NASA Technical Memorandum 4698 

:--{IGOr 0 

rjlulq7 

Performance Trend of Different Algorithms for 

Structural Design Optimization 

Surya N. Patnaik, Rula M. Coroneos, James D. Guptill, 

and Dale A. Hopkins 

OCTOBER 1996 

National Aeronautics and 

Space Administration 

- - ----

----~ l 

NASA Technical Memorandum 4698 

:--{IGOr 0 

rjlulq7 

Performance Trend of Different Algorithms for 

Structural Design Optimization 

Surya N. Patnaik, Rula M. Coroneos, James D. Guptill, 

and Dale A. Hopkins 

OCTOBER 1996 

National Aeronautics and 

Space Administration 

- - ----

----~ l 
._._

NASA Technical Memorandum 4698 

Performance Trend of Different Algorithms for 

Structural Design Optimization 

Surya N. Patnaik 

Ohio Aerospace Institute 

Brook Park, Ohio 

Rula M. Coroneos, James D. Guptill, and Dale A. Hopkins 

Lewis Research Center 

Cleveland, Ohio 

National Aeronautics and 

Space Administration 

Office of Management 

Scientific and Technical 

Information Program 

1996 

._._

NASA Technical Memorandum 4698 

Performance Trend of Different Algorithms for 

Structural Design Optimization 

Surya N. Patnaik 

Ohio Aerospace Institute 

Brook Park, Ohio 

Rula M. Coroneos, James D. Guptill, and Dale A. Hopkins 

Lewis Research Center 

Cleveland, Ohio 

National Aeronautics and 

Space Administration 

Office of Management 

Scientific and Technical 

Information Program 

1996 
Performance Trend of Different Algorithms for Structural Design Optimization 

Surya N. Patnaik 

Ohio Aerospace Institute 

Brook Park, Ohio 44142 

Rula M. Coroneos, James D. Guptill, and Dale A. Hopkins 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135 

Summary 

Nonlinearprograrnming algorithms play an important role in 

structural design optimization. Fortunately, several algorithms 

with computer codes are available. At NASA Lewis Research 

Center, a project was initiated to assess performance of different optimizers through the development of a computer code 

CometBoards. This paper summarizes the conclusions of that 

research. CometBoards was employed to solve sets of small, 

medium and large structural problems, using different optimizers 

on a Cray-YMP8E18128 computer. The reliability and efficiency of the optimizers were determined from the performance of these problems. For small problems, the performance 

of most of the optimizers could be considered adequate. For 

large problems however, three optimizers (two sequential 

quadratic programming routines, DNCONG ofIMSL and SQP 

of IDE SIGN, along with the sequential unconstrained minimizations technique SUMT) outperformed others. At optimum, 

most optimizers captured an identical number of active displacement and frequency constraints but the number of active 

stress constraints differed among the optimizers. This discrepancy can be attributed to singularity conditions in the optimization and the alleviation of this discrepancy can improve the 

efficiency of optimizers. 

Introduction 

Nonlinear programming algorithms play an important role 

in structural design optimization. Fortunately, several algorithms 

with computer codes have been developed during the past few 

decades. To assess performance of different optimizers, a 

project was initiated at NASA Lewis Research Center and a 

computer code called CometBoards, which is an acronym for 

Comparative Evaluation Test Bed of Optimization and Analysis 

Routines for the Design of Structures (ref. I), was developed. 

Because licenses for some of the optimization codes has 

expired, numerical results are provided herein for six optimizers. 

CometBoards incorporates about a dozen popular optimization 

codes. These are: the feasible directions method (FD; ref. 2); 

fully utilized design (FUD; ref. 3); genetic algorithm (GENMO; 

ref. 4); generalized reduced gradient method (GRG; ref. 5); the 

DNCONG of the IMSL routine (ref. 6); modified feasible 

direction method (MFD; ref. 7); NPSOL, which is available in 

the NAG library (ref. 8); the optimality criteria methods (OC; 

ref. 3); the reduced gradient method (RG; ref. 9); the sequential 

linear programming method (SLP; ref. 2); the sequential quadratic programming technique (SQP of IDESIGN; ref. 10); the 

sequence of unconstrained minimizations technique (SUMT; 

ref. 11); and the cascade strategy, which includes more than one 

optimization algorithm (ref. 12). CometBoards was employed 

to solve a set of 41 structural problems by using its eight 

optimizers on a Cray-YMP8E18128 computer. The reliability 

and efficiency of the eight optimizers were ascertained on the 

basis of the performance of these problems. The problems were 

solved for multiple load conditions, and behavior constraints 

were imposed on stresses, displacements, and frequencies. The 

examples were selected so that at optimum, numerous stress, 

displacement, and frequency constraints were active. Initial 

design, upper and lower bounds, and convergence parameters 

were specified to ensure that the evaluation had no bias towards 

any particular optimizer or any particular problem. The eight 

optimizers might have been updated during the time 

CometBoards was developed, but any such improvements were 

not accounted for. 

Evaluations of optimizers that are available in the literature 

(refs. 13 to 21) deal broadly with individual code validation by 

their developers. The studies lacked uniformity because problems and computational platforms differed and the evaluations 

were over a decade old. For example, Arora, (refs. 14 and 15), 

the developer of SQP of IDE SIGN, compared his algorithm to 

the NAGINPSOL optimizer. Most of Arora's problems were 

trusses for stress and displacement constraints and were sol ved 

on a PRIME 750 computer. Schittkowski, who is the developer 

of the DNCONG optimization routine in the IMSL library, 

essentially validated his code (refs. 16 and 17) by solving many 

theoretical examples on a Telefunken-TR-400 computer. 

Venkayya (ref. 18), one of the developers of ASTROS, in 

which OC and FD optimizers are used (ref. 19), attempted an 

evaluation of a few practical problems on a VAX 11/785 

computer. An intermediate complexity wing problem, used by 

Venkayya with stress and displacement constraints (ref. 18), is 

Performance Trend of Different Algorithms for Structural Design Optimization 

Surya N. Patnaik 

Ohio Aerospace Institute 

Brook Park, Ohio 44142 

Rula M. Coroneos, James D. Guptill, and Dale A. Hopkins 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135 

Summary 

Nonlinearprograrnming algorithms play an important role in 

structural design optimization. Fortunately, several algorithms 

with computer codes are available. At NASA Lewis Research 

Center, a project was initiated to assess performance of different optimizers through the development of a computer code 

CometBoards. This paper summarizes the conclusions of that 

research. CometBoards was employed to solve sets of small, 

medium and large structural problems, using different optimizers 

on a Cray-YMP8E18128 computer. The reliability and efficiency of the optimizers were determined from the performance of these problems. For small problems, the performance 

of most of the optimizers could be considered adequate. For 

large problems however, three optimizers (two sequential 

quadratic programming routines, DNCONG ofIMSL and SQP 

of IDE SIGN, along with the sequential unconstrained minimizations technique SUMT) outperformed others. At optimum, 

most optimizers captured an identical number of active displacement and frequency constraints but the number of active 

stress constraints differed among the optimizers. This discrepancy can be attributed to singularity conditions in the optimization and the alleviation of this discrepancy can improve the 

efficiency of optimizers. 

Introduction 

Nonlinear programming algorithms play an important role 

in structural design optimization. Fortunately, several algorithms 

with computer codes have been developed during the past few 

decades. To assess performance of different optimizers, a 

project was initiated at NASA Lewis Research Center and a 

computer code called CometBoards, which is an acronym for 

Comparative Evaluation Test Bed of Optimization and Analysis 

Routines for the Design of Structures (ref. I), was developed. 

Because licenses for some of the optimization codes has 

expired, numerical results are provided herein for six optimizers. 

CometBoards incorporates about a dozen popular optimization 

codes. These are: the feasible directions method (FD; ref. 2); 

fully utilized design (FUD; ref. 3); genetic algorithm (GENMO; 

ref. 4); generalized reduced gradient method (GRG; ref. 5); the 

DNCONG of the IMSL routine (ref. 6); modified feasible 

direction method (MFD; ref. 7); NPSOL, which is available in 

the NAG library (ref. 8); the optimality criteria methods (OC; 

ref. 3); the reduced gradient method (RG; ref. 9); the sequential 

linear programming method (SLP; ref. 2); the sequential quadratic programming technique (SQP of IDESIGN; ref. 10); the 

sequence of unconstrained minimizations technique (SUMT; 

ref. 11); and the cascade strategy, which includes more than one 

optimization algorithm (ref. 12). CometBoards was employed 

to solve a set of 41 structural problems by using its eight 

optimizers on a Cray-YMP8E18128 computer. The reliability 

and efficiency of the eight optimizers were ascertained on the 

basis of the performance of these problems. The problems were 

solved for multiple load conditions, and behavior constraints 

were imposed on stresses, displacements, and frequencies. The 

examples were selected so that at optimum, numerous stress, 

displacement, and frequency constraints were active. Initial 

design, upper and lower bounds, and convergence parameters 

were specified to ensure that the evaluation had no bias towards 

any particular optimizer or any particular problem. The eight 

optimizers might have been updated during the time 

CometBoards was developed, but any such improvements were 

not accounted for. 

Evaluations of optimizers that are available in the literature 

(refs. 13 to 21) deal broadly with individual code validation by 

their developers. The studies lacked uniformity because problems and computational platforms differed and the evaluations 

were over a decade old. For example, Arora, (refs. 14 and 15), 

the developer of SQP of IDE SIGN, compared his algorithm to 

the NAGINPSOL optimizer. Most of Arora's problems were 

trusses for stress and displacement constraints and were sol ved 

on a PRIME 750 computer. Schittkowski, who is the developer 

of the DNCONG optimization routine in the IMSL library, 

essentially validated his code (refs. 16 and 17) by solving many 

theoretical examples on a Telefunken-TR-400 computer. 

Venkayya (ref. 18), one of the developers of ASTROS, in 

which OC and FD optimizers are used (ref. 19), attempted an 

evaluation of a few practical problems on a VAX 11/785 

computer. An intermediate complexity wing problem, used by 

Venkayya with stress and displacement constraints (ref. 18), is 
also included in our test bed with the addition of frequency 

constraints. Ragsdell's evaluation (refs. 20 and 21) includes 

mostly simple mechanical application problems. Our current 

paper differs from those available in the literature in several 

respects: (1 ) a single tool, CometBoards, evaluates all eight 

optimizers on a common Cray-YMP computer; (2) solutions to 

a set of problems, which were grouped into categories of small, 

medium, and large, are used; and (3) design parameters were 

selected to ensure that the evaluation had no bias towards 

problems or optimizers. In brief, the comprehensive evaluation 

presented in this paper does not duplicate previous work. This 

paper presents a brief theory of optimization methods, a 

description of CometBoards, a summary of the numerical 

examples and their solutions, discussion, and conclusions. 

Symbols 

FD feasible directions 

FUD fully utilized design 

GENMO genetic algorithm 

GRG generalized reduced gradient method 

IMSL international mathematical subroutine library 

MFD modified feasible direction method 

NPSOL nonlinear programming package of the systems 

optimization laboratory 

OC optimality criteria 

RG reduced gradient 

SLP sequential linear programming 

SQP sequential quadratic programming 

SUMT sequence of unconstrained minimizations technique 

Theory of Optimization Methods 

Structural design can be formulated as: Find the n design 

variables x, within prescribed upper and lower bounds (x f s;,Xj 

s;,xF, i = 1,2, .. . ,n) which make a scalar objective functionfi x) an extremum (here, minimum weight) subject to: a set of m j 

inequality constraints g.( x) ~ 0, (j = 1,2, . . . ,m) and me equality .L J 

constraints gj+mi (x) = 0 (j = 1,2, . . . ,m,). 

Stress, displacement, and frequency behavior constraints were 

investigated in this study. A cursory account of representative 

2 

optimization methods available in CometBoards is provided 

herein. Readers may refer to specified references for details. 

(I) The sequence of unconstrained minimizations technique 

(SUMT), as implemented in the code NEWSUMT, is available 

in CometBoards. In NEWSUMT, the penalty function has been 

modified to improve efficiency and a modified Newton's 

approach is used to calculate the direction vector while a golden 

section technique is used to determine step length. 

(2) Sequential linear programming (SLP), as implemented 

in design optimization tools (DOT 2.0) is available in 

CometBoards. From the original nonlinear problem, a linear 

programming subproblem is obtained by linearizing a set of 

critical constraints and the objective function around a design 

point. The linearization process and linear solution sequence is 

repeated until convergence is achieved. 

(3) The method of feasible directions (FD), as implemented 

in DOT 2.0, is available in CometBoards. In FD, a usable 

feasible direction is used. A minimum along the search direction is generated by polynomial approximation. 

