- Email: [email protected]
Automatic shape optimization of three-dimensional shell structures with large shape changes
Compufers d Sfrucrures Vol. 49. No. I. pp. 167478, Printed in Great Britain.
1993 0
56.00 + 0.00 00457949/93 1993 Pergamon Press Ltd
AUTOMATIC SHAPE OPTIMIZATION OF THREE-DIMENSIONAL SHELL STRUCTURES WITH LARGE SHAPE CHANGES A. A. GATEst and M. L. ACCOR~I$ tNava1 Underwater Systems Center, Code 412, Bldg 96, Rm A02, New London, CT 06320-5594,U.S.A. SDepartment of Civil Engineering, University of Connecticut, Storm, CT 06269-3037, U.S.A. (Received 13 March 1992) Abstract-A
finite element-based shape optimization program has been developed for three-dimensional shell structures which allows for large shape changes. The new shape optimization program has been achieved by linking together adaptive mesh generation, substructuring, and linear and nonlinear optimization techniques to a commercial finite element analysis program (MSC/NASTRAN). The program has the capability of optimizing shapes by allowing multiple edges to move. A new procedure was developed to determine the best edge to move at each iteration of the shape optimization. The position of this edge is then determined using nonlinear optimization techniques. This process is incrementally continued until the edges are at their minimum positions, or the change in objective function is less than a tolerance value. This program is presently being applied to weight minimization.
INTRODUCTION During the last decade significant progress has been made in the development of finite element-based shape optimization. An excellent survey of this work is presented by Haftka and Grandhi [l]. The need for automatic mesh generation that insures accurate finite element results during the shape optimization process was clearly identified. The design element concept, introduced by Imam [2], has been used with automatic mesh generation to perform shape optimization. Botkin [3] used this concept with regionalized mesh generation for automatic shape optimization of two-dimensional plate structures. Later, Botkin and Bennett [4] extended this work to three-dimensional assemblages of plate structures. In more recent work, the need to tie shape optimization capabilities to commercially available finite element programs has been identified. Yang [5] developed a modular program to perform shape optimization of three-dimensional solid structures using NASTBAN. Kodiyalam et al. [6] describe a shape optimization capability that was recently added to MSC/NASTRAN. In this work, the shape optimization capability is limited to moving outer grid point locations. Eventually the process must be stopped due to distortion of the outer elements and a new mesh must be generated externally. In this paper, a modular program that uses a commercial finite element code (MSCjNASTBAN) to perform automatic shape optimization of threedimensional plate structures is described. The program uses substructuring and automatic mesh generation to allow for multiple edge movement and CA.3 49,1--L
large shape changes during the optimization process. A new method was developed to determine which edge is best to move for shape optimization, then a nonlinear optimization method is used to determine the optimum position of this edge. This process is performed repeatedly with different edges until the final optimum shape is found. SURSTRUCTURING AND AUTOMATIC MESH GENERATION
A shell shape is defined as a three-dimensional assemblage of facetted plates with edges that are allowed to move and edges that are not allowed to move during the optimization process. Movement of edges is permitted in the two directions perpendicular to the edge, as shown in Fig. 1, which allows for optimization of surface locations of the structure. Each edge that is allowed to move is considered as a substructure during the shape optimization process. Each edge is specified by NDP design points which determine a polynomial function which defines the position of that edge. The edges that are not allowed to move are also defined by points that remain fixed during the shape optimization process. After all the edges are defined by design points, an automatic mesh generator is used to create the finite element model using quadrilateral elements, as shown in Fig. 2. The coordinates of the design points for each edge are stored in separate files (edge files) and the commands to generate the mesh are kept in another file. During the shape optimization process, the edge files are automatically edited and a new finite element
A. A. GATESand M. L. Accoasi
168 Edge3
(1b) represent restrictions on allowable stresses, strains, and displacements of the structure. The geometric constraints (lc) are restrictions on the shape of the structure. The objective function and constraint functions for each edge i can be approximated using a Taylor series expansion in the design points Xi = {X, , A’,, . . .}. A linear expansion is written as
jm) Edge
I
+
Fig. I. Three-dimensional shell structure with multiple edges allowed to move. model is generated. The skew angles and aspect ratios of each element are automatically calculated to check for element distortion. If they are acceptable, the shape optimization program continues. Since multiple edges are allowed to move in the shape optimization process, a program was developed to determine the best edge to optimize during each iteration. This requires information from the initial design of the shape and the data from a linear optimization on each edge. A decision is made by considering the initial weight, optimized weight based on a linear approximation, initial constraint values, and constraint values based on a linear approximation. The best edge is then optimized using quadratic approximation functions and a nonlinear optimization program. PROBLEM
STATEMENT
AND SOLUTION
xf;(xp) + vfi(xp)(x,
- xp,,
(2)
where Xp are the design points for the initial shape and Vfi are the gradients of the objective or constraint functions. To calculate the first (constant) term of (2) a finite element analysis on the initial shape is performed and the objective and constraint values are found. To generate the second (linear) terms of (2), each edge is perturbed NDP times and the corresponding analyses are performed. These results are then used to solve for the function gradients in (2). A quadratic Taylor series could also be used to approximate the objective and constraint functions, however, additional finite element analyses are required. The quadratic expansion is written as f;(X,) zjxxp)
+ Vfi(XS).(Xi - xp, +~Xi-X~)‘.H~(Xp).(Xi-X~)~
(3)
where Hi is the Hessian matrix. An additional NDP (NDP + 1)/2 perturbed designs and analyses are required to determine the Hessian matrix. Once the linear (2) or quadratic (3) approximate functions have been generated, they are used within an optimization program to determine the value of the objective and
PROCEDURES
A brief presentation of the shape optimization problem is presented. Additional details can be found in the book by Vanderplaats [‘I]. The structure to be optimized had NE edges that are allowed to move in the optimization process and each edge is defined by NDP design points. The shape optimization problem is defined for each edge i = 1, NE using the edge design points Xi = {xj}i, j = 1, NDP as follows: Minimize
Fi(Xi)
(la)
k = l,M
(lb)
Design’ point
‘Curve
fit
edges
subject to &(xi)GO,
where Fi(Xi) is the objective function for each edge as a function of the design points defining that edge and g&(X,) are the constraint functions for each edge as a function of the design points defining that edge. For the current application, the objective (la) is to minimize the weight of the structure. The constraints
Fig. 2. Edge design points and automatic mesh generation.
