basic rule is to always start a new local search from the point that has the largest distance from the current best local minimum point. A direct benefit of this is to minimize the number of repeated visits to the same region of attraction. A gradient-based optimization tool is used as the local search engine. Specifically, the powerful multidisciplinary design optimization capability in MSC.Nastran is used here while the effective optimization engine is DOT from VR&D [3,4] Asymptotically correct condition or a user specified time budget is used to terminate the global search process.

The four major aspects of the method are first presented in a Methodology section. This is followed by some discussion of how the method has been implemented and the paper is concluded with several examples that demonstrate the developed technique

METHODOLOGY

In the multi-start method [1], the design space, S is represented by a set of points uniformly distributed over S. Assume there are n continuous design variables and each design variable starts from m levels n

(values). This requires m sample points to cover the design space in a full factorial design. The following table was specified to select a level for a given number of design variables: n 1 2 3 >3

m 10 5 3 2

For the case of level = 2, two further strategies are employed. The first relates to where to set the design value in the design space. The second provides an option to set a full or factorial (orthogonal array) design.

Select the two levels either at the corners or interiors With m=2, the user can select whether the design variables are set at the corners (user specified lower and upper limits of a design variable) or at interior points determined by Golden ratio. The Golden ratio numbers are 0.618 and 0.382 in the unit length space so that the two levels are:

x x

1

x 0.618 x x ,x

0.382

i 2

i

U

i

i

x 0.382 x

0.618

i

L

where

U

L

(2)

i

i

L

(3)

u

i

are lower and upper bounds for the i-

th design variable. Sometimes, it may be more useful to use the interior representation since the interior points cover more inner region. A local search starting from there may have a better chance of finding a local minima.

Further reduce the design space using the orthogonal array (OA) When the number of design variables is large, e.g., 100, the full factorial design with two levels will need


2100=12E30 points to represent the design space. Considering that the practical industry applications often involve hundreds of design variables, the algorithm has an option to reduce design space using the orthogonal array techniques [5]. An orthogonal array (OA) is one of many approaches in DOE to study the design space, usually containing the large number of design variables, with a small number of experiments. It provides an elegant means to selecting a small but intelligent subset of the design space to significantly reduce the number of experiment runs.

monitor these regions. , Therefore, the method in this paper does not include any explicit evaluation of regions of attractions. Instead, two heuristic techniques are employed to speed up the global search process. The first one is to always start a local search (LS) from the point that has the largest distance (LD) from the current best local search (BLS) point while the second is to start an LS from a reflection point of the LD when it is near the end of the global search process.

Start an LS from an LD point An orthogonal array is a design matrix whose column corresponds to a factor (or design variable) and whose row corresponds to an experiment (or run). The number of columns of an OA represents the maximum number of factors that can be studied using that array. The orthogonal arrays have two important properties: 1) the size reduction property that an orthogonal array can represent a full factorial design of size of 2n with at least N experiments (runs) where N is a multiplier of 4 and n=N-1; and 2) the orthogonal property that states that the cells in each column sum to zero and the dot product of any column pair is zero [5]. The concept of orthogonal arrays has been widely used in various applications. It has recently been used in the discrete design variable processing capability in Sol 200 [6].

Local Search Phase During a local search phase, the existing multidisciplinary optimization capability available in MSC.Nastran is used to find a local minimum [3,4]. Whether a global search procedure is efficient depends greatly on the local search phase. There is an interesting aspect of the local search is worth discussing. A local search engine usually has a difficult time to solve certain optimization problems that are poorly conditioned. According to [7], the local search engine like DOT is less effective to solve the problems In these cases, the global optimization procedure can be used as a workaround to solve such type of problems by starting from multiple starting points and therefore increases the chance that the true global optimum will be found.

