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Research Papers

Boundary Slope Control in Topology Optimization for Additive Manufacturing: For Self-Support and Surface Roughness

[+] Author and Article Information
Cunfu Wang

Department of Mechanical Engineering,
University of Wisconsin - Madison,
Madison, WI 53706
e-mail: cwang392@wisc.edu

Xiaoping Qian

Department of Mechanical Engineering,
University of Wisconsin - Madison,
Madison, WI 53706
e-mail: qian@engr.wisc.edu

William D. Gerstler

GE Global Research,
1 Research Cir,
Niskayuna, NY 12309
e-mail: gerstler@ge.com

Jeff Shubrooks

Raytheon Company,
870 Winter Street,
Waltham, MA 02451
e-mail: shubrooks@raytheon.com

1Corresponding author.

An earlier version of this paper appeared in ASME 2018 Manufacturing Science and Engineering Conference.

Manuscript received July 30, 2018; final manuscript received June 4, 2019; published online June 26, 2019. Assoc. Editor: Sam Anand.

J. Manuf. Sci. Eng 141(9), 091001 (Jun 26, 2019) (15 pages) Paper No: MANU-18-1575; doi: 10.1115/1.4043978 History: Received July 30, 2018; Accepted June 04, 2019

This paper studies how to control boundary slope of optimized parts in density-based topology optimization for additive manufacturing (AM). Boundary slope of a part affects the amount of support structure required during its fabrication by additive processes. Boundary slope also has a direct relation with the resulting surface roughness from the AM processes, which in turn affects the heat transfer efficiency. By constraining the minimal boundary slope, support structures can be eliminated or reduced for AM, and thus, material and postprocessing costs are reduced; by constraining the maximal boundary slope, high-surface roughness can be attained, and thus, the heat transfer efficiency is increased. In this paper, the boundary slope is controlled through a constraint between the density gradient and the given build direction. This allows us to explicitly control the boundary slope through density gradient in the density-based topology optimization approach. We control the boundary slope through two single global constraints. An adaptive scheme is also proposed to select the thresholds of these two boundary slope constraints. Numerical examples of linear elastic problem, heat conduction problem, and thermoelastic problems demonstrate the effectiveness and efficiency of the proposed formulation in controlling boundary slopes for additive manufacturing. Experimental results from metal 3D printed parts confirm that our boundary slope-based formulation is effective for controlling part self-support during printing and for affecting surface roughness of the printed parts.

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Figures

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Fig. 1

Support structures and surface roughness in AM depend on the boundary slope angle α: (a) boundary slope angle α, (b) small α with support and high surface roughness, and (c) large α without support and low surface roughness

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Fig. 2

The boundary slope angle α and non-self-supporting and smooth boundaries; α¯l is the lower bound for self-support and α¯u is the upper bound for surface roughness

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Fig. 3

Design of a 2D plate with and without boundary slope constraints: (a) design specification, (b) No α¯l, α¯u constraints, C=0.496, (c) α¯l=45deg, C=0.579, (d) α¯u=135deg, C=0.515, and (e) α¯l=45deg, α¯u=135deg, C=0.613

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Fig. 4

Directional density gradient and its Heaviside projection for the optimized design in Fig. 3(b): (a) b⋅∇γ¯, (b) b⋅∇γ¯∥∇γ¯∥, (c) H45degl(b⋅∇γ¯∥∇γ¯∥), and (d) H45degl(b⋅∇γ¯∥∇γ¯∥)b⋅∇γ¯

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Fig. 5

Measurements of the unprintability and smoothness of the solution in Fig. 3(a): (a) the projected density γ¯ by Heaviside projection, (b) the unprintable and rough boundaries H45degl(b,∇γ¯), and (c) the printable and smooth boundaries H135degu(b,∇γ¯)

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Fig. 6

Optimized designs of the 2D plate with a different selection of P¯α¯l: (a) prescribed P¯α¯l=1, C=0.538, (b) prescribed P¯α¯l=0.6, C=0.562, and (c) prescribed P¯α¯l=0.2, C=0.579

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Fig. 7

The relationship between unprintability U and the given P¯α¯l

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Fig. 8

Design of the 2D MBB beam without grayness constraint used in Ref. [46]. The volume fraction of the solid material is θ¯=0.5: (a) design specification, (b) RAMP, no α constraints, C=177.24, (c) RAMP, α¯l=45deg, C=221.20, and (d) SIMP, α¯l=45deg, C=216.38.