Automatic shape optimization for three-dimensional shell structures constraint functions at the optimum location for each edge. SHAPE OPTIMIZATION
Linear approximate functions (2) are generated by perturbing the design points on each edge and by performing the corresponding finite element analyses. For each edge, the linear approximate functions are optimized using the simplex method and are evaluated at their optimum position. The objective and constraint values for each edge at their optimum position are then used to determine the best edge to optimize using a program developed by the authors [8]. Before performing a nonlinear optimization on the best edge, an analysis is performed with the best edge set at the limit of the perturbed design range which produces the minimum weight. If no constraints are violated at this limit, then the design points for the
PROGRAM
The shape optimization program is shown in Fig. 3. For the initial shape and after each iteration, the input data and edge data are checked to determine if any errors will occur during linear optimization, edge selection, or nonlinear optimization. If so, the program stops and provides the user with an explanation. The design points are checked to determine if they are at their geometric limits. If so, the program eliminates that edge from the optimization process.
I
Adoptive mesh generotlon
I
1 Select
(
Cptimizotion
I
1
Fimt?
1
1
best edge tooptimize
I
I
1
Mesh generotlon ond FE onolysis
Form new quadratic
1 Quodrotlc approximate function ond nonlinear optimlzotion
I t
-
I
Mesh generation FE analysis
169
on x*
Fig. 3. Shape optimization flowchart.
Replace xo with x* improve &sign
I
A. A. GATIZSand M. L. ACCORSI
170
Edae 3
I
Stress constraints -33,000
psi 5 umx 5
30,000 psi Arrows indicate directional movement
-33,OOOpsi 5 ornln 5
30,000 psi
Edge
Strain constraints
-o.ol%~,,
50.01
Displacement constraints
I 4
-1.0 inch5&,~l,Oinch 5 -1.0 inch5 8,,,
IO IO
sl.Oinch
Fig. 4. Bracket with applied load and constraint conditions.
! :
!
!
3 3
5
2 z
8 8
12 12
Y Y
Fig. 5. Edges of bracket allowed to move.
Fig. 6. Evolution of shape.
171
Automatic shape optimization for thmedimensional shell structures Table 1. Shape optimization results for example 1 FEAs Iteration 0 1 2 3 4 5 Allowable
SWCd
Total FEAs FEAs saved
47 20
FEAs
3 3 11 3
13 11 9 5 9
edge are set at this limit and nonlinear optimization is not needed for that iteration. Since nonlinear optimization is not used, fewer finite element analyses are required to position the edge for that iteration. If there is a constraint violation at the perturbed design range limit, then nonlinear optimization is performed to find the optimum position of the edge within the perturbed design range. For nonlinear optimization, additional perturbed designs of the best edge and additional finite element analyses are performed and used with the previous linear function information to generate quadratic approximate functions (3). Nonlinear optimization Shape max. stress ys itemtion
lterat ion
Max. stress (psi) 26005 26177 26235 26235 27271 29870 30000
weight (lb) 42.43 40.11 38.40 36.03 31.85 28.87
Min. stress (psi) -26005 - 26075 -26068 -26067 -31225 -33281 -33000
is then performed using a modified method of hyperspheres developed by the authors [8]. To check the predicted optimum shape, a final finite element analysis is performed using the design point locations from the nonlinear optimization program. The results of this finite element analysis and the quadratic approximate function evaluations for objective and constraints are input into a convergence checking program. If the accuracy of the quadratic approximate functions is poor, new quadratic functions are generated using the information from the final finite element analysis at the predicted optimum point. If the accuracy is found acceptable, then convergence of the shape optimization process is evaluated by considering the magnitude of change in the objective function, and by checking if the design points for all the edges are at their limit values. If convergence has not occurred, the design points for the edge under consideration are permanently replaced with the optimum points and another iteration is started. The shape optimization program uses an IRIS workstation as the master computer and a Cray X-MP as a slave. The IRIS sends input files to the Cray, the Cray performs the required finite element analyses, and the IRIS retrieves the results and performs the optimization procedures. This capability allows for shape optimization of large complex models.
