The inclusion of body forces has received a good deal of attention in boundary element research. The consideration of such forces is essential in the desgin of high performance components such as fan and turbine disks in a gas turbine engine. Due to their critical performance requirements, optimal shapes are often desired for these components. The boundary element method (BEM) offers the possibility of being an efficient method for such iterative analysis as shape optimization. The implicit-differentiation of the boundary integral equations is performed to obtain the sensitivity equations. The body forces are accounted for by either the particular integrals for uniform body forces or by a surface integration for non-uniform body forces. The corresponding sensitivity equations for both these cases are presented. The validity of present formulations is established through a close agreement with exact analytical results

Aithal, R.

Cheng, Jizu

Kane, James H.

Saigal, Sunil

English

NASA Technical Reports Server

N89- 25187 
TREATMENT OF BODY FORCES IN 
BOUNDARY ELEMENT DESIGN SENSITIVITY ANALYSIS* 
Sunil Saigal, R. Aithal and Jim Cheng 
Mechanical Engineering Department 
Worcester Polytechnic Institute 
Worcester, Massachusetts 
*Sponsor: NSF grant MSM - 870742 
759 
https://ntrs.nasa.gov/search.jsp?R=19890015816 2020-03-20T01:50:20+00:00Z
INTRODUCTION 
The inclusion of body forces has received a good deal of attention in boundary element research. 
The consideration of such forces is essential in the design of high performance components such as 
fan and turbine disks in a gas turbine engine. Due to their critical performance requirements, optimal 
shapes are often desired for these components. The boundary element method (BEM) offers the 
possibility of being an efficient method for such iterative analysis as shape optimization. 
A survey of efforts in the area of sensitivity analysis in BEM was given by Mota, Soares and Choi 
[l]. The shape sensitivity using a finite-difference formulation was given by Wu [2] and using the 
implicit-differentiation formulation by Barone and Yang [3], Saigal et al. [4-61, and Rice and 
Mukherjee. * Mukherjee and Chrmdra [7] presented a BEM sensitivity formulation for materially 
nonlinear problems. The treatment of body forces for sensitivity analysis has not received much 
attention. 
In this paper, the implicit-differentiation of the boundary integral equations [8] is performed to 
obtain the sensitivity equations. The body forces are accounted for by either the particular integrals 
[9, lo] for uniform body forces or by a surface integration [I 11 for non-uniform body forces. The 
corresponding sensitivity equations for both these cases are presented. The validity of present 
formulations is established through a close agreement with exact analytical results. 
* Rice, J.S. and Mukherjee, S. ,  "Design Sensitivity Coefficients for Axisymmetric 
I Elasticity Problems by Boundary Element Methods", private communication. 
BOUNDARY ELEMENT ANALYSIS EQUATIONS 
Including the effect of temperature variation Q, the stress tensor CY.. is given in equation (l), and the 
equation of equilibrium is given in equation (2). Starting with a weak statement of equation (2) and 
using the divergence theorem twice the integral equation (3) is obtained. Assuming steady state 
condition, using the divergence theorem, and applying Green‘s second function leads to equation (4) 
where the thermal effects have been reduced to a boundary integral form. 
1J 
E 
( 1 4 4  &ekk- 
E vE 
( l+v)( 1 -2v) Oii = - (l+v) eij + 
E, v, and a are the modulus of elasticity, Poisson’s ratio, and coefficient of thermal expansion, 
respectively. Tij and Uij are the fundamental (Kelvin) solutions for traction and displacement, 
