Optimality criteria method for topology optimization under multiple constraints
Introduction
Structural topology optimization received considerable attention in recent years. Several approaches based on a density-like function were proposed [1], [2], but resulted in optimization models with rather large number of design variables. Non-linear mathematical programming for such problems, on the other hand, is costly and time consuming. An attractive alternative is the optimality criteria method, which solves the optimality conditions directly if closed-form expressions can be derived [3]. The existing framework of optimality criteria method, however, is limited to the optimization of a simple energy functional (compliance [4] or eigenfrequencies) with a single constraint on material resource, as pointed out in the Refs. [2], [5], [6].
The present paper extends the optimality criteria method to problems with multiple constraints, and provides a generalized way to extract the optimality criteria. In the same spirit, the optimality criteria for tunnel support [7] and compliant mechanism [8] were established and solved.
Two types of problems exist in the literatures on topology optimization. One is to minimize a performance function, subject to equilibrium equations and the constraint on material resource. The other is to minimize the material resource, subject to equilibrium equations, and performance functions. The present work will focus on the second type of problems, namely to minimize the material volume under multiple displacement constraints.
Section snippets
Optimality criteria under multiple constraints
The problem of topology optimization under multiple constraints can be stated as follows:such thatwhere μ denotes a design variable with the lower bound μmin and the upper bound μmax, ρ(μ) the local density of the material, the material stiffness, the design domain, Γt the traction boundary, t the traction acting on the structure, and the actual and the kinematically admissible displacements of the structure, hα
Multiple displacement constraints
Consider a short cantilever beam of in-plane dimension 80L×80L with multiple displacement constraints, as shown in Fig. 1a. Two upward point forces of intensity P=0.01E0L are applied at the upper corner A and the lower corner B of the right side of the beam. The displacement constraints arewhere uA2, uB2 are the vertical displacements of points A and B.
The domain is divided into 6400 elements. The following artificial material model is used:where
Formulation
The works of Sigmund [13], Sigmund and Gibiansky [14], Silva et al. [15] and Neves et al. [16] promote the investigation on topology optimization design of material cells. The early work of Sigmund [13] employed the optimality criteria method. His subsequent works on continuum-type problems, however, addressed the difficulty on using optimality criteria method to reach satisfying results. We will show that the framework of Section 2 gives the remedy to this difficulty.
For a structure consisting
Conclusions
1. We revise the optimality criteria method to accommodate multiple constraints. The gradient-split Taylor series expansion is employed to present the relationship between constraints and design variables in an explicit form. The computational cost for updating the Lagrangian multipliers is proved to be rather minor when the number of displacement constraints is small. The computational burden mainly comes from the analyses of various adjoint structures.
2. The power of the method is illustrated
Acknowledgments
The authors would like to thank for the support by National Natural Science Foundation of China.
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