COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Topological Optimization Analysis of 3-D Continuum Structure
with Stress and Displacement Constraints”
H. L. Ye*, Y. K. Sui
Numerical Simulation Center for Engineering, Beijing University of Technology, Beijing, 100022 China
Email: [email protected], [email protected]
Abstract
It is well known that topology optimization of 3D continuum structure is one of the major challenges
because of difficulty to establish a good geometric model which comprising a large number of design
variables, and complexity of optimization algorithm. On the other hand, the problem under multiple load
case is not easy to be approached than one under single load case, because the former becomes a multiple
objective problem based on compliance objective function. In order to overcome these difficulties, the
optimal topology model of 3D continuum structure is established based on ICM (Independent Continuous
Mapping) method, which refers to weight as objective and subjected to stress constraints and displacement
constraints with multi-load-cases. A globalization of stress constraints is proposed by virtue of the von
Mises’ yield criteria in theory of elastic failure. Thus, transformation of all elements’ stress constraints into a
structural energy constraint is achieved, namely, a global constraint substitutes for lots of local constraints.
As a result, the numbers of constraints is reduced，and the complexity of the sensitivity analysis is decreased.
For global displacement constraints, an explicit expression of displacement with respect to the topological
variables is formulated by using of unit virtual load method. In order to decrease the error of numerical
calculation generated by the order magnitude between different physical quantities, the optimal model that
normalizes with two types of dimensionless constraints is further derived for continuum structure with stress
constraints and displacement constraints. Furthermore, the best path transmitted force in the multiple load
cases is selected successfully. The dual quadratic programming is applied for to solve the optimal model of
continuum. Consequently, the number of design variables is dramatically decreased; the efficiency of
computation is improved. In addition, the present optimal model and its algorithm have been implemented
by means of the MSC/Patran software platform using PCL. Several numerical examples indicate that the
method is effective and efficient. As an example, Figure 1 illustrates the background structure of an elastic
body with size 10×0. 6×2 m3. The two corner points on the bottom side are fixed. Four central forces P1
= P2 = P3 = P4 = 450 kN are located in the middle of top of beam. The allowable stress is 50 Mpa, and the
displacement constraint value of the four nodal points where the forces are located is less than 0.8 mm along
with the up-to-down direction. The optimal topology configuration of the structures is illustrated in Figure 2.
Figure 1: Finite element model
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
”
Figure 2: Optimal topology configuration
Supported by the National Natural Science Foundation of China (10472003), Beijing Natural Science Foundation (3002002),
Beijing Educational Committee (KM200410005019) and the Foundation of Beijing University of Technology for PhD.
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