respectively. p and q are the load point and the field point, respectively, and R is the distance 
between these points. ui, ti , and Fi are the components of displacement, traction and body force, 
respectively. 
GRAVITATIONAL AND CENTRIFUGAL FORCE SENSITIVITY 
If the total displacement is written as a sum of a complementary and a particular integrd component 
761 
as in equation (5 ) ,  then in the absence of temperature variation, the last two terms in equation (4) 
drop out giving equation (6). Discretizing the boundary using boundary elements with 
displacements and tractions interpolated as shown in equation (7), we get the matrix relationship 
(8).Substituting from equation (5) into equation (8), we obtained a relationship in equation (9) 
including the effect of particular solutions due to body forces. Particular solutions (up) and (tp) 
were given by Banerjee and co-workers [9- 101 at SUNY - Buffalo. Implicit differentiation of equation (9) 
with respect to the design variable XL results in the sensitivity equation (10). The contribution of 
the body forces is included in the vector (fp} given in equation (1 1). 
Discretizing equation (6) using interpolation functions for displacements and tractions 
u = [H]{u} ; t = [Hl{t} 
The matrix form of equation (6) is obtained as 
Substituting equation (5) in equation (8) 
Differentiating with respect to the design variable, X, 
Where 
(7) 
The superscripts c and p refer to the complementary and particular soultions, respectively. [ H 3 is 
a matrix of interpolation functions. ( ),L denotes the derivative of ( ) with respect to the design 
variable X, 
THERMOELASTIC SENSITIVITY 
For the case of temperature variation Q and temperature gradient @,n, the tern with volume integral 
in equation (4) drops out. Then using the interpolation given in equation (7), we get the matrix 
relationship given in the equation (12). Implicit -differentiation of equation (12) leads to an equation 
similar to equation (10) but with a different definition for vector I@}. This relationship is given in 
equations( 13) and (14). The matrix [VI involves thermoelastic kernels which include elliptic 
integrals of the first and the second kind. The present sensitivity analysis requires derivatives of 
these elliptic integrals which are easily determined through chain rule of differentiation, 
{PI = [VI,L{TI + [VI{TI,L 
(T) is the vector of nodal temperatures. 
763 
SEMI- ANALYTICAL SENSITIVITY FORMULATION 
The sensitivities can now be obtained using equations (10) and (1 1) for centrifugal and gravitational 
body forces, and using equations (13) and (14) for thermal body forces. We, however, need to 
determine sensitivity matrices such as [F),L and [G],L; and sensitivity vectors such as   up},^ and 
{ tP) ,L. In the semi-analytical approach, the design variable XL is first perturbed by an amount 
AXL. The system matrices [F(XL+AXL)] , [G(XL+AXL)J, etc., are generated based on the new 
geometry. The sensitivities are then simply obtained using forward-difference relationships shown 
in equations (15) and (16). It is noted that the sensitivity results will depend on the perturbation step 
size AXkHowever, this step will result in substantial simplification of the implementation of the 
sensitivity algorithm. 
FULL ANALYTICAL SENSITIVITY FORMULATION 
I 
I For the full-analytical approach, the sensitivity matrices and vectors are directly calculated from their 
analytical expressions given in equations (17) and (18). These expressions, however, need the 
sensitivities of geometry quantities such as x , ~ ,  y , ~  , n,L, etc. The initial geometry is first used for 
solution of vectors ( u )  and { t )  in equations (8) or (12). This geometry is then changed through a ! 
