# Low Weight Structures

Design of optimal microstructures in the low porosity limit.

The *homogenization method* links macroscopic optimal design problems to optimal microstructures.
In the case of the minimum compliance problem, for instance, it is known that optimal designs involve locally optimal multi-layered laminated composites.
In some cases, it is even known that optimal microstructures *require* multiple scales.
In two dimensions, it is known that when the volume fraction of material converges to zero, the properties of these laminates converge to that of the well-known Michell truss structures, but that this is not true in three dimensions.
In collaboration with R.V.~Kohn, I studied the problem of optimal low weight microstructures.
We devised a simple class of microstructures, the Single Scale Laminates, made of arrays of beams in 2D and walls in 3D.
These structures are *extremal* in the sense that no stiffer structure is possible using the same total mass.
They are also *universal* in the sense that given any Hooke’s law there exists a single-scale laminate of the same weight that is stiffer.
We proved a superposition theorem showing that at first order in the volume fraction, the Hooke’s law of a Single Scale Laminate is simply the sum of the Hooke’s law of its layers.
Using Single Scale Laminates, the problem of finding the microstructure of minimum weight with a given stiffness becomes a simple semi-definite optimization problem (Bourdin & Kohn, 2008).

*In 2D, a triangular lattice (left) and a Kagomé lattice (center) are both examples of single-scale laminates; since they have the same layer directions and thicknesses, the Superposition Principle shows they have the same Hooke’s law. For a general single-scale laminate, the thicknesses of the layers can depend on their orientation; thus the structure shown on the right is also a 2D single-scale laminate.*

### References

- Bourdin, B., & Kohn, R. V. (2008). Optimization of Structural Topology in the High-Porosity Regime.
*J. Mech. Phys. Solids*,*56*, 1043–1064. DOI:10.1016/j.jmps.2007.06.002 Download

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