Keywords: non-physical, transport equations, heat transfer, finite elements.

Sophisticated computational approaches exist for the accurate and efficient modelling of solidification processes. Models and methods have been developed to account for complex boundary conditions, moving boundaries, varying thermo-physical properties, macro and micro transport processes used in the prediction of defects such as segregation and porosity. Fundamental to the many approaches, in the absence of convection, is the efficient solution to the diffusion equation. Many of the numerical methods for the solution of the diffusion equation can be viewed as classical and are commonly employed in commercial software. However, the solidification of a pure or eutectic substance is of particular interest here as it gives rise to a material discontinuity. Discontinuous behaviour provides substantial obstacles to the efficient application of mesh based numerical techniques. Accounting for strong discontinuities is presently of particular interest to the finite element research community with for example the development of cohesive and enriched elements to cater for material separation. Although strong discontinuities are of importance, of equal if not of greater interest and the focus in this paper, are weak discontinuities, which are present at any material change.

In this paper a new formulation for the solution of the discontinuous isothermal solidification problem is presented. The formulation has similarities with the now classical capacitance and source methods traditionally used in commercial software. However, the new approach focuses on the solution of the governing enthalpy-transport equation rather than the governing parabolic partial differential heat equation. The advantage is that discontinuous physics can be accounted for without approximation and the arbitrariness common to classic approaches is avoided. Also introduced, is the concept of non-physical enthalpy, which unlike physical enthalpy has numerical values that are not moving-frame invariant. Understanding the behaviour of the non-physical enthalpy is central to the successful treatment of discontinuities. A particular drawback is that non-physical enthalpy is non-intuitive and new mathematical constructs are required to describe its behaviour. This involves the introduction of transport equations which provide the new concept of relative moving-frame invariance for the non-physical enthalpy. The principal advantage however is that a unified methodology is established for the treatment of discontinuities. This is shown to establish real rigour and in many respects the formulation highlights the erroneous choices made with established classical approaches and casts in a totally new light a somewhat traditional problem. The new methodology is applied to a range of simple problems to provide an in-depth treatment and for ease of understanding but also to best describe the behaviour of the non-intuitive non-physical enthalpy. The method's remarkable accuracy and stability is demonstrated in the paper.

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