Shape min. stress M iteration Shape weight ys iterotion -22 -24 5 Y %
-26m-•-26 -3O-
60
2 $
SO
-32-
40.-m-.
-34-36
-
-30
-
-400
-m-
1
30
.-.
20 1 I
I
I
2
I
I
3
4
5
Itemtion
Fig. 7. Maximum and minimum principal stresses versus iteration.
‘O_, 0
I
3
2
4
ltemtion
Fig. 8. Shape weight versus iteration.
5
A. A. GATESand M. L. Acco~sl
172
SHAPE OPTIMIZATION EXAMPLES
Table 2(b). Comparison approximate
Two example problems are presented to demonstrate the capability of the new shape optimization program. The first problem is a bracket subject to a uniformly distributed load along one edge and fixed at the base as shown in Fig. 4. The constraint conditions and edges allowed to move are illustrated in Figs 4 and 5, respectively. The first example demonstrates the program capability to optimize a shape with multiple edges allowed to move, allow two-dimensional edge movement, and to use edges defined by two or four design points. Edges 1, 2, and 3 are allowed in-plane movement only. Edge 4 is allowed to move in-plane and perpendicular to the plane it occupies. The in-plane movement is defined as edge 4 and the out-of-plane movement is defined as edge 5. Currently, each optimization iteration locates an edge in one direction only, and therefore and edge with two directions of movement must be defined twice. Figure 6 is the evolution of the shape changing to its optimum with respect to the loading and constraints. Figure 7 is the maximum and minimum principal stress versus iteration. Up to the third iteration, the movement of the edges had little effect on the constraints which is a result of the logic used in the best edge subprogram. Figure 8 is the shape weight versus iteration which appears almost linear. Edges 3, 4, and 5 moved to their geometric limits. Edge 1 was found to be within 10% of its geometric limit. Edge 2 was positioned by the nonlinear optimizer. The total number of finite element analyses performed was 47 and the total number of finite element analyses saved by bypassing nonlinear optimization ‘was 20. Table 1 contains the numerical results from the initial design to the optimum shape. Table 2 is a list of the maximum and minimum stresses, strains, and displacements from the finite element analysis and approximate function values at
Table 2(a). Comparison of finite element results and approximate function values at the predicted optimum, maximum values
Average constraint error = 0.726% Allowable
Maximum stress (psi) 30000
Maximum strains 0.01
FEA
Approximate
29870 29209 26072 24917 24066 23737
29999 28862 25444 25111 25498 23954
0.00384 0.00370 0.00358
0.00384 0.00364 0.00356
Maximum displacements (in) 1.0 0.2435 0.243 1
0.2439 0.2436
FEA
Approximate
Minimum stresses (psi) -33000 -26041 -26050 - 26670 -27901 -32090 -33281
- 25688 -26046 - 27885 -28358 - 32292 -32999
Minimum strains -0.01
-0.00363 -0.00377 -0.00422
-0.00366 -0.00381 -0.00422
Minimum displacements (in) -1.0
-o.OoOO -0.0000
-o.oooo -o.om
the predicted optimum shape. The average error between the approximate function values and finite element analysis was 0.7% at the predicted optimum shape. Table 3. Comparison of approximated and finite element results at the predicted optimum, maximum, and minimum values Average constraint error = 1.08% Allowable
FEA
Approximate
41804 41747 41173 40780 40356 39330
40996 40959 40446 40113 39768 38648
0.00838 0.00828 0.00754
0.00826 0.00817 0.00735
Maximum stresses (psi) 41000
Maximum strains 0.01
Maximum displacements (in) 1.0
Average constraint error = 0.726%
Allowable
of finite element results and function values at the predicted optimum, minimum values
Allowable
0.016844 0.016447
0.016843 0.016445
Average constraint error = 1.08% FEA ADDroximate
Minimum stresses (psi) -41000
Minimum strains -0.01
-23155 - 23290 -24319 - 24529 -29616 -31637
-23332 -23561 - 24422 -24731 - 29439 -31258
-0.00720 - 0.00745 - 0.00747
- 0.00704 -0.00724 -0.00725
Minimum displacements (in) -1.00 -0.00606 -0.00629
- 0.00620 -0.00641
Automatic shape optimization for three-dimensional
shell structures
173
300 psi I
3000 Ibs Arrow
indicate
direction
of movement
Distributed Stmss
w-5’
constmints
‘500
-41,000
psi
5
crnal
5 41,000
-41,000
psi
5
umln
5 41,OOOpsi
Strain
lb/in
psi
x,
hl”
h0*
I
6.3 6.3
3 3
IO IO
Directionof movement * 2
constraint5
-0.01 s cmx s 0.01
-0.015 %I”
5
Displacement
-1.Oinch
We
0.0 I
constraints
5 BYmax
- 1.0 inch5
Fig. 10. Ages of stiffened cylinder allowed to move.
3,
5
1.0 inch
,,,fn 5
1.0 inch
Fig. 9. Stiffened cylinder with applied loads and constraint conditions.
