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

The development of efficient mesh-based methods for modelling moving boundary problems continues to be of interest to the research community. Moving boundary problems offer substantial challenges and many numerical approaches have been proposed. In the area of solidification modelling some of the numerical approaches can now be viewed as classical and are commonly employed in commercial codes. The methods can be classified into two groups; front tracking (adaptive) and fixed domain methods. Front tracking (adaptive) methods provide for a accurate description of isothermal solidification but at the cost of complex meshing and re-meshing strategies, generally needed to cater for phase-front distortion, element birth and collapse. Although adaptive methods provide for high accuracy in the presence of a material discontinuity their complexity has resulted in the favouring of fixed-domain approaches. Fixed domain methods tend to be more versatile and easier to implement but it is evident from the literature they can suffer inaccuracy particularly when material discontinuities are present. In order to account for the poor performance displayed by fixed-domain methods in the presence of discontinuities an alternative approach has recently being proposed involving the use of non-physical variables for the precise removal of discontinuities. The method itself can be categorised as a fixed-domain method as no mesh modification is required although a material discontinuity must be tracked. The method is founded on the solution of weighted transport equations, where unlike differential equations readily cater for material discontinuities.

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. A recent innovation for accounting for weak discontinuities has been the discovery of non-physical variables which are founded and defined using transport equations.

This paper is concerned with the application of the non-physical approach to solidification modelling in the presence of more than one material discontinuity. A typical feature of the enthalpy-temperature response in solidification is discontinuities at phase transition temperatures as a consequence of phase change and latent heat release. In these circumstances, depending on the conditions that prevail, an element in a finite element mesh can have more than one discontinuity present. A methodology that can cater for multiple discontinuities is presented in this paper. The non-physical approach permits the precise removal of weak discontinuities arising in the governing transport equations. In order to facilitate the application of the approach the finite element equations are presented in the form of weighted transport equations. The method utilises a non-physical form of enthalpy that possesses a remarkable source distribution like property at a discontinuity. It is demonstrated in the paper that it is through this property that multiple discontinuities can be exactly removed from an element so facilitating the use of continuous approximations. The new methodology is applied to a range of simple problems to provide an in-depth treatment and for ease of understanding to demonstrate the methods remarkable accuracy and stability.

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