Eulerian and Lagrangian control volume capacitance methods for coupled solidification and deformation modelling

Practical solidification modelling requires methods that are efficient and accurate. Current approaches to solidification modelling, using the finite element method (FEM), are principally founded on capacitance methods. Unfortunately they suffer from a major drawback in that energy is not correctly transported through elements and so providing a source of inaccuracy. This has been addressed recently with the discovery of the control volume capacitance method (CVCM). This paper is concerned with the development and application of the CVCM to problems where mass transport and solidification are combined. The approach adopted is founded on theory that describes energy transfer through a control volume (CV) moving relative to the transporting mass. Equivalent governing partial differential equations are established, which are designed to be transformable into a finite element system that is commonly used to model transient heat-conduction problems, Different choices of capacitance are possible resulting in different forms of equivalent equation. In particular, two forms of transport equation are investigated, which apply to either Eulerian or Lagrangian descriptions of the problem. This approach circumvents the need to use the methods of Bubnov-Galerkin and Petrov-Galerkin and thus eliminates many of the stability problems associated with these approaches. An integration scheme is described that accurately caters for enthalpy fluxes generated by mass transport. The CV approach is tested against known analytical solutions and is show to be accurate, stable and computationally competitive.