(4) SQP of IDE SIGN, DNCONG ofIMSL, and NPSOL in 

NAG, three implementations of the sequential quadratic programming technique, are available in CometBoards. In this 

technique, the original nonlinear problem is solved through a 

sequence of quadratic subproblems. In SQP of IDESIGN, a 

Lagrangian function is approximated. The step length is obtained 

by minimizing a composite descent function. DNCONG of 

IMSL uses quasi-Newton updates for the Hessian of the 

Lagrangian function while the constraints are linearized 

(ref. 22). The step length for an augmented Lagrangian is 

calculated using a bisection method (ref. 23). NPSOL in NAG 

also uses an augmented Lagrangian. The search direction is 

generated through a quadratic subproblem while step length is 

calculated using an augmented Lagrangian, which is designed 

to avoid discontinuities as much as possible. 

(5) The reduced gradient method (RG), as implemented in 

the code OPT, has been incorporated into CometBoards. This 

method partitions the design variable into decision and slave 

variables and a reduced gradient is used to generate a search 

direction. A line search is carried out by bounding the minimum 

and then calculating the minimum within some tolerance. 

(6) The optimality criteria method (OC), available in 

CometBoards, can be considered as a variant of the Lagrange 

mUltiplier approach applied to structural design problems. In 

OC, an iterative scheme is followed to update the multipliers 

and the design variables separately. 

Description of CometBoards 

The basic organization of CometBoards is depicted in figure 1. 

The central executive with command level interface (fig. 1) 

links the three modules (optimizer, analyzer, and data input) of 

the code to formulate and solve an optimization problem. The 

analyzer options are the displacement method (refs. 8 and 20), 

also included in our test bed with the addition of frequency 

constraints. Ragsdell's evaluation (refs. 20 and 21) includes 

mostly simple mechanical application problems. Our current 

paper differs from those available in the literature in several 

respects: (1 ) a single tool, CometBoards, evaluates all eight 

optimizers on a common Cray-YMP computer; (2) solutions to 

a set of problems, which were grouped into categories of small, 

medium, and large, are used; and (3) design parameters were 

selected to ensure that the evaluation had no bias towards 

problems or optimizers. In brief, the comprehensive evaluation 

presented in this paper does not duplicate previous work. This 

paper presents a brief theory of optimization methods, a 

description of CometBoards, a summary of the numerical 

examples and their solutions, discussion, and conclusions. 

Symbols 

FD feasible directions 

FUD fully utilized design 

GENMO genetic algorithm 

GRG generalized reduced gradient method 

IMSL international mathematical subroutine library 

MFD modified feasible direction method 

NPSOL nonlinear programming package of the systems 

optimization laboratory 

OC optimality criteria 

RG reduced gradient 

SLP sequential linear programming 

SQP sequential quadratic programming 

SUMT sequence of unconstrained minimizations technique 

Theory of Optimization Methods 

Structural design can be formulated as: Find the n design 

variables x, within prescribed upper and lower bounds (x f s;,Xj 

s;,xF, i = 1,2, .. . ,n) which make a scalar objective functionfi x) an extremum (here, minimum weight) subject to: a set of m j 

inequality constraints g.( x) ~ 0, (j = 1,2, . . . ,m) and me equality .L J 

constraints gj+mi (x) = 0 (j = 1,2, . . . ,m,). 

Stress, displacement, and frequency behavior constraints were 

investigated in this study. A cursory account of representative 

2 

optimization methods available in CometBoards is provided 

herein. Readers may refer to specified references for details. 

(I) The sequence of unconstrained minimizations technique 

(SUMT), as implemented in the code NEWSUMT, is available 

in CometBoards. In NEWSUMT, the penalty function has been 

modified to improve efficiency and a modified Newton's 

approach is used to calculate the direction vector while a golden 

section technique is used to determine step length. 

(2) Sequential linear programming (SLP), as implemented 

in design optimization tools (DOT 2.0) is available in 

CometBoards. From the original nonlinear problem, a linear 

programming subproblem is obtained by linearizing a set of 

critical constraints and the objective function around a design 

point. The linearization process and linear solution sequence is 

repeated until convergence is achieved. 

(3) The method of feasible directions (FD), as implemented 

in DOT 2.0, is available in CometBoards. In FD, a usable 

feasible direction is used. A minimum along the search direction is generated by polynomial approximation. 

(4) SQP of IDE SIGN, DNCONG ofIMSL, and NPSOL in 

NAG, three implementations of the sequential quadratic programming technique, are available in CometBoards. In this 

technique, the original nonlinear problem is solved through a 

sequence of quadratic subproblems. In SQP of IDESIGN, a 

Lagrangian function is approximated. The step length is obtained 

by minimizing a composite descent function. DNCONG of 

IMSL uses quasi-Newton updates for the Hessian of the 

Lagrangian function while the constraints are linearized 

(ref. 22). The step length for an augmented Lagrangian is 

calculated using a bisection method (ref. 23). NPSOL in NAG 

also uses an augmented Lagrangian. The search direction is 

generated through a quadratic subproblem while step length is 

calculated using an augmented Lagrangian, which is designed 

to avoid discontinuities as much as possible. 

(5) The reduced gradient method (RG), as implemented in 

the code OPT, has been incorporated into CometBoards. This 

method partitions the design variable into decision and slave 

variables and a reduced gradient is used to generate a search 

direction. A line search is carried out by bounding the minimum 

and then calculating the minimum within some tolerance. 

(6) The optimality criteria method (OC), available in 

CometBoards, can be considered as a variant of the Lagrange 

mUltiplier approach applied to structural design problems. In 

OC, an iterative scheme is followed to update the multipliers 

and the design variables separately. 

Description of CometBoards 

The basic organization of CometBoards is depicted in figure 1. 

The central executive with command level interface (fig. 1) 

links the three modules (optimizer, analyzer, and data input) of 

the code to formulate and solve an optimization problem. The 

analyzer options are the displacement method (refs. 8 and 20), 
CometBoards 

- Displacement - Analysis data 

- Force (IFM) - Design data 

- Others - Optimization data 

GRG 

IMSL 

MFD 

NPSOL 

OC 

RG 

SLP 

SQP 

SUMT 

Central executive via 

command level 

interface 

Results 

Figure 1 .-Comparative evaluation test bed of optimization and analysis 

routines for the design of structures (CometBoards). 

the integrated force method (refs. 3 and 25), and the simplified 

force method (ref. 3), etc. There are three input data files, one 

for analysis (anldat), one for design (dsgndat), and one for 

optimization (optdat). CometBoards has considerable flexibility in solving a design problem by choosing anyone of the 

available optimizers and anyone of three analyzers. A more 

detailed description of CometBoards can be found in the 

reference 1, User's Manual. 

Example Problems 

The numerical test bed of CometBoards includes over 41 

problems, most of which were taken from the literature (refs. 1, 

3, 18, and 26 to 31). Minimum weight was the objective and a 

linking strategy was followed to reduce the number of design 

variables. Stress, displacement, and frequency behavior constraints were considered. Multiple static load conditions and 

consistent elemental mass for dynamic analyses were also considered. The load conditions, mass distributions, and behavior 

limitations were specified to ensure that several types ofbehavior constraints were active at the optimum. The initial design of 

unity was considered for all problems unless otherwise specified. Each problem had a consistent set of upper and lower 

bounds specified. Typically, default optimization parameters 

and convergence criteria specified in the individual codes were 

used. These parameters, however, were changed when convergence difficulty was encountered. Results for all 41 examples 

are summarized in table I. The normalized optimum weight and 

the normalized Cray-YMP8E/8 128 CPU time for a select set of 

14 examples are gi ven in table II and depicted in figures 2 to 5. 

The weight was normalized with respect to the optimum weight 

obtained for the best feasible design. A brief description of the 

14 examples follows. 

Examples PIa to PId: 3-Bar Truss 

The popular 3-bar truss (refs. 3, 26, 30), as shown in figure 6, (with modulus E= 30 000 ksi, and density p =0.11b/in.3) 

was subjected to a single load condition. It had three design 

variables, and six constraints (3 stress, two displacement and 

one frequency). Optimum weight and CPU time are depicted in 

table II (PIa, Plb, PIc, Pld), and figures 2 and 5. The optimum 

weight was 92.87 lb and one stress, one displacement, and one 

frequency constraints were active. Five optimizers (SUMT, 

SQP, IMSL, NPSOL and RG) performed satisfactorily. OC 

was inadequate, yielding a 38.6 percent over-design. The 

problem was solved again for three different initial designs (the 

SUMT optimum design, 150 percent of SUMT optimum, and 

50 percent of SUMT optimum). Results followed the pattern of 

the earlier problem where the initial design was unity. The CPU 

times on the Cray-YMP computer required for different 

optimizers are depicted in figure 5. For unit ini tial design, SQP 

required the least CPU time of 0.14 sec, while RG was most 

expensive at 3.18 sec. 

Example P2: Tapered IO-Bar Truss 

A tapered 1 O-bar aluminum truss (ref. 3), shown in figure 7, 

was subjected to two load conditions. It had 10 design variables, 

3 

--------------- 

CometBoards 

- Displacement - Analysis data 

- Force (IFM) - Design data 

- Others - Optimization data 

GRG 

IMSL 

MFD 

NPSOL 

OC 

RG 

SLP 

SQP 

SUMT 

Central executive via 

command level 

interface 

Results 

Figure 1 .-Comparative evaluation test bed of optimization and analysis 

routines for the design of structures (CometBoards). 

the integrated force method (refs. 3 and 25), and the simplified 

force method (ref. 3), etc. There are three input data files, one 

for analysis (anldat), one for design (dsgndat), and one for 

optimization (optdat). CometBoards has considerable flexibility in solving a design problem by choosing anyone of the 

available optimizers and anyone of three analyzers. A more 

detailed description of CometBoards can be found in the 

reference 1, User's Manual. 

Example Problems 

The numerical test bed of CometBoards includes over 41 

problems, most of which were taken from the literature (refs. 1, 

3, 18, and 26 to 31). Minimum weight was the objective and a 

linking strategy was followed to reduce the number of design 

variables. Stress, displacement, and frequency behavior constraints were considered. Multiple static load conditions and 

consistent elemental mass for dynamic analyses were also considered. The load conditions, mass distributions, and behavior 

limitations were specified to ensure that several types ofbehavior constraints were active at the optimum. The initial design of 

unity was considered for all problems unless otherwise specified. Each problem had a consistent set of upper and lower 

bounds specified. Typically, default optimization parameters 

and convergence criteria specified in the individual codes were 

used. These parameters, however, were changed when convergence difficulty was encountered. Results for all 41 examples 

are summarized in table I. The normalized optimum weight and 

the normalized Cray-YMP8E/8 128 CPU time for a select set of 

14 examples are gi ven in table II and depicted in figures 2 to 5. 

The weight was normalized with respect to the optimum weight 

obtained for the best feasible design. A brief description of the 

14 examples follows. 

Examples PIa to PId: 3-Bar Truss 

The popular 3-bar truss (refs. 3, 26, 30), as shown in figure 6, (with modulus E= 30 000 ksi, and density p =0.11b/in.3) 

was subjected to a single load condition. It had three design 

variables, and six constraints (3 stress, two displacement and 

one frequency). Optimum weight and CPU time are depicted in 

table II (PIa, Plb, PIc, Pld), and figures 2 and 5. The optimum 

weight was 92.87 lb and one stress, one displacement, and one 

frequency constraints were active. Five optimizers (SUMT, 

SQP, IMSL, NPSOL and RG) performed satisfactorily. OC 

was inadequate, yielding a 38.6 percent over-design. The 

problem was solved again for three different initial designs (the 

SUMT optimum design, 150 percent of SUMT optimum, and 

50 percent of SUMT optimum). Results followed the pattern of 

the earlier problem where the initial design was unity. The CPU 

times on the Cray-YMP computer required for different 

optimizers are depicted in figure 5. For unit ini tial design, SQP 

required the least CPU time of 0.14 sec, while RG was most 

expensive at 3.18 sec. 

