How does the initial cell configuration influence the final topology in a metamaterial generation process?

Guevara-Corzo, Jeffrey; García-Sánchez, Jesús; Quintero-Ramírez, Carolina; Begambre-Carrillo, Oscar

doi:10.1590/1679-78257904

Jeffrey Guevara-Corzo ^**Corresponding author.

conceptualization

investigation

writing and coding

School of Mechanical Engineering, Universidad Industrial de Santander, Bucaramanga 680002, Santander, Colombia. Email: jeffrey.guevara1@correo.uis.edu.co

https://orcid.org/0000-0002-8929-5903

Jesús García-Sánchez

writing

review

editing

Institute of Mechanical Engineering, Universidade Federal de Itajubá, Itajubá 37500-903, Minas Gerais, Brazil. Email: jesus@unifei.edu.br

https://orcid.org/0000-0003-0806-1660

Carolina Quintero-Ramírez

writing

review

editing

Department of Structural Engineering, Universidade de São Paulo, São Carlos 13566-590, São Paulo, Brazil. Email: carolqr@sc.usp.br

https://orcid.org/0000-0002-1993-2305

Oscar Begambre-Carrillo

review

resources

supervision and writing

School of Civil Engineering, Universidad Industrial de Santander, Bucaramanga 680002, Santander, Colombia. Email: ojbegam@uis.edu.co

https://orcid.org/0000-0002-2895-9374

*Corresponding author.

Editor: Marco L. Bittencourt

Figures (16)
Tables (6)

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Figure 1
Flowchart of the proposed mechanical metamaterial cell generation strategy.

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Figure 2
Representation of Three Load Cases Used in Numerical Homogenization: a) Undeformed state; b) ε1 = [1,0,0]T ; c) ε2 = [0,1,0]T and d) ε3 = [0,0,1]T

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Figure 3
Initial Cell Configurations studied. (a) – (c): 2D cases with 10, 20, and 30 mm circular void diameters, respectively; (d) – (e): 3D cases initial cell cross-section with 10, 20, and 30 mm spherical void diameters, respectively.

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Figure 4
Final Topology for 2D Cases. VD: void diameter; VF: volume fraction; OF: objective function.

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Figure 5
Value of the OF (F1 to F4) and volume fractions (30% to 50%) for the 3D Cells: a) Void Diameter of 10 mm; b) Void Diameter of 20 mm, and c) Void Diameter of 30 mm.

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Figure 6
Selection and Post-processing for 2D with 50% of VF, and 20 mm VD.

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Figure 7
Evolution of the OF during the 2D TO process for the selected cases. a) F1. b) F2. b) F3 and d) F4. All with VF=50% and VD=20 mm

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Figure 8
Final Topology for 3D Cases. VD: void diameter; VF: volume fraction; OF: objective function.

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Figure 9
Comparison of F3 and F7 using isotropy restriction and OC algorithm.

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Figure 10
Comparison of F3 and F7 using isotropy restriction and MMA algorithm.

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Figure 11
Comparison of four 3D final topologies obtained in this work without isotropy constrains with those reported by Lu et al. (2022)Lu, C., Hsieh, M., Huang, Z., Zhang, C., Lin, Y., Shen, Q., Zhang, L. (2022). Architectural design and additive manufacturing of mechanical metamaterials: a review. Engineering, 17, 44-63.

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Figure 12
Value of the OFs (F1 to F7) and VF (30% to 50%) for the 3D Cells: a) VD of 10 mm; b) VD of 20 mm, and c) VD of 30 mm.

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Figure 13
Selection and Post-Processing for the 3D Results.

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Figure 14
Evolution of the OF during the 3D TO process. a) F1 (VF=50%, VD=20mm). b), c), d), e) and g) F2 to F5 and F7 (VF=50%, VD=10mm). f) F6 (VF=40%, VD=20mm).

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Figure 15
Pareto Diagrams of standardized effects. a) 2D cases; b) 3D cases.

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M i n → F x , St. E ρ e = ρ e P E 0 , e = 1,2 , … , n K X i = f i , i = 1,2 , … , n g j x ≥ 0 , j = 1,2 , … , J 1 Υ ∑ e = 1 n v e ρ e ≥ V , e = 1,2 , … , n ρ m i n ≤ ρ e ≤ 1 , e = 1,2 , … , n

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Table 1
Objective Functions used for 2D and 3D scenarios. OF: objective function

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Table 2
Value of the ANSYS Homogenized Elastic Tensor (HET) for 2D Cases. HET is in Pa

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Table 3
Elastic Tensor Values Normalized (TVN) for 2D Cases. TVN was calculated with respect to the highest tensor value for each case.