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Fig. 9

The convergence history for the design in Fig. 8(c)

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Fig. 10

Optimized designs of the MBB beam with the volume fraction θ¯=0.3. hmax is the mesh size, and r is the filter radius: (a) no α constraint, hmax = 0.01, r = 0.175, C=300, (b) α¯l=45deg, hmax = 0.01, r = 0.175, C=1965, (c) α¯l=45deg, hmax = 0.01, r = 0.35, C=3456, and (d) α¯l=45deg, hmax = 0.005, r = 0.175, C=550

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Fig. 11

Optimized design of the MBB beam with the volume fraction θ¯=0.1. hmax is the mesh size and r is the filter radius: (a) no α constraint, hmax = 0.0025, r = 0.1, C=1752 and (b) α¯l=45deg, hmax = 0.0025, r = 0.1, C=22,626.

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Fig. 12

Optimal design of 2D heat conduction: (a) design specification and (b) optimized design C = 4836.25

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Fig. 13

Optimized designs of the 2D heat conduction problem with boundary slope constraints: (a) α¯l=30deg, P¯α¯l=0.039, C = 4876.64, (b) α¯l=45deg, P¯α¯l=0.038, C = 4830.45, (c) α¯l=60deg, P¯α¯l=0.037, C = 4983.33, (d) α¯u=150deg, P¯α¯u=0.064, C = 4906.73, (e) α¯u=135deg, P¯α¯u=0.256, C = 5081.01, (f) α¯u=120deg, P¯α¯u=0.063, C = 5459.38, (g) α¯l=30deg, α¯u=150deg, P¯α¯l=0.038, P¯α¯u=0.25, C = 4913.21, (h) α¯l=45deg, α¯u=135deg, P¯α¯l=0.037, P¯α¯u=0.26, C = 4892.91, and (i) α¯l=60deg, α¯u=120deg, P¯α¯l=0.039, P¯α¯u=0.252, C = 6162.64

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Fig. 14

Optimized designs of the 2D heat conduction problem based on the SIMP scheme without the grayness constraint in Ref. [46]: (a) no α¯l, C=5959.29 and (b) α¯l=45deg, C=6645.03

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Fig. 15

Optimal design of 3D heat conduction: (a) design specification and (b) optimized design C = 2774.04

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Fig. 16

Optimized designs of the 3D heat conduction problem with boundary slope constraints: (a) α¯l=30deg, P¯α¯l=0.006, C = 2794.30, (b) α¯l=45deg, P¯α¯l=0.006, C = 2807.88, (c) α¯l=60deg, P¯α¯l=0.006, C = 2834.56, (d) α¯u=150deg, P¯α¯u=0.031, C = 2826.04, (e) α¯u=135deg, P¯α¯u=0.031, C = 2847.46, (f) α¯u=120deg, P¯α¯u=0.031, C = 2897.73, (g) α¯l=30deg, α¯u=150deg, P¯α¯l=0.005, P¯α¯u=0.031, C = 2834.22, (h) α¯l=45deg, α¯u=135deg, P¯α¯l=0.006, P¯α¯u=0.028, C = 2862.83, and (i) α¯l=60deg, α¯u=120deg, P¯α¯l=0.007, P¯α¯u=0.031, C = 2909.41

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Fig. 17

Bottom view and cross-sectional view of the optimized designs in Figs. 15(b),16(b),16(e), and 16(h): (a) no α¯l,α¯u constraints, (b) α¯l=45deg, (c) α¯u=135deg, and (d) α¯l=45deg, α¯u=135deg

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Fig. 18

Design specifications of the thermoelastic problems with uniformly distributed temperature: (a) 2D and (b) 3D

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Fig. 19

Optimized design of the 2D thermoelastic problem with uniformly distributed temperature: (a) no α¯l, α¯u constraints, C = 27,249.3, (b) α¯l=45deg, C = 32,138.4, (c) α¯u=135deg, C = 31,244.5, and (d) α¯l=45deg, α¯u=135deg, C = 45,392.3

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Fig. 20

Optimized designs of the 3D thermoelastic problem under uniformly distributed temperature: (a) no α¯l,α¯u constraints, C = 4.26, (b) α¯u=135deg, C = 4.26, (c) α¯l=45deg, C = 4.29, and (d) α¯l=45deg, α¯u=135deg, C = 4.29

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Fig. 21

Optimization history for 3D thermoelastic structures under uniform temperature: (a) without overhang constraints and (b) with lower and upper bounds on the overhang angle

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Fig. 22

3D printed parts for the designs in Fig. 20: (a) no α¯l,α¯u constraints, (b) α¯l=45deg, (c) α¯u=135deg, and (d) α¯l=45deg, α¯u=135deg

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Fig. 23

Different views of the part in Fig. 22(b): (a) bottom view of the printed part, (b) magnified regions A and B, and (c) printed part and STL model near region C

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