Apply heuristic techniques to speed up global search With the approximated global search space in place, we are ready to launch the local searches. As stated earlier, the subsequent local searches should follow a path that minimizes the number of repeated visits to the same R

* x,

the region of attraction. Although

The basic idea is to start an LS from a point with the largest distance from the BLS. The rationale is that given the current BLS, (i.e., the region of attraction), the chance to initiate a new region of attraction will be highest if a new location is started as far away from the current BLS. Therefore, after a BLS is found, the distance between all the active design points and the BLS is computed and the distance values are ranked in the ascending order. The design point with the largest distance (LD) is selected as the next starting point for the subsequent LS. The distance function in the n-dimensional space is calculated by N

d

( xi

x BLS ) 2

(4)

i

where xi is taken from the i-th factor in a particular row (run or sample point) of the orthogonal array while x BLS is taken from the i-th factor of the BLS point. When an LS based on the LD point produces a better BLS, it indicates that a new local minimum or a new region of attraction has been found. The BLS is then updated and a new LS is initiated from the next LD point. This process is repeated until the procedure is terminated with some kind of stopping criteria (See next section).

Start an LS from an RD point Before the global search is terminated at the point described above, we may jump-start the global search with the hope that the new starting point is located in a new region of attraction. This new point is obtained using a reflection of the LD point with respect to the BLS by x

i

2x

r

i

xd , i

1, 2,..., N

(5)

blm i

where x r is the i-th dimension of the reflection point,

x

i

i

is the i-th dimension of the BLS point and x d is the i-th dimension of the LD point. Note that a blm

knowledge of the regions of attraction is very valuable, it is also expensive to explicitly calculate and 4 American Institute of Aeronautics and Astronautics

check is made to ensure all the lower and upper bounds are enforced for the reflection point.

Stop Global Optimization Procedure Three major stopping criteria are used in the GO procedure. Two are based on user time budget criteria while the third is based on the asymptotically correct condition. All three criteria use parameters that can be changed by the user. Maximum allowable number of FE analyses (500) This is a grand stopping criterion to limit the total number of FE analyses in a GO procedure. Maximum allowable times - (24 Hr) This is a grand stopping criterion based on the time budget. When the GO procedure exceeds the given time budget, it is terminated after the current local search is complete. Maximum allowable number of consecutive LS’s that do not produce a better BLS - (10). This is an asymptotically correct condition that states that if an improved design is not found after a specified number of local searches , the current best local search point is the global solution.

Determining a Unique Local Minimum Point and Selecting the Global Optimum from Multiple Local Minima According to [3], a local minimum is at hand when the following condition is satisfied: the relative change in the objective is less than CONV1 (0.001) or the absolute change in the objective is less than CONV2 (1.E-20) and the maximum constraint is less than GMAX (0.005). Since two unique local minima from two separate local searches may be numerically identical due to numerical tolerance overlapping, they should only be counted as one local minimum. In this paper, two local minimum point are said to be unique when their relative difference is greater than 2*CONV1 and their absolute difference is greater than 2*CONV2. When all the unique local minima are found, the global solution can be easily selected from the best of all the local minima.

IMPLEMENTATION Figure 1 provides a design flow diagram for the GO procedure. The procedure has been implemented using the Client Server concept. The procedure is developed in a set of Fortran routines that act as a client program while the Nastran is treated as a server. The communication between the client and server is done through the MSC.Toolkit [8].

Since the GO procedure involves multiple local searches, a summary file is provided that shows all the starting points, records the objective, maximum constraint value and design variables for each local search. In addition, a separate file is created that lists the local minima found for monitoring the process. An XY-plot based on this file can be created using any type of spreadsheet program. In addition, the results for all the local minima are also stored for further study.

