Global multiscale design optimization of procedural lattice structures for fatigue enhancement

Abstract

This paper introduces procedural generation of lattice structures inside predefined volumes for fatigue life enhancement of multiscale constructed structural components. Global optimization is applied to search the design space such that the mass and fatigue damage is minimized given the orientation of the microscopic lattice unit cells and the volume fraction of lattice material. A two-step optimization procedure is applied to avoid local minima. The macro-scale distribution of homogenized lattice material is first determined using a design search method. The second stage employs global optimization with surrogate modeling. Training data is initialized by hypercube sampling to fit a surrogate model so that a genetic algorithm can locate the global minima of the global design space using the lattice rotation and solid volume fraction. New results are applied to the procedural methods to create new testable models which optimize the microscale for minimum mass. New data points are used to enrich the surrogate model database and the model is rebuilt. The procedure iterates until a convergence criterion is met. A case study is presented to demonstrate the proposed methodology to which the mass was reduced by 87%, while fatigue performance was conserved for long life cycles of 10⁶. A comparison between lattice modeled with linear beam elements versus lattice material as a three-dimensional solid is presented to validate the optimal solution. It is found that stresses were underestimated by at most a factor of four, while the difference in deflections was maintained within 8.1%.

Appendices

Appendix

A: Cubic lattice properties

Cubic lattice relative properties (Poisson ratio = 0.3)
Density	Radius	D/L	1st lame	2nd lame	Poisson ratio	Modulus
0.00E+ 00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	1.53E+00	0.00E+00
4.67E−03	2.00E−02	4.00E−02	3.06E−05	4.11E−06	1.53E+00	4.61E−06
1.37E−02	4.00E−02	8.00E−02	1.66E−04	3.65E−05	1.45E+00	4.03E−05
3.55E−02	6.00E−02	1.20E−01	7.45E−04	2.58E−04	1.35E+00	2.79E−04
5.63E−02	8.00E−02	1.60E−01	1.62E−03	7.20E−04	1.29E+00	7.68E−04
8.44E−02	1.00E−01	2.00E−01	3.19E−03	1.79E−03	1.21E+00	1.87E−03
1.15E−01	1.20E−01	2.40E−01	5.47E−03	3.67E−03	1.15E+00	3.80E−03
1.55E−01	1.40E−01	2.80E−01	9.47E−03	7.61E−03	1.09E+00	7.76E−03
2.01E−01	1.60E−01	3.20E−01	1.51E−02	1.40E−02	1.03E+00	1.41E−02
2.40E−01	1.80E−01	3.60E−01	2.14E−02	2.22E−02	9.86E−01	2.21E−02
2.88E−01	2.00E−01	4.00E−01	3.05E−02	3.49E−02	9.46E−01	3.45E−02
3.38E−01	2.20E−01	4.40E−01	4.28E−02	5.34E−02	9.11E−01	5.23E−02
3.83E−01	2.40E−01	4.80E−01	5.65E−02	7.51E−02	8.84E−01	7.31E−02
4.44E−01	2.60E−01	5.20E−01	7.88E−02	1.11E−01	8.61E−01	1.07E−01
4.90E−01	2.80E−01	5.60E−01	1.01E−01	1.46E−01	8.49E−01	1.41E−01
5.48E−01	3.00E−01	6.00E−01	1.34E−01	1.96E−01	8.43E−01	1.89E−01
5.98E−01	3.20E−01	6.40E−01	1.72E−01	2.51E−01	8.44E−01	2.42E−01
6.47E−01	3.40E−01	6.80E−01	2.17E−01	3.12E−01	8.50E−01	3.01E−01
6.95E−01	3.60E−01	7.20E−01	2.71E−01	3.81E−01	8.61E−01	3.69E−01
7.35E−01	3.80E−01	7.60E−01	3.31E−01	4.51E−01	8.74E−01	4.38E−01
7.87E−01	4.00E−01	8.00E−01	4.16E−01	5.41E−01	8.93E−01	5.27E−01
8.27E−01	4.20E−01	8.40E−01	5.03E−01	6.25E−01	9.12E−01	6.12E−01
8.63E−01	4.40E−01	8.80E−01	5.88E−01	7.01E−01	9.29E−01	6.89E−01
8.93E−01	4.60E−01	9.20E−01	6.72E−01	7.71E−01	9.45E−01	7.61E−01
9.18E−01	4.80E−01	9.60E−01	7.48E−01	8.29E−01	9.59E−01	8.21E−01
9.43E−01	5.00E−01	1.00E+00	8.23E−01	8.83E−01	9.71E−01	8.78E−01
1.00E+00	5.40E−01	1.08E+00	1.00E+00	1.00E+00	1.00E+00	1.00E+00

