The document discusses optimizing truss structures using Voronoi tessellations and evolutionary optimization. It aims to 1) Create a mathematical model of randomized Voronoi tessellations in MatLab, 2) Transfer the model to ANSYS and NASTRAN for finite element analysis and genetic algorithm optimization to minimize mass under stress and deflection constraints, and 3) Additively manufacture and test optimal designs. Results show the optimized designs closely resemble natural rounded shapes and provide insight into additive manufacturing capabilities. Future work involves optimizing additional natural structures and properties.

1. Optimal Design of Truss Structures, Voronoi Cells, and Evolutionary Optimization
Sean Kearns - University of Colorado at Boulder
Patrick Hull, PhD - ES21
Background
By reducing vehicular structural mass, increased
payload mass may be allocated. Structural
optimization typically minimizes mass while assuming
certain deflection, and stress constraints. The
benchmark optimization problem focuses on the
cross-sectional area of each bar (refer to Fig. 3).
Here, techniques such as geometric sizing,
computational stress analysis, and evolutionary
computation are used to determine the optimal
topology of randomized Voronoi tessellations.
A Voronoi tessellation is a semi-regular tessellation
of polygons such that every
polygon, or Voronoi cell
contains all points closest
to the generating point, or
Voronoi center than any other.
The boundary of a Voronoi cell
is constructed from the
bisections between each
Voronoi center.
Subject to the design space of discretized blocks
presented in figure 2, the benchmark structural
optimization problem uses the Ten-Bar truss as seen in
figure 3, and is the structure we will be comparing to.
Objectives
• Create mathematical model of randomized
Voronoi tessellations in MatLab.
• Produce a data transfer file to ANSYS, and
NASTRAN.
• Use geometric sizing, Finite Element Analysis,
and genetic algorithms to optimize node
structures’ mass, stress, and deflection.
• Additively manufacture optimal designs.
• Publish paper on our findings.
Conclusions
• Patrick Hull’s compliant mechanism optimization solution,
and the randomized Voronoi Tessellation solution closely
resemble rounded, and smooth shapes that you might
find in nature.
• Alternative manufacturing methods offer practical
solutions to natural structures without sacrificing
structural integrity.
• Provides engineers with a greater insight on additive
manufacturing capabilities.
Results
Solution comparisons for optimal cross-section areas of bars (in²) for each
bar length (Fig. 3) from previous researchers on the benchmark structural
optimization problem (Hull 2006).
Future Work
• Develop, and optimize model of randomized Delaunay
mesh (dual of Voronoi structure).
• Continuation of optimizing natural dynamic structures,
and respective properties.
• Enhance additive manufacturing abilities by
introducing new structures, and modeling
natural structures (observe figure 6).
• Create structures and topology optimization
team at Marshall.
Acknowledgments
I would like to thank my mentor Patrick for his contributions during
this project, and the lessons he taught me along the way. Also, a
special thanks to Alex Few, Adam Burt, and John Seixal for their
assistance, and instruction during my time at MSFC.
Visual outputs from ANSYS and
NASTRAN
Fig. 3 Standard Ten-Bar truss with 2 load conditions , and two boundary conditions.
Fig. 1 Voronoi Tessellation of 16 nodes.
360 in 360 in
360in
P1 P2
Fig. 4 Approximate optimized shape using geometric sizing,
finite element analysis, and genetic algorithms with ANSYS
(Hull). 25,000 PSI applied to all members, and 10,000 pound
load conditions at P1, and P2
Fig. 5 Ten-Bar truss optimization from NASTRAN. Lowest stress
to highest denoted in false color scale from blue to red.
Bar
Number
Length of
bars (in)
NASTRAN
Fig. 5
Schmitt-
Miura
Schmitt-
Farchi
Venkayya Haug-
Arora
Hull et al
Fig. 4
1 360 24.37 24.43 24.25 23.4 23.27
2 509 20.818 21.06 20.69 21.08 21.2
3 360 30.62 30.66 33.42 30.41 30.03
4 509 8.4155 8.58 8.39 8.69 7.47
5 360 0.1 0.1 0.1 0.1 0.1
6 360 0.22981 0.1 0.1 0.1 0.1
7 360 0.16575 0.1 0.1 0.1 0.1
8 360 14.997 14.59 14.26 14.9 15.29
9 509 0.23011 0.1 0.1 0.19 0.1
10 509 20.44 21.06 20.69 21.08 21.2
Total Mass
(lb)
5078 5074 5108 5053 5010 5133
References
Hull P, Tinker M, Doizer G (2006) Evolutionary optimization of a geometrically
refined truss. Springer 31:311-319.
Fig. 6 Natural bubble
interaction structure
Fig. 2
4
7
2
5
1 8
6
10
3
9