EXAMPLES Problem 1 The first problem is originated from Ref. 9 but Ref.10 provides its definition as well as solution: Minimize 2

f

x1 , x 2

(4

2 .1 x 1

4

2

x1 / 3 ) x1

2

x1 x 2

(4

2

4 x2 ) x2

Subject to:

g x1 , x 2 1

x1 , x 2

sin 4 x1

2 sin

2

2 x2

0

1

The design space for this problem is shown in Fig. 2a. Since the constraint is a sinusoidal function, the design space consists of many disjoint regions, each having a circle-looking shape. According to [10], the global optimum occurs at the point xmin=(0.109,0.623) with fmin=-0.9711. Figure 2b shows 25 starting points and the global minimum obtained by the GO procedure. The starting points are marked by no where n is the local search sequence order. For example, the first local search starts from point 1o that has the largest distance from the initial local search point, Oo. The global solution is obtained by 22nd local search starting from (0.0,0.5) and arriving at x*=(0.10969,-0.62356) with f*=0.97113. Although this problem shows the multiple disjoint feasible regions, the method is still able to find the global solution with only 80 function evaluations. It should be pointed out that the accuracy of the method depends on the number of levels selected in approximating the design space. The problem is solved using the defaulted level=5. When level=3 and level=4 are used, the resulting global solutions are f=-0.767 and f=-0.891, respectively.

Problem 2 This problem, taken from Ref.10, seeks to minimize the weight of a tube subject to constraints on shear stress and wall thickness.


Minimize 0.25

2 ( x1

of the engine for the frequency range from 3.0Hz to 20.Hz.

2 x2 )

Subject to 16 x1 1000 ( x1 4

16 x 2

500 ( x1 4

3 0.5( x1

0

x2 4 )

x2 )

x1 , x2

x2 4 )

0

0

0

100

2.1 10 6 . According to where 7.82 10 , [10], the global optimum occurs at the point x min (31.053,25.053) with f min 0.00206753 . In the neighborhood of this point, the iso-objective curve is almost tangent to the boundary of feasible region (See Figure 3a). This phenomenon shows that the design space is almost flat near the global solution so that the local optimizer tends to stop prematurely. However, this type of problem is much easier to solve by the GO procedure than the first problem since it does not have disjoint regions. When multiple local searches are launched from those well balanced starting points, the chance to find the global solution is very high. 6

When the GO procedure is used to solve the tube problem, it obtains the global solution at x min

( 31.082, 25.094) with fmin 0.002059 . The weight obtained here is smaller than the weight reported in Ref. 10, but this is felt to be due to the numerical tolerance used for defining an active constraint.

Figure 3b plots the global search path for the tube problem. Although the design space is flat, the global solution is found with 9 local searches and total 81 function evaluations within 1% error. We notice that the global solution is obtained by the local search from a starting feasible design. We could increase the level from 3 to 5. However, the high resolution in approximating the design space does not lead to any further improved design. Interestingly, the method of Ref. 10 required 1353 analyses to obtain a solution within 1% error, removing the algorithm of that paper from consideration as a “practical” GO optimization procedure.

The design task is to minimize the RMS function by varying x,y, z locations, rates of springs and dampers of engine mount, transmission mount and left/right toque strut mounts. A total 20 design variables are defined. First, a single unconstrained minimization problem is performed based on the original user defined design model. The RMS value is reduced from 27.03 to 9.79. Figure 4c shows the actual acceleration responses. Subsequent local searches are performed starting from the LD point from the initial local search. The global search space consists of 33 starting design points. Number 33 is determined as follows: 20 design variables produce a requirement for at least 24 points according to the OA table. Then, this number is increased to match the closest power of 2, in this case, 32. Finally, the center of the design space is added to come up 33 points. The job finds 20 unique local minima. It uses 21 local searches and a total of 560 finite element analyses. Notice that after the global solution is found at local search 11, the next 10 consecutive local searches do not find any better design, so the procedure is stopped. The total elapsed time is less than 2 hours on a single Linux machine. Considering the complexity of this dynamic response optimization task that covers both sizing and shape variables, the procedure is quite efficient in finding these local minima. Figure 4d plots the history of this global search result. Each square symbol represents the final RMS value from a local search. A number appearing next to the symbol indicates a local search sequence number. A special symbol, a big dot represents the global solution obtained by the 11th local search. Those squares with sequence number larger than 11 represent local minima with higher values. Finally, Figure 4e shows the acceleration response plots for the global and the first local minima (solid line vs. dashed line). With the help of the GO procedure, the acceleration response is further reduced from 4.78 to 2.88, additional 40% reduction.

