Optimality of thermal expansion bounds in three dimensions

In this short note, we use topology optimization to design multi-phase isotropic three-dimensional composite materials with extremal combinations of isotropic thermal expansion and bulk modulus. In so doing, we provide evidence that the theoretical bounds for this combination of material properties are optimal. This has been shown in two dimensions, but not heretofore in three dimensions. We also show that restricting the design space by enforcing material symmetry by construction does not prevent one from obtaining extremal designs.