Wednesday, December 19, 2012

Octet Truss for Topology Optimization

 Random Octet Truss Array (on shapeways)
This post demonstrate a work-flow for topology optimization using open source tools (with mixed success). The approach uses an unpenalized method (see wiki) that maps the material density output from the optimizer to unit-cells based on the octet truss (inspired by this white paper, see pp10). As some further motivation for this approach, the students working on the record-setting human powered helicopter demonstrated that multi-scale trusses (trusses with elements made of smaller trusses) were a very efficient structural concept (see the comments for further references on multi-scale structures). One of the benefits of not penalizing (using variable density solutions rather than trying to achieve predominantly solid-void solutions) is that we don't need to spend time doing parameter continuation on the penalization exponent.

Work-flow Steps

1. Define design domain
3. Run optimizer (ToPy)
4. Map optimized density to octet truss unit cells, generate mged (BRLCAD) script
5. Run mged script, export stl of the geometry
6. 3D Print It!
For demonstration purposes I'll just use the dogleg example from ToPy which takes care of steps 1 and 2. The input deck for that example must be modified to change the penalization and gray-scale filtering (GSF) options. Here's a gist that shows the modified input deck (note that P_FAC=1, and I have commented out the continuation and GSF parameters).

The output of running the optimizer this way is a variable density solution.
The vizualizations show a peel-away (iso-volumes) with various density thresholds to give an idea of the density variation through the part. There are still solid (red) and void (not shown) cells in the solution, but there are also a significant number of intermediate density cells (compare to this penalized solution which results in predominantly solid-void solutions).

The limits (either fully solid or completely void) are based on the manufacturing constraints for Shapeway's Strong and Flexible material (manufacturing constraints provide a natural way to regularize the problem so we can get unique solutions). On the low density side, the minimum diameter of the truss members is 0.7 mm. The high density limit is determined by the ratio of volume in the octet mesh to the corresponding unit cube. When this ratio reaches unity the octet cell is replaced by a cube of solid material. Shapeways recommends at least 0.5mm clearance and feature size, so clearance between truss members smaller than this is unlikely to be cleared of unconsolidated print material. Shapeways recommends an aspect ratio of 10 (length to thickness) for members. This gives a minimum octet truss cell made of members 0.7 mm in diameter and 7 mm in length; just to be on the safe side I bumped the minimum up to 0.4 mm radius and 8mm length. The maximum bounding box for a print is 650x350x550 mm.

This gist shows a python script which defines functions to generate an octet truss unit cell parameterized by the solid fraction. The solid geometry is generated in BRLCAD. This gist creates an array of octet truss primitives. The script is just a really crude way to automate mged through its command line interface (here's an example of scripting BRLCAD in Perl that uses a similar approach).

I attempted to run through this work-flow with the ToPy dogleg example shown above, but it resulted in a 7GB database file, and my poor little laptop ran out of memory trying to create an stl. I guess I have another thing to try out on GCE or AWS now. There are also file size limitations on Shapeways, so even after I generate the stl it's likely that I'll have to do some decimation to get the file size down.

Limitations of this Implementation

The connection between density and stiffness of the octet truss cell is currently not well established. A more accurate optimization would take in to account how the stiffness of the octet truss varies as the diameter of the members varies (stiffness vs. density). This would require a bit of finite element analysis on some trusses of varying thickness with periodic boundary conditions (e.g. section 3: Microstructure Approaches in this review).

As the figure above shows, only the radius of the octet truss members is varied, but not their position. Thus, the bounding box for each radius is different. It would be more consistent to adjust the position of the members to remain within a constant bounding box as the radius is varied.

The approach outlined above does not scale to 3D problems very well with the present tools.

1. Here's a paper that analyzes the effective macroscopic properties (strenght / density variation) as a function of the octet truss unit cell's member length and radius: Effective properties of the octet-truss lattice material

Abstract. The effective mechanical properties of the octet-truss lattice structured material have been investigated both experimentally and theoretically. Analytical and FE calculations of the elastic properties and plastic yielding collapse surfaces are reported. The intervention of elastic buckling of the struts is also analysed in an approximate manner. Good agreement is found between the predictions of the strength and experimental observations from tests on the octet-truss material made from a casting aluminium alloy. Moreover, the strength and stiffness of the octet-truss material are stretching-dominated and compare favourably with the corresponding properties of metallic foams. Thus, the octet-truss lattice material can be considered as a promising alternative to metallic foams in lightweight structures.

2. 244GB of RAM, dual processor large memory instances, \$3.50 per hour for Linux instances: EC2 for In-Memory Computing - The High Memory Cluster Eight Extra Large Instance

3. Abstract: Matrix methods of linear algebra are used to analyse the structural mechanics of the periodic pinjointed truss by application of Bloch’s theorem. Periodic collapse mechanisms and periodic states of self-stress are deduced from the four fundamental subspaces of the kinematic and equilibrium matrix for the periodic structure. The methodology developed is then applied to the Kagome lattice and the triangular–triangular (T–T) lattice. Both periodic collapse mechanisms and collapse mechanisms associated with uniform macroscopic straining are determined. It is found that the T–T lattice possesses only macroscopic strain-producing mechanisms, while the Kagome lattice possesses only periodic mechanisms which do not generate macroscopic strain. Consequently, the Kagome lattice can support all macroscopic stress states. The macroscopic stiffness of the Kagome and T–T trusses is obtained from energy considerations. The paper concludes with a classification of collapse mechanisms for periodic lattices.
The Structural Performance of the Periodic Truss