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Table 4
Value of the ANSYS Homogenized Elastic Tensor (HET) for 3D Cases. HET is in Pa.

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Table 5
Elastic Tensor Values Normalized (TVN) for 3D Cases. TVN was calculated with respect to the highest tensor value for each case.

2D OF	3D OF
$F_{1} = E_{1111}^{H}$	$F_{1} = E_{1111}^{H}$
$F_{2} = E_{1111}^{H} + E_{2222}^{H}$	$F_{2} = E_{1111}^{H} + E_{2222}^{H}$
$F_{3} = E_{1111}^{H} + E_{2222}^{H} + E_{1122}^{H} + E_{2211}^{H}$	$F_{3} = E_{1111}^{H} + E_{2222}^{H} + E_{3333}^{H}$
$F_{4} = E_{1212}^{H}$	$F_{4} = E_{1111}^{H} + E_{2222}^{H} + E_{3333}^{H}$ $E_{1122}^{H} + E_{2233}^{H} + E_{1133}^{H}$
	$E_{2211}^{H} + E_{3322}^{H} + E_{3311}^{H}$
	$F_{5} = E_{1212}^{H}$
	$F_{6} = E_{1212}^{H} + E_{2323}^{H}$

HET	$F_{1}$	$F_{2}$	$F_{3}$	$F_{4}$
$E [1,1]$	8.89E+10	4.60E+10	2.53E+10	1.75E+10
$E [2,1]$	4.14E+09	4.75E+09	2.02E+10	1.68E+10
$E [2,2]$	2.24E+10	4.60E+10	2.53E+10	1.75E+10
$E [3,3]$	3.73E+08	9.76E+08	4.16E+09	1.48E+10

HET	$F_{1}$	$F_{2}$	$F_{3}$	$F_{4}$
$E [1,1]$	1.00	1.00	1.00	1.00
$E [2,1]$	0.05	0.10	0.80	0.96
$E [2,2]$	0.25	1.00	1.00	1.00
$E [3,3]$	0.00	0.02	0.16	0.85

	$F_{1}$	$F_{2}$	$F_{3}$	$F_{4}$	$F_{5}$	$F_{6}$	$F_{7}$
$E [1,1]$	1.07E+11	1.00E+11	6.80E+10	2.42E+10	1.00E+11	3.58E+09	7.42E+09
$E [2,1]$	1.77E+10	2.79E+10	1.54E+10	1.17E+10	2.79E+10	3.38E+09	3.59E+09
$E [3,1]$	1.77E+10	7.96E+09	1.54E+10	1.17E+10	7.96E+09	7.03E+07	3.59E+09
$E [2,2]$	3.30E+10	8.89E+10	6.80E+10	2.38E+10	8.89E+10	7.16E+09	7.42E+09
$E [3,2]$	2.61E+10	4.14E+09	1.54E+10	1.15E+10	4.14E+09	3.38E+09	3.59E+09
$E [3,3]$	3.30E+10	2.24E+10	6.80E+10	2.38E+10	2.24E+10	3.58E+09	7.42E+09
$E [4,4]$	2.25E+10	3.09E+10	1.41E+10	9.10E+09	3.09E+10	2.81E+09	3.29E+09
$E [5,5]$	1.42E+10	3.74E+08	1.41E+10	9.04E+09	3.74E+08	2.81E+09	3.29E+09
$E [6,6]$	2.25E+10	8.42E+09	1.41E+10	9.06E+09	8.42E+09	9.28E+07	3.29E+09

	$F_{1}$	$F_{2}$	$F_{3}$	$F_{4}$	$F_{5}$	$F_{6}$	$F_{7}$
$E [1,1]$	1.00	1.00	1.00	1.00	1.00	0.50	1.00
$E [2,1]$	0.17	0.28	0.23	0.48	0.28	0.47	0.48
$E [3,1]$	0.17	0.08	0.23	0.48	0.08	0.01	0.48
$E [2,2]$	0.31	0.89	1.00	0.98	0.89	1.00	1.00
$E [3,2]$	0.24	0.04	0.23	0.48	0.04	0.47	0.48
$E [3,3]$	0.31	0.22	1.00	0.99	0.22	0.50	1.00
$E [4,4]$	0.21	0.31	0.21	0.38	0.31	0.39	0.44
$E [5,5]$	0.13	0.00	0.21	0.37	0.00	0.39	0.44
$E [6,6]$	0.21	0.08	0.21	0.37	0.08	0.01	0.44

[1] *Corresponding author.

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How does the initial cell configuration influence the final topology in a metamaterial generation process?

Abstract

Graphical Abstract