An investigation into using the Kreisselmeier-Steinhauser (KS) function in Simultaneous Analysis and Design (SAND) was performed in an effort to improve the performance and decrease the computational effort required for the minimum weight optimization of structural systems subjected to simultaneous strength, displacement, and size side constraints. Use of the KS function decreases the number of constraints in an optimization by combining them into a small number of constraint functions or even into a single constraint function, thereby reducing the number of constraint gradient evaluations and improving the performance of SAND. Three different formulations were tried in which the linear and nonlinear constraint equations were combined in various ways using the KS function. The standard three-bar, ten-bar, and seventy-two bar trusses were analyzed and optimized using these formulations. Specific and general programs were coded to perform SAND optimizations on these structures using the KS formulations. All programs utilized the optimizer NPSOL implemented on a SUN-SPARC IPX to optimize the respective structures. For the evaluated truss structures, the use of Kreisselmeier-Steinhauser functions in SAND decreased the required workspace for all cases and the computer time for some cases while yielding excellent accuracy.

Simultaneous Analysis and Design (SAND) was first suggested by Fox and Schmit² in the 1960?s to analyze and optimize a given structure in a single step, using the displacements as design variables and the global stiffness equations as nonlinear equality constraints. They used the conjugate gradient (CG) minimization technique for solving linear analysis problems. The CG method was not very effective in dealing with the equality equilibrium constraints generated from the ill-conditioned stiffness matrices formed during the finite element modeling of the structure. Haftka⁸ used a preconditioned conjugate gradient technique and element-by-element formulations in SAND. His research showed that the speed of convergence can be improved by using these formulations. In the 1980?s, Smaoui and Schmit¹⁰ worked on optimization of geometrically nonlinear structures using a generalized gradient algorithm.

Ringertz¹¹ conducted research on using the sequential quadratic programming method for optimization of structures with nonlinear response. He used two different formulations for the equilibrium constraints.

In 1996, Striz, Wu, and Sobieszczanski-Sobieski⁶ developed SAND-MM and SAND-LMM approaches to optimize structures. SAND-MM is a mixed method SAND approach which uses the element stiffness equations as nonlinear equality constraints instead of the global stiffness equations, in conjunction with the linear nodal force equilibrium equations. This method adds the element forces as variables to the system. SAND-LMM (local mixed method) is a variation of the SAND-MM method. It applies the element equilibrium equations in their local element form as nonlinear equality constraints and uses the nodal displacements for each element as design variables rather than the global displacements. This approach requires the displacements for all elements to be equal at a nodal point, resulting in additional linear constraint equations. Execution times for the optimization of large truss structures using the SAND-MM and SAND-LMM methods were longer when compared to the SAND method. This was due to the increase in the number of constraint gradient evaluations. The investigators suggested the use of the Kreisselmeier-Steinhauser function approach in order to decrease the computational effort by reducing the constraint system to a smaller set or even to a single cumulative equation. The present research represents a continuation of this work and uses the Kreisselmeier-Steinhauser function in an effort to investigate a possible reduction in the computational effort required for solving structural optimization problems using the SAND approach.

The structures under consideration are modeled by the finite element method. They are optimized for minimum weight subject to simultaneous stress, displacement, and size side constraints. The stress constraints are framed in terms of the displacement design variables. The global stiffness equations are used as nonlinear equality constraints.

Simultaneous Analysis And Design Formulation

In Simultaneous Analysis and Design, the element sizes along with the displacements are used as design variables. Here, structures are optimized for minimum weight. This objective function is expressed in terms of the design variables. The structure is subjected to linear and nonlinear constraints. The stress constraints are framed in terms of the displacement design variables and are linear in nature. The global stiffness equations generated during the finite element modeling of the structure are used as nonlinear equality constraints. Side constraints are imposed on the member sizes and nodal displacements (design variables).

Analyzing any structure using the finite element method generates the following equilibrium equations:

where [K] is the global stiffness matrix for the structure, written in terms of the member sizes, {U} is the displacement vector, and {F} is the external force vector.

The following mathematical formulation is used for the structural design problem of truss structures using the SAND approach:

Optimize {X} such that the weight of the structure is minimized subject to stress and equilibrium constraints, i.e. :

Minimize

W = g _iX_iL_i (2)

subject to

where

{X} is the set of design variables (bar areas and nodal displacements)
L_i is the length of bar element i
bl_k is the lower limit on design variable k
bu_k is the upper limit on design variable k
nelm is the total number of bar elements in the structure
g _i is the weight density of bar element i
i represents bar areas
j represents displacements
k represents bar areas and displacements
l represents free degrees of freedom
s _l is the lower limit of the stress in bar element i
s _u is the upper limit of the stress in bar element i
H_l(X) are the global stiffness constraints associated with the free degrees of freedom

KS functions are used to reduce the number of constraints in the optimization to a small number of constraints or even to a single constraint by closely and smoothly approximating the constraint envelope. Three different formulations have been considered in which the linear and nonlinear constraints have been combined in various ways in KS functions as discussed below.

