An optimization capability for involute structures has been developed. Its key feature is the use of global material geometry variables which are so chosen that all combinations of design variables within a set of lower and upper bounds correspond to manufacturable designs. A further advantage of global variables is that their number does not increase with increasing mesh density. The accuracy of the sensitivity derivatives has been verified both through finite difference tests and through the successful use of the derivatives by an optimizer. The state of the art in composite design today is still marked by point design algorithms linked together using ad hoc methods not directly related to a manufacturing procedure. The global design sensitivity approach presented here for involutes can be applied to filament wound shells and other composite constructions using material form features peculiar to each construction. The present involute optimization technology is being applied to the Space Shuttle SRM nozzle boot ring redesigns by PDA Engineering

Hart, J. K.

Stanton, E. L.

English

NASA Technical Reports Server

N89-25199 - ern 
INVOLUTE COMPOSITE DESIGN EVALUATION 
USING GLOBAL DESIGN SENSITIVITY DERIVATIVES 
J. K. Hart and E. L. Stanton 
PDA Engineering 
Costa Mesa, California 
99 1 
https://ntrs.nasa.gov/search.jsp?R=19890015828 2020-03-20T01:49:29+00:00Z
Involute Composite Design Overview 
The strong interaction between material architecture, processing and structural 
performance for nozzle components was described at an earlier NASA symposium for 
laminated involute composites. Since that meeting the Space Shuttle SRM nozzle has test 
fired involute nozzle components and progress has been made in analyzing their 
sensitivity to ply pattern design. The parameters that control ply pattern shape [ 1,2] also 
control tooling for the manufacture of involute composite structures. In the current 
CAD/CAM idiom these parameters might be called material form features and they 
provide a basis for global composite design sensitivity derivatives. They are not to be 
confused with laminate point design parameters that ignore ply continuity constraints 
present in finite dimension structural components with curvature. We first define the 
involute design problem and illustrate these commonly used approaches for composite 
shell structures. Then analytic sensitivity derivatives are developed and used to analyze 
test rings and cones with maximum stress failure criteria. 
START LINE r s k )  
I 992 
Involute Exit Cone 
Ply Material Distribution 
The figure below illustrates the orientation of the plies in a conical section of an 
involute exit cone. The intersection of a ply with a plane defined by a constant Z 
coordinate (e.g., curves AD and BC) is an involute curve. Each ply can be mapped 
to an adjacent ply by a rotation of 0 degrees about the axis of symmetry, where 
and N is the number of plies in the involute structure. 
993 
Involute Design Practice 
Industrial practice in the U.S. for involute ply pattern design at one time was limited to 
either ID or 0Dstart.lines partly because of the geometric complexity of the problem. 
Both of these design approaches have one straight edge which also makes layup and 
inspection easier. On the negative side these patterns are more difficult to form to shape 
and do not insure fiber continuity in critical stress regions. An alternative approach used 
by PDA places thestartline near the midsurface of the net part to insure fiber continuity. 
Typically a great many patterns are examined by trial and error using CAE tools in 
designing involute composite structures. Low manufacturing risk and high margins of 
safety during a motor firing are the figures of merit. The shape of the component, hence 
weight, is prescribed in most cases and rarely is this shape open to significant change. The 
shape of the ply pattern in contrast is open to wide variations and suggests the need for 
design sensitivity analyses to improve trial and error procedures and ultimately to 
automate the procedure. 
I O.D. Start-line Ply Pattern 
PDA Start-line Ply Pattern 
I.D. Start-line Ply Pattern 
994 
Involute Design Variables 
The design space includes the ply count and the ply thickness product Nt, the 
helix angle Q0, and up to 6 variables defining the start line, which is a ply 
meridian lying in the r-z plane. These variables determine at each node three 
