A graph-theoretic-based method for analyzing conduction problems

A new graph-theoretical approach is presented for heat diffusion analysis given an arbitrary geometry and spatially varying parameters. A weighted directed graph with positive communication or conduction weights is proposed for heat transfer analysis, where the communication weights of the graph are determined based on the positions of the nodes and the spatially varying thermal and material properties of the domain. Transient Neumann and Dirichlet conditions are considered at the boundary nodes of the graph. Then, the temperature at each interior node is updated based on the temperatures at its in-neighbor nodes, where the in-neighbor vertices of interior nodes and the conduction weights are assigned based on the conduction graph. A set of coupled first-order ordinary differential equations determines the transient temperatures at the interior nodes for a prescribed boundary condition. The proposed method can be applied to both steady-state and transient heat diffusion analysis.