COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Structural Topology Optimization Using Level Set Method
Michael Yu Wang *
Department of Automation & Computer-Aided Engineering, The Chinese University of Hong Kong , Shatin, NT, Hong
Kong, China
Email: yuwang@acae.cuhk.edu.hk
Abstract In this paper, we specifically put forth the level set method, in which the shape of a structure is
employed and modified directly such that certain design objectives are obtained. It yields a complete solution
space and allows for the application to complex engineering problems, including material, boundary and geometric
behavior, linear or nonlinear. A bridge type structure is presented here to illustrate the technique.
Key words: topology optimization, level-set method, bridge type structure
INTRODUCTION
Structural optimization, in particular the shape and topology optimization, plays an important role in structural design
and analysis. Optimization techniques could deliver a design with optimal performance requirements such as strength,
stiffness, weight, natural frequency, or buckling. Particularly, topology optimization has the greatest potential for the
improvement in structural performance, since topology optimization procedures permit changes in the connectivity of
the geometry of the structure during the design process. Over the past decade, there have been some extensive
developments of various approaches to topology optimization [1]. Here, we specifically put forth the level set method,
in which the shape of a structure is employed and modified directly such that certain design objectives are obtained.
The level-set method is a versatile and efficient technique for problems involving the motion of curves and surfaces
[2]. It is by now a classical tool in many fields of applications such as fluid mechanics and image processing [2]. For
structural topology optimization, the technique makes use of the classical shape sensitivity in an Eulerian framework.
One attractive attribute of the method is that it gives a natural way of describing closed boundaries (curves or surfaces)
and allows for automatic changes of topology, such as merging and breaking of boundaries, with calculations easily
made on a fixed rectilinear grid. Over the recent years, the technique has been developed for many topology
optimization problems, including solid structures [3] materials [4], and compliant mechanisms [5].
THE LEVEL SET METHOD
The general problem of structure optimization is specified as
Minimize J ( u, Ω ) = ∫ F ( u ) dΩ
Ω
Ω
subject to:
∫
Ω
(1)
dΩ ≤ M
In this standard notion, the goal of optimization is to find a structure Ω ⊆ R d (d = 2 or 3) as a minimizer for the design
criterion J ( u , Ω ) with respect to a specific function described by F ( u ) . The inequality describes the limit on the
amount of material in terms of the maximum admissible volume M of the design. The structure is either linear or
nonlinear with displacement field u solved from its elastic equilibrium equation. The design function F ( u ) may
involve any physical or geometric quantity of the structure, such as the mean compliance, stress, or natural frequency.
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The key concept of level set technique is to embed the structural boundary Γ as the zero level-set of the implicit
function, Φ : R d a R , such that Γ = { x : Φ ( x ) = 0} . As illustrated in Fig. 1 for a two-dimensional structure, the
boundary curves are embedded in three-dimensional function Φ ( x ) with a fixed topology. The embedding Φ of d + 1
dimension accommodates not only the shape boundary but also the global and regional attribute of the shape, i.e., the
shape interior.
Figure 1: The 2D geometric shape is embedded within a level set function
The topology optimization of (1) then can be described as a dynamic process of level set changing in pseudo-time t.
The surface of the embedding function may move up and down on a fixed coordinate system without ever altering its
topology. The structural boundary embedded on Φ ( x ) can undergo drastic topological changes. However, there is no
need to directly track these structural topological changes. The evolution of the implicit function Φ ( x ) is described by
the Hamilton-Jacobi convection equation
∂Φ ( x )
∂t
+ ∇Φ ( x ) ⋅ V ( x ) = 0
(2)
where V = dx dt is the velocity function.
Several features and advantages of this method representing the solid shape include:
(1) First, level set models are topologically flexible. The scalar function Φ would allow the boundary models to easily
change the structural topology while undergoing optimization in that they can form holes, split to form multiple
boundaries, or merge with other boundaries to form a single surface.
(2) The level set equation will not change the parameterization of the solid shape; the level set formulation is a
parameterization free formulation.
(3) The level set model provides a natural way to accommodate a convergent sequence for the optimal solution with a
suitable choice of the velocity field.
(4) Further, a number of numerical techniques have been developed to make the initial value problem of (2)
computationally robust and efficient [2].
ILLUSTRATION EXAMPLE
A bridge type structure is presented here to illustrate the technique. A rectangular design domain of L long and H high
with a ratio of L : H = 12 : 6 represents a bridge loaded vertically at the center point of its bottom with P = 30 N as
shown in Fig. 2. The left bottom corner of the beam is fixed, while it is simply supported at the right bottom corner.
Only 31% of the material in the design domain is allowed to be used for the bridge structure, and the goal of the design
is to find the structure with highest stiffness or lowest mean compliance. The initial design and some intermediate and
the final optimization results are shown in Fig. 3.
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Figure 2: A bridge type structure with fixed-simple supports. (a) Initial design
(b – g) Intermediate results. (h) Final solution of optimal structure
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CONCLUSION
Topology optimization using level-set method is a novel technique for structure optimization. It yields a complete
solution space and allows for the application to complex engineering problems, including material, boundary and
geometric behavior, linear or nonlinear. It is suitable for complex problems and has been successfully used in industry
for engineering designs.
REFERENCES
1. Bendsoe MP, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer, Berlin,
Germany, 2003.
2. Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. Springer, New York, USA, 2003.
3. Wang MY, Wang X, Guo D. A level set method for structural topology optimization. Computer Methods in
Applied Mechanics and Engineering, 2003; 192(1): 227-246.
4. Wang MY, Zhou SW. Synthesis of shape and topology of multi-material structures with a phase-field method.
Journal of Compute-Aided Materials Design, 2004; 11(2-3): 117-138.
5. Wang MY, Chen SK, Wang X, Mei Y. Design of multimaterial compliant mechanisms using level-set methods.
Journal of Mechanical Design, Trans. of ASME, 2005; 127(5): 941-956.
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