Element‐free Galerkin method for thermosolutal convection and macrosegregation

Abstract

SUMMARYIn the present work, the element-free Galerkin (EFG) method is applied to a continuum solidification model that calculates thermosolutal convection and macrosegregation during dendritic solidification of multicomponent alloys. Simulations for directional solidification of a binary Pb-Sn alloy and a Ni-base quaternary alloy have been performed in a rectangular two-dimensional domain. In both calculations, the alloy melt is cooled from below and the growth of the mushy zone is followed in time. The formation of macrosegregation defects known as 'freckles' has been successfully simulated using the meshless EFG method. A varying degree of sensitivity of results to the number and distribution of meshfree particles was obtained. The potential of the method for a broader range of solidification models is discussed.

Numerical methods for improved efficiency in macrosegregation modeling

“…The element‐free Galerkin (EFG) method used in this work was first introduced in 24 and is presented here in the context of a generic transport equation where c is the dependent variable, D is a diffusion coefficient and s represents a source.…”

“…For stabilization the MLS functions W j are modified using a Petrov–Galerkin treatment of the convective terms. Full details are given in 24.…”

“…Three different combinations of algorithms have been implemented using the projection method described above and the Petrov–Galerkin stabilization of the convective terms: Galerkin finite element method using bilinear and tri‐linear isoparametric elements. Mesh‐less element free Galerkin method. Mesh adaptation with linear triangular elements. Additional precautions must be taken to avoid locking due to the equal order interpolation of velocity and pressure that does not satisfy the LBB consistency condition; these are discussed in 23, 24, 58. The time variable is discretized using a first‐order backward Euler scheme, and systems of equations are solved by means of a pre‐conditioned conjugate gradient method.…”

“…Literature reviews have been presented in 20–23. So far, the models have been based on finite differences; finite volumes or finite elements, defined over a single two‐dimensional domain discretized using a fixed computational mesh, only very recently models utilizing mesh‐less methods and adaptive mesh refinement have been developed 24, 25.…”

“…The third area addressed in this work is the use of adaptive meshing to resolve the region containing the solidification front and those regions where freckles are developing, the motivation is the same that led to the implementation of mesh‐free methods, together with the fact that at this point the latter have not realized the desired improvements in efficiency 24. The literature on adaptive mesh refinement is vast, and no attempt will be made here to cover it, we only address the few publications that treat the equations of our interest.…”