The second example demonstrates the use of the program for aerospace applications and exercises the program’s ability to perform multi-edge and surface optimization. The problem is a rib-stiffened cylinder
with an internal support and external braces subjected to internal pressure and additional internal and external loads as shown in Fig. 9. Symmetry of the model is used. Edges 1 and 2 are allowed to move in the z direction which will change the shape of the cylinder, ribs, brace, and internal support as shown in Fig. 10. Edge 3 is allowed to move in the x direction. The internal support is defined by two edge numbers (4 and 5) to allow for movement in two directions as shown. The constraint conditions are given in Fig. 9. Figure 11 is the initial finite element model consisting of 547 facetted plate elements defined by 680 nodes. It took approximately 6 h to
Table 4. Surface and shape optimization results
stress (psi)
Min. stress (psi)
31194 31192 31192 31186 31183 31184 27436 23308 30615 40880 41804
-19703 -23461 -23423 -37192 -35208 -35398 -26425 - 27341 -22934 -31190 -31637
41ooo
41000
Max.
FEAs Iteration
SaVed
0 1 2 3 4 5 6 7 8 9
3 3 3 3 3 3 3 3 3
Allowable Total FEAs FEAs saved
72 27
FEAs 12 11 9 9 7 5 5 5 3 6
Weight (lb) 103.68 103.27 loo.47 100.21 100.01 99.52 91.50 84.86 82.55 79.97 79.96
Fig. 11. Initial finite element model of stiffened cylinder.
174
A. A. GAIESand M. L. Accoas~
Optimum
Fig. 12. Evolution of shell shape
build the model and generate the necessary input files for this problem. This problem confirmed the program’s unique ability to perform automatic adaptive mesh generation as shown in Figs 12-15. After manually generating an initial model, all of the remaining models needed for optimization were automatically generated with
no user interaction. The ribs, internal and external support geometry depend on the position of the cylindrical surface, so that changing edges 1 and 2 affects the entire model. Table 3 is a list of the finite element analysis and approximate function results at the predicted optimum shape. Table 4 contains the results from the initial
Automatic shape optimization for three-dimensional shell structures
175
Initial
Optimum
Fig. 13. Evolution of internal ribs.
design to the optimum shape. Figures 16 and 17 are plots of the minimum and maximum principal stresses versus iteration and weight versus iteration, respectively. The program generated 72 different finite element models automatically which would be very time consuming to do manually. The program also extracted the correct information, generated
approximate functions, and performed 29 linear optimizations and one nonlinear optimization. This task is extremely difficult to do manually and subject to human error. The substructuring program made decisions based on numerical values; decisions made manually by a user based on shape would be very difficult due to the complexity of the problem.
A. A. GAG and M. L. Acco~s~
176
lnitlal
Optimum
Fig. 14. Evolution of internal support and external brace.
Automatic shape optimization for threedimensional shell structures
Initial
Optimum
Fig. 15. Evolution of entire shape.
A. A. GAsxa and M. L. Accoast MMmm /manlmum stnss m
WeightH itoration
lteratkn
wright -m-m.
I
600
2
3
4
6 6 Itoration
“I
6
9
Fig. 16, Maximum and minimum principal stresses versus iteration.
Fig. 17. Shape weight versus iteration.
CONCLUSIONS
the Hessian matrix may be sufkient to perform nonlinear optimization with reasonable accuracy. Both techniques would substantially reduce the number of finite element analyses required.
Development of an automatic adaptive mesh generating program has allowed for successful investigation of shape optimization for shell structures ~dergoing Iarge shape changes. ~ubst~ctu~ng also becomes an important issue since selecting the correct edge may produce a substantial savings in cost and in some cases reduce the constraint values allowing for additional optimization, as demonstrated in the second example for iterations 5 and 6. Further investigation of the program is necessary to determine how much a shape oan be changed during each iteration while maintaining acceptable approximate function accuracy. The use of linear and nonlinear optimization in the shape optimization program may be too excessive. However, this was done to insure the best possible chance of success for this type of shape optimi~at~on program, Addi~ona~ consideration is needed to minimize the number of finite element analyses which are needed to generate data for the edge selection program and nonlinear optimizer. For example, each edge could be perturbed in the direction of decreasing cost rather than performing finear op~rni~~on for the subst~~t~ng program. Ako using the linear and diagonal terms in
REFERENCES
T. Haftka and R. V. Grandhi, Structural shape optimization-a survey. Comput. Meth. appl. Mech.
I. R.
Engng 57,91-106 (1986). 2. M. H. Imam, Three-dimensional shape optimization. Int. 3. Pbner. Me&. &gng 18, 661-673 (1982). 3. M. E. R&kin, Shape op~~~t~on of plate and shell structures. &AA frill Zo, 26g-273 (1982). 4. M. E. Botkin and J. A. Bennett, Shape optimization of three-dimensional folded plate structures. AIAA Jnl23,
18041810
(1985).
5. R. 1. Yang, A three-dimensional
shape optimization system-shop3d. Comput. Srruct. 31,8%X90 (1939). 6. S. Kodiyaksm, G. N. Vanderpktats and H. Miura, Structerat shape opti~~t~Qn with MSCINA~~N. Comput. Struct, 40, 821-829 (FM).
7. G. N. Vanderplaats, Numerical Optimization Techniques ,for Engineering Design with Appliccatians. McGraw-Hill
(1984). 8. A. A. Gates, Deveiopment of an automatic shape optimization program for shell structures undergoing large geometry changes. Ph.D. dissertation, University of Connecticut, Storm (1992).