Example P2: Tapered IO-Bar Truss 

A tapered 1 O-bar aluminum truss (ref. 3), shown in figure 7, 

was subjected to two load conditions. It had 10 design variables, 

3 

--------------- 
TABLE I - SUMMARY FOR 41 TEST BED PROBLEMS 

Problem Problem description and Constraints Active constraints for ontimization codes 

number number of design variables spec ified SUMT SOP IMSL NPSOL RG ex: 

Pi a 3-bar truSS (3 my. ID = I) 3S.20. IF IS. !D. IF IS. !D. IF IS. !D. IF IS. 10. IF IS. 10. IF ' IF 

Plb 3-bar truSS (3 IDY. ID = OPT) 3S. 20. IF IS. 10, IF IS. 10. IF IS. ID, IF IS. !D, IF IS, !D. IF ' IF 

Pi c 3-bar truSS (3 my. ID = 1.5 x OPT) 3S.20. IF IS. 10. IF IS. 10. IF IS. !D. IF IS. ID. IF IS. !D, IF ' IF 

Pld 3-bar truSS (3 IDY. ID - 0.5 x OPT) 3S.20. IF IS. !D. IF IS. !D. IF IS. 10. IF IS. 10. IF IS. 10. IF ' IF 

P2 Tapered I O-bar truss (10 lOY) 20S, 40. IF 7S, 20. IF 8S, 20. IF 8S. 20. IF 8S. 20. IF ' IS. IF '5S. 20 . IF 

P3 Tapered cantilever beam (8 IDY) 16S. 40. IF 6S 6S 6S 6S 6S 4S 

P4 25-bar truss (8 LOY) 50S. 36D 40 40 40 4D ' 20 (a) 

P5 165 feet tall antenna tower (6 LDY) 504S. 240. IF '7S. 12D. IF 6S. 120. IF 6S. 12D. IF 6S, 12D, IF ' IF (a) 

P6 60-bar trussed ring (25 LDY) 180S.3D. IF 21S, ID. IF 21S. !D. IF 21S. ID. IF 19S. 10, IF '18S. ID. IF ' lOS, IF 

P7 Geodesic dome (12 LDY) 252S. ID. IF 162S. ID. IF 168S. ID. IF 156S. 10. IF 162S. 10. IF '18S.ID ' 12S 

P8a Intermediate Complexity Wing (57 LDY, m =1.0) 316S,4D. IF 106S, 10 117S. ID. IF 11 7S. ID. IF 75S. ID. IF ' 6S ' 19S 

P8b Intermediate Complexity Wing (57 LOY. ID =OPT) 316S.4D.IF (a) 106S, ID 117S, 10. IF 11 7S, 10, IF 106S. 10 '42S 

P8c Intermediate Complexity Wing (57 LOY, ID =1.5 x OPT) 316S,4D, IF 109S. 10 ' 1I5S, 10. IF '88S. 10. IF '2S '7S (a) 

PSd Intermediate Comolexitv Win2 (57 LOY. ID =0.5 x OPT) 316S.4D. IF II7S. 10. IF '99S. 10. IF (a) ' 14S '7S (a) 

P9 I O-bar truss (10 my) 20S. 4D. IF 7S.2D. IF 8S.2D. IF 8S.20. IF 8S.20 . IF 'IF ' 5S. 2D. IF 

PIO lO-bar truss (5 LOY) 20S.40. IF 6S.2D. IF 6S, 20. IF 6S.20. IF 6S. 2D. IF '3S. IF 6S. 2D. IF 

PI I Stiffened 10-bar truss (IS my) 36S. 4D. IF 85 9S 8S 9S ' IS 5S 

PI2 Stiffened 10-bar truss 12 LDY) 36S.4D. IF 2S 2S 2S 2S ' IF 2S 

PI 3 Cantilever membrane ( 8 my) 16S. 4D. IF 6S 6S 6S 6S 6S 4S 

PI4 Cantilever membrane ( I LDY) 16S.4D, IF IS IS IS IS IS IS 

PIS Cantilever membrane 16-quad elements (16 IDY) 32S.4D. IF IS.4D (a) 4D ' 4S '5S 40 

PI6 Cantilever membrane 32-quad elements (16 LOY) 64S.4D. IF IS.4D 4D 4D ' 9S ' 4S 4D 

PI7 Cantilever membrane 48-quad elements ( 16 LOY) 96S.4D, IF 13S.40 ' 13S.4D 15S.4D '85 '3S 13S, 20 

PI 8 Cantileve r membrane 64-auad elements (I6 LOY) 128S.4D. IF 3'S.4D '29S. 4D 31S. 4D ' 16S ( a) 23S.20 

PI9 60-bar trussed ring (25 LOY) 180S 3SS ' 35S 38S 38S '1 4S 40S 

P20 6O-bar trussed ring (25 LOY) 30 ID 10 ID ID ' ID ID 

P21 6O-bar trussed ring (25 LOY) IF (a) IF IF IF ' IF ' IF 

P22 60-bar trussed ring (25 LDY) 180S. 240 28S.10 '30S,10 27S.I D 29S,1O '20S ISS, 10 

P23 60-bar trussed rin o 125 LOY) 24D. IF ' ID. IF ID. IF ID. IF ' ID. IF ' IF ' 10. IF 

P24 Stiffened 60-bar trussed ring (49 LOY) 252S 75S 75S 75S 75S 75S '59S 

P25 Stiffened 60-bar trussed ring (49 LDV) 3D ID 'ID 10 (a) ID ID 

P26 Stiffened 60-bar trussed ring (49 LOY) IF IF IF IF (a) IF IF 

P27 Stiffened 60-bar trussed ring (49 LOY) 252S.240 75S. ID 75S, ID 75S. ID 75S, 10 76S.ID ' 17S 

P2S Stiffened 60-bar trussed ring (49 LOY) 24D. IF 10. IF 10. IF ID. IF ' ID ID. IF 10. IF 

P29 Stiffened 6O-bar trussed rin~ (49 LDY) 252S. 3D. IF 44S. IF 46S. IF 46S. IF 46S. IF 47S. IF '3S 

P30 Stiffened ring (24 IDY) 72S 28S 28S 28S 2SS 27S 2SS 

P31 Stiffened ring (24 my) 30 ID 10 ID (a) ID ID 

P32 Stiffened ring (24 my) IF IF IF IF (a) 'IF IF 

P33 Stiffened ring (24 IDY) 72S.24D 28S. 10 27S.ID 2SS.10 28S. 10 25S.10 ' ISS. ID 

P34 Stiffened ring (24 my) 24D. IF 10. IF ID. IF ID. IF ' 20 !D. IF 10. IF 

P35 Stiffened ring (24 IDY) 72S. 30 . IF 17S. IF 17S. IF 17S. IF 17S. IF 17S. IF ' 16S. IF 

' Optimum weight obtained differs by more than 5 percent or constraint violation more than I percent (see ref. I). 

IDV: Independent design variable ID: Initial design LOV: Linked design variable OPT: SUMT optimum design 0 : Oisplacement constraints F: Frequency constraints 

s: Stress constraint s 

4 

TABLE I - SUMMARY FOR 41 TEST BED PROBLEMS 

Problem Problem description and Constraints Active constraints for ontimization codes 

number number of design variables spec ified SUMT SOP IMSL NPSOL RG ex: 

Pi a 3-bar truSS (3 my. ID = I) 3S.20. IF IS. !D. IF IS. !D. IF IS. !D. IF IS. 10. IF IS. 10. IF ' IF 

Plb 3-bar truSS (3 IDY. ID = OPT) 3S. 20. IF IS. 10, IF IS. 10. IF IS. ID, IF IS. !D, IF IS, !D. IF ' IF 

Pi c 3-bar truSS (3 my. ID = 1.5 x OPT) 3S.20. IF IS. 10. IF IS. 10. IF IS. !D. IF IS. ID. IF IS. !D, IF ' IF 

Pld 3-bar truSS (3 IDY. ID - 0.5 x OPT) 3S.20. IF IS. !D. IF IS. !D. IF IS. 10. IF IS. 10. IF IS. 10. IF ' IF 

P2 Tapered I O-bar truss (10 lOY) 20S, 40. IF 7S, 20. IF 8S, 20. IF 8S. 20. IF 8S. 20. IF ' IS. IF '5S. 20 . IF 

P3 Tapered cantilever beam (8 IDY) 16S. 40. IF 6S 6S 6S 6S 6S 4S 

P4 25-bar truss (8 LOY) 50S. 36D 40 40 40 4D ' 20 (a) 

P5 165 feet tall antenna tower (6 LDY) 504S. 240. IF '7S. 12D. IF 6S. 120. IF 6S. 12D. IF 6S, 12D, IF ' IF (a) 

P6 60-bar trussed ring (25 LDY) 180S.3D. IF 21S, ID. IF 21S. !D. IF 21S. ID. IF 19S. 10, IF '18S. ID. IF ' lOS, IF 

P7 Geodesic dome (12 LDY) 252S. ID. IF 162S. ID. IF 168S. ID. IF 156S. 10. IF 162S. 10. IF '18S.ID ' 12S 

P8a Intermediate Complexity Wing (57 LDY, m =1.0) 316S,4D. IF 106S, 10 117S. ID. IF 11 7S. ID. IF 75S. ID. IF ' 6S ' 19S 

P8b Intermediate Complexity Wing (57 LOY. ID =OPT) 316S.4D.IF (a) 106S, ID 117S, 10. IF 11 7S, 10, IF 106S. 10 '42S 

P8c Intermediate Complexity Wing (57 LOY, ID =1.5 x OPT) 316S,4D, IF 109S. 10 ' 1I5S, 10. IF '88S. 10. IF '2S '7S (a) 

PSd Intermediate Comolexitv Win2 (57 LOY. ID =0.5 x OPT) 316S.4D. IF II7S. 10. IF '99S. 10. IF (a) ' 14S '7S (a) 

P9 I O-bar truss (10 my) 20S. 4D. IF 7S.2D. IF 8S.2D. IF 8S.20. IF 8S.20 . IF 'IF ' 5S. 2D. IF 

PIO lO-bar truss (5 LOY) 20S.40. IF 6S.2D. IF 6S, 20. IF 6S.20. IF 6S. 2D. IF '3S. IF 6S. 2D. IF 

PI I Stiffened 10-bar truss (IS my) 36S. 4D. IF 85 9S 8S 9S ' IS 5S 

PI2 Stiffened 10-bar truss 12 LDY) 36S.4D. IF 2S 2S 2S 2S ' IF 2S 

PI 3 Cantilever membrane ( 8 my) 16S. 4D. IF 6S 6S 6S 6S 6S 4S 

PI4 Cantilever membrane ( I LDY) 16S.4D, IF IS IS IS IS IS IS 

PIS Cantilever membrane 16-quad elements (16 IDY) 32S.4D. IF IS.4D (a) 4D ' 4S '5S 40 

PI6 Cantilever membrane 32-quad elements (16 LOY) 64S.4D. IF IS.4D 4D 4D ' 9S ' 4S 4D 

PI7 Cantilever membrane 48-quad elements ( 16 LOY) 96S.4D, IF 13S.40 ' 13S.4D 15S.4D '85 '3S 13S, 20 

PI 8 Cantileve r membrane 64-auad elements (I6 LOY) 128S.4D. IF 3'S.4D '29S. 4D 31S. 4D ' 16S ( a) 23S.20 

PI9 60-bar trussed ring (25 LOY) 180S 3SS ' 35S 38S 38S '1 4S 40S 

P20 6O-bar trussed ring (25 LOY) 30 ID 10 ID ID ' ID ID 

P21 6O-bar trussed ring (25 LOY) IF (a) IF IF IF ' IF ' IF 

P22 60-bar trussed ring (25 LDY) 180S. 240 28S.10 '30S,10 27S.I D 29S,1O '20S ISS, 10 

P23 60-bar trussed rin o 125 LOY) 24D. IF ' ID. IF ID. IF ID. IF ' ID. IF ' IF ' 10. IF 

P24 Stiffened 60-bar trussed ring (49 LOY) 252S 75S 75S 75S 75S 75S '59S 

P25 Stiffened 60-bar trussed ring (49 LDV) 3D ID 'ID 10 (a) ID ID 

P26 Stiffened 60-bar trussed ring (49 LOY) IF IF IF IF (a) IF IF 

P27 Stiffened 60-bar trussed ring (49 LOY) 252S.240 75S. ID 75S, ID 75S. ID 75S, 10 76S.ID ' 17S 

P2S Stiffened 60-bar trussed ring (49 LOY) 24D. IF 10. IF 10. IF ID. IF ' ID ID. IF 10. IF 

P29 Stiffened 6O-bar trussed rin~ (49 LDY) 252S. 3D. IF 44S. IF 46S. IF 46S. IF 46S. IF 47S. IF '3S 

P30 Stiffened ring (24 IDY) 72S 28S 28S 28S 2SS 27S 2SS 

P31 Stiffened ring (24 my) 30 ID 10 ID (a) ID ID 

P32 Stiffened ring (24 my) IF IF IF IF (a) 'IF IF 

P33 Stiffened ring (24 IDY) 72S.24D 28S. 10 27S.ID 2SS.10 28S. 10 25S.10 ' ISS. ID 

P34 Stiffened ring (24 my) 24D. IF 10. IF ID. IF ID. IF ' 20 !D. IF 10. IF 

P35 Stiffened ring (24 IDY) 72S. 30 . IF 17S. IF 17S. IF 17S. IF 17S. IF 17S. IF ' 16S. IF 

' Optimum weight obtained differs by more than 5 percent or constraint violation more than I percent (see ref. I). 

