Discrete thermal element modelling of heat conduction in particle systems: Basic formulations
The paper presents a new numerical method for the resolution of the heat problem in a system of discrete and 2D circular neighboring particles. One particle exchanges heat along some arcs of its boundary, the remaining part being assumed isolated. The temperature \(T_{i}\) on the boundary of a particle may be computed in terms of the resultant flux \(Q_{i}\) through this arc and of the temperature \(T_{\text{o}}\) at the center of the particle, using Dini's integral representation, as \(T_{i}=\sum_{j=1,\dots ,n}h_{ij}Q_{j}-T_{\text{o}}\), where the \(h_{ij}\) are positive. The discrete thermal element method (DTEM) consists to introduce the vectors \(\mathbf{T} _{c}=(T_{1},\dots ,T_{n})^{T}\) of temperatures and \(\mathbf{Q} _{c}=(Q_{1},\dots ,Q_{n})^{T}\) of resultant fluxes and the thermal resistance matrix \(\mathbf{H}_{c}=(h_{ij})_{i,j=1,\dots ,n}\), and to write the system \(\mathbf{T}_{c}-\mathbf{e}T_{\text{o}}=\mathbf{H}_{c}\mathbf{Q} _{c}\), with \(\mathbf{e}=(1,\dots ,n)^{T}\). The authors assume that \(\mathbf{ H}_{c}\) is invertible, then eliminate \(T_{\text{o}}\), and approximate \( h_{ii} \) and \(h_{ij}\), \(i\neq j\). In the last parts of the paper, the authors present numerical computations based on this approach for a system of 12 or 1084 particles and compare their results to that obtained with the FEM method. They prove that the error using the DTEM method is smaller than that obtained using this FEM method.
scientific article; zbMATH DE number 5276062
scientific article