A non‐physical enthalpy method for the numerical solution of isothermal solidification

Abstract

SUMMARYIn 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 not only 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. Demonstrated in the paper is the methods' remarkable accuracy and stability.

A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

“…The proposed method is able to track irregular phase-change fronts and permits the creation/removal of a phase inside another. Figure 12-F compares the position of the phase front along the corner diagonal with the reference solution obtained from [39]. The influence of the additional heat sources is evident at t > 0.008s, where the phase front is retarded due to the effect of the nearby positive heat sources.…”

“…This problem is the extension of the 1D isothermal phase change for a semi-infinite rectangular corner [42] [23] [39]. Here, the problem is slightly modified by adding localized heat sources.…”

Dynamic buckling of an imperfect elastoplastic beam subjected to an impulsive load analyzed using transport equations

“…Davey used the transport equations in the field of modeling heat transfer in the presence and absence of boundary and material discontinuities [1][2][3][4][5][6][7]. The performance of the capacity control methods for convection-solidification problems was investigated by Davey [1].…”

“…Rodriguez et al [4][5] used finite difference method to solve the transfer equations governing on diffusion-convection equations and made some improvement on the standard control volume capacitance method. Application of nonphysical variables in isothermal solidification problem was studied by Davey and Mondragon [6]. The source term for the removal of the discontinuity of the transport equations was selected.…”

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