Temporal Discretisation for Heat Conduction

Abstract

Problems where the temperature field at various points in the domain varies with time are referred to as transient problems, as opposed to steady state problems, where the temperature remains constant at a given point in the domain, for all times. In structural mechanics transient problems are analogous with dynamics and steady state problems are analogous with statics. The finite element discretisation discussed in the previous chapter was limited to the heat conduction equations without the term containing the temporal derivative. Although, most real life heat transfer problems are time-dependent, for many engineering problems it is sufficient to calculate a steady spatial temperature field. For example, the temperature field for electrical or mechanical machinery in operational conditions may be calculated as a steady state problem governed by the steady heat conduction equation and appropriate boundary conditions, using the procedure outlined in the previous chapter. However, there are other problems where the transient effects cannot be ignored. For example, it may be required to calculate the temperature field for machinery subjected to time-dependent or cyclic thermal loading. Other examples are phase change problems (solidification, melting etc.). For such problems the complete heat conduction equations including the temporal derivative term must be used. Therefore, a temporal discretisation of the transient heat conduction equations is required in addition to the spatial discretisation.