Euler angles a, y, and which rotate the reference frame into the material 
frame. From these angles a strain transformation matrix is calculated and the 
distribution of these matrices within a finite element is used to calculate the 
element stiffness matrix. This is a global relation independent of finite 
element mesh. 
DESIGN VARIABLES - (X) 
(X) = (Nt,@,,R,(Z)) 
DEPENDENT VARIABLES - Eule Angle 
STIFFNESS MATRIX 
995 
Design Sensitivity Formulation 
The direct method of design sensitivity analysis is used. The governing 
equation for linear statics (equation 1) is differentiated to obtain equation 2. 
Equation 3 is obtained by solving equation 2 for (dU/dX). The remainder of 
the effort is directed toward evaluating (dU/dX). The finite element analysis 
already produces the factored stiffness matrix so it is only necessary to 
evaluate the part of equation 3 in parentheses. The element stiffness matrix 
in equation 4 and the element thermal load vector in equation 5 (only the 
thermal load is sensitive to material geometry orientation) may be 
differentiated to obtain equation 6. The differentiation is simplified because 
there is no shape sensitivity: the derivative of the strain-displacement 
transformation matrix [B] is zero. The new matrices [Q] and [R] are 
functions of the elasticity matrix [C], the strain transformation matrix [D] 
and its inverse, and the corresponding derivatives, all given in equations 7 
and 8. The integrands in equation 6 are developed in closed form. All 
sensitivity calculations and all finite element analyses are performed by a test 
version of the P/COMPOSITE module in PATRAN. 
Differentiate [KI{U} = {F} 
to obtain 
Then 
Given 
and 
then 
where 
and 
(3) 
(4) 
996 
Vector and Matrix Sensitivity Integration 
Originally the sensitivity integration (equation 6) was performed by 
inserting the ( U }  vector inside the second integral and making the 
substitution 
( E )  = [BINJ) 
to convert the matrix integration to a vector integration. The economy of 
this approach is evident, but finite difference tests show a failure to converge 
manifested by a "plateau" phenomenon for step sizes below a certain 
threshold. The onset of this deviation occurs at a step size that is too large to 
be attributable to round-off error. Because of this error, it was decided that 
matrix integration would be used for all sensitivity calculations. 
The accompanying graph was generated for the helix angle design variable in 
a 439 degree of freedom test cone problem, and the error shown is typical. 
C O M P R R I S O N  O F  A N R L Y T I C  AND F I N I T E  D I F F E R E N C E  
D E R I V A T I V E S  H I T H  R E S P E C T  TO H E L I X  A N G L E  
FOR VECTOR AND M A T R I X  I N T E G R A T I O N  
LEGEND I 
10-2 10-1 100 101 102 
H E L I X  RNGLE INCREMENT ( D E G R E E S )  
997 
Formulation of Optimization Problem 
The objective of the optimization is to minimize risk. To this end, the shape 
(and thus the weight) of the part are fixed and the optimization is used to find 
the ply pattern design furthest from the constraint surfaces, subject to 
manufacturing and side constraints. The mathematical formulation of the 
optimization problem is given below. The objective function is a slack 
variable p which represents the load margin (i.e., the distance between the 
load index and unity) to be maximized. The slack variable is added to each 
response constraint gj. In addition, there are manufacturing constraints hk 
which do not require the buffer of the slack variable. The Method of 
Feasible Directions algorithm [3] in MICRODOT is used to solve the primal 
form of the optimization problem. Dual methods are not used because the 
number of constraints is much greater than the number of design variables. 
Approximation concepts [4, 5 ,  and 61 are used to formulate the sequence of 
approximate problems. 
p -+ max 
Subject to gj(X) + p IO 
hk(X) 5 0 
e<xi<xF 
j = 12, ..., m 
k = 1,2, ..., 
i = 1,2, ..., n 
I 998 
Manufacturing Constraints on Design Parameters 
TRANSITION REGION 
CONE 
t s- ARC LENGTH 
Two of the manufacturing constraints alluded to on the previous page are 
shown in the picture below. The central angle is the angle subtended by a ply 
as it extends from the inner radius to the outer radius, and the arc angle is the 
angle between a ply surface tangent and the circumferential direction. Good 
