Solution of free‐boundary problems using finite‐element/Newton methods and