perturbation AXL of the design variable. Only the geometry sensitivities are then calculated using 
forward-difference approximation. These geometry sensitivities are needed for evaluating terms in 
equations (17) and (18). 
* T  
N 1  
[F],L = C 
[G],L = C 
C [t I,J HI J + [;IT[ HI J,L I dc 
C [U I,L[ HI J + [u*IT[ HI J,L I dc 
0 j =  1 
* T  
N 1  
0 j = l  
The superscripts T and * refer to the transpose of the matrix and the fiindamental solutions, 
respectively . 
3-D Centrifugal Loading Particular Integral Sensitivities: 
h, p are the Lame's constants. xLdenotes the derivative of coordinate x with respect to the design 
variable X, and yL and zL have similar definitions.The corresponding traction sensitivities can be 
found using the constitutive relation. 
765 
SINGULAR TERMS IN SYSTEM SENSITIVITY MATRIX 
For determining terms in [F],L an extension of rigid body technique used for singular terms in m 
is used. This extension is based on the fact that the sensitivities corresponding to rigid body 
displacements and tractions are zero leading to a row-sum type property for [FJ,L. Thus from 
equation (lo), the singular terms for 3-D can be obtained as given in equation (21). For 2-D, these 
terms are similarly obtained. For axisymmetric case, a rigid body motion in Z - direction and an 
inflation mode in the radial direction are used. 
1 
0 
1 
0 
1 
0 
n- 1 n 
j = 1,3;.- j = 2,4;- 
766 
STRESS SENSITIVITY RECOVERY 
The solution of sensitivity equations yields the boundary traction sensitivities only. The stress 
sensitivities at other locations can be obtained directly using differentiated elasticity equations 
describing stress-Strain relationships. These relations are given in equation (22) for the 
axisynimetric case. 
I$., nz are the components of the outward normal in the r and z directions, respectively. 
J,L 1 
e22,L = - 2 u2,5 i- 7 2 ’ 5  L
J 
u2,kL denotes the mixed derivative with respect to the dimensionless coordinates 5 and the design 
variable X,. 
767 
NUMERICAL EXAMPLES 
The above formulations were applied to a series of selected examples to determine the design 
sensitivities for displacements and tractions. These examples include: (a) a rotating circular disk of 
constant thickness with a central hole shown in Figure 1 analyzed using two-dimensional elements, 
(b) a rotating circular disk with hyperbolic varying thickness and with a central hole shown in 
Figure 2 analyzed using axisymmetric elements, (c) a hollow cylinder under plane strain shown in 
Figure 3 subjected to pressure and temperature change and analyzed using axisymmetric elements, 
(d) a pressurized hollow cylinder under temperature variation shown in Figure 4 and analyzed using 
axisymmetric elements, (e) a solid circular bar (Figure 5) under self-weight analyzed using three-dimensional 
elements, and (f) a rotating circular disk (Figure 6) analyzed using three-dimensional elements. For examples 
I without temperature variation the material data used were E=30x107 psi, v 4 . 3 ;  and for examples 
with temperature variation the material data used were E=l psi, v=0.3, and a=0.02 /OF. The results 
obtained from the present formulations were compared with the exact solutions to check these 
formulations. For exact sensitivities the elasticity solutions were first expressed in terms of the 
design variable and then differentiated with respect to this design variable. A good comparison of 
the present results was seen from the results presented in the following pages. 
Design Sensitivity Analysis of a Rotating Circular Disk. 
Sensitivity 
Radial Displacement x (10 -3) Traction (x lo3) 
Location- Analytical This Analytical This 
Study Study 
A 6.7678 6.7677 4.8061 4.7625 
B 6.5929 6.5929 8.7301 8.6993 
C 2.8329 2.8330 3.6905 3.6923 
D 2.6684 2.6685 3.8066 3.8103 
E 6.7678 6.7678 
