Topology optimization based on reduction methods with applications to multiscale design and additive manufacturing

doi:10.1007/s11465-019-0564-8

Front. Mech. Eng.

2020, Vol. 15

Issue (1) : 151-165 https://doi.org/10.1007/s11465-019-0564-8

RESEARCH ARTICLE

Topology optimization based on reduction methods with applications to multiscale design and additive manufacturing

Emmanuel TROMME^1,²(

), Atsushi KAWAMOTO², James K. GUEST¹

¹. Department of Civil Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
². Toyota Central R&D Labs., Inc., Nagakute, Aichi 480-1192, Japan

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Abstract

Advanced manufacturing processes such as additive manufacturing offer now the capability to control material placement at unprecedented length scales and thereby dramatically open up the design space. This includes the considerations of new component topologies as well as the architecture of material within a topology offering new paths to creating lighter and more efficient structures. Topology optimization is an ideal tool for navigating this multiscale design problem and leveraging the capabilities of advanced manufacturing technologies. However, the resulting design problem is computationally challenging as very fine discretizations are needed to capture all micro-structural details. In this paper, a method based on reduction techniques is proposed to perform efficiently topology optimization at multiple scales. This method solves the design problem without length scale separation, i.e., without iterating between the two scales. Ergo, connectivity between space-varying micro-structures is naturally ensured. Several design problems for various types of micro-structural periodicity are performed to illustrate the method, including applications to infill patterns in additive manufacturing.

Keywords multiscale topology optimization micro-structure additive manufacturing reduction techniques substructuring static condensation super-element

Corresponding Author(s): Emmanuel TROMME

Just Accepted Date: 18 September 2019 Online First Date: 15 November 2019 Issue Date: 21 February 2020

Cite this article:

Emmanuel TROMME,Atsushi KAWAMOTO,James K. GUEST. Topology optimization based on reduction methods with applications to multiscale design and additive manufacturing[J]. Front. Mech. Eng., 2020, 15(1): 151-165.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0564-8
https://academic.hep.com.cn/fme/EN/Y2020/V15/I1/151

Fig.1 Design domain discretization with a zoom on a unit-cell/super-element. H is the size of the square unit cell and h is the size of the finite element mesh. It must be noted that the method is not restricted to square unit cell.

Fig.2 Tree structure of a multi-level reduction.

Fig.3 Flowchart of the optimization process.

Fig.4 Design problems. (a) Double clamped beam; (b) cantilever beam with a distributed load; (c) Michell beam; (d) invert compliant mechanism.

Fig.5 Three different types of periodicity. (a) Fully periodic structure; (b) user-defined selective periodicity; (c) fully free-form periodicity (no imposed periodicity). The unit cells containing design variables are shaded and letters identify the unit cell micro-structures.

Fig.6 Optimized micro-structures of the double clamped beam design problem. (a) M=2, C=9.24; (b) M=4, C=11.13; (c) M=8, C=12.29; (d) M=16, C=14.02; (e) M=64, C=16.75.

Fig.7 Optimized designs of the double clamped beam. (a) M=2; (b) M=4; (c) M=64.

Fig.8 Optimized micro-structures and structures of the cantilever beam design problem, where M is the number of unit cells along the vertical axis. (a) M=2, C=1.098×10^–2; (b) M=4, C=1.240×10^–2; (c) M=8, C=1.298×10^–2; (d) M=16, C=1.336×10^–2.

Tab.1 Computation time using multi-level reduction for the double clamped beam having 26229762 dofs

Fig.9 Cantilever beam optimized designs-Layered micro-structure periodicity. (a) M=4, Thickness=

{14, 14, 14, 14}

, C=8.672×10^–3; (b) M=16, Thickness=

{416, 816, 416}

, C=1.071×10^–2; (c) M=18, Thickness=

{318, 1218, 318}

, C=1.145×10^–2; (d) M=18, Thickness=

{618, 618, 618}

, C=1.064×10^–2.

Fig.10 Optimized designs for the double clamped beam (a, b and c) and the compliant inverter mechanism (d) without imposing any periodicity on the unit cell. (a) M=16, h=H/10, C=9.02; (b) M=16, h=H/40, C=9.87,

r min ? = 8 h

; (c) M=16, h=H/40, C=9.66; (d) M=10, h=H/12,

C pen = − 2.046

C unpen = − 0.236

Fig.11 Optimized design of the double clamped beam considering local volume constraints to promote lattice-like features (M=16, h=H=40). (a) C=9.95, LVF=0.9; (b) C=10.33, LVF=0.7; (c) C=10.91, LVF=0.5.

Fig.12 Optimized design of the compliant inverter considering local volume constraints (M=10, h=H/12). (a)

C pen = − 1.973

C unpen = − 0.537

, LVF=0.9; (b)

C pen = − 1.901

C unpen = − 0.355

, LVF=0.8; (c)

C pen = − 1.733

C unpen = − 0.215

, LVF=0.7.

Fig.13 Optimized designs of the double clamped beam with minimum and maximum local volume constraints (M=8, h=H/40). (a) C=10.685, LVF_m=0.05; (b) C=11.431, LVF_m=0.2; (c) C=14.717, LVF_m=0.3.

Fig.14 Optimized designs of the Michell beam with minimum and maximum local volume constraints (M=20, h =H/10). (a) C=43.804, Max LVF=0.9; (b) C=46.785, Max LVF =0.7; (c) C=64.112, Max LVF=0.5.

Fig.15 Double clamped beam re-design (M=16, h=H/40, r_min=8h for left column and r_min=2h for right column). (a) C=11.20, V_f=0.25; (b) C=11.06, V_f=0.25; (c) C=14.34, V_f=0.2; (d) C=13.92, V_f=0.2; (e) C=20.00, V_f=0.15; (f) C=18.83, V_f=0.15.

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