A numerical framework for the design of Joule-heating circuits to thermally activate smart materials

Joule-heating circuits, when embedded in a matrix material, enable fast heat generation and transfer, and are suitable for quick activation of thermally responsive materials. However, current computational tools for predicting the heat diffusion in such applications often lack robustness and efficiency, thus rendering the design of embedded Joule-heating circuits difficult. In this work, we propose a numerical framework based on a Green's function (GF) formalism to address this issue. Direct temperature solutions are yielded by summing the contributions of the initial condition, heat source and boundary conditions which all can be robustly configured. In addition, we exploit multiple periodicities to greatly reduce the computational labor: the multidimensional GFs can be pre-calculated by decomposition into the products of one-dimensional ones, and the pre-calculated GFs are repeatedly reused in a recursive time-marching scheme. The framework is first validated for a 2D problem with spatial and temporal results from experiments as well as finite-element simulations, showing more than 68% economy of computation time. We further employ the proposed framework to guide the design of embedded Joule-heating circuits by imposing a design threshold and comparing the simulated results in terms of heating time and temperature distribution for different wire densities. Moreover, the developed framework can also be extended to solve 3D problems which is demonstrated by an application in predicting the shape recovery response time of a programmed shape memory polymer sheet.