Interpolation by dual kriging of a moving flow front and conservation of the fluid mass

摘要

One of the main issues when simulating the filling of a mould cavity is not so much solving the equations governing the flow, but rather to make sure that the calculation is accurate, although the shape of the geometrical domain changes constantly. In order to improve the accuracy of the numerical approximation, it is interesting to refine locally the mesh in the neighbourhood of the flow front and project on the new mesh the different scalar fields involved in the calculation. Very often a significant inaccuracy may result from these operations. The numerical error is cumulative and relatively small errors at each time step can lead to significant discrepancies at the end of the simulation. For all these reasons, errors on the conservation of the fluid mass represent a major challenge in injection moulding simulations. In this paper, a new approach based on dual kriging interpolation is proposed to project the flow front from one mesh to another, while conserving automatically the fluid mass in the cavity. The advantage of this methodology dwells in its generality, as it is valid both in the two and three-dimensional cases. It solves a major problem that occurs commonly in the numerical approximation of moving boundary problems, namely the conservation of the fluid mass. It consists of following precisely the displacement of the flow front by updating at each time t a saturation function f(x,t) defined at all points x in the cavity. At any given time, this function is one in the saturated domain, zero in the empty zone and is bounded between zero and one to reflect partial saturation in the vicinity of the flow front. An interpolation of the saturation function based on dual kriging provides an implicit equation of the moving front. From this equation the moving front can then be interpolated on the new mesh, so as to conserve exactly the fluid mass in the mould. After describing the main equations of dual kriging, the article presents the algorithm devised to conserve the fluid mass at each calculation step. Numerical experiments are carried out to validate these concepts for three-dimensional fluid fronts. The conservation of fluid mass is verified for multiple or merging flow fronts, obstacles, ribbed connections and parts of complex geometry.