Problem 3

Problem 4

Figure 4 presents a generic engine model that represents a V6 engine and automatic transmission as well as other components such as engine mount, transmission mount, crankshaft, drive axles, and flywheel. Fig.4a is the front view and Fig.4b is the side view of the generic engine model. The frequency dependent loading conditions roughly simulate freeway hop and suspension hop. A Root Mean Square (RMS) function is defined using the z-components of the acceleration response at the center of gravity (CG)

Figure 5a shows the Intermediate Complexity Wing (ICW) model that has been used in numerous design optimization studies. The model includes spars and ribs that are modeled with rods and shear panels. The top and bottom wing skins are modeled by composite elements with 0/+45/-45/90 laminates. The design task is to minimize the volume of the wing by varying the spar and rib thicknesses and cross sectional area and the wing skin laminate thicknesses. A total of 153 design variables are defined. Design constraints are imposed that place conditions on a failure index based


on Tsai-Wu theory, element stress and the second natural frequency. The purpose in presenting this problem is to show that the GO method is very efficient for problems with more than 100 design variables. It has been observed in previous ICW design studies that small changes in the local optimization algorithm or in the design task can result in significantly different final designs. The reason for these differences is not fully understood, but the existence of these different final designs makes the ICW a strong candidate for application in a GO algorithm. The results obtained from the GO do confirm that a variety of final designs can be obtained for the same design task starting from different initial designs. Figure 5b shows the global search history. From a total of 12 local searches, 10 unique local minima were found. Local searches 5 and 12 produce two non-unique local minima. The global search process is terminated after 10 consecutive local searches do not produce an improved design. The very first local search produced a local minimum point at f=592.76 while the global solution obtained by the second local search has f=530. The eigenvalue constraint is active for these two solutions. Two sets of 153 design variables are compared (not shown here) and a difference of more than 300% is observed. In addition, the local minimum design with the highest volume occurs at local search 8 with f=864.17. Although the maximum constraint still corresponds to a failure index, the second eigenvalue is far way from the lower bound (77150 vs. 63200).

CONCLUSIONS AND RECOMMENDATIONS The practical GO procedure presented in this paper is effective and efficient. It replaces tedious and errorprone manual multiple starts with the automated process. It can solve large or small scale optimization problems with multiple local minima and is able to find the global solution in a robust and efficient way. As demonstrated by the examples, the technique is useful when it is known that the design space is flat and a very precise optimum is desired and when, for whatever reason, the optimizer is inconsistent in arriving at the same optimal design when starting from different initial designs. The results presented in this paper are very promising. However, more work could be done to further advance the technology. Candidates for future work include: 1. Develop the algorithm to track and monitor the status of a region of attraction to screen

out those local searches that may produce repeated local minimum. 2. Support restart schemes so that the previous search information can be utilized for the subsequent searches. 3. Enable launching local search jobs on distributed multiple processors 4. Apply the tool to solve various automotive industry applications such as the road response optimization problems presented in [11] and NVH applications.

ACKNOWLEDGMENTS The GO procedure presented here is based on a project funded by General Motors Corp. Dr. Wayne Nack was instrumental in this endeavor and his efforts are greatly appreciated. The efforts of several MSC team members: Steve Wilder, Vinh Lam and Jesse Ou are also acknowledged.

REFERENCES [1] Arora, J.S. et al, “Global Optimization Methods for Engineering Applications – A Review,” Journal of Structural Optimization, Volume 9, Number ¾, July, 1995. [2] Gary P. et. al, “A Survey of Global Optimization Methods,” Sandia National Laboratories, Albuquerque, NM 87185. http://www.cs.sandia.gov/opt/survey/, [3] Moore, G., MSC/Nastran Design Sensitivity and Optimization Use’s Guide, The MSC.Software Corporation, 1994. [4] VR&D, “User’s Manual for DOT-Design Optimization Tool,” Version 5.0, 1999. [5] Schmidt, S.R. and Launsby, R.G., “Under- standing Industrial Designed Experiments,” 4th edition, Air Academy Press, Colorado Sprint, CO 80920, 1997. [6] MSC.Nastran 2001 Release MSC.Software Corporation, 2001.

