Deep material network for thermal conductivity problems: Application to woven composites

The thermal conductivity of materials dictates their functionality and reliability, especially for materials with complex microstructural topologies, such as woven composites. In this paper, we develop a physics-informed machine-learning architecture built specifically for solving thermal conductivity problems. Originally developed for mechanical problems, we extend and develop a deep material network (DMN) that incorporates (i) principles from thermal homogenization directly into the network architecture in which nodes propagate heat flux and temperature gradient (as opposed to stress and strain in the original 'mechanical' DMN) and (ii) nodal rotations to capture the topological complexity of the materials' microstructure. The result is a 'thermal' DMN better suited for thermal conductivity problems than the 'mechanical' deep material network. We demonstrate the ability of this 'thermal' DMN to act as an accurate reduced order model with a significantly smaller number of degrees of freedom on two different woven microstructures examples. Our results show that the 'thermal' DMN can not only accurately predict the averaged effective thermal conductivity of these complex weaved composite structures but also the distribution of local heat flux and temperature gradients. Based on these performances, we show how this 'thermal' DMN can be exercised for rapid uncertainty and sensitivity analyses to assess microstructure effects and variability of the properties of the composite's constituents, a task that would be otherwise computationally prohibitive with direct numerical simulations. Based on its architecture, the 'thermal' DMN opens possibilities for multiscale, multiphysics simulations for a heterogeneous structure, especially when coupled with its mechanical counterpart.