Kreisselmeier-Steinhauser Formulation 1

This formulation can be considered as an extension to the SAND approach. It aims at improving the SAND methodology by reducing the number of linear equations (stress constraints) by combining them into a single nonlinear inequality equation. When the

stress inequality constraints are combined into a single constraint using the KS function, the following equation is obtained:

(6)

where
s (X_i) £ 0 are the stress constraints
{X} is the set of design variables (bar areas and
nodal displacements)
r is the KS tolerance parameter
nelm is the number of elements in the structure

The mathematical formulation used for this approach can be written as follows:

Optimize {X} such that the weight of the structure is minimized subject to the equilibrium and KS constraints or :

Minimize

W = g _iX_iL_i (7)

where KS(s (X_i)) is the KS function obtained by combining the linear constraint equations and H_l(X) are the global stiffness constraints associated with the unrestrained degrees of freedom.

In this formulation, the number of linear constraint equations is reduced to zero. On the other hand, the number of nonlinear constraints is increased by one. Overall, the total number of constraint equations used in this formulation is less than the number of constraint equations used in the standard SAND approach.

Kreisselmeier-Steinhauser Formulation 2

This formulation uses the Kreisselmeier-Steinhauser function to combine the global stiffness equations into one equation. When these nonlinear equality constraints are combined together using the KS function, the following equation is obtained

The mathematical formulation used for this approach can be written as follows:

Optimize {X} such that the weight of the structure is minimized subject to the linear stress constraints and the cumulative KS constraint or :

Minimize

W = g _iX_iL_i (12)

subject to

-¥ £ £ TOL (15)

Here, the number of nonlinear equations is decreased to one cumulative KS constraint. The number of linear constraints remains the same.

Kreisselmeier-Steinhauser Formulation 3

In this formulation, the global stiffness equations and the stress constraints are combined into a single cumulative constraint. This formulation reduces the problem to a single constraint optimization problem. When the KS function is used on the linear and nonlinear constraints, the following equation is obtained:

The mathematical formulation for this approach can be written as follows:

Optimize {X} such that the weight of the structure is minimized subject to one cumulative KS constraint or :

Minimize

W = g _iX_iL_i (17)

-¥ £ £ TOL (19)

where is the KS function obtained by combining the linear and nonlinear equations, and m is the number of free degrees of freedom in the structure.

The standard three-bar, ten-bar, and seventy-two bar truss structures were analyzed and optimized during this research. The geometric dimensions for these structures are specified in Figure 1. A description of the material properties for the elements along with the bounds imposed on the element sizes, the nodal displacements, and the allowable stresses is given in Table 1.

A description of the number of design variables and constraints used in the different formulations for the structures under consideration is given in Table 2. It can be observed that the number of linear and nonlinear constraints decreases for the KS formulations.

Table 1 Element Properties and Bounds on Design Variables for Truss Structures

	3-Bar Truss Structure		10-Bar Truss Structure		72-Bar Truss Structure
Material Used	Aluminum		Aluminum		Aluminum
Young?s Modulus (psi)	1E7		1E7		1E7
Specific Mass (lbf/In3)	0.1		0.1		0.1
Maximum Area (in.2)	2.0	9.0		2.0
Minimum Area (in.2)	0.1		0.1		0.1
Maximum Displacement (in.)	0.5		10.0		0.25
Minimum Displacement (in.)	-0.5		-10.0		-0.25
Max. Allowable Stress (psi)	20000		25000		25000
Min. Allowable Stress (psi)	-20000		-25000		-25000

Table 2 Numbers of Design Variables and Constraints for Truss Structures

Structure	Three-bar Truss				Ten-bar Truss					Seventy-two bar Truss
Method	SAND	KS 1	KS 2	KS3		SAND	KS 1	KS 2	KS 3		SAND	KS 1	KS 2	KS 3
Areas	3	3	3	3		10	10	10	10		72	72	72	72
Displacements	2	2	2	2		8	8	8	8		48	48	48	48
Linear Constraints	6	0	6	0		20	0	20	0		144	0	144	0
Non-linear Constraints	2	3	1	1		8	9	1	1		48	49	1	1

* KS1, KS2, KS3 refer to the KS Formulations 1,2, and 3, respectively.