IDV: Independent design variable ID: Initial design LOV: Linked design variable OPT: SUMT optimum design 0 : Oisplacement constraints F: Frequency constraints 

s: Stress constraint s 

4 
TABLE fl .-OPTIMUM WEIGHT AND CRAY-YMP 8EJ8128 CPU TIME FOR 

SELECTED SET OF EXAMPLE PROBLEMS 

Optimization methods 

Problem SUMT SQP IMSL NPSOL RG 

number Weight b 

PIa 1.001 

Plb 1.001 

PIc 1.001 

Pld 1.001 

P2 1.000 

P3 0.999 

P4 1.000 

P5 "0.940 

P6 1.000 

P7 1.021 

P8a 1.004 

P8b 0.790 

P8c 1.000 

P8d 1.000 

"Infeasible design . 

bNormalized weight. 

CPU 

1.799 

7.833 

1.588 

1.926 

1.500 

1.268 

2.533 

1.065 

1.605 

0.538 

1.116 

1.680 

1.000 

1.000 

Weight b CPU Weight b 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.01 9 1.000 1.019 

1.000 1.000 1.000 

1.000 1.000 1.015 

1.000 1.000 1.000 

1.000 1.000 1.000 

"0.998 0.571 "1.077 

"0.98 1 0.244 (Failed) 

CPU Weight b CPU Weight b CPU 

1.972 1.000 2.076 1.000 22.069 

9.633 1.000 155.267 1.000 5.567 

1.662 1.000 31.818 1.000 2.000 

2.733 1.000 14.444 1.000 16.452 

1.133 1.000 1.022 (Fai led) ---

0.976 1.00 1 3.169 1.000 6.121 

1.539 1.000 3.759 (Failed) ---

l.l 34 1.017 5.026 (Fai led) ---

1.1 20 1.000 3.899 1.832 20.716 

0.550 1.016 0.658 2.022 0.066 

1.695 1.037 7.712 (Fai led) ---

5.884 1.000 3.323 1.004 0.089 

0.602 1.346 0.244 1.471 0.627 

--- "0.50 1 0.028 (Failed) ---

OC 

Weight b CPU 

1.386 10.257 

1.386 49.233 

1.386 10.000 

1.386 10.985 

1.056 9.324 

1.028 16.834 

(Failed) ---

2.773 4.62 

"1.041 8.279 

2.976 4.456 

1.201 1.571 

1.350 8.076 

1.000 0.666 

1.000 0.276 

5 

TABLE fl .-OPTIMUM WEIGHT AND CRAY-YMP 8EJ8128 CPU TIME FOR 

SELECTED SET OF EXAMPLE PROBLEMS 

Optimization methods 

Problem SUMT SQP IMSL NPSOL RG 

number Weight b 

PIa 1.001 

Plb 1.001 

PIc 1.001 

Pld 1.001 

P2 1.000 

P3 0.999 

P4 1.000 

P5 "0.940 

P6 1.000 

P7 1.021 

P8a 1.004 

P8b 0.790 

P8c 1.000 

P8d 1.000 

"Infeasible design . 

bNormalized weight. 

CPU 

1.799 

7.833 

1.588 

1.926 

1.500 

1.268 

2.533 

1.065 

1.605 

0.538 

1.116 

1.680 

1.000 

1.000 

Weight b CPU Weight b 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.000 1.000 1.000 

1.01 9 1.000 1.019 

1.000 1.000 1.000 

1.000 1.000 1.015 

1.000 1.000 1.000 

1.000 1.000 1.000 

"0.998 0.571 "1.077 

"0.98 1 0.244 (Failed) 

CPU Weight b CPU Weight b CPU 

1.972 1.000 2.076 1.000 22.069 

9.633 1.000 155.267 1.000 5.567 

1.662 1.000 31.818 1.000 2.000 

2.733 1.000 14.444 1.000 16.452 

1.133 1.000 1.022 (Fai led) ---

0.976 1.00 1 3.169 1.000 6.121 

1.539 1.000 3.759 (Failed) ---

l.l 34 1.017 5.026 (Fai led) ---

1.1 20 1.000 3.899 1.832 20.716 

0.550 1.016 0.658 2.022 0.066 

1.695 1.037 7.712 (Fai led) ---

5.884 1.000 3.323 1.004 0.089 

0.602 1.346 0.244 1.471 0.627 

--- "0.50 1 0.028 (Failed) ---

OC 

Weight b CPU 

1.386 10.257 

1.386 49.233 

1.386 10.000 

1.386 10.985 

1.056 9.324 

1.028 16.834 

(Failed) ---

2.773 4.62 

"1.041 8.279 

2.976 4.456 

1.201 1.571 

1.350 8.076 

1.000 0.666 

1.000 0.276 

5 
6 

2.0 

1.8 

1.6 

~1.4 

E 

::J 

E 

R 

o 

II 1.2 

G 

.c 

-

OJ 'w 

3: 

"0 1.0 Q) 

.t::l 

~ 

E 

o 

z 0.8 

0.6 

0.4 

0.2 

-

~ 

-

-

-

-

-

-

-

SUMT 

Problem set 

0 P1a 

0 P1b 

0 P1c 

P1d 

0 P2 

P3 

0 P4 

0 P5 

0 P7 

0 P9 I 

I 0 P10 

1II~ 0 P11 

P12 

I 0 P13 

I1III11 

P14 

0 P15 

0 P16 

I11II 

0 P17 

0 P18 

sap IMSL NPSOL RG OC 

Figure 2.-Performance of different optimizers for small problems. 

6 

2.0 

1.8 

1.6 

~1.4 

E 

::J 

E 

R 

o 

II 1.2 

G 

.c 

-

OJ 'w 

3: 

"0 1.0 Q) 

.t::l 

~ 

E 

o 

z 0.8 

0.6 

0.4 

0.2 

-

~ 

-

-

-

-

-

-

-

SUMT 

Problem set 

0 P1a 

0 P1b 

0 P1c 

P1d 

0 P2 

P3 

0 P4 

0 P5 

0 P7 

0 P9 I 

I 0 P10 

1II~ 0 P11 

P12 

I 0 P13 

I1III11 

P14 

0 P15 

0 P16 

I11II 

0 P17 

0 P18 

sap IMSL NPSOL RG OC 

Figure 2.-Performance of different optimizers for small problems. 
Problem set 

2.0 n 

0 P6 

0 P19 

1.B -

P20 

1.6 -

E • P21 

::> 

E 1.4 - E • P22 

0 

II P23 

G 1.2 -

.c 

-

OJ "a; 0 P30 

=: 1.0 - "0 

(l) 

~ 

0 P31 

""iii 

E O.B - ... 0 P32 0 

z 

0.6 - 0 P33 

P34 

0.4 -

P35 

0.2 w. ~~ ¥WL- ~~ ~ 

SUMT sap IMSL NPSOL RG OC 

Figure 3.-Performance of different optimizers for medium problems. 

2.0 

Problem set 

1.B 0 PBa 

1.6 0 PBb 

E 

::> 

E 1.4 

PBc 

°li 

0 PBd 

T""" 1.2 

.c 

- P24 OJ 

.a; 

:= 1.0 "0 P25 Q) 

~ 

""iii 

E O.B 0 P26 

0 

z 

0.6 0 P27 

0 P2B 

0.4 

0 P29 

0.2 

SUMT sap IMSL NPSOL RG OC 

Figure 4.- Performance of different optimizers for large problems. 

7 

Problem set 

2.0 n 

0 P6 

0 P19 

1.B -

P20 

1.6 -

E • P21 

::> 

E 1.4 - E • P22 

0 

II P23 

G 1.2 -

.c 

-

OJ "a; 0 P30 

=: 1.0 - "0 

(l) 

~ 

0 P31 

""iii 

E O.B - ... 0 P32 0 

z 

0.6 - 0 P33 

P34 

0.4 -

P35 

0.2 w. ~~ ¥WL- ~~ ~ 

SUMT sap IMSL NPSOL RG OC 

Figure 3.-Performance of different optimizers for medium problems. 

2.0 

Problem set 

1.B 0 PBa 

1.6 0 PBb 

E 

::> 

E 1.4 

PBc 

°li 

0 PBd 

T""" 1.2 

.c 

- P24 OJ 

.a; 

:= 1.0 "0 P25 Q) 

~ 

""iii 

E O.B 0 P26 

0 

z 

0.6 0 P27 

0 P2B 

0.4 

0 P29 

0.2 

SUMT sap IMSL NPSOL RG OC 

Figure 4.- Performance of different optimizers for large problems. 

7 
2.0 

E 1.S ::J 

E 

:g 

0 

... 

Q) 

E 

'';::; 1.0 ::J 

a. 

u 

-0 

Q) 

.!::! 

cti 

E 

0 

z O.S 

0.0 

8 

Problem set 

D P1a 

D P1b 

P1c 

P1d 

• P2 

• P3 

0 P4 

0 PS 

D P6 

D P7 

P8a 

P8b 

P8c 

0 P8d 

SUMT sap IMSL NPSOL RG OC 

Figure S.-Cray-YMP cpu time for different optimization methods for 14 problems. 

1---100 in·- - ... ·+I .... --·100 in.----J 

4 

100 in. 3 

Figure 6.- Three-bar truss. (Elements are circled, nodes 

are not.) 

2.0 

E 1.S ::J 

E 

:g 

0 

... 

Q) 

E 

'';::; 1.0 ::J 

a. 

u 

-0 

Q) 

.!::! 

cti 

E 

0 

z O.S 

0.0 

8 

Problem set 

D P1a 

D P1b 

P1c 

P1d 

• P2 

• P3 

0 P4 

0 PS 

D P6 

D P7 

P8a 

P8b 

P8c 

0 P8d 

SUMT sap IMSL NPSOL RG OC 

Figure S.-Cray-YMP cpu time for different optimization methods for 14 problems. 

1---100 in·- - ... ·+I .... --·100 in.----J 

4 

100 in. 3 

Figure 6.- Three-bar truss. (Elements are circled, nodes 

are not.) 
t Y 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

100 t16 

T~ L Ii 

>.='------{ 

5 

~ .. ---162.5 in.,--- .r----162.5 in.;------~/ 

Figure 7.- Tapered ten-bar truss. (Elements are circled, nodes are not.) 

t Y 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

100 t16 

T / ~~~5 :--_-!4 (j) T T' ® @ 50 in. 

/ ~ ________________ ~~ ________________ ~-L ________ • 

/ 11 2 3 x 

fooI / .. f----162.5 in.c-- - .r-- --162.5 in.- -----+1--/ 

Figure 8.-Tapered cantilever beam modeled with eight triangular membrane elements. 

(Elements are circled, nodes are not.) 

and 25 behavior constraints (20 stress, four displacement, and 

one frequency). The optimum weight was 3326.74 Ib with 11 

active constraints (8 stress, two displacement, and one frequency). Four optimizers (SUMT, SQP, IMSL, and NPSOL) 

converged for this example. Optimizer RG failed, and OC was 

marginal at 5.6 percent over-design. Cray-YMP CPU time 

varied between 1.28 sec for NPSOL and 1.91 sec for SUMT. 

Example P3: Tapered Cantilever Beam 

The cantilever truss of example P2, was modeled next using 

8 triangular membrane elements, as shown in figure 8. The 

loads and constraints were identical to example P2. The eight 

thicknesses of the elements were considered the 8 design 

-----------

variables. The problem had 21 constraints (1 6 von Mises stress, 

four displacement and one frequency). Optimum results 

obtained are given in table II. The optimum weight was 

1440.24 Ib with six active stress constraints. Five optimizers 

(SUMT, SQP, IMSL, NPSOL, and RG) performed well while 

OC produced a 2.8-percent over-design (see table IT). CrayYMP CPU time varied from 1.79 sec for IMSL to 11.22 sec for 

RG. 

Example P4: 2S-Bar Truss 

A 25-bar aluminum truss (refs. 26 and 27), as shown in figure 9, had 8 linked design variables, and was subjected to two 

load conditions. It had a total of 86 behavior constraints, 

9 

t Y 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

100 t16 

T~ L Ii 

>.='------{ 

5 

~ .. ---162.5 in.,--- .r----162.5 in.;------~/ 

Figure 7.- Tapered ten-bar truss. (Elements are circled, nodes are not.) 

t Y 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

100 t16 

T / ~~~5 :--_-!4 (j) T T' ® @ 50 in. 