design practice for nozzle components dictates that the central angle should 
not exceed 120 degrees and the arc angle should not exceed 10-15 degrees. 
I CONEKYLINOER 
999 
First Optimization Sequence For Involute Ring Problem 
The first optimization problem is an axisymmetric carbon-carbon cylinder 
having 82 degrees of freedom. The design variables are the helix angle @ and 
the slope m. Five iterations are required to increase the margin of safety 
from .708 at the starting point to 1.055 at the optimum. 
F I R S T  O P T I M I Z A T I O N  P R T H  F O R  I N V O L U T E  R I N G  
M A R G I N  O F  S A F E T Y  SHOWN F O R  EACH I T E R A T I O N  
I N T E R N R L  PRESSURE L O A D .  M Q X  S T R E S S  C R I T E R I O N  
- 4 2 0  
- 350 
- 2 8 0  
.2 10 
- 140 LEGEND 
+ O P T I H I Z R T I B N  PATH 
-32.0 -28.0 -24.0 -26.0 -16 .0  -12.0 -8.00 
H E L I X  RNGLE [ D E G R E E S )  
1000 
Summary Table for First Optimization Path 
The iteration history of the first optimization path is shown in Table 1. In the 
course of the optimization the critical stress component varies from fill 
tension in the outer element at the beginning to in-plane shear on the inner 
radius at the optimum. As the optimization proceeds, the constraint tolerance 
is reduced from .03 to the value of .001 required for convergence. The finite 
difference tests on the sensitivity derivatives were used to select move limits 
that would predict the response to within about 10 percent. The optimization 
results indicate that this choice was conservative enough. 
TABLE 1 
~~ 
.4144 .7077 -20.000O .4000 13 T2 
.4682 .8806 -30.000' .3000 lo" . 1  .03 25 T2 3 3  15 
.4938 .9755 -30.556' .2022 100 .1  .03 4 s12 9 2 31 
.5100 1.0409 -30.071' .1478 lo" .1  .008 3 s12 15 2 5 
.5127 1.0522 -30.440' .lo19 lo" .1 .008 3 s12 9 3  4 
.5133 1.0547 -30.297' .lo86 10" . 1  .001 3 s12 3 2  2 
1001 
Second Optimization Sequence For Involute Ring Problem 
A different starting point is used for the same optimization problem. Here 
four iterations are required to increase the margin of safety from .563 at the 
starting point to 1.055 at the optimum. Note that the same optimum is 
reached in both cases. 
SECOND O P T I M I Z R T I O N  P R T H  F O R  I N V O L U T E  R I N G  
I N T E R N R L  PRESSURE L O R D .  M R X  S T R E S S  C R I T E R I O N  
M A R G I N  OF S A F E T Y  SHOWN A T  EACH I T E R A T I O N  
-420 
.350 
.280 
.2 10 
-32.0 -28.0 -24.0 -20.0 -16.0 -12.0 -8.00 
H E L I X  FlNGLE [ D E G R E E S )  
1002 
Sununary Table for Second Optimization Path 
The iteration history of the second optimization path is shown in Table 2. 
Although the margin of safety shows greater improvement along this path, 
fewer iterations are required. Throughout the optimization the critical 
region is along the inner radius. The critical stress component varies from 
fill tension at the beginning to in-plane shear at the optimum. As before, the 
constraint tolerance is reduced from .03 to the value of ,001 required for 
convergence. Again, the move limits appear conservative enough. 
TABLE 2 
8 c) 
.M Y 
E 
cwc E m  c w m  
cw .s 8 Most Critical 08 0 8  0 8  
*a $ 8  0 C Constraint g y g-3 &*ti 
h 27 @ b"c 8 *e '$$ Design Variables Move Limits 
Node Component Z <  Z w  Z w  oz E m  @" 
~~ 
.3602 .5629 -10.000' 0.0000 2 n 
.4184 .7195 -20.000' .0390 100 .1 .03 2 n 4 2 29 
.5118 1.0482 -30.000' .1390 100 .1 .03 1 T2 4 3 27 
.5131 1.0537 -30.247' .1144 100 .1 .001 3 s 12 3 2  5 
.5134 1.0551 -30.324' .lo57 100 .1 .001 3 s12 3 2  2 
1003 
Involute Test Cone Optimization 
The second optimization problem is an axisymmetric carbon-carbon test 
cone tested and analyzed by Stanton and Kipp [7] and having 439 degrees of 
freedom. The model is subjected to an axial load along the aft rim and is 
constrained axially along the forward rim. The design variables are the 
product of the ply count and the ply thickness Nt, the helix angle Q0, and four 
additional variables controlling the start line. The initial design is the same 
as the final design selected by Stanton and Kipp: it is therefore expected that 
the initial design is close to optimal. Four iterations are, in fact, required to 
increase the margin of safety from 2.653 at the starting point to 2.798 at the 
optimum. 
I 
I 
I 
I 1.970 i n .  
10.870 in-- >I 
I I 15.00" 
I 
I 
I 
I I 
I I 
- _ _  / 
r( 1.970 in .  I . 5  
A 
I 1004 
Reference and Optimized Ply Pattern 
