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Optimal Design of Stiff Structures subjected to Body Forces
P.VasundharaKumari
PakeerurajuPodugu
Chauhan Rupesh
Scientist 'C'
Product Design &
Engineering Division
Defence Research &
Development Laboratory
(DRDL)
Kanchanbagh,
Hyderabad-500 058 (India)
Scientist 'D'
Product Design &
Engineering Division
Defence Research &
Development Laboratory
(DRDL)
Kanchanbagh,
Hyderabad-500 058 (India)
Scientist ‘C'
Product Design &
Engineering Division
Defence Research &
Development Laboratory
(DRDL)
Kanchanbagh,
Hyderabad-500 058 (India)
Keywords: Topology Optimization, Structural optimization, Inertial forces
Abstract
In this paper, we present a methodology for designing a stiffest structure in presence of body forces to get an optimal material
distribution thereby reducing the design cycle time. In general, excessive deformations of the structure will lead to uncertainties in the
measurements/outputs of the experiment. This kind of problems is usually dealt by minimizing the mean compliance with volume
constraint but without considering the body forces to make the problem convex i.e. to have a unique solution. But, presence of body
forces will make the problem difficult for the required convergence. One such application is test fixture for a rocket motor which is made
stiffer in presence of body forces and thrust loads. This methodology can be extended to similar structural requirements. In this work,
HyperMesh and OptiStruct modules of HyperWorks software are used to get optimal distribution of the material.
Introduction
Stiff structures are generally preferred for many applications such as bridges, tall buildings, aerospace
applications, automobiles etc. Optimal stiff structures are usually obtained using different metrics such as
mean compliance (i.e. work done by the external forces), maximum deflection, maximum stress and
fundamental frequency. These metrics are used in optimization in the following ways.
•
Minimize the mean compliance (maximum global stiffness) [1]
•
Minimize the maximum deflection
•
Minimize the maximum stress
•
Maximize the fundamental frequency
Among these ways, most commonly followed method is to minimize the mean compliance with a volume
constraint. In the absence of density dependent body forces and design dependent loads, this problem is a
convex problem i.e. it has a unique solution. But the presence of body forces, for example self-weight and
inertial loads, will lead to difficulties such as non-monotonous behaviour of the compliance, possible
unconstrained character of the optimum and the parasitic effect for low densities when using the power
model (SIMP) [2].
Test fixtures for characterizing rocket motors are generally designed for stiffness such that misalignment
due to elastic deflectionis minimized. There can be deviation between the geometric centreline of a rocket
nozzle and the true position of the line of thrust of the rocket, called “gas misalignment”, which is to be within
permissible limit [3]. It is little difficult to measure the gas misalignment due to lack of adequate equipment.
The misalignment is estimated by measuring the side forces. A facility is required to measure thrust line
angle and thrust-line displacement with reasonable accuracy. Six-component force is required to define
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completely the magnitude and position of a force in space. Three types of arrangements can be considered
in designing of a thrust-misalignment test setup.
1. The rocket motor is mounted horizontally, as in a conventional static firing, with the thrust reaction
and three other reactions horizontal and the remaining two vertical.
2. The rocket motor is mounted vertically with the thrust distributed among three vertical reactions, and
the three other reactions in a horizontal plane.
3. The rocket motor mounted vertically, with the thrust reaction in a single force, the remaining five
reactions being distributed horizontally in two planes.
For the problem at hand, configurations 1 and 3 were considered. Process methodology including
optimization problem formulation is detailed in the next section. Subsequently results & discussion and
conclusions are explained.
Optimization Process Methodology
The continuum element method is used in topology optimization. Here the domain is discretized into a
number of elements as it is usually done for finite element analysis (FEA). The optimization algorithm
decides where to keep material and where not to keep so that a desired objective is met. This is basically a
discrete optimization problem. The most common approach to solve this problem by optimization is by
replacing the integer variables with continuous variables and then pushing these values to either zero or one
by introducing penalty. This is called the Simple Isotropic Material with Penalization (SIMP) approach. The
is given by
young's modulus of element,
is Young's modulus of the base material
is fictitious density of element and
n is the penalty number
Penalty number of 3 generally gives good results using SIMP approach.
where
Problem formulation
The optimization problem can be mathematically defined as
Subjected to
Where
is volume of the structure
is the prescribed/specified volume
is global stiffness matrix
is global displacement vector
is force vector due to body forces
is force vector due to other loads
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Solution Methodology
Two rocket motor mounting arrangements are studied and their details are explained as follows. Design
domains for horizontal and vertical mounting arrangements of the rocket motors are shown in Figure 1 and
Figure 2. They are meshed with hexahedral elements. Thrust and side forces are applied and bottom
surface is fixed. Acceleration due to gravity is considered.