/ ~ ________________ ~~ ________________ ~-L ________ • 

/ 11 2 3 x 

fooI / .. f----162.5 in.c-- - .r-- --162.5 in.- -----+1--/ 

Figure 8.-Tapered cantilever beam modeled with eight triangular membrane elements. 

(Elements are circled, nodes are not.) 

and 25 behavior constraints (20 stress, four displacement, and 

one frequency). The optimum weight was 3326.74 Ib with 11 

active constraints (8 stress, two displacement, and one frequency). Four optimizers (SUMT, SQP, IMSL, and NPSOL) 

converged for this example. Optimizer RG failed, and OC was 

marginal at 5.6 percent over-design. Cray-YMP CPU time 

varied between 1.28 sec for NPSOL and 1.91 sec for SUMT. 

Example P3: Tapered Cantilever Beam 

The cantilever truss of example P2, was modeled next using 

8 triangular membrane elements, as shown in figure 8. The 

loads and constraints were identical to example P2. The eight 

thicknesses of the elements were considered the 8 design 

-----------

variables. The problem had 21 constraints (1 6 von Mises stress, 

four displacement and one frequency). Optimum results 

obtained are given in table II. The optimum weight was 

1440.24 Ib with six active stress constraints. Five optimizers 

(SUMT, SQP, IMSL, NPSOL, and RG) performed well while 

OC produced a 2.8-percent over-design (see table IT). CrayYMP CPU time varied from 1.79 sec for IMSL to 11.22 sec for 

RG. 

Example P4: 2S-Bar Truss 

A 25-bar aluminum truss (refs. 26 and 27), as shown in figure 9, had 8 linked design variables, and was subjected to two 

load conditions. It had a total of 86 behavior constraints, 

9 
l 

Figure 9.-Twenty-five bar-truss. (Elements are circled, 

nodes are not.) 

(50 stress and 36 displacement). Fouroptimizers (SUMT, SQP, 

IMSL, and NPSOL) converged to an optimum weight of 

544.73 Ib with fo ur active displacement constraints (tables I 

and II). Optimizers RG and OC failed. Cray-YMP CPU time 

ranged from 1.64 sec for SQP to 6.15 sec for NPSOL. 

Example P5: 165-Ft-Tall Antenna Tower 

A 165-ft-tall steel antenna tower with 252 members, as 

depicted in figure 10, (ref. 26), had six linked design variables 

and was subjected to two load conditions. Its overhead dish 

antenna was modeled as a lumped mass for freq uency calcul ations. It had a total of 529 behavior constraints (504 stress, 24 

displ acement, and one frequency). Three optimizers (SQP, 

IMSL, and NPSOL) converged to an optimum solution of 

5299.84 lb with small deviations (table II). At optimum, six 

stress, 12 displacement, and one freq uency constraints were 

active. Optimizers RG and OC fa iled while SUMT produced a 

six percent under-design. The-Cray-YMP CPU time varied 

between 376.83 sec for SQP and 1893.80 sec for NPSOL. 

Example P6: 60-Bar Trussed Ring 

A 60-bar trussed aluminum ring (ref. 3) was subjected to 

three load conditions and had two lumped masses, as depicted 

in figure 11. It had a total of 184 constraints (180 stress, three 

displacement, one frequency) and 25 linked design variables. 

The optimum weight was 414.51 Ibs, and at optimum, 22 stress, 

one displacement, and one frequency constraints were active. 

Four optimizers (SUMT, SQP, IMSL, and NPSOL) converged 

(table II). Optimizer RG fai led, whereas OC produced a 4.1 percent over-design with a 1.1 percent constraint violation. CrayYMP CPU solution time ranged from 36.96 sec for SQP to 

144.11 sec for NPSOL. 

10 

z 

k x 

Figure 1 O.-One-hundred-sixty-five-ft tall 

antenna tower. 

Example P7: Geodesic Dome 

A geodesic dome (refs. 28 and 29), shown in figure 12 

with a diameterof240 in. and height of 30 in., was subj ected to 

a single load condition. It was modeled using 156 bars and 

96 tri angul ar membrane elements. The bars were made 

of a material with modulus E = 30 000 ksi, and density p = 

0.1 Ib/in.

3 Membranes were made of aluminum, with modulus 

E = 10 000 ksi, and density p = 0.1 Ib/i n.

3 The bar areas and 

membrane thicknesses were grouped to obtain eight and fo ur 

linked design variables, respectively. The dome had a total of 

254 constraints, (156 stresses for bars, 96 von Mises stresses for 

membranes, one displacement, and one frequency). The optimum weight obtained was 1022.67 lb with 170 active constraints, (168 stress constraints, one displacement, and one 

frequency (table I). Four optimizers (SUMT, SQP, IMSL, and 

NPSOL) converged with small deviations. Optimizers RG and 

OC failed. The Cray-YMP CPU time varied between 448.32 sec 

for SUMT to 548.36 sec for NPSOL. 

l 

Figure 9.-Twenty-five bar-truss. (Elements are circled, 

nodes are not.) 

(50 stress and 36 displacement). Fouroptimizers (SUMT, SQP, 

IMSL, and NPSOL) converged to an optimum weight of 

544.73 Ib with fo ur active displacement constraints (tables I 

and II). Optimizers RG and OC failed. Cray-YMP CPU time 

ranged from 1.64 sec for SQP to 6.15 sec for NPSOL. 

Example P5: 165-Ft-Tall Antenna Tower 

A 165-ft-tall steel antenna tower with 252 members, as 

depicted in figure 10, (ref. 26), had six linked design variables 

and was subjected to two load conditions. Its overhead dish 

antenna was modeled as a lumped mass for freq uency calcul ations. It had a total of 529 behavior constraints (504 stress, 24 

displ acement, and one frequency). Three optimizers (SQP, 

IMSL, and NPSOL) converged to an optimum solution of 

5299.84 lb with small deviations (table II). At optimum, six 

stress, 12 displacement, and one freq uency constraints were 

active. Optimizers RG and OC fa iled while SUMT produced a 

six percent under-design. The-Cray-YMP CPU time varied 

between 376.83 sec for SQP and 1893.80 sec for NPSOL. 

Example P6: 60-Bar Trussed Ring 

A 60-bar trussed aluminum ring (ref. 3) was subjected to 

three load conditions and had two lumped masses, as depicted 

in figure 11. It had a total of 184 constraints (180 stress, three 

displacement, one frequency) and 25 linked design variables. 

The optimum weight was 414.51 Ibs, and at optimum, 22 stress, 

one displacement, and one frequency constraints were active. 

Four optimizers (SUMT, SQP, IMSL, and NPSOL) converged 

(table II). Optimizer RG fai led, whereas OC produced a 4.1 percent over-design with a 1.1 percent constraint violation. CrayYMP CPU solution time ranged from 36.96 sec for SQP to 

144.11 sec for NPSOL. 

10 

z 

k x 

Figure 1 O.-One-hundred-sixty-five-ft tall 

antenna tower. 

Example P7: Geodesic Dome 

A geodesic dome (refs. 28 and 29), shown in figure 12 

with a diameterof240 in. and height of 30 in., was subj ected to 

a single load condition. It was modeled using 156 bars and 

96 tri angul ar membrane elements. The bars were made 

of a material with modulus E = 30 000 ksi, and density p = 

0.1 Ib/in.

3 Membranes were made of aluminum, with modulus 

E = 10 000 ksi, and density p = 0.1 Ib/i n.

3 The bar areas and 

membrane thicknesses were grouped to obtain eight and fo ur 

linked design variables, respectively. The dome had a total of 

254 constraints, (156 stresses for bars, 96 von Mises stresses for 

membranes, one displacement, and one frequency). The optimum weight obtained was 1022.67 lb with 170 active constraints, (168 stress constraints, one displacement, and one 

frequency (table I). Four optimizers (SUMT, SQP, IMSL, and 

NPSOL) converged with small deviations. Optimizers RG and 

OC failed. The Cray-YMP CPU time varied between 448.32 sec 

for SUMT to 548.36 sec for NPSOL. 
7 

4 

10 

o Location of lumped mass 

Figure 11.-Sixty-bar trussed ring. (Elements are circled, nodes are not.) 

11 

7 

4 

10 

o Location of lumped mass 

Figure 11.-Sixty-bar trussed ring. (Elements are circled, nodes are not.) 

11 
z 

~x 

Figure 12.-Geodesic dome. (All boundary nodes are fully restrained. Supports are shown for three 

sides only. Supports for other sides are not shown.) 

z 

x y 

o 

Figure 13.-lntermediate complexity wing. (Representative elements are circled, 

nodes are not.) 

Examples P8a to P8d: Intermediate Complexity Wing 

An intermediate complexity wing (refs. 3 and 18), shown in 

figure 13, was modeled with a total of 158 elements consisting 

of 39 bars, two triangular membranes, 62 quadrilateral membranes, and 55 shear panels. The wing is made of aluminum 

with modulus E = 10 500 ksi, and density p = 0.1 Ib/in.

3 The 

elements were grouped to obtain 57 linked design variables. 

The wing, which was subjected to two load conditions, had a 

total of 321 behavior constraints, (316 stress, four displacement, and one frequency). The optimum design fo r this problem was obtained from four different initial points, (1) initial 

design of unity; (2) initial design equal to the SUMT optimum 

12 

design; (3) initial design equal to 150 percent of the SUMT 

optimum design; and (4) initial design which is infeasible at 

50 percent lower than the SUMT optimum design. Results 

obtained for all fo ur cases are summarized in table II (P8a to 

P8d). The optimum design was 387 .76 1b and there were a total 

of 119 active constraints, (117 stress, one displacement, and 

one frequency). For initial design of unity (see table II, problem 

P8a), optimizers SUMT, SQP, IMSL, and NPSOL reached the 

optimum within a 3.7-percent error margin. Optimizer RG 

failed to solve the problem. Optimizer OC also failed to converge to the optimum (producing 20. I-percent over-design). 

Cray-YMP CPU time varied between 1075.21 sec for SQP to 

8292.35 sec for NPSOL. 

z 

~x 

Figure 12.-Geodesic dome. (All boundary nodes are fully restrained. Supports are shown for three 

sides only. Supports for other sides are not shown.) 

z 

x y 

o 

Figure 13.-lntermediate complexity wing. (Representative elements are circled, 

nodes are not.) 

Examples P8a to P8d: Intermediate Complexity Wing 

An intermediate complexity wing (refs. 3 and 18), shown in 

figure 13, was modeled with a total of 158 elements consisting 

of 39 bars, two triangular membranes, 62 quadrilateral membranes, and 55 shear panels. The wing is made of aluminum 

with modulus E = 10 500 ksi, and density p = 0.1 Ib/in.

3 The 

elements were grouped to obtain 57 linked design variables. 

The wing, which was subjected to two load conditions, had a 

total of 321 behavior constraints, (316 stress, four displacement, and one frequency). The optimum design fo r this problem was obtained from four different initial points, (1) initial 

design of unity; (2) initial design equal to the SUMT optimum 

12 

design; (3) initial design equal to 150 percent of the SUMT 

optimum design; and (4) initial design which is infeasible at 

50 percent lower than the SUMT optimum design. Results 

obtained for all fo ur cases are summarized in table II (P8a to 

P8d). The optimum design was 387 .76 1b and there were a total 

of 119 active constraints, (117 stress, one displacement, and 

one frequency). For initial design of unity (see table II, problem 

P8a), optimizers SUMT, SQP, IMSL, and NPSOL reached the 

optimum within a 3.7-percent error margin. Optimizer RG 

failed to solve the problem. Optimizer OC also failed to converge to the optimum (producing 20. I-percent over-design). 

Cray-YMP CPU time varied between 1075.21 sec for SQP to 

8292.35 sec for NPSOL. 
Discussion 

For the purpose of this discussion, the 41 problems of the test 

bed are grouped as small, medium, and large. For the small 

problems (Group I), the number of linked design variables 

ranged between three and 19. Group I contains a total of 

19 problems, which are designated as PIa to P5, P7, and P9 to 

PI8. The normalized optimum weight for small problems 

obtained by each optimizer is depicted in figure 2. For medium 

problems (Group II), the number of linked design variables 

ranged between 20 and 39. There are 12 medium problems, 

which are designated as P6, PI9 to P23, and P30 to P35. The 

normalized optimum weight for the medium problems obtained by each optimizer is illustrated in figure 3. Problems with 

more than 40 independent design variables are referred to as 

large problems (Group III). There are 10 large problems which 

are designated as P8a to P8d, and P24 to P29. The normalized 

optimum weight for large problems obtained by each optimizer 

is depicted in figure 4. 

The following discussion is divided into five categories: (1) 

convergence to the optimum weight, (2) number of active 

constraints at optimum, (3) Cray-YMP8E18128 CPU time 

required to solve the problem, (4) singularity in structural 

optimization, and (5) default optimization parameters. 

