In this paper, a method is proposed for the design optimization of structural components where both shape and topology are optimized. The boundaries of the shape of the structure are represented using contours of a “shape density” function. The contour of the density function corresponding to a threshold value is defined as the boundary of the shape. The shape density function is defined over a feasible domain and is represented by a continuous piece-wise interpolation over the finite elements used for structural analysis. The values of the density function at the nodes serve as the design variables of the optimization problem. The advantage of this shape representation is that both shape and topology of the structure can be modified and optimized by the optimization algorithm. Unlike previous methods for shape and topology optimization, the material is not modeled as porous or composite using the homogenization method. Instead the material properties of the structure are assumed to depend on the density function and many approximate material property-density relations have been studied. The shape and topology of structural components are optimized with the objective of minimizing the compliance subject to a constraint on the total mass of the structure.

Issue Section:
Research Papers
Topics:
Shapes, Topology, Density, Optimization, Design, Structural elements (Construction), Composite materials, Finite element analysis, Interpolation, Materials properties, Optimization algorithms, Structural analysis
1.
Allaire, G., and Kohn, R. V., 1992, “Topology Optimization and Optimal Shape Design Using Homogenization,” Topology Design of Structures, Bendsoe and Mota Soares, eds., Kluwer Academic Publishers.
2.
Bendsoe, M. P., Diaz, A., and Kikuchi, N., 1992, “Topology and Generalized Layout Optimization of Elastic Structures,” Topology Design of Structures, Bendsoe and Mota Soares, eds., Kluwer Academic Publishers.
3.
Bendsoe
M. P.
, and
Rodrigues
H. C.
,
1991
, “
Integrated Topology and Boundary Shape Optimization of 2-D Solids
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
87
, pp.
15
–
34
.
4.
Bendsoe
M. P.
, and
Kikuchi
N.
,
1988
, “
Generating Optimal Topologies in Structural Design Using a Homogenization Method
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
71
, pp.
197
–
224
.
5.
Haftka, R. T., and Gurdal, Z., 1992, Elements of Structural Optimization, 3rd Edition, Kluwer Academic Publishers.
6.
Haftka
R. T.
, and
Grandhi
R. V.
,
1986
, “
Structural Shape Optimization—A Survey
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
57
, pp.
91
–
106
.
7.
Hemp, W. S., 1973, “Michell’s Structural Continua,” Optimal Structures, Clarendon Press, Oxford.
8.
Kirsch, U., 1981, Optimum Structural Design: Concepts, Methods and Applications, McGraw-Hill Book Company, New York.
9.
Kohn
R. V.
, and
Strang
G.
,
1986
, “
Optimal Design and Relaxation of Variational Problems
,”
Communications in Pure and Applied Mathematics
, Vol.
39
, pp.
113
–
137
(Part I), pp. 139–182 (Part II), and pp. 333–350 (Part III).
10.
Kumar, A. V., and Gossard, D. C., 1992, “Geometric Modeling for Shape and Topology Optimization,” Fourth IFIP WG 5.2, Geometric Modeling for Product Realization, Wilson, P. R., et al., eds.
11.
Kumar, A. V., 1993, “Shape and Topology Synthesis of Structures using a Sequential Optimization Algorithm,” Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA.
12.
Papalambros
P.
, and
Chirehdast
M.
,
1990
, “
An Integrated Environment for Structural Configuration Design
,”
Journal of Engineering Design
, Vol.
1
, No.
1
, pp.
73
–
96
.
13.
Strang
G.
, and
Kohn
R., V.
,
1986
, “
Optimal Design in Elasticity and Plasticity
,”
International Journal for Numerical Methods in Engineering
, Vol.
22
, pp.
183
–
188
.
14.
Suzuki
K.
, and
N. Kikuchi
N.
,
1991
, “
A Homogenization Method for Shape and Topology Optimization
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
93
, pp.
291
–
318
.
15.
Zienkiewicz, 1967, The Finite Element Method, 3rd Edition, McGraw-Hill Book Co., U.K.
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