So the idea behind my phd thesis is the following. If it is true that some network can optimize their energy dissipation, is it true also that they could optimize other properties, say their elastic moduli, electric conductivity etc? If so, can we derive bounds for these properties? And what about the inner structure of these materials?

The answer in short is the following:

- Take a volume of material V, organized in slender rods/pipes in a bigger volume V’.
- Consider that some quantities is flowing through these rods, and that there exists a potential describing this process. For instance: electrical charge (with electrical potential), fluid mass (with pressure) or forces and torques (with the displacements as potential). In each case there is a proper module describing the behaviour of the initial material, for instance the electrical conductivity of the each rod.
- Then it is possible to prove that there exists a maximum a class of isotropic networks such that some “module” (electrical conductivity, fluid conductivity or elastic moduli) are bigger than any other material using the same fluid, conductivity or elastic moduli.
- The modules of the optimal network only depends on their density V/V’ and their proper module.
- All optimal elastic networks are also optimal from the electrical and mass conductivity point of view, the reverse is not true.
- This class of networks can be embodied in different geometries but very strict geometrical conditions have to be fulfilled. These conditions are necessary and sufficient.
- These conditions can be used analytically and numerically to find new optimal networks.

This is possible through the applications a theorem known as the Principle of Minimum Potential Energy, widely used — in a different way — in mechanics. The details can be found here and here.

Now, what do these networks look like?

Next: Simple 2d networks

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