Convergence to Optimum Weight 

The normalized optimum weights for all 41 problems, obtained by the six optimizers are depicted in figures 2 to 4 for 

small, medium, and large problems, respectively. In these 

figures, unity represents optimum weight and more than unity 

indicates over-design, while less than unity is infeasible design. 

For the purpose of comparison, a solution with constraint 

violation of less than one percent and weight which is within 

one percent of the best feasible design is considered optimum. 

A design is acceptable when the constraint violation is less than 

one percent and the weight is within five percent of the 

minimum obtained by the eight optimizers. Convergence to the 

optimum solution for each of the six optimizers follows. 

(1 ) SUMT converged to optimum solution for 35 of 41 

examples, which consisted of 17 small, nine medium, and nine 

large problems. SUMT failed for four problems. These are: one 

small problem (P5), two medium problems (P21 and P23), and 

one large problem (P8b). For both medium problems, the 

SUMT solution was more than one percent infeasible. For the 

large problem, SUMT gave an under-design of more than five 

percent. 

(2) SQP of IDESIGN, successfully solved 32 of 41 

examples, which consisted of 15 small, 10 medium, and seven 

large problems. This optimizer failed to give a feasible optimum design for three small problems (PIS, P17, and PI8), two 

medium problems (P19 and P22), and three large problems 

(P8c, P8d, and P2S). 

(3) IMSL optimizer DNCONG successfully solved 37 of 41 

examples, which consisted of 17 small, 12 medium, and eight 

large problems. DNCONG of IMSL failed to optimize the 

intermediate complexity wing (problems P8c and P8d). 

(4) NPSOL successfully solved 25 of 41 examples, which 

consisted of 13 small, eight medium, and four large problems. 

This optimizer failed (with an infeasible design over one 

percent) for: four small problems (P15 to PI8); four medium 

problems (P23, P3I, P32, and P34); and four large problems 

(P8d, P25, P26, and P28). It produced more than five percent 

over-design for large problem P8c. 

(S) RG successfully solved 13 of 41 examples, which consisted of seven small, four medium, and two large problems. 

RG failed for 12 small problems. It also failed for seven 

medium problems and three large problems. The optimizer RG 

failed with weIl over 100 percent error in the optimum weight 

for 15 problems. 

(6) OC successfully solved 16 of 41 examples, which consisted of six small, five medium, and five large problems. OC 

failed fornine small, two medium, and five large problems with 

an error in the optimum weight exceeding fi ve percent, as weIl 

as for three medium problems with an infeasible design greater 

than one percent. 

Number of Active Constraints at Optimum 

The number of active constraints at the optimum for all 

examples is given in table I. Typically, differentoptimizers produced identical numbers of active frequency and active displacement constraints. However, the number of active stress 

constraints generated depended on the optimizer of choice. For 

example, with the geodesic dome problem (P7), the number of 

active stress constraints produced were 168 by SQP of IDE SIGN , 

162 by SUMT and NPSOL, and IS6 by IMSL. Consider also 

the set of five examples depicted in table III that failed to 

converge, which produced minimum weights between 3.2 to 

12.7 percent over- or under-designs. These examples produced 

correct numbers of displacement and frequency constraints, but 

failed to produce the correct numbers of active stress constraints. The deficiency in the number of active stress constraints ranged between three for problem P2 to 42 for problem 

P8a. For these problems the failure of the optimizers could be 

attributed to their inability to produce the correct number of 

active stress constraints. This aspect is also described in the 

section entitled, "Singularity in Structural Optimization" of 

this paper. 

CPU Time Required For the Solution 

The normalized CPU times on a Cray-YMP8E/8128 computer were recorded for a set of 14 examples. The normalization 

was with respect to SQP of IDESIGN except for problems 

P8c and P8d, which were normalized with respect to SUMT 

13 

Discussion 

For the purpose of this discussion, the 41 problems of the test 

bed are grouped as small, medium, and large. For the small 

problems (Group I), the number of linked design variables 

ranged between three and 19. Group I contains a total of 

19 problems, which are designated as PIa to P5, P7, and P9 to 

PI8. The normalized optimum weight for small problems 

obtained by each optimizer is depicted in figure 2. For medium 

problems (Group II), the number of linked design variables 

ranged between 20 and 39. There are 12 medium problems, 

which are designated as P6, PI9 to P23, and P30 to P35. The 

normalized optimum weight for the medium problems obtained by each optimizer is illustrated in figure 3. Problems with 

more than 40 independent design variables are referred to as 

large problems (Group III). There are 10 large problems which 

are designated as P8a to P8d, and P24 to P29. The normalized 

optimum weight for large problems obtained by each optimizer 

is depicted in figure 4. 

The following discussion is divided into five categories: (1) 

convergence to the optimum weight, (2) number of active 

constraints at optimum, (3) Cray-YMP8E18128 CPU time 

required to solve the problem, (4) singularity in structural 

optimization, and (5) default optimization parameters. 

Convergence to Optimum Weight 

The normalized optimum weights for all 41 problems, obtained by the six optimizers are depicted in figures 2 to 4 for 

small, medium, and large problems, respectively. In these 

figures, unity represents optimum weight and more than unity 

indicates over-design, while less than unity is infeasible design. 

For the purpose of comparison, a solution with constraint 

violation of less than one percent and weight which is within 

one percent of the best feasible design is considered optimum. 

A design is acceptable when the constraint violation is less than 

one percent and the weight is within five percent of the 

minimum obtained by the eight optimizers. Convergence to the 

optimum solution for each of the six optimizers follows. 

(1 ) SUMT converged to optimum solution for 35 of 41 

examples, which consisted of 17 small, nine medium, and nine 

large problems. SUMT failed for four problems. These are: one 

small problem (P5), two medium problems (P21 and P23), and 

one large problem (P8b). For both medium problems, the 

SUMT solution was more than one percent infeasible. For the 

large problem, SUMT gave an under-design of more than five 

percent. 

(2) SQP of IDESIGN, successfully solved 32 of 41 

examples, which consisted of 15 small, 10 medium, and seven 

large problems. This optimizer failed to give a feasible optimum design for three small problems (PIS, P17, and PI8), two 

medium problems (P19 and P22), and three large problems 

(P8c, P8d, and P2S). 

(3) IMSL optimizer DNCONG successfully solved 37 of 41 

examples, which consisted of 17 small, 12 medium, and eight 

large problems. DNCONG of IMSL failed to optimize the 

intermediate complexity wing (problems P8c and P8d). 

(4) NPSOL successfully solved 25 of 41 examples, which 

consisted of 13 small, eight medium, and four large problems. 

This optimizer failed (with an infeasible design over one 

percent) for: four small problems (P15 to PI8); four medium 

problems (P23, P3I, P32, and P34); and four large problems 

(P8d, P25, P26, and P28). It produced more than five percent 

over-design for large problem P8c. 

(S) RG successfully solved 13 of 41 examples, which consisted of seven small, four medium, and two large problems. 

RG failed for 12 small problems. It also failed for seven 

medium problems and three large problems. The optimizer RG 

failed with weIl over 100 percent error in the optimum weight 

for 15 problems. 

(6) OC successfully solved 16 of 41 examples, which consisted of six small, five medium, and five large problems. OC 

failed fornine small, two medium, and five large problems with 

an error in the optimum weight exceeding fi ve percent, as weIl 

as for three medium problems with an infeasible design greater 

than one percent. 

Number of Active Constraints at Optimum 

The number of active constraints at the optimum for all 

examples is given in table I. Typically, differentoptimizers produced identical numbers of active frequency and active displacement constraints. However, the number of active stress 

constraints generated depended on the optimizer of choice. For 

example, with the geodesic dome problem (P7), the number of 

active stress constraints produced were 168 by SQP of IDE SIGN , 

162 by SUMT and NPSOL, and IS6 by IMSL. Consider also 

the set of five examples depicted in table III that failed to 

converge, which produced minimum weights between 3.2 to 

12.7 percent over- or under-designs. These examples produced 

correct numbers of displacement and frequency constraints, but 

failed to produce the correct numbers of active stress constraints. The deficiency in the number of active stress constraints ranged between three for problem P2 to 42 for problem 

P8a. For these problems the failure of the optimizers could be 

attributed to their inability to produce the correct number of 

active stress constraints. This aspect is also described in the 

section entitled, "Singularity in Structural Optimization" of 

this paper. 

CPU Time Required For the Solution 

The normalized CPU times on a Cray-YMP8E/8128 computer were recorded for a set of 14 examples. The normalization 

was with respect to SQP of IDESIGN except for problems 

P8c and P8d, which were normalized with respect to SUMT 

13 
TABLE llI.-FOUR EXAMPLES THAT FAILED TO REACH OPTIMUM WEIGHT 

VERSUS BEST FEASIBLE DESIGN 

Problem Optimization Percent Number of active constraints at 

number method over-design optimum versus best feasible design 

Frequency Stress Displacement 

P2 OC 56 I 5 2 

vs. SQP 0 I 8 2 

P6 RG 83.2 I 18 I 

vs. SQP 0 I 21 I 

P8a NPSOL 31 I 75 1 

vs. IMSL 0 I 117 I 

P8c IMSL 7.7 I 88 I 

vs. SUMT 0 I 109 I 

TABLE IV.-PROBLEMS WITH ACTIVE CONSTRAINTS EXCEEDING 

THE NUMBER OF DESIGN VARIABLES 

[SO mgu Iamy · can occur In eac h 0 f t h ese problems.] 

Problem Description Number of Number of active 

number design constraints 

variables at optimum 

P2 Tapered ten-bar truss 10 II 

P5 Antenna tower 6 20 

P7 Geodesic dome 12 170 

P8a Intermediate complexity wing 57 119 

P9 Ten-bar truss 10 II 

PIO Ten-bar truss 5 9 

P17 Cantilever membrane 16 19 

PI8 Cantilever membrane 16 35 

PI9 Sixty-bar trussed ring 25 38 

P22 Sixty-bar trussed ring 25 30 

P24 Stiffened ring 49 75 

P27 Stiffened sixty-bar trussed ring 49 76 

P30 Stiffened ring 

P33 Stiffened ring 

(table II and fig. 5). CPU time differed among optimizers. Even 

for a small problem (P 1 a), nonnalized CPU time differed from 

1.0 for SQP to 22.069 for RG. For a medium problem (P6), 

normalized time differed between 1.0 for SQP to 3.899 for 

NPSOL. For a large problem (P8a), normalized CPU time 

varied from 1.695 for IMSL to 1.116 for SUMT and 1.000 for 

SQP of IDE SIGN . We observed that variation in CPU time was 

rather mild for large problems. 

Singularity in Structural Optimization 

Singularity was identified for three situations (refs. 3 and 

30): 

14 

24 28 

24 28 

(l ) the number of active constraints exceed the number of 

design variables. Out of the 41 problems the 14 examples listed 

in table IV are prone to this type of singularity. 

(2) linear functional dependencies among a small number of 

active stress constraints. This type of singularity is suspected to 

have occurred for some of the examples given in table III. 

(3) linear functional dependencies among a small number of 

active stress and displacement constraints. The identification of 

this type of singularity by mere inspection may be difficult. 

Singularity alleviation as discussed in references 8, 26, and 

27 can reduce computation and improve reliabili ty of optimizers 

(fig. 14). 

TABLE llI.-FOUR EXAMPLES THAT FAILED TO REACH OPTIMUM WEIGHT 

VERSUS BEST FEASIBLE DESIGN 

Problem Optimization Percent Number of active constraints at 

number method over-design optimum versus best feasible design 

Frequency Stress Displacement 

P2 OC 56 I 5 2 

vs. SQP 0 I 8 2 

P6 RG 83.2 I 18 I 

vs. SQP 0 I 21 I 

P8a NPSOL 31 I 75 1 

vs. IMSL 0 I 117 I 

P8c IMSL 7.7 I 88 I 

vs. SUMT 0 I 109 I 

TABLE IV.-PROBLEMS WITH ACTIVE CONSTRAINTS EXCEEDING 

THE NUMBER OF DESIGN VARIABLES 

[SO mgu Iamy · can occur In eac h 0 f t h ese problems.] 

