Optimal design of manufacturable three-dimensional composites with multifunctional characteristics

We present an optimization method to design three-dimensional composite microstructures with multifunctional characteristics. To illustrate the fascinating types of microstructures that can arise in multifunctional optimization, we apply our methodology to the study the simultaneous transport of heat and electricity in three-dimensional, two-phase composites. We assume that phase 1 has a high thermal conductivity but low electrical conductivity and phase 2 has a low thermal conductivity but high electrical conductivity. The objective functions consist of different combinations of the dimensionless effective thermal and electrical conductivities. When the sum of the effective thermal and electrical conductivities is maximized, we find that the optimal three-dimensional microstructures are triply periodic bicontinuous composites with interfaces that are the Schwartz primitive (P) and diamond (D) minimal surfaces. Maximizing the effective thermal conductivity and minimizing the effective electrical conductivity results in a special dispersion of inclusions in a connected matrix. The effective properties of both the bicontinuous and singly connected microstructures lie on known optimal cross-property bounds. When the sum of the effective thermal and electrical conductivities is minimized, the result is the three-dimensional checkerboard, which is the optimal single-scale microstructure. It is important to note that current fabrication techniques enable one to manufacture all of the aforementioned optimal single-scale composites.