Cubic lattice curve fitted effective stresses at yielding (Poisson ratio = 0.3)
Density	${\sigma }_{xx}^{y}$	${\sigma }_{yy}^{y}$	${\sigma }_{zz}^{y}$	${\sigma }_{xy}^{y}$	${\sigma }_{yz}^{y}$	${\sigma }_{xz}^{y}$	${\sigma }_{\mathrm{avg}}^{y}$
0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
4.67E−03	1.60E−03	1.60E−03	1.60E−03	1.10E−04	1.12E−04	1.11E−04	8.54E−04
1.37E−02	4.77E−03	4.77E−03	4.77E−03	3.19E−04	3.19E−04	3.20E−04	2.55E−03
3.55E−02	1.22E−02	1.22E−02	1.22E−02	1.23E−03	1.20E−03	1.20E−03	6.69E−03
5.63E−02	1.82E−02	1.82E−02	1.81E−02	1.87E−03	1.87E−03	1.88E−03	1.00E−02
8.44E−02	2.64E−02	2.64E−02	2.64E−02	3.15E−03	3.15E−03	3.15E−03	1.48E−02
1.15E−01	3.48E−02	3.49E−02	3.48E−02	4.69E−03	4.68E−03	4.67E−03	1.98E−02
1.55E−01	4.56E−02	4.57E−02	4.55E−02	7.55E−03	7.55E−03	7.53E−03	2.66E−02
2.01E−01	5.91E−02	5.94E−02	5.91E−02	1.12E−02	1.12E−02	1.11E−02	3.52E−02
2.40E−01	6.97E−02	7.00E−02	6.97E−02	1.46E−02	1.46E−02	1.46E−02	4.22E−02
2.88E−01	8.30E−02	8.32E−02	8.29E−02	1.96E−02	1.96E−02	1.96E−02	5.13E−02
3.38E−01	9.86E−02	9.88E−02	9.86E−02	2.64E−02	2.64E−02	2.64E−02	6.25E−02
3.83E−01	1.11E−01	1.12E−01	1.11E−01	3.28E−02	3.29E−02	3.28E−02	7.22E−02
4.44E−01	1.30E−01	1.30E−01	1.30E−01	4.40E−02	4.40E−02	4.40E−02	8.69E−02
4.90E−01	1.44E−01	1.45E−01	1.44E−01	5.38E−02	5.38E−02	5.38E−02	9.91E−02
5.48E−01	1.67E−01	1.67E−01	1.67E−01	6.93E−02	6.93E−02	6.93E−02	1.18E−01
5.98E−01	1.87E−01	1.87E−01	1.87E−01	8.52E−02	8.52E−02	8.52E−02	1.36E−01
6.47E−01	2.09E−01	2.09E−01	2.09E−01	1.03E−01	1.03E−01	1.03E−01	1.56E−01
6.95E−01	2.33E−01	2.33E−01	2.33E−01	1.26E−01	1.26E−01	1.26E−01	1.79E−01
7.35E−01	2.56E−01	2.56E−01	2.56E−01	1.51E−01	1.51E−01	1.51E−01	2.03E−01
7.87E−01	2.87E−01	2.87E−01	2.87E−01	1.92E−01	1.92E−01	1.92E−01	2.39E−01
8.27E−01	3.21E−01	3.21E−01	3.21E−01	2.31E−01	2.31E−01	2.31E−01	2.76E−01
8.63E−01	3.85E−01	3.85E−01	3.85E−01	2.97E−01	2.97E−01	2.97E−01	3.41E−01
8.93E−01	4.58E−01	4.58E−01	4.58E−01	3.79E−01	3.79E−01	3.79E−01	4.18E−01
9.18E−01	5.35E−01	5.35E−01	5.35E−01	4.68E−01	4.68E−01	4.68E−01	5.02E−01
9.43E−01	6.42E−01	6.42E−01	6.42E−01	5.90E−01	5.90E−01	5.90E−01	6.16E−01
1.00E+00	1.00E+00	1.00E+00	1.00E+00	1.00E+00	1.00E+00	1.00E+00	1.00E+00

1. Fitted curves are not accurate for high density values (> 0.9) due to downplaying the effects of stress concentrations.

B: Pareto frontier values

Optimization	Log(damage)	Damage	Volume fraction	Distance to utopia
A	− 1.53E+01	2.27E−07	9.98E−02	1.20E+01
B	− 2.20E+01	2.74E−10	2.00E−01	5.27E+00
C	− 2.50E+01	1.33E−11	3.00E−01	2.26E+00
D	− 2.61E+01	4.57E−12	4.00E−01	1.24E+00
E	− 2.68E+01	2.33E−12	5.00E−01	7.05E−01
F	− 2.70E+01	1.81E−12	6.00E−01	6.48E−01
G	− 2.71E+01	1.64E−12	7.00E−01	7.15E−01
H	− 2.73E+01	1.43E−12	8.00E−01	8.00E−01
I	− 2.73E+01	1.42E−12	8.99E−01	8.99E−01