NPSOL⁷ on a SUN-SPARC IPX was used as an optimizer by the specific and general programs to solve the structures under consideration for minimum weight subject to stress, displacement, and size side constraints. For the different SAND and KS formulations, specific programs were written to optimize only specific truss structures as well as general programs capable of optimizing arbitrary truss structures of any size. Information about the required workspace, the number of function evaluations, the number of calls to the function and constraint evaluation routine, and the CPU time required to optimize the structure, was recorded from these programs for the respective structures.

Selected optimization results obtained for the three-bar, ten-bar, and seventy-two bar structures using the KS formulations are shown in Tables 3, 4, and 5,

respectively. From these results it is observed that, for the structures evaluated, the workspace decreased by varying amounts when the KS function was used in SAND. The column "Number of Calls to Function and Constraint Evaluation Routine" gives information on the number of times the constraint gradient matrix was calculated. It can be observed that the number of constraint gradient evaluations for the KS formulations is larger than that for the standard SAND formulation. However, the number of constraints in the matrix is lower for the KS formulations than for the standard SAND formulation.

A prudent choice of the initial design variables will decrease the CPU time required for optimization of the structures. Therefore, values for the initial design variables are calculated by performing a finite element analysis of the structure. Scaling the constraint equations was found to decrease the CPU times required for the optimizations.

Table 3 Optimization Results for Three-Bar Truss Structure using Specific Programs

Method	Stress as a Function of	Workspace	Number of Major Iterations	Number of Calls to Function and Constraint Evaluation Routine	Time (S)	Accuracy
KS Formulation 1	displacement	243	22	45	0.5	exact
KS Formulation 2	displacement	277	51	76	1.0	~1% off
KS Formulation 3	displacement	181	65	138	1.0	~1% off
SAND	displacement	308	5	9	0.6	exact
SAND*	displacement	219	10	11	1.8	exact
SAND-MM*	displacement	763	587	1749	21.3	exact
SAND-MM*	element forces	824	119	873	5.5	exact
SAND-LMM*	displacement	1207	587	1738	22.8	exact
SAND* (MINOS)	displacement	3146	22	253	2.0	exact

* Results obtained by Wu⁶

Table 4 Optimization Results for Ten-Bar Truss Structure using Specific Programs

Method	Stress as a Function of	Workspace	Number of Major Iterations	Number of Calls to Function and Constraint Evaluation Routine	Time (S)	Accuracy
KS Formulation 1	displacement	1521	44	113	2.3	exact
KS Formulation 2	displacement	1645	138	285	3.2	exact
KS Formulation 3	displacement	1065	96	220	2.4	exact
SAND	displacement	2044	15	24	2.1	exact
SAND*	displacement	1774	7	12	2.8	exact
SAND-MM*	displacement	7185	8	12	4.9	exact
SAND-MM*	element forces	7688	49	49	16.9	exact
SAND-LMM*	displacement	15970	28	28	16.1	exact
SAND* (MINOS)	displacement	4432	14	109	2.6	exact

* Results obtained by Wu⁶

Table 5 Optimization Results for Seventy-Two Bar Truss Structure using General Programs

Method	Stress as a Function of	Workspace	Number of Major Iterations	Number of Calls to Function and Constraint Evaluation Routine	Time (s)	Accuracy
KS Formulation 1	displacement	43989	93	154	398	exact
KS Formulation 2	displacement	50325	260	889	2805	~5% off
KS Formulation 3	displacement	31461	96	610	770	exact
SAND	displacement	62692	65	177	471	exact
SAND*	displacement	53292	21	32	64.0	exact
SAND-MM*	displacement	1.02e6	5	8	2539**	exact
SAND* (MINOS)	displacement	16728	6	525	13.4	exact

* Results obtained by Wu⁶

** Optimal result is reached with the initial design variable values chosen very close to the optimal values.

From the results obtained for all structures, it can be observed that KS formulation 3 is the most efficient in terms of the required workspace since it consistently led to the largest reduction in workspace compared to the standard SAND approach. KS formulation 1 seems to be the most efficient for lower CPU times, since it always led to a decrease in CPU time for the standard SAND approach. A comparison with the results obtained for the SAND-MM and SAND-LMM approaches developed by Wu^5,6 indicates that the KS formulations are, for the most part, more efficient in terms of workspace and CPU time required to perform optimizations. Similar observations can be made from the results obtained for the ten-bar (Table 4) and seventy-two bar (Table 5) truss structures. The comparison between the standard SAND formulations of the present study and Wu is somewhat skewed, since the constraints were treated differently in the two versions of the program. All results are quite accurate and within the tolerance limits of the KS function.

