Modeling Transient Heat Transfer Using SPH and Implicit Time Integration

In this article, a two-dimensional transient heat conduction problem is modeled using smoothed particle hydrodynamics (SPH) with a Crank-Nicolson implicit time integration technique. The main feature of this work is that it applies implicit time stepping, an unconditionally stable Crank-Nicolson approach, in the thermal conduction simulation of liquid-phase diffusion (LPD) semiconductor crystal growth. This SPH simulation is compared with the equivalent finite-volume results. As well, two transient thermal conduction test problems are simulated using both explicit and Crank-Nicolson schemes, and their results compared with the analytical solutions. One of the current drawbacks of SPH is that explicit time-stepping algorithms, such as predictor-corrector methods or leapfrog methods, require extremely small time steps for a stable simulation. Using implicit time integration opens SPH up to a much larger class of practical problems in applied mechanics.