1993 0
56.00 + 0.00 00457949/93 1993 Pergamon Press Ltd
AUTOMATIC SHAPE OPTIMIZATION OF THREE-DIMENSIONAL SHELL STRUCTURES WITH LARGE SHAPE CHANGES A. A. GATEst and M. L. ACCOR~I$ tNava1 Underwater Systems Center, Code 412, Bldg 96, Rm A02, New London, CT 06320-5594,U.S.A. SDepartment of Civil Engineering, University of Connecticut, Storm, CT 06269-3037, U.S.A. (Received 13 March 1992) Abstract-A
finite element-based shape optimization program has been developed for three-dimensional shell structures which allows for large shape changes. The new shape optimization program has been achieved by linking together adaptive mesh generation, substructuring, and linear and nonlinear optimization techniques to a commercial finite element analysis program (MSC/NASTRAN). The program has the capability of optimizing shapes by allowing multiple edges to move. A new procedure was developed to determine the best edge to move at each iteration of the shape optimization. The position of this edge is then determined using nonlinear optimization techniques. This process is incrementally continued until the edges are at their minimum positions, or the change in objective function is less than a tolerance value. This program is presently being applied to weight minimization.
INTRODUCTION During the last decade significant progress has been made in the development of finite element-based shape optimization. An excellent survey of this work is presented by Haftka and Grandhi [l]. The need for automatic mesh generation that insures accurate finite element results during the shape optimization process was clearly identified. The design element concept, introduced by Imam [2], has been used with automatic mesh generation to perform shape optimization. Botkin [3] used this concept with regionalized mesh generation for automatic shape optimization of two-dimensional plate structures. Later, Botkin and Bennett [4] extended this work to three-dimensional assemblages of plate structures. In more recent work, the need to tie shape optimization capabilities to commercially available finite element programs has been identified. Yang [5] developed a modular program to perform shape optimization of three-dimensional solid structures using NASTBAN. Kodiyalam et al. [6] describe a shape optimization capability that was recently added to MSC/NASTRAN. In this work, the shape optimization capability is limited to moving outer grid point locations. Eventually the process must be stopped due to distortion of the outer elements and a new mesh must be generated externally. In this paper, a modular program that uses a commercial finite element code (MSCjNASTBAN) to perform automatic shape optimization of threedimensional plate structures is described. The program uses substructuring and automatic mesh generation to allow for multiple edge movement and CA.3 49,1--L
large shape changes during the optimization process. A new method was developed to determine which edge is best to move for shape optimization, then a nonlinear optimization method is used to determine the optimum position of this edge. This process is performed repeatedly with different edges until the final optimum shape is found. SURSTRUCTURING AND AUTOMATIC MESH GENERATION
A shell shape is defined as a three-dimensional assemblage of facetted plates with edges that are allowed to move and edges that are not allowed to move during the optimization process. Movement of edges is permitted in the two directions perpendicular to the edge, as shown in Fig. 1, which allows for optimization of surface locations of the structure. Each edge that is allowed to move is considered as a substructure during the shape optimization process. Each edge is specified by NDP design points which determine a polynomial function which defines the position of that edge. The edges that are not allowed to move are also defined by points that remain fixed during the shape optimization process. After all the edges are defined by design points, an automatic mesh generator is used to create the finite element model using quadrilateral elements, as shown in Fig. 2. The coordinates of the design points for each edge are stored in separate files (edge files) and the commands to generate the mesh are kept in another file. During the shape optimization process, the edge files are automatically edited and a new finite element
A. A. GATESand M. L. Accoasi
168 Edge3
(1b) represent restrictions on allowable stresses, strains, and displacements of the structure. The geometric constraints (lc) are restrictions on the shape of the structure. The objective function and constraint functions for each edge i can be approximated using a Taylor series expansion in the design points Xi = {X, , A’,, . . .}. A linear expansion is written as
jm) Edge
I
+
Fig. I. Three-dimensional shell structure with multiple edges allowed to move. model is generated. The skew angles and aspect ratios of each element are automatically calculated to check for element distortion. If they are acceptable, the shape optimization program continues. Since multiple edges are allowed to move in the shape optimization process, a program was developed to determine the best edge to optimize during each iteration. This requires information from the initial design of the shape and the data from a linear optimization on each edge. A decision is made by considering the initial weight, optimized weight based on a linear approximation, initial constraint values, and constraint values based on a linear approximation. The best edge is then optimized using quadratic approximation functions and a nonlinear optimization program. PROBLEM
STATEMENT
AND SOLUTION
xf;(xp) + vfi(xp)(x,
- xp,,
(2)
where Xp are the design points for the initial shape and Vfi are the gradients of the objective or constraint functions. To calculate the first (constant) term of (2) a finite element analysis on the initial shape is performed and the objective and constraint values are found. To generate the second (linear) terms of (2), each edge is perturbed NDP times and the corresponding analyses are performed. These results are then used to solve for the function gradients in (2). A quadratic Taylor series could also be used to approximate the objective and constraint functions, however, additional finite element analyses are required. The quadratic expansion is written as f;(X,) zjxxp)
+ Vfi(XS).(Xi - xp, +~Xi-X~)‘.H~(Xp).(Xi-X~)~
(3)
where Hi is the Hessian matrix. An additional NDP (NDP + 1)/2 perturbed designs and analyses are required to determine the Hessian matrix. Once the linear (2) or quadratic (3) approximate functions have been generated, they are used within an optimization program to determine the value of the objective and
PROCEDURES
A brief presentation of the shape optimization problem is presented. Additional details can be found in the book by Vanderplaats [‘I]. The structure to be optimized had NE edges that are allowed to move in the optimization process and each edge is defined by NDP design points. The shape optimization problem is defined for each edge i = 1, NE using the edge design points Xi = {xj}i, j = 1, NDP as follows: Minimize
Fi(Xi)
(la)
k = l,M
(lb)
Design’ point
‘Curve
fit
edges
subject to &(xi)GO,
where Fi(Xi) is the objective function for each edge as a function of the design points defining that edge and g&(X,) are the constraint functions for each edge as a function of the design points defining that edge. For the current application, the objective (la) is to minimize the weight of the structure. The constraints
Fig. 2. Edge design points and automatic mesh generation.