Figure 1: Design domain for horizontal mounting
Figure 2: Design domain for Vertical mounting
The parameters used in FEA and Optimization are shown in
Table 1.
Table 1: Parameters used in FEA and Optimization
Material
Steel
Young’s Modulus
200 GPa
3
Density
7800 kg/m
Poisson’s Ratio
0.3
As the deflections are small, linear FEA is used to solve the equilibrium equations and the optimization is
solved using gradient-based optimization technique available in OptiStruct module of HyperWorks and the
results are presented in the next section.
Results & Discussions
Optimal material distribution for both of the mounting arrangements are shown in Figure 3 and Figure 4.
Optimum solution were obtained after 116 and 91 iterations for horizontal and vertical mounting
arrangements respectively.
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Figure 3: Optimal material distribution for horizontal
mounting
Figure 4: Optimal material distribution for vertical
mounting
Deformed profile for both the mounting schemes are shown in Figure 5 and Figure 6.
Figure 5: Deflection plot for horizontal mounting
Figure 6: Deflection plot for vertical mounting
Von-Mises stress distribution for both the mounting schemes are shown in Figure 7andFigure 8.
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Figure 7: Von-Mises stress plot for horizontal
mounting
Figure 8: Von-Mises stress plot for vertical
mounting
From optimal topology as shown in Figure 3 and Figure 4, they are engineered with hollow standard
sections for ease of manufacturing. Engineered configurations for both the mounting schemes are shown in
Figure 9 and Figure 10.
Figure 10: Engineered configuration for vertical
Figure 9: Engineered configuration for horizontal
mounting
mounting
As the optimal topology is altered, engineered configurations are analysed and verified for deflections and
stresses. Deformed profiles for both the mounting schemes are shown in Figure 11 and Figure 12.
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Figure 11: Deflection plot for horizontal mounting
Figure 12: Deflection plot for vertical mounting
Von-Mises stress distribution for both the mounting schemes are shown in Figure 13 and Figure 14.
Figure 13: Von-Mises stress plot for horizontal
Figure 14: Von-Mises stress plot for vertical
mounting
mounting
From the results, it was observed that maximum deflections for both the configurations are within the
specified limits (upto 1.0 mm).
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Benefits Summary
In conventional design, finding the material distribution, meeting the design specifications simultaneously, is
iterative and time consuming. But the optimization based design hinders the evolution. With this approach,
the design cycle time is considerably reduced.
Challenges
Presence of density dependent body forces will make any optimization algorithm difficult for convergence.
Many times the solution diverges [2]. The gradient based optimization algorithm in OptiStruct module of
Altair HyperWorks handled the non-convex problem successfully.
Conclusions
The optimization based design helped in obtaining geometric configuration for both the mounting
arrangements. The methodology developed here will be implemented for similar structural requirements.
ACKNOWLEDGEMENTS
The authors would like to thank Shri A. Balasubramanian, Scientist‘F’, Head PDED and Dr. K Ramesh
Kumar, Scientist‘G’, DPTT for motivation and providing the required facilities.
REFERENCES
[1]
[2]
[3]
[4]
[5]
Bendsøe, M.P., and Singmund, O., “Topology Optimization – Theory, Methods and Applications”, Springer Verlag, New York,
2003.
M. Bruyneel and P. Duysinx, “Note on topology optimization of continuum structures including self-weight”, Structural
Multidisciplinary Optimization (2005) 29, pp 245-256.
W.L. Rogers, “Determination of thrust alignment in Rocket Engines”, Journal of the American Rocket Society, Vo. 23 No. 6 (1953),
pp. 355-359
Peter W Christensen and Anders Klarbring, “An Introduction to Structural Optimization”, Springer Science, 2009
Optistruct Optimization –Analysis, Concept and Optimization , HyperWorks 11.0
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