Problem Description Number of Number of active 

number design constraints 

variables at optimum 

P2 Tapered ten-bar truss 10 II 

P5 Antenna tower 6 20 

P7 Geodesic dome 12 170 

P8a Intermediate complexity wing 57 119 

P9 Ten-bar truss 10 II 

PIO Ten-bar truss 5 9 

P17 Cantilever membrane 16 19 

PI8 Cantilever membrane 16 35 

PI9 Sixty-bar trussed ring 25 38 

P22 Sixty-bar trussed ring 25 30 

P24 Stiffened ring 49 75 

P27 Stiffened sixty-bar trussed ring 49 76 

P30 Stiffened ring 

P33 Stiffened ring 

(table II and fig. 5). CPU time differed among optimizers. Even 

for a small problem (P 1 a), nonnalized CPU time differed from 

1.0 for SQP to 22.069 for RG. For a medium problem (P6), 

normalized time differed between 1.0 for SQP to 3.899 for 

NPSOL. For a large problem (P8a), normalized CPU time 

varied from 1.695 for IMSL to 1.116 for SUMT and 1.000 for 

SQP of IDE SIGN . We observed that variation in CPU time was 

rather mild for large problems. 

Singularity in Structural Optimization 

Singularity was identified for three situations (refs. 3 and 

30): 

14 

24 28 

24 28 

(l ) the number of active constraints exceed the number of 

design variables. Out of the 41 problems the 14 examples listed 

in table IV are prone to this type of singularity. 

(2) linear functional dependencies among a small number of 

active stress constraints. This type of singularity is suspected to 

have occurred for some of the examples given in table III. 

(3) linear functional dependencies among a small number of 

active stress and displacement constraints. The identification of 

this type of singularity by mere inspection may be difficult. 

Singularity alleviation as discussed in references 8, 26, and 

27 can reduce computation and improve reliabili ty of optimizers 

(fig. 14). 
400 

:B 300 ~ 

.c 

Cl 'w 

~ 200 

217.olb- / 

/ 

/ 

00 L(~~----~----~-----L----~----~ 

400 

,Q 300 ~ 

.c 

Cl 

'iii 

~ 200 

o 10 20 30 40 50 

Number of iterations 

~ 217.0 Ib / 

1o ~(b_)----~----~----~----~--~ 

o 10 20 30 40 50 

Number of iterations 

Figure 14.-Convergence characteristics of a three-bar 

truss, showing the merit function (weight) versus the 

number of iterations (breaks in the graph indicate weight 

up to the order of 1 ()5lb). (a) Solution when singularity is 

disregarded. (b) Solution when singularity is alleviated. 

Default Optimization Parameters 

Default parameters (such as convergence criteria, step length, 

stop criteria, active constraint region, iteration limitations, etc.) 

specified by individual optimization codes, were used to solve 

the problems. When a problem failed, the default parameters 

were changed in an attempt to successfully solve the problem. 

In the solution of the 41 test bed problems, it was necessary to 

change the default optimization parameters quite often in order 

to reach the correct solution. On an overall basis, default 

parameters of SUMT, SQP, and IMSL algorithms were adequate for the solution of most problems. Most of the default 

parameters for RG and NPSOL were changed to improve their 

performances. 

Concluding Remarks 

None of the optimizers could successfully solve all the 

problems. Most optimizers, however, can solve at least one 

third of the examples. For large problems, the Cray-YMP CPU 

time was comparable among the optimizers that succeeded. 

Alleviation of singularity can improve the optimizer efficiency. 

A single winner which can be called most reliable and 

efficient could not be identified. Overall, three optimizers 

(IMSL, SUMT, and SQP of IDE SIGN) scored high marks. For 

small problems, four optimizers (IMSL, SUMT, SQP of 

IDESIGN, SLP, and NPSOL) satisfactorily solved more than 

fifty percent of the problems. For medium problems, four 

optimizers (IMSL, SQP of IDESIGN, SUMT, and NPSOL) 

produced correct solutions for at least half of the problems. For 

large problems three optimizers (IMSL, SUMT, and SQP of 

IDESIGN) were found to be reliable and efficient. 

References 

I. Guptill. 1.D., et al.: CometBoards User's Manual, NASA TM-4537, 1996. 

2. Vanderplaats, G.N.: DOT User's Manual, Version 2.04. VMA Engineering, 1989. 

3. Patnaik, S.N.; Guptill, J.D.; and Berke, L.: Merits and Limitations of 

Optimality Criteria Method for Structural Optimization. NASA 

TP-3373, 1993. 

4. Belegundu, A., et al.: Multiobjective Optimization of Laminated Ceramic 

Composites Using Genetic Algorithms. AlAAfUSAFfNASAlISSMO 

Proceedings of Symposium on Multidisciplinary Analysis and Optimization, 1994. pp. 1015-1022. 

5. Lasdon. L.; and Waren, A.D.: GRG2 User's Guide, University of Texas at 

Auston, Texas, 1986. 

6. IMSL User's Manual: MATHILIBRARY: FORTRAN Subroutines for 

Mathematical Applications. Vol. 3, chapt. 8, 1987, p. 903. 

7. Belegundu, A.D.; Berke. L.; and Patnaik, S.N.: An Optimization Algorithm Based on the Method of Feasible Directions. Structural Optimization 

J., vol. 9, 1995, pp. 83-88. 

8. NAG FORTRAN Library Manual, MARK 15: E04UCF - NAG Fortran 

Library Routine Document, vol. 4, 1991. 

9. Gabriele, G.A.; and Ragsdell, K.M.: OPT - A Nonlinear Programming 

Code in FORTRAN Implementing the Generalized Reduced Gradient 

Method, User's Manual. University of Missouri-Columbia. 1984. 

10. Arora, J.S.: IDESIGN User's Manual, Version 3.S.2. Optimal Design 

Laboratory, University of Iowa.lune 1989. 

II . Miura, H.; and Schmit. lr .. L.A.: NEWSUMT: A FORTRAN Program for 

Inequality Constrained Function Minimization, User's Guide. (NASA 

Contract NGR-OS-OO7-337.) ASA CR-159070. 1979. 

12. Patnaik, et al.: A Cascade Optimization Strategy for Solution of Difficult 

Multidisciplinary Design Problems. AIAA-96-1 628, 1996. 

13 Thanedar, P.B .. et a1.: Perfonnance of Some SQP Algorithms on Structural 

Design Problems-Sequential Quadratic Programming. In!. 1. Numer. 

Meth. Eng., vol. 23. Dec. 1986. pp. 2187-2203. 

14. Tseng, C.H.; and Arora, 1.S.: On Implementation of Computational 

Algorithms for Optimal Design 1: Preliminary Investigation. In!. 1. 

umer. Meth. Eng .. vol. 26. 1988. pp. 1365-1382. 

15. Tseng. C.H.: and Arora, J.S.: On Implementation of Computational 

Algorithms for Optimal Design 2: Extensive Numerical Investigation. 

Int. J. umer. Meth. Eng., vol. 26.1988, pp. 1383-1402. 

16. Schittkowski, K.: A umerical Comparison of 13 Nonlinear Programming 

Codes With Randomly Generated Test Problems. Numer. Optimization 

Dynam. Syst., orth-Holland Publishing Co .• Amsterdam, 1980, 

pp.2 13-234. 

17. Hock, W.; and Schitlkowski. K.: A Comparative Perfonnance Evaluation 

of 27 Nonlinear Programming Codes. CO, vol. 30. 1983, pp. 335-358. 

15 

400 

:B 300 ~ 

.c 

Cl 'w 

~ 200 

217.olb- / 

/ 

/ 

00 L(~~----~----~-----L----~----~ 

400 

,Q 300 ~ 

.c 

Cl 

'iii 

~ 200 

o 10 20 30 40 50 

Number of iterations 

~ 217.0 Ib / 

1o ~(b_)----~----~----~----~--~ 

o 10 20 30 40 50 

Number of iterations 

Figure 14.-Convergence characteristics of a three-bar 

truss, showing the merit function (weight) versus the 

number of iterations (breaks in the graph indicate weight 

up to the order of 1 ()5lb). (a) Solution when singularity is 

disregarded. (b) Solution when singularity is alleviated. 

Default Optimization Parameters 

Default parameters (such as convergence criteria, step length, 

stop criteria, active constraint region, iteration limitations, etc.) 

specified by individual optimization codes, were used to solve 

the problems. When a problem failed, the default parameters 

were changed in an attempt to successfully solve the problem. 

In the solution of the 41 test bed problems, it was necessary to 

change the default optimization parameters quite often in order 

to reach the correct solution. On an overall basis, default 

parameters of SUMT, SQP, and IMSL algorithms were adequate for the solution of most problems. Most of the default 

parameters for RG and NPSOL were changed to improve their 

performances. 

Concluding Remarks 

None of the optimizers could successfully solve all the 

problems. Most optimizers, however, can solve at least one 

third of the examples. For large problems, the Cray-YMP CPU 

time was comparable among the optimizers that succeeded. 

Alleviation of singularity can improve the optimizer efficiency. 

A single winner which can be called most reliable and 

efficient could not be identified. Overall, three optimizers 

(IMSL, SUMT, and SQP of IDE SIGN) scored high marks. For 

small problems, four optimizers (IMSL, SUMT, SQP of 

IDESIGN, SLP, and NPSOL) satisfactorily solved more than 

fifty percent of the problems. For medium problems, four 

optimizers (IMSL, SQP of IDESIGN, SUMT, and NPSOL) 

produced correct solutions for at least half of the problems. For 

large problems three optimizers (IMSL, SUMT, and SQP of 

IDESIGN) were found to be reliable and efficient. 

References 

I. Guptill. 1.D., et al.: CometBoards User's Manual, NASA TM-4537, 1996. 

2. Vanderplaats, G.N.: DOT User's Manual, Version 2.04. VMA Engineering, 1989. 

3. Patnaik, S.N.; Guptill, J.D.; and Berke, L.: Merits and Limitations of 

Optimality Criteria Method for Structural Optimization. NASA 

TP-3373, 1993. 

4. Belegundu, A., et al.: Multiobjective Optimization of Laminated Ceramic 

Composites Using Genetic Algorithms. AlAAfUSAFfNASAlISSMO 

Proceedings of Symposium on Multidisciplinary Analysis and Optimization, 1994. pp. 1015-1022. 

5. Lasdon. L.; and Waren, A.D.: GRG2 User's Guide, University of Texas at 

Auston, Texas, 1986. 

6. IMSL User's Manual: MATHILIBRARY: FORTRAN Subroutines for 

Mathematical Applications. Vol. 3, chapt. 8, 1987, p. 903. 

7. Belegundu, A.D.; Berke. L.; and Patnaik, S.N.: An Optimization Algorithm Based on the Method of Feasible Directions. Structural Optimization 

J., vol. 9, 1995, pp. 83-88. 

8. NAG FORTRAN Library Manual, MARK 15: E04UCF - NAG Fortran 

Library Routine Document, vol. 4, 1991. 

9. Gabriele, G.A.; and Ragsdell, K.M.: OPT - A Nonlinear Programming 

Code in FORTRAN Implementing the Generalized Reduced Gradient 

Method, User's Manual. University of Missouri-Columbia. 1984. 

10. Arora, J.S.: IDESIGN User's Manual, Version 3.S.2. Optimal Design 

Laboratory, University of Iowa.lune 1989. 

II . Miura, H.; and Schmit. lr .. L.A.: NEWSUMT: A FORTRAN Program for 

Inequality Constrained Function Minimization, User's Guide. (NASA 

Contract NGR-OS-OO7-337.) ASA CR-159070. 1979. 

12. Patnaik, et al.: A Cascade Optimization Strategy for Solution of Difficult 

Multidisciplinary Design Problems. AIAA-96-1 628, 1996. 

13 Thanedar, P.B .. et a1.: Perfonnance of Some SQP Algorithms on Structural 

Design Problems-Sequential Quadratic Programming. In!. 1. Numer. 

Meth. Eng., vol. 23. Dec. 1986. pp. 2187-2203. 

14. Tseng, C.H.; and Arora, 1.S.: On Implementation of Computational 

Algorithms for Optimal Design 1: Preliminary Investigation. In!. 1. 

umer. Meth. Eng .. vol. 26. 1988. pp. 1365-1382. 

15. Tseng. C.H.: and Arora, J.S.: On Implementation of Computational 

Algorithms for Optimal Design 2: Extensive Numerical Investigation. 

Int. J. umer. Meth. Eng., vol. 26.1988, pp. 1383-1402. 

16. Schittkowski, K.: A umerical Comparison of 13 Nonlinear Programming 

Codes With Randomly Generated Test Problems. Numer. Optimization 

Dynam. Syst., orth-Holland Publishing Co .• Amsterdam, 1980, 

pp.2 13-234. 

17. Hock, W.; and Schitlkowski. K.: A Comparative Perfonnance Evaluation 

of 27 Nonlinear Programming Codes. CO, vol. 30. 1983, pp. 335-358. 

15 
18. Canfield, R.A.; Gran<ihi, R.V.; and VenJeayya. V.B.: Optimum Design of 

Structures With Multiple Constraints. AIAA J., vol. 26, Jan. 1988. 

pp. 78-35 . 