NPSOL was used during the present research for all optimization purposes because all the matrices generated using the KS formulations were dense. The optimization code MINOS¹² which is capable of handling both sparse and dense matrices during optimization was used by Wu⁶ to optimize truss structures using the SAND, SAND-MM, and SAND-LMM approaches. A comparison between the results obtained for the standard SAND formulation using MINOS and NPSOL is presented. It can be observed that, for the three-bar and ten-bar structures, the required workspace for MINOS is larger than that for NPSOL whereas, for the seventy-two bar structure, MINOS requires a smaller workspace than NPSOL. The CPU times required for the SAND optimizations using NPSOL are observed to be less than those for MINOS except for the KS formulation 2 in the ten-bar truss and for the seventy-two bar structure.

Using the Kreisselmeier-Steinhauser function in SAND reduces the number of constraints in the optimization formulation. This causes a decrease in the workspace for all the KS formulations and, for some of the KS 1 results, of the CPU time required to optimize the structures, thereby reducing the computational effort and improving the performance to some extent. KS formulation 1 was observed to be the most efficient in terms of improved CPU times. KS formulation 3

was the most efficient in terms of workspace required for optimization. Overall, the resulting gains were less than expected. For larger structures, taking advantage of the sparsity of the formulation by using a sparse system optimizer such as MINOS seems to improve performance more than the KS formulations.

The present research investigated the usefulness of the KS function to optimize truss structures only. A logical extension of this research would be to optimize frame structures and other built-up finite element structures using these formulations. In order to make SAND more efficient, an investigation into the use of the max norm in SAND is recommended, which could possibly reduce the number of constraints involved in the optimization process.

The authors would like to acknowledge helpful technical discussions with Dr. Jarek Sobieski and Sharon Padula at NASA Langley Research Center.

1. Balling, R.J., and Sobieszczanski-Sobieski, J., " Optimization of Coupled Systems: A Critical Overview of Approaches", AIAA-94-4330, Proceedings, AIAA/NASA/USAF/ISSMO 5th Symposium on Multidisciplinary Analysis and Optimization, Panama City, Florida, September 1994, pp. 753-767.

2. Fox, R.L., and Schmit, L.A., "Advances in the Integrated Approach to Structural Synthesis", Journal of Spacecraft, Vol. 3, No. 6, 1966, pp. 855-866.

3. Kreisselmeier, G., and Steinhauser, R., "Systematic Control Design by Optimizing a Vector Performance Index", Proceedings, IFAC Symposium on Computer-Aided Design of Control Systems, Zürich, Switzerland, 1979, pp. 113-117.

4. Sobieszczanski-Sobieski, J., "A Technique for Locating Function Roots and for Satisfying Equality Constraints in Optimization", Structural Optimization, Vol. 4, 1992, pp. 241-243.

5. Striz, A.G., Wu, Z., and Sobieszczanski-Sobieski, J., "An Efficiency Study of the Simultaneous Analysis and Design of Structures", Proceedings, 1st World Congress of the International Society for Structural and Multidisciplinary Optimization, Goslar, Germany, May 1995, pp. 401-406.

6. Wu, Z., "An Efficiency Study of Simultaneous Analysis and Design", M.S. Thesis, The University of Oklahoma, January 1996.

7. Gill, P.E., Murray, W., Saunders, M.A., and Wright, M.H., User's Guide for NPSOL (Version 4.0), Technical Report 86-2, January 1986, Stanford University, Stanford, California.

8. Haftka, R.T., and Gürdal, Z., "Elements of Structural Optimization", 3rd Edition, Kluwer Academic Publishing, Dordrecht, The Netherlands, 1993, pp. 400-406.
   
   
9. Haftka, R.T., "Simultaneous Analysis and Design", AIAA Journal, Vol. 23, No. 7, July 1985, pp. 1099-1103.

10. Smaoui, H., and Schmit, L.A., "An Integer Approach to the Synthesis of Geometrically Non-Linear Structures", International Journal for Numerical Methods in Engineering, Vol. 26, 1988, pp. 550-570.

11. Ringertz, U.T., "Optimization of Structures with Nonlinear Response", Engineering Optimization, Vol. 14, 1989, pp. 179-188.
    
    
12. Murtagh, B.A., and Saunders, M.A., "MINOS 5.4 User?s Guide, Technical Report SOL 83-20R", December 1983, Stanford University, Stanford, California.