Stability of the mixed time partitioning methods in relation to the size of time step

Abstract

AbstractIn this paper we focus on stability of a mixed time partitioning methods in relation to time step size which is using in numerical modelling of two‐component alloys solidification. We present the numerical integration methods to solve solidification problems in a fast and accurate way. Our approach exploits the fact that physical processes inside a mould are of different nature than those in a solidifying cast. As a result different time steps can be used to run computations within both sub‐domains. Because processes that are modeled in the cast sub‐domain are more dynamic they require very fine‐grained time step. On the other hand a heat transfer within the mould sub‐domain is less intense, and thus coarse‐grained step is sufficient to guarantee desired precision of computations. We propose using a fixed time step in the cast and its integer multiple in other parts of mould. We use one‐step explicit and implicit time integration Θ schemes. These time integration schemes are applied to equations obtained after spatial discretization. The implicit scheme is unconditionally stable, but stability of the explicit scheme depends on the size of time step. Critical time step size can be determined on the basis of eigenvalues of the amplification matrix that depend on the material properties, size and type of the finite element. In this work we present the manner of determining the critical time step and its affect on the course of numerical simulation of solidification. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)