Elsevier

Computer-Aided Design

Volume 81, December 2016, Pages 70-80
Computer-Aided Design

Minimum void length scale control in level set topology optimization subject to machining radii

https://doi.org/10.1016/j.cad.2016.09.007Get rights and content

Highlights

  • This paper presents a minimum length scale control method for structural topology optimization.

  • Two lower bounds are concurrently applied which corresponds to the rough and finish machining operations, respectively.

  • The rough machining lower bound is satisfied by developing a signed distance-related constraint.

  • The finish machining lower bound is addressed through the curvature flow control.

Abstract

This paper presents a minimum void length scale control method for structural topology optimization. Void length scale control has been actively investigated for decades, which intends to ensure the topology design manufacturable given the machining tool access. However, only a single lower bound has been applied in existing methods, which does not fit the multi-stage rough-to-finish machining. To fix this issue, the proposed minimum void length scale control method employs double lower bounds which corresponds to the rough and finish machining operations, respectively. This method has been implemented under the level set framework. For technical details, the rough machining lower bound is satisfied by developing a signed distance-related constraint, which ensures enough space for the rough machining tool movement and thus, guarantees the machining efficiency. The finish machining lower bound is addressed through the curvature flow control, which ensures the small features manufacturable and also a good finish dimension and surface. Through a few numerical case studies, it is proven that the minimum void length scale can be effectively controlled without sacrificing much of the structural performance.

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, Φ(X):RnR, represents any structure in the implicit form, as: {Φ(X)>0,XΩ/ΩΦ(X)=0,XΩΦ(X)<0,XD/Ω where Ω represents the material domain, D indicates the entire design domain, and thus D/Ω 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. Line1:Min.J(u,Φ)=D12De(u)e(u)H(Φ)dΩLine2:s.t.a(u,v,Φ)=l(v,Φ),vUadLine3:DH(Φ)dΩVmaxLine4:1R1<κ(X)0,for anyXΩvorΩbvLine5:κ=n=(Φ(X)|Φ(X)|)Line6:Φ(Pvi)K2,i=1,2,,nLine7:a(u,v,Φ)=DDe(u)e(v)H(Φ)dΩLine8:l(v,Φ)=DpvH(Φ)dΩ+Dτvδ(Φ)|Φ|dΩ.

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: Dv=103D in which Dv 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).

References (53)

  • M. Zhou et al.

    Minimum length scale in topology optimization by geometric constraints

    Comput Methods Appl Mech Engrg

    (2015)
  • S. Chen et al.

    Shape feature control in structural topology optimization

    Comput-Aided Des

    (2008)
  • X. Guo et al.

    Explicit feature control in structural topology optimization via level set method

    Comput Methods Appl Mech Engrg

    (2014)
  • J. Liu et al.

    A novel CACD/CAD/CAE integrated design framework for fiber-reinforced plastic parts

    Adv Eng Softw

    (2015)
  • J. Luo et al.

    A new level set method for systematic design of hinge-free compliant mechanisms

    Comput Methods Appl Mech Engrg

    (2008)
  • Q. Xia et al.

    Constraints of distance from boundary to skeleton: For the control of length scale in level set based structural topology optimization

    Comput Methods Appl Mech Engrg

    (2015)
  • N. Gardan et al.

    Topological optimization of internal patterns and support in additive manufacturing

    J Manuf Syst

    (2015)
  • Y. Wang et al.

    Length scale control for structural optimization by level sets

    Comput Methods Appl Mech Engrg

    (2016)
  • J. Liu et al.

    A survey of manufacturing oriented topology optimization methods

    Adv Eng Softw

    (2016)
  • M.P. Bendsøe et al.

    Topology optimization

    (2004)
  • T. Borrvall et al.

    Topology optimization of fluids in Stokes flow

    Int J Numer Methods Fluids

    (2003)
  • S.-H. Ha et al.

    Topological shape optimization of heat conduction problems using level set approach

    Numer Heat Transfer B

    (2005)
  • T. Yamada et al.

    A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects

    J Mech Des

    (2011)
  • N.P. van Dijk et al.

    Level-set methods for structural topology optimization: a review

    Struct Multidiscip Optim

    (2013)
  • G.I.N. Rozvany

    A critical review of established methods of structural topology optimization

    Struct Multidiscip Optim

    (2009)
  • O. Sigmund et al.

    Topology optimization approaches

    Struct Multidiscip Optim

    (2013)
  • Cited by (26)

    • Hole control methods in feature-driven topology optimization

      2023, Computer Methods in Applied Mechanics and Engineering
    • Review on design and structural optimisation in additive manufacturing: Towards next-generation lightweight structures

      2019, Materials and Design
      Citation Excerpt :

      The authors optimized a cantilever beam for self-supported AM (admissible inclination angle of 45°), using a multi-LS interpolation approach (mutual dependence between adjacent layers), to illustrate their method (see Fig. 8a). Similarly, the skeletonisation of 2D LS topologies was utilized in [149] to constrain the minimum hole size and control the number of holes in the topology (based on [150]) to ensure better manufacturability. In [151], it was applied to avoid small struts by controlling the minimum or maximum length scale of the features i.e. the distance between skeleton and boundary.

    • Piecewise length scale control for topology optimization with an irregular design domain

      2019, Computer Methods in Applied Mechanics and Engineering
    View all citing articles on Scopus
    View full text