F 6.7678 6.7676 
E Location coordinates (r, z) in inches are: A(4.333, O.), B(5.061, O.), C(17.061, O.), D(19.0, O.), 
E(3.864, 1.035), F(3.967,0.522) 
12 ELEMENTS 8 ELEMENTS 
12 ELEMENTS 
12 ELEMENTS 
B 
0 =10 RADISEC 
4-4 IN- * 
16 IN 
FIGURE 1 
Design Sensitivity Analysis of a Rotating Disk 
with Hyperbolic Varying Thickness 
Exact 
5.07485 
5.66207 
5.99760 
6.16066 
6.20577 
6.17095 
6.08307 
5.%129 
5.81938 
5.66731 
5.51231 
5.35968 
5.21325 
5.07583 
4.94945 
4.83557 
4.73521 
4.64908 
4.57767 
4.52126 
4.48003 
4.45403 
4.44327 
4.44769 
4.46718 
RADIUS 
(inch) 
4.00 
4.25 
4.50 
4.75 
5.00 
5.25 
5.50 
5.75 
6.00 
6.25 
6.50 
6.75 
7.00 
7.25 
7.50 
7.75 
8.00 
8.25 
8.50 
6.75 
9.00 
9.25 
9.50 
9.75 
10.00 
Mesh B 
3.9470 
5.4810 
5.6433 
5.&450 
5.8609 
5.8355 
5.7574 
5.6542 
5.5318 
5.4024 
5.2668 
5.1372 
5.0076 
4.8915 
4.7787 
4.6831 
4.5934 
4.5222 
4.4589 
4.4141 
4.3782 
4.35% 
4.3511 
4.3540 
4.4120 
~ 
SENSITIVITY 
Exact 
2.05931 
2.05819 
2.04760 
2.02989 
2.00685 
1.97984 
1.94995 
1.91803 
1.88478 
1.85078 
1.81653 
1.78246 
1.74893 
1.71628 
1.68481 
1.65480 
1.62650 
1.60016 
1.57602 
1.55428 
1.53517 
1.51889 
1.50565 
1.49564 
1.48906 
Mesh A Mesh B 
1.9987 1.9986 
I 1.9887 
1.9725 1.9725 
-- 1.9522 
1.92% 1.9294 
__ 1.9046 
1.8789 1 .8785 
-- 1.8512 
1.8236 1 .8229 
- 1.7940 
1.7653 1.7645 
-- 1.7348 
1.7060 1.7052 _ _  1.6759 
1.6482 1.6474 
-- I ,6198 
1.5945 1.5937 
_- 1.5692 
1.5474 1.5466 
_- 1.5261 
1.5089 1.5081 
-- 1.4928 
1.4811 1.4803 _ _  1.4710 
1.4660 1.4652 
HOOP STR. x 10-3 
Note: Exact solution is for the assumption of plane stress. 
Mesh A 15 element model; Mesh B 30 element model 
I 
I 
D 
I 
I 
I 
I 
I 2ELEM 
+ I 
r C 
1 ELEM 
Ro= 10 
DESIGN VARIABLE: INNER RADIUS, Ri 
FIGURE 2 
Design Sensitivity Analysis of a Plane Strain Hollow 
Cylinder Under Pressure and Temperature Variation 
Exact 
I 
This 
Studv 
SENSITIVITY I 
RADIUS I RADIAL DISPLACEMENT I RADIAL STRESS 
(inch) 
3.0 
3.5 
4.0 
4.5 
5.0 
5.5 
6.0 
Exact 
3.6305 
3.6694 
3.6168 
3.5302 
3.4341 
3.3396 
3.2513 
Study 
3.6315 
3.6712 
3.6174 
3.5309 
3.4350 
3.3402 
3.2523 
Exact 
0.000 
-0.1058 
-0.1181 
-0.0971 
-0.0653 
-0.0318 
0.000 
T 
P 
This 
Studv 
0.000 
-0.1074 
-0.1186 
-0.0975 
-0.0654 
-0.0318 
0.000 
CIRCUMFERENTIAL 
STRESS 
0.5955 
0.6995 
0.7112 
0.6903 
0.6591 
0.6264 
0.5955 
+ T =3.0 
1 -  1 1 1  I b R  
I 1 
= 5.0 F 
= 1.0 PSI 
0 
4 U b 
3.0 IN 3.0 IN 
0.5956 
0.6993 
0.7110 
0.6902 
0.6594 
0.6265 
0.5956 
FIGURE 3 
771 
RADIUS 
(inch) 
1 .oo 
1.125 
1.25 
1.375 
1.50 
1.625 
1.75 
1.875 
2.00 
Pressurized Hollow Sphere Under Temperature Variation 
RADIAL DISPLACEMENT 
Exact 
1.5126 
1.758 
1.8563 
1.8985 
1.9247 
1.9526 
1.9896 
2.0384 
2.0996 
This 
Study 
1.4992 
1.7448 
1.8436 
1.887 
1.9144 
1.9433 
1.9813 
2.031 1 
2.0952 
P = 5.0 
T = 6.0 
772 
Z 
4 
SENSITIVITY 
RADIAL STRESS 
Exact 
0.000 
-0.754 
-0.901 
-0.815 
-0.653 
-0.471 
-0.297 
-0.139 
0.000 
This 
Studv 
0.0003 
-0.618 
-0.927 
-0.791 
-0.642 
-0.466 
-0.290 
-0.128 
0.0003 
L 4ELEM 
4 A- _-  D 
1.0 IN 2.0 IN 
FIGURE 4 
CIRCUMFERENTIAL 
!?l-REs 
~ 
Exact 
1.500 
1.857 
1.923 
1.879 
1.801 
1.715 
1.634 
1.562 
1.500 
This 
Study 
1.495 
1.886 
1.91 0 
1.876 
1.796 
1.709 
1.630 
1.561 
1.496 
Solid Circular Bar under Self-weight 
~ ~~ 
Location Sensitivity of displacements in Z direction ( lo3) Sensitivity of Stress in Z direction 
x=3.536 Exact Full- Semi- Exact Full- Semi- 
y =3.536 Analytical Analytical Analytical Analytical 
A(z=O.O) 0.0 0.0 0.0 2318.4 2320.4 235 1.6 
B(z=5.0) 6.7620 6.76 13 6.7614 1738.8 1747.6 1758.3 
1159.2 1188.7 1 189.8 C(z=lO.O) 11.592 1 1.589 11.589 
D(z=15.0) 14.490 14.488 14.488 579.6 587.6 589.4 
E(z=20.0) 15.456 15.448 15.448 0.0 0.0 0.0 
20 IN 
FIGURE 5 
773 
Three Dimensional Rotating Circular Disk 