The reference ply pattern designed by Stanton and Kipp is shown below. The 
warp aligned test cone design with the start line following the midsurface of 
the shell was considered radical when it was first discussed with 
manufacturers. The design goal at that time was to develop the full strength 
of the carbon-carbon material in the critical cone-cylinder transition under 
axial load. That too was considered impractical but both are common 
practice today. 
It can be seen that the shape of the optimized ply pattern in the forward 
region has changed considerably due to changes in the start line. In addition, 
the width of the ply pattern has been uniformly reduced due to the increase in 
the ply count and the ply thickness. 
REFERENCE PLY PA'ITERN 
OPTIMIZED PLY PAITERN 
1005 
Summary Table For Test Cone Optimization Path 
The optimization history for the test cone problem is given below. 
(Max Stress Failure Criteria) 
Most 
critical 
Constraint 
Design Variables B z 
B 
Nt +o Y1 m y2 5 2 2  8 3  E v ,  % lz 
~~ ~ 
0 .7263 2.6536 2.2500 O.oo00' 2.3200 3.500~10-2 2.6225 .7500 48 C1 
1 .7328 2.7418 2.2446 6.0972O 2.3136 7.883~10-3 2.5725 .7640 53 C1 
2 .7360 2.7884 2.3803 4.7420' 2.2636 2.217~10-3 2.6225 .7776 48 C1 
3 .7366 2.7970 2.3814 5.6872' 2.2389 l.OOOxlO-3 2.6419 .7839 53 C1 
4 .7367 2.7983 2.3815 5.6538' 2.2382 l.OOOxlO-3 2.6431 .7841 53 C1 
~~ 
0 
1 .4 100 .05 .05 .05 .2 .oo 1 2 3 14 
2 .4 100 .05 .os .os .2 .oo 1 2 4 32 
3 .4 100 .05 .05 .025 .2 .oo 1 3 2 43 
4 .4 100 .05 -05 .025 .2  .0002 3 2 6 
1006 
Conclusions 
An optimization capability for involute structures has been developed. Its 
key feature is the use of global material geometry variables which are so 
chosen that all combinations of design variables within a set of lower and 
upper bounds correspond to manufacturable designs. A further advantage of 
global variables is that their number does not increase with increasing mesh 
density. The accuracy of the sensitivity derivatives has been verified both 
through finite difference tests and through the successful use of the 
derivatives by an optimizer. 
The state of the art in composite design today is still marked by point design 
algorithms linked together using ad hoc methods not directly related to a 
manufacturing procedure. The global design sensitivity approach presented 
here for involutes can be applied to filament wound shells and other 
composite constructions using material form features peculiar to each 
construction. The present involute optimization technology is being applied 
to the Space Shuttle SRM nozzle boot ring redesigns by PDA Engineering. 
- A design sensitivity capability using global material 
geometry variables has been developed. 
- The number of global variables is insensitive to finite 
element mesh density. 
- The sensitivity derivatives have been used successfully in 
an optimization context. 
- The sensitivity integral calculations in vector form 
yielded a tangible error not shared by the corresponding 
calculations in matrix form. 
- The global variable approach can be applied to other 
composite constructions using material form features 
peculiar to each construction. 
1007 
' I .  
I References 
1. Savage, E.E., "The Geometry of Involute Construction", JANNAF 
Rocket Nozzle Technology Subcommittee Meeting 1979, CPIA 
Publication 310, 1980, pp. 293-308. 
2. Pagano, N.J., "General Relations for Exact and Inexact Involute Bodies of 
3. Vanderplaats, G.N., "A Robust Feasible Directions Algorithm for Design 
Synthesis" , Proceedings, AI A A/ AS ME/ASCE/AHS , 24 th Structures, 
Structural Dynamics and Materials Conference, Lake Tahoe, Nevada, 
May 2-4, 1983. 
Revolution", AFWAL-TR-82-4053, April 1982. 
4. Schmit, L.A. and Farshi, B., "Some Approximation Concepts for 
Structural Synthesis", AIAA Journal, Vol. 12, No. 5 ,  May 1974, pp. 
692-699. 
5. Schmit, L.A. and Miura, H., "Approximation Concepts for Efficient 
Structural Synthesis", NASA CR-2552, March 1976. 
I 
I 
I 6. Lust, R.V. and Schmit, L.A., "Alternative Approximation Concepts for 
Space Frame Synthesis", NASA CR-172526, March 1985. 
7. Stanton, E.L. and Kipp, T.E., "Nonlinear Mechanics of Two-Dimensional 
Carbon-Carbon Composite Structures and Materials", AIAA Journal, 
Vol. 23, August 1985, pp. 1278-1284. 
I 
1008 


Involute composite design evaluation using global design sensitivity derivatives

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