Minimum void length scale control in level set topology optimization subject to machining radii
Introduction
Topology optimization has been actively investigated in the past few decades, which is now a major structural design methodology. Compared to trial and errors, topology optimization employs the automatic sensitivity-driven design loops which ensures outstanding efficiency and optimality; in addition, the method does not require much about the initial guess to produce creative design solutions, which makes it widely accepted by the design community.
Currently, SIMP (Solid Isotropic Material with Penalization) [1], ESO (Evolutionary Structural Optimization) [2], and level set [3], [4] are the main topology optimization methods. These methods all have their unique characteristics and at the same time, are tightly associated. A broad range of design problems governed by different physical disciplines have been solved through these methods, i.e. solid mechanics [1], [2], [3], [4], [5], [6], fluid dynamics [7], [8], and thermal dynamics [9], [10], [11], etc. A few comprehensive literature surveys can be found in [12], [13], [14].
On the other hand, topology optimization is still poorly developed given certain engineering backgrounds. A critical challenge is the manufacturability. In the past two decades, quite a few research publications have been dedicated to addressing this challenge. Length scale control is concerned for machining parts, which has been addressed based on both SIMP [15], [16], [17], [18], [19], [20], [21], [22], [23], [24] and level set [25], [26], [27], [28], [29], [30]; No interior void and undercut restrictions are necessary for injection molding/casting parts, which have also been addressed based on SIMP [31], [32], [33], [34], [35], [36] and level set [37], [38], [39]; recently, manufacturability of 3D printed parts has also attracted plenty of concerns [40], [41], [42], [43].
In this work, we focus on the length scale control of machining parts. As mentioned in the last paragraph, length scale control has been realized through both SIMP and level set. For SIMP, both the void and solid phases have been effectively controlled about the maximum and minimum length scales. For level set, the length scale control has been mainly implemented on the solid phase, but not the void, even though it is technically realizable. On the other hand, the existing minimum void length scale control methods only employ one lower bound, while in practice, the machining process is conducted from rough-to-finish by utilizing several machining tools. Therefore, a better method is needed which employs multiple lower bounds corresponding to the different-sized machining tools. Hence, a major contribution of this paper is the minimum void length scale control subject to multiple lower bounds, and this research is conducted under the level set framework.
Section snippets
Literature survey
Length scale control has been a long-lasting and challenging research problem which intends to guarantee the topology design manufacturable. To be specific, the void length scale should be larger than the minimum machining tool size, and the component length scale should not be too small because of the induced machining difficulties.
The pioneering works were the filtering method [44] and the local gradient constraint method [45], which were developed to eliminate the checker-board patterns
Basic introduction to level set function
Level set function, , represents any structure in the implicit form, as: where represents the material domain, indicates the entire design domain, and thus represents the void.
Generally, the level set field satisfies the signed distance regulation through solution of Eq. (2), through which absolute of the level set value at any point represents its shortest distance to the structural boundary and the sign indicates the point to be either
Optimization problem and its solution
A typical compliance minimization topology optimization problem including the void length scale constraints is formulated in Eq. (8). The structural compliance is to be minimized subject to the material volume fraction constraint.
In Eq. (8), line 1
Numerical examples
In this section, a few numerical examples will be studied to prove the effectiveness of the proposed minimum void length scale control method. All the implementations are based on Matlab.
For all the numerical examples, the finite element analysis (FEA) is performed based on fixed quadrilateral meshes and the artificial weak material is employed for voids in order to avoid the stiffness matrix singularity, which is: in which is the elasticity tensor of the void.
The volume constraint
Conclusion
In this paper, the minimum void length scale control is well addressed under the level set framework. Innovatively, two lower bounds are concurrently applied which correspond to the different machining tool radii of the rough-to-finish machining process. The derived optimal design demonstrates the outstanding characteristics that, both rough and finish machining operations can be effectively performed, through which both the machining efficiency and quality can be guaranteed.
As for the side
Acknowledgments
The authors would like to acknowledge the support from China Scholarship Council (CSC) (2011637036) and Natural Sciences and Engineering Research Council of Canada (NSERC) (RGPIN-2014-05641).
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