19. Neill, D.l.; Johnson, E.H.; Herendeen, D.L.: Automated Structural Optimization System (ASTROS). AFWAL-TR-88-3028, Wright Patterson 

Air Force Base vol. II - User's Manual, April 1988. 

20. Ragsdell. K.M.: A Survey of Some Useful Optimization Methods, Proceedings of the Design Engineering Technical Conference, A.H. Soni 

and H. Atmaran, eds., New York 1974. pp. 129-135. 

21. Ragsdell, K.M.: The Evaluation of Optimization Software for Engineering 

Design. Lecture Notes in Economics and Mathematical Sys., SpringerVerlag, vol. 199, New York 1982. 

22. Pshenichny, B.: Algorithms for the General Problem of Mathematical 

Programming, Kibemetica, no. 5, 1970, pp. 120-125. 

23. Schittkowski, K.: NLPQL: A FORTRAN Subroutine Solving Constrained 

Nonlinear Programming Problems. Ann. Oper. Res .. vol. 5, 1986, 

pp. 485-500. 

24. Venkayya, V.B.; and Tischler, V.A.: ANALYZE: Analysis of Aerospace 

Structures With Membrane Elements. Tech. Rep. AFFDL-TR-78-170, 

1978. 

16 

25. Patnaik, S.N.; and Gallagher, R.H.: Gradients of Behaviour Constraints 

and Reanalysis Via the Integrated Force Method. lnt. J. Numer. Meth. 

Eng., vol. 23,1986, pp. 2205-2212. 

26. Patnaik, S.N.; and Srivastava, N.K.: On Automated Optimum Design of 

Trusses. Compo Methods Appl. Mech. Eng., 1976, pp. 245-265. 

27. Vanderplaats, G.N.; and Moses. F.: Automated Design of Trusses for 

Optimum Geometry. J. Struct. Div. Amer. Soc. Civil Eng. Proc., vol. 98, 

1972, pp. 671-690. 

28. Berke, L; and Khot, N.S.: Use of Optimality Criteria Methods for Large 

Scale Systems. Air Force Flight Dynamics Lab .• Wright-Patterson Air 

Force Base, 1974. 

29. Khot, N.S.: Optimization of Structures for Strength and Stability Requirements. Air Force Flight Dynamics Lab .. Wright-Patterson Air Force 

Base, 1973. 

30. Patnaik, S.N.; Guptill, J.D.; and Berke, L.: Singularity in Structural 

Optimization. Int. J. Num. Meth. Eng., vol. 36.1993, pp. 931-944. 

31. Gendy, A.S., etal.: LargeScale Structural Optimization with Substructuring 

in a Parallel Computational Environment. OAIIOSCINASA Proceedings of Symposium on Application of Parallel and Distributed Computing, 

1994, pp. 89-105. 

18. Canfield, R.A.; Gran<ihi, R.V.; and VenJeayya. V.B.: Optimum Design of 

Structures With Multiple Constraints. AIAA J., vol. 26, Jan. 1988. 

pp. 78-35 . 

19. Neill, D.l.; Johnson, E.H.; Herendeen, D.L.: Automated Structural Optimization System (ASTROS). AFWAL-TR-88-3028, Wright Patterson 

Air Force Base vol. II - User's Manual, April 1988. 

20. Ragsdell. K.M.: A Survey of Some Useful Optimization Methods, Proceedings of the Design Engineering Technical Conference, A.H. Soni 

and H. Atmaran, eds., New York 1974. pp. 129-135. 

21. Ragsdell, K.M.: The Evaluation of Optimization Software for Engineering 

Design. Lecture Notes in Economics and Mathematical Sys., SpringerVerlag, vol. 199, New York 1982. 

22. Pshenichny, B.: Algorithms for the General Problem of Mathematical 

Programming, Kibemetica, no. 5, 1970, pp. 120-125. 

23. Schittkowski, K.: NLPQL: A FORTRAN Subroutine Solving Constrained 

Nonlinear Programming Problems. Ann. Oper. Res .. vol. 5, 1986, 

pp. 485-500. 

24. Venkayya, V.B.; and Tischler, V.A.: ANALYZE: Analysis of Aerospace 

Structures With Membrane Elements. Tech. Rep. AFFDL-TR-78-170, 

1978. 

16 

25. Patnaik, S.N.; and Gallagher, R.H.: Gradients of Behaviour Constraints 

and Reanalysis Via the Integrated Force Method. lnt. J. Numer. Meth. 

Eng., vol. 23,1986, pp. 2205-2212. 

26. Patnaik, S.N.; and Srivastava, N.K.: On Automated Optimum Design of 

Trusses. Compo Methods Appl. Mech. Eng., 1976, pp. 245-265. 

27. Vanderplaats, G.N.; and Moses. F.: Automated Design of Trusses for 

Optimum Geometry. J. Struct. Div. Amer. Soc. Civil Eng. Proc., vol. 98, 

1972, pp. 671-690. 

28. Berke, L; and Khot, N.S.: Use of Optimality Criteria Methods for Large 

Scale Systems. Air Force Flight Dynamics Lab .• Wright-Patterson Air 

Force Base, 1974. 

29. Khot, N.S.: Optimization of Structures for Strength and Stability Requirements. Air Force Flight Dynamics Lab .. Wright-Patterson Air Force 

Base, 1973. 

30. Patnaik, S.N.; Guptill, J.D.; and Berke, L.: Singularity in Structural 

Optimization. Int. J. Num. Meth. Eng., vol. 36.1993, pp. 931-944. 

31. Gendy, A.S., etal.: LargeScale Structural Optimization with Substructuring 

in a Parallel Computational Environment. OAIIOSCINASA Proceedings of Symposium on Application of Parallel and Distributed Computing, 

1994, pp. 89-105. 
REPORT DOCUMENTATION PAGE 

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October 1996 Technical Memorandum 

4_ TITLE AND SUBTITLE 5_ FUNDING NUMBERS 

Performance Trend of Different Algorithms for Structural Design Optimization 

6. AUTHOR(S) WU - 505-63-5B 

Surya N. Patnaik, Rula M. Coroneos, James D. Guptill, and Dale A. Hopkins 

7_ PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8_ PERFORMING ORGANIZATION 

REPORT NUMBER 

National Aeronautics and Space Administration 

Lewis Research Center E-9690 

Cleveland, Ohio 44135-3191 

9_ SPONSORINGIMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING 

AGENCY REPORT NUMBER 

National Aeronautics and Space Administration 

Washington, D_C. 20546-0001 NASA TM-4698 

11. SUPPLEMENTARY NOTES 

Surya N. Patnaik, Ohio Aerospace Institute, 22800 Cedar Point Road, Cleveland, Ohio 44142 and NASA Resident 

Research Associate at Lewis Research Center; Rula M. Coroneos, James D_ Guptill, and Dale A. Hopkins, NASA Lewis 

Research Center. Responsible person, Surya N_ Patnaik, organization code 5210, (216) 962-3135. 

12a_ DISTRIBUTION/AVAILABILITY STATEMENT 12b_ DISTRIBUTION CODE 

Unclassified -Unlimited 

Subject Category 39 

This publication is available from the NASA Center for Aerospace Information, (301) 621-0390. 

13_ ABSTRACT (Maximum 200 words) 

Nonlinear programming algorithms play an important role in structural design optimization. Fortunately, several algorithms with computer codes are available_ At NASA Lewis Research Center, a project was initiated to assess performance 

of different optimizers through the development of a computer code CometBoards. This paper summarizes the conclusions of that research. CometBoards was employed to solve sets of small, medium and large structural problems, using 

different optimizers on a Cray-YMP8E/8128 computer. The reliability and efficiency of the optimizers were determined 

from the performance of these problems. For small problems, the performance of most of the optimizers could be 

considered adequate. For large problems however, three optimizers (two sequential quadratic programming routines, 

DNCONG of IMSL and SQP of IDESIGN, along with the sequential unconstrained minimi zations technique SUMT) 

outperformed others. At optimum, most optimizers captured an identical number of active displacement and frequency 

constraints but the number of active stress constraints differed among the optimizers. This discrepancy can be attributed to 

singularity conditions in the optimization and the alleviation of this discrepancy can improve the efficiency of optimizers. 

14_ SUBJECT TERMS 15. NUMBER OF PAGES 

Nonlinear programming algorithms; Structural design optimization; Sequential uncon- 19 

strained minimization; Quadratic programming; Reduced gradient; Feasible directions; 16_ PRICE CODE 

Singularity condition A03 

17_ SECURITY CLASSIFICATION 18_ SECURITY CLASSIFICATION 19_ SECURfTY CLASSIFICATION 20. LIMITATION OF ABSTRACT 

OF REPORT OFTHIS PAGE OF ABSTRACT 

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Prescribed by ANSI Std. Z39-1B 

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REPORT DOCUMENTATION PAGE 

Form Approved 

OMB No. 0704-0188 

Public. reporting ~rch~": for this collection of information ~ estimated. to .average 1 hour per respo ns~ . including the time for reviewing instructions, searching existing data sources. 

gathering and malnta ",n~ the data needed, and completIng and revIewIng the collection of information. Send comments regarding this burden estimate or any other aspect of this collE7ct'o~ of IOform~tlon , IOcludJ."g suggestions for reducing this burde~ , to Washmgton Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson 

DavIs H'ghway, SUIte 1204. Arlington. VA 22202-4302, and to the OffIce of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503. 

L AGENCY USE ONLY (Leave blank) rREPORT DATE 13

_ REPORT TYPE AND DATES COVERED 

October 1996 Technical Memorandum 

4_ TITLE AND SUBTITLE 5_ FUNDING NUMBERS 

Performance Trend of Different Algorithms for Structural Design Optimization 

6. AUTHOR(S) WU - 505-63-5B 

Surya N. Patnaik, Rula M. Coroneos, James D. Guptill, and Dale A. Hopkins 

7_ PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8_ PERFORMING ORGANIZATION 

REPORT NUMBER 

National Aeronautics and Space Administration 

Lewis Research Center E-9690 

Cleveland, Ohio 44135-3191 

9_ SPONSORINGIMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING 

AGENCY REPORT NUMBER 

National Aeronautics and Space Administration 

Washington, D_C. 20546-0001 NASA TM-4698 

11. SUPPLEMENTARY NOTES 

Surya N. Patnaik, Ohio Aerospace Institute, 22800 Cedar Point Road, Cleveland, Ohio 44142 and NASA Resident 

Research Associate at Lewis Research Center; Rula M. Coroneos, James D_ Guptill, and Dale A. Hopkins, NASA Lewis 

Research Center. Responsible person, Surya N_ Patnaik, organization code 5210, (216) 962-3135. 

12a_ DISTRIBUTION/AVAILABILITY STATEMENT 12b_ DISTRIBUTION CODE 

Unclassified -Unlimited 

Subject Category 39 

This publication is available from the NASA Center for Aerospace Information, (301) 621-0390. 

13_ ABSTRACT (Maximum 200 words) 

Nonlinear programming algorithms play an important role in structural design optimization. Fortunately, several algorithms with computer codes are available_ At NASA Lewis Research Center, a project was initiated to assess performance 

of different optimizers through the development of a computer code CometBoards. This paper summarizes the conclusions of that research. CometBoards was employed to solve sets of small, medium and large structural problems, using 

different optimizers on a Cray-YMP8E/8128 computer. The reliability and efficiency of the optimizers were determined 

from the performance of these problems. For small problems, the performance of most of the optimizers could be 

considered adequate. For large problems however, three optimizers (two sequential quadratic programming routines, 

DNCONG of IMSL and SQP of IDESIGN, along with the sequential unconstrained minimi zations technique SUMT) 

outperformed others. At optimum, most optimizers captured an identical number of active displacement and frequency 

constraints but the number of active stress constraints differed among the optimizers. This discrepancy can be attributed to 

singularity conditions in the optimization and the alleviation of this discrepancy can improve the efficiency of optimizers. 

14_ SUBJECT TERMS 15. NUMBER OF PAGES 

Nonlinear programming algorithms; Structural design optimization; Sequential uncon- 19 

strained minimization; Quadratic programming; Reduced gradient; Feasible directions; 16_ PRICE CODE 

Singularity condition A03 

17_ SECURITY CLASSIFICATION 18_ SECURITY CLASSIFICATION 19_ SECURfTY CLASSIFICATION 20. LIMITATION OF ABSTRACT 

OF REPORT OFTHIS PAGE OF ABSTRACT 

Unclassified Unclassified Unclassified 

NSN 7540-01-280-5500 Sta nda rd Form 298 (Rev. 2-89) 

Prescribed by ANSI Std. Z39-1B 

298-102