Guide,

The

[7] Rastogi, N. et. al, “Discrete Optimization Capability in GENESIS Structural Analysis and Optimization Software,” The Proceedings of 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, AAIAA 2002-5646, September 2002, Atlanta, GA.


[8] MSC.Nastran Toolkit User’s MSC.Software Corporation, 2001.

Guide,

The

. [9] Gomez, S. and Levy, A.,“The tunneling method for solving the constrained global optimization problem with several non-connected feasible regions,” in A. Dold and B.Eckmann (eds.), Lecture Notes in Mathematics, vol. 909, pp.34-47, 1982. [10] Jones, D. R. and McDonald, G.C., “DIRECT: A Global Optimization Algorithm for Computer Aided Engineering,” Research Report no. OS-271, General Motors Corp., January 20, 1994

[11] Chou, D., Johnson, E., Nack, W and Schwerzler, D, “Road Response Optimization”, 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, April 7-10, 2003, Norfolk, VA.

Start first LS, Initialize BLS

Create Global search space with OA

Better BLS?

Y

Set BLS, NFEAruns, and NLSruns = 1

Y

NFEA> GFEA? N

N Y stop

Is mirror Point?

Start LS with Mirror Point?

Y

Compute distances of all points from BLS.

N N

Y

NFEA> GFEA?

stop

N

Y NLSruns >GmxLS?

Compute a mirror point

N

Start a new LS


stop

Figure 1 Design Flow Chart for GO Procedure Notation BLS – Current best design from local search GFEA – Maximum allowable number of finite analyses GmxLS – Maximum allowable number of consecutive local searches not producing any improved design LS - Local search

Figure 2a An Example Showing Many Disjoint Feasible Regions

1.0

2 o

+0.5

0.0

7

o

20o

13o

9o

12o

4o

Figure 2a

15o

14o

24o 925o 23o American Institute of Aeronautics and Astronautics

21o

16

o

17o

Oo

Very First Starting Point

Figure 2b Illustration of Starting points and the Path to Global Solution


Figure 3a The Tube Problem Showing Flat Design Space

100. Figure 2a

75.

9o

50 .

f*=2.0659-03

25 .

x*=(31.082,25.094)

0.0 0.0

25 .

50 .

75.

Figure 3b The Search Path to the Global Solution


100.

CG

Transmission Mount

Engine Mount Figure 4a. Front View of The Generic Engine Model

Engine Outline

Transmission Outline

CG

Transmission Mount Figure 4.b Side View of the Generic Engine Model Engine Mount

Figure 4.b Side View of the Generic Engine Model


25.0 Initial design from first local search

Z-Component Acceleration at CG

Final design from first local search

20.0

15.0

10.0

5.0

0.0 0.0

5.0

10.0

15.0 Frequency

20.0

25.0

Figure 4c Acceleration Response Plot

35 30

19

RMS value

25 17

14

20 2

15 10

10

6 3

21

13

4 5

1 7

15

8

12 9

5

18

20

16

11

0 0

5

10

15

20

Local Search Sequence Order

Figure 4d Illustration of Global Search History


25

6.0 Local minimum from first search Global solution

z-component acceleration at CG

5.0

4.0

3.0 S 2.0

1.0

0.0 0.0

5.0

10.0

15.0 Frequency

Figure 4e Comparison of Acceleration Responses


20.0

25.0

Figure 5a Model of Intermediate Complexity Wing

900 8

850 800

Volume

750 700 650

6

600

3

1

4 9

10

7

550

11

5

2

12

500 0

2

4

6

8

10

12

Local Search Sequence Order

Figure 5b Illustration of Global Search History


14