Modeling heat transfer subject to inhomogeneous Neumann boundary conditions by smoothed particle hydrodynamics and peridynamics

• Published in: International journal of heat and mass transfer Vol. 139; pp. 948 - 962
• Main Authors: Wang, Jianqiang, Hu, Wei, Zhang, Xiaobing, Pan, Wenxiao
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.08.2019; Elsevier BV
• Subjects: Boundary conditions; Computational fluid dynamics; Computer simulation; Conduction heating; Conductive heat transfer; Cracks; Fluid flow; Fluid mechanics; Fluxes; Free convection; Heat transfer; Inhomogeneous Neumann boundary condition; Kernels; Mathematical models; Natural convection; Numerical analysis; Numerical methods; Peridynamics; Robin boundary condition; Smooth particle hydrodynamics; Smoothed particle hydrodynamics; Taylor series; Thermal diffusion; Thermodynamics; Robin boundary condition; Smoothed particle hydrodynamics; Inhomogeneous Neumann boundary condition; Peridynamics; Natural convection; Heat transfer
• DOI: 10.1016/j.ijheatmasstransfer.2019.05.054
• ISSN: 0017-9310

•A new method is introduced for imposing inhomogeneous Neumann and Robin BCs in smoothed particle hydrodynamics (SPH) and peridynamics for modeling heat transfer problems.•The new method can be employed for solving various transient or steady heat transfer problems subject to linear or nonlinear fluxes going through boundaries.•The numerical solutions by the new method converge to the classical solutions with notably improved accuracy.•The new method can be extended in SPH for solving other classical PDEs subject to inhomogeneous Neumann or Robin BCs, e.g., mass transfer in reactive transport, and is transferable to other numerical/modeling frameworks that also rely on nonlocal formulations. Nonzero fluxes going through boundaries/interfaces are normally observed in heat transfer, which in general can be described as inhomogeneous Neumann boundary conditions (BCs). Both smoothed particle hydrodynamics (SPH) and peridynamics have been employed for modeling heat transfer or thermal diffusion processes. The former is a numerical method used to approximate the solutions of classical heat diffusion PDEs. The latter provides a nonlocal model for heat diffusion. They both employ a nonlocal formulation, which requires a full support of the nonlocal kernel to ensure accuracy. In this work, we propose a new, higher-order method to enforce inhomogeneous Neumann BCs in SPH and peridynamic model for heat transfer problems. In that, fictitious layers of (ghost) particles are needed to guarantee full support of the nonlocal kernel. The temperature is extrapolated to the ghost particles based on the Taylor expansion and the BC to be imposed. By such, no additional term is introduced into the heat equation; meanwhile, the numerical solutions converge to the classical solutions with notably improved accuracy. To validate, assess, and demonstrate the proposed method, we simulate different transient or steady heat transfer problems subject to linear or nonlinear BCs, including heat conduction, natural convection, and presence of insulated cracks. The numerical results are compared with the exact solutions of classical PDEs, solutions of other numerical methods, or experimental data.