~ ~~~~ 
Location Displacement Sensitivity L+ L( Radial Stress Sensitivity (104) 
y=o.o 
z=1.5 
Exact Full- Semi- Exact Full- Semi- 
Analytical Analytical Analytical Analytical 
I x= 4.0 
x= 5.0 
x= 6.0 
x= 7.0 
x= 8.0 
x=16.0 
x=17.0 
x=18.0 
x=19.0 
x=20.0 
Q,6768 
0.6613 
0.62 16 
0.5768 
0.5329 
0.298 1 
0.2840 
0.2736 
0.2668 
0.2640 
0.6775 
0.6610 
0.6212 
0.5765 
0.5326 
0.297 8 
0.2838 
0.2733 
0.2667 
0.2639 
0.6735 
0.657 1 
0.6179 
0.5737 
0.5302 
0.2969 
0.2829 
0.2725 
0.2660 
0.263 1 
0.0840 
0.8564 
0.9650 
0.90 18 
0.8004 
0.3740 
0.3691 
0.37 15 
0.3807 
0.3960 
0.1907 
0.8413 
0.9674 
0.8966 
0.8061 
0.3814 
0.3649 
0.3782 
0.3752 
0.4083 
0.2023 
0.8511 
0.9644 
0.8945 
0.8032 
0.3800 
0.3635 
0.3770 
0.3740 
0.4083 
Z 
I X 
R = 4.0 IN 16.0 IN 
FIGURE 6 
CONCLUSIONS 
The treatment of body forces of the centrifugal, gravitational, and thermal types in the 
implicit-differentiation formulation for the design sensitivity analysis of two-dimensional, 
axisymmetric, and three-dimensional problems is presented. The particular integral sensitivity 
expressions for the gravitational and centrifugal type body forces are developed. The thermoelastic 
sensitivity kernels are given for the thermal type body forces. A semi-analytical and a full-analytical 
approach for determining the sensitivity system matrices are used. A wide range of problems are 
solved for design sensitivities due to body forces and the results are validated through comparisons 
with exact analytical solutions. 
REFERENCES 
1. Mota, Soares, C.A., and Choi, K.K., "Boundary Elements in Shape Optimal 
Design of Structures", The Optimum Shape, International Symposium, 
General Motors Research Labs, Warren, Michigan (J.A. Bennett and M.E. 
Botkin, eds.), Plenum Press, New York, 1986, pp. 199-231. 
2. Wu, S.J., "Application of the Boundary Element Method for Structural Shape 
Optimization", Ph.D. Thesis, The University of Missouri, Columbia, 1986. 
3. Barone, M.R. and Yang, R.-J., "Boundary Integral Equations for Recovery of 
Design Sensitivities in Shape Optimization", AIAA Tournal, Vol. 26, No. 5, 
May 1988, pp. 589-594. 
4. Kane, J.H. and Saigal, S., "Design Sensitivity Analysis of Solids Using BEM", 
Journal of Engineering Mechanics, Vol. 114, No. 10, October 1988, pp. 1703- 
1721. 
5. Saigal, S., Borggard, J.T., and Kane, J.H., "Boundary Element Implicit 
Differentiation Equations for Design Sensitivities of Axisymmetric 
Structures", International Tournal of Solids and Structures, June 1989. 
6.  Aithal, R., Saigal, S., and S. Mukherjee, "On the Three-Dimensional 
Boundary Eleinetit Implici t-Differentiation Formulation for Structural Design 
Sensitivity Aiialysis",Computers and Mathematics With Applications, 1989. 
775 
7. Mukherjee, S. and Chandra, A., "A Boundary Element Formulation for Design 
Sensitivities in Materially Nonlinear Problems", Acta Mechanica, 1989. 
8. Banerjee, P.K. and Butterfield, R., Boundarv Elenien t Methods in Enpineering 
Science, McGraw-Hill, 1981. 
9. Pape, D.A., and Banerjce,P.K., "Treatment of Body Forces in 2D Elastostatic 
BEM using Particular Integrals", Journal of Applied Mechanics, Vol. 54, 1987, 
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10. Henry, D.P., Jr., Pape, D.A., and Banerjee, P.K., "New Axisymmetric BEM 
Formulation for Body Forces using Particular Integrals", Journal of 
Engineerin? Mechanics., Vol. 113, No. 5, 1987, pp. 671-688 
11. Rizzo, F.J., and Shippy, D.J., "An Advanced Boundary Integral Equation 
Method for Three-Dimensional Thermoelas tici ty", International Journal for 
Numerical Methods in EnEineering Vol. 11,1977, pp. 1753-1768. 
776 


Treatment of body forces in boundary element design sensitivity analysis

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