Automatic shape optimization for three-dimensional shell structures constraint functions at the optimum location for each edge. SHAPE OPTIMIZATION
Linear approximate functions (2) are generated by perturbing the design points on each edge and by performing the corresponding finite element analyses. For each edge, the linear approximate functions are optimized using the simplex method and are evaluated at their optimum position. The objective and constraint values for each edge at their optimum position are then used to determine the best edge to optimize using a program developed by the authors [8]. Before performing a nonlinear optimization on the best edge, an analysis is performed with the best edge set at the limit of the perturbed design range which produces the minimum weight. If no constraints are violated at this limit, then the design points for the
PROGRAM
The shape optimization program is shown in Fig. 3. For the initial shape and after each iteration, the input data and edge data are checked to determine if any errors will occur during linear optimization, edge selection, or nonlinear optimization. If so, the program stops and provides the user with an explanation. The design points are checked to determine if they are at their geometric limits. If so, the program eliminates that edge from the optimization process.
I
Adoptive mesh generotlon
I
1 Select
(
Cptimizotion
I
1
Fimt?
1
1
best edge tooptimize
I
I
1
Mesh generotlon ond FE onolysis
Form new quadratic
1 Quodrotlc approximate function ond nonlinear optimlzotion
I t
-
I
Mesh generation FE analysis
169
on x*
Fig. 3. Shape optimization flowchart.
Replace xo with x* improve &sign
I
A. A. GATIZSand M. L. ACCORSI
170
Edae 3
I
Stress constraints -33,000
psi 5 umx 5
30,000 psi Arrows indicate directional movement
-33,OOOpsi 5 ornln 5
30,000 psi
Edge
Strain constraints
-o.ol%~,,
50.01
Displacement constraints
I 4
-1.0 inch5&,~l,Oinch 5 -1.0 inch5 8,,,
IO IO
sl.Oinch
Fig. 4. Bracket with applied load and constraint conditions.
! :
!
!
3 3
5
2 z
8 8
12 12
Y Y
Fig. 5. Edges of bracket allowed to move.
Fig. 6. Evolution of shape.
171
Automatic shape optimization for thmedimensional shell structures Table 1. Shape optimization results for example 1 FEAs Iteration 0 1 2 3 4 5 Allowable
SWCd
Total FEAs FEAs saved
47 20
FEAs
3 3 11 3
13 11 9 5 9
edge are set at this limit and nonlinear optimization is not needed for that iteration. Since nonlinear optimization is not used, fewer finite element analyses are required to position the edge for that iteration. If there is a constraint violation at the perturbed design range limit, then nonlinear optimization is performed to find the optimum position of the edge within the perturbed design range. For nonlinear optimization, additional perturbed designs of the best edge and additional finite element analyses are performed and used with the previous linear function information to generate quadratic approximate functions (3). Nonlinear optimization Shape max. stress ys itemtion
lterat ion
Max. stress (psi) 26005 26177 26235 26235 27271 29870 30000
weight (lb) 42.43 40.11 38.40 36.03 31.85 28.87
Min. stress (psi) -26005 - 26075 -26068 -26067 -31225 -33281 -33000
is then performed using a modified method of hyperspheres developed by the authors [8]. To check the predicted optimum shape, a final finite element analysis is performed using the design point locations from the nonlinear optimization program. The results of this finite element analysis and the quadratic approximate function evaluations for objective and constraints are input into a convergence checking program. If the accuracy of the quadratic approximate functions is poor, new quadratic functions are generated using the information from the final finite element analysis at the predicted optimum point. If the accuracy is found acceptable, then convergence of the shape optimization process is evaluated by considering the magnitude of change in the objective function, and by checking if the design points for all the edges are at their limit values. If convergence has not occurred, the design points for the edge under consideration are permanently replaced with the optimum points and another iteration is started. The shape optimization program uses an IRIS workstation as the master computer and a Cray X-MP as a slave. The IRIS sends input files to the Cray, the Cray performs the required finite element analyses, and the IRIS retrieves the results and performs the optimization procedures. This capability allows for shape optimization of large complex models.
Shape min. stress M iteration Shape weight ys iterotion -22 -24 5 Y %
-26m-•-26 -3O-
60
2 $
SO
-32-
40.-m-.
-34-36
-
-30
-
-400
-m-
1
30
.-.
20 1 I
I
I
2
I
I
3
4
5
Itemtion
Fig. 7. Maximum and minimum principal stresses versus iteration.
‘O_, 0
I
3
2
4
ltemtion
Fig. 8. Shape weight versus iteration.
5
A. A. GATESand M. L. Acco~sl
172
SHAPE OPTIMIZATION EXAMPLES
Table 2(b). Comparison approximate
Two example problems are presented to demonstrate the capability of the new shape optimization program. The first problem is a bracket subject to a uniformly distributed load along one edge and fixed at the base as shown in Fig. 4. The constraint conditions and edges allowed to move are illustrated in Figs 4 and 5, respectively. The first example demonstrates the program capability to optimize a shape with multiple edges allowed to move, allow two-dimensional edge movement, and to use edges defined by two or four design points. Edges 1, 2, and 3 are allowed in-plane movement only. Edge 4 is allowed to move in-plane and perpendicular to the plane it occupies. The in-plane movement is defined as edge 4 and the out-of-plane movement is defined as edge 5. Currently, each optimization iteration locates an edge in one direction only, and therefore and edge with two directions of movement must be defined twice. Figure 6 is the evolution of the shape changing to its optimum with respect to the loading and constraints. Figure 7 is the maximum and minimum principal stress versus iteration. Up to the third iteration, the movement of the edges had little effect on the constraints which is a result of the logic used in the best edge subprogram. Figure 8 is the shape weight versus iteration which appears almost linear. Edges 3, 4, and 5 moved to their geometric limits. Edge 1 was found to be within 10% of its geometric limit. Edge 2 was positioned by the nonlinear optimizer. The total number of finite element analyses performed was 47 and the total number of finite element analyses saved by bypassing nonlinear optimization ‘was 20. Table 1 contains the numerical results from the initial design to the optimum shape. Table 2 is a list of the maximum and minimum stresses, strains, and displacements from the finite element analysis and approximate function values at
Table 2(a). Comparison of finite element results and approximate function values at the predicted optimum, maximum values
Average constraint error = 0.726% Allowable
Maximum stress (psi) 30000
Maximum strains 0.01
FEA
Approximate
29870 29209 26072 24917 24066 23737
29999 28862 25444 25111 25498 23954
0.00384 0.00370 0.00358
0.00384 0.00364 0.00356
Maximum displacements (in) 1.0 0.2435 0.243 1
0.2439 0.2436
FEA
Approximate
Minimum stresses (psi) -33000 -26041 -26050 - 26670 -27901 -32090 -33281
- 25688 -26046 - 27885 -28358 - 32292 -32999
Minimum strains -0.01
-0.00363 -0.00377 -0.00422
-0.00366 -0.00381 -0.00422
Minimum displacements (in) -1.0
-o.OoOO -0.0000
-o.oooo -o.om
the predicted optimum shape. The average error between the approximate function values and finite element analysis was 0.7% at the predicted optimum shape. Table 3. Comparison of approximated and finite element results at the predicted optimum, maximum, and minimum values Average constraint error = 1.08% Allowable
FEA
Approximate
41804 41747 41173 40780 40356 39330
40996 40959 40446 40113 39768 38648
0.00838 0.00828 0.00754
0.00826 0.00817 0.00735
Maximum stresses (psi) 41000
Maximum strains 0.01
Maximum displacements (in) 1.0
Average constraint error = 0.726%
Allowable
of finite element results and function values at the predicted optimum, minimum values
Allowable
0.016844 0.016447
0.016843 0.016445
Average constraint error = 1.08% FEA ADDroximate
Minimum stresses (psi) -41000
Minimum strains -0.01
-23155 - 23290 -24319 - 24529 -29616 -31637
-23332 -23561 - 24422 -24731 - 29439 -31258
-0.00720 - 0.00745 - 0.00747
- 0.00704 -0.00724 -0.00725
Minimum displacements (in) -1.00 -0.00606 -0.00629
- 0.00620 -0.00641
Automatic shape optimization for three-dimensional
shell structures
173
300 psi I
3000 Ibs Arrow
indicate
direction
of movement
Distributed Stmss
w-5’
constmints
‘500
-41,000
psi
5
crnal
5 41,000
-41,000
psi
5
umln
5 41,OOOpsi
Strain
lb/in
psi
x,
hl”
h0*
I
6.3 6.3
3 3
IO IO
Directionof movement * 2
constraint5
-0.01 s cmx s 0.01
-0.015 %I”
5
Displacement
-1.Oinch
We
0.0 I
constraints
5 BYmax
- 1.0 inch5
Fig. 10. Ages of stiffened cylinder allowed to move.
3,
5
1.0 inch
,,,fn 5
1.0 inch
Fig. 9. Stiffened cylinder with applied loads and constraint conditions.
The second example demonstrates the use of the program for aerospace applications and exercises the program’s ability to perform multi-edge and surface optimization. The problem is a rib-stiffened cylinder
with an internal support and external braces subjected to internal pressure and additional internal and external loads as shown in Fig. 9. Symmetry of the model is used. Edges 1 and 2 are allowed to move in the z direction which will change the shape of the cylinder, ribs, brace, and internal support as shown in Fig. 10. Edge 3 is allowed to move in the x direction. The internal support is defined by two edge numbers (4 and 5) to allow for movement in two directions as shown. The constraint conditions are given in Fig. 9. Figure 11 is the initial finite element model consisting of 547 facetted plate elements defined by 680 nodes. It took approximately 6 h to
Table 4. Surface and shape optimization results
stress (psi)
Min. stress (psi)
31194 31192 31192 31186 31183 31184 27436 23308 30615 40880 41804
-19703 -23461 -23423 -37192 -35208 -35398 -26425 - 27341 -22934 -31190 -31637
41ooo
41000
Max.
FEAs Iteration
SaVed
0 1 2 3 4 5 6 7 8 9
3 3 3 3 3 3 3 3 3
Allowable Total FEAs FEAs saved
72 27
FEAs 12 11 9 9 7 5 5 5 3 6
Weight (lb) 103.68 103.27 loo.47 100.21 100.01 99.52 91.50 84.86 82.55 79.97 79.96
Fig. 11. Initial finite element model of stiffened cylinder.
174
A. A. GAIESand M. L. Accoas~
Optimum
Fig. 12. Evolution of shell shape
build the model and generate the necessary input files for this problem. This problem confirmed the program’s unique ability to perform automatic adaptive mesh generation as shown in Figs 12-15. After manually generating an initial model, all of the remaining models needed for optimization were automatically generated with
no user interaction. The ribs, internal and external support geometry depend on the position of the cylindrical surface, so that changing edges 1 and 2 affects the entire model. Table 3 is a list of the finite element analysis and approximate function results at the predicted optimum shape. Table 4 contains the results from the initial
Automatic shape optimization for three-dimensional shell structures
175
Initial
Optimum
Fig. 13. Evolution of internal ribs.
design to the optimum shape. Figures 16 and 17 are plots of the minimum and maximum principal stresses versus iteration and weight versus iteration, respectively. The program generated 72 different finite element models automatically which would be very time consuming to do manually. The program also extracted the correct information, generated
approximate functions, and performed 29 linear optimizations and one nonlinear optimization. This task is extremely difficult to do manually and subject to human error. The substructuring program made decisions based on numerical values; decisions made manually by a user based on shape would be very difficult due to the complexity of the problem.
A. A. GAG and M. L. Acco~s~
176
lnitlal
Optimum
Fig. 14. Evolution of internal support and external brace.
Automatic shape optimization for threedimensional shell structures
Initial
Optimum
Fig. 15. Evolution of entire shape.
A. A. GAsxa and M. L. Accoast MMmm /manlmum stnss m
WeightH itoration
lteratkn
wright -m-m.
I
600
2
3
4
6 6 Itoration
“I
6
9
Fig. 16, Maximum and minimum principal stresses versus iteration.
Fig. 17. Shape weight versus iteration.
CONCLUSIONS
the Hessian matrix may be sufkient to perform nonlinear optimization with reasonable accuracy. Both techniques would substantially reduce the number of finite element analyses required.
Development of an automatic adaptive mesh generating program has allowed for successful investigation of shape optimization for shell structures ~dergoing Iarge shape changes. ~ubst~ctu~ng also becomes an important issue since selecting the correct edge may produce a substantial savings in cost and in some cases reduce the constraint values allowing for additional optimization, as demonstrated in the second example for iterations 5 and 6. Further investigation of the program is necessary to determine how much a shape oan be changed during each iteration while maintaining acceptable approximate function accuracy. The use of linear and nonlinear optimization in the shape optimization program may be too excessive. However, this was done to insure the best possible chance of success for this type of shape optimi~at~on program, Addi~ona~ consideration is needed to minimize the number of finite element analyses which are needed to generate data for the edge selection program and nonlinear optimizer. For example, each edge could be perturbed in the direction of decreasing cost rather than performing finear op~rni~~on for the subst~~t~ng program. Ako using the linear and diagonal terms in
REFERENCES
T. Haftka and R. V. Grandhi, Structural shape optimization-a survey. Comput. Meth. appl. Mech.
I. R.
Engng 57,91-106 (1986). 2. M. H. Imam, Three-dimensional shape optimization. Int. 3. Pbner. Me&. &gng 18, 661-673 (1982). 3. M. E. R&kin, Shape op~~~t~on of plate and shell structures. &AA frill Zo, 26g-273 (1982). 4. M. E. Botkin and J. A. Bennett, Shape optimization of three-dimensional folded plate structures. AIAA Jnl23,
18041810
(1985).
5. R. 1. Yang, A three-dimensional
shape optimization system-shop3d. Comput. Srruct. 31,8%X90 (1939). 6. S. Kodiyaksm, G. N. Vanderpktats and H. Miura, Structerat shape opti~~t~Qn with MSCINA~~N. Comput. Struct, 40, 821-829 (FM).
7. G. N. Vanderplaats, Numerical Optimization Techniques ,for Engineering Design with Appliccatians. McGraw-Hill
(1984). 8. A. A. Gates, Deveiopment of an automatic shape optimization program for shell structures undergoing large geometry changes. Ph.D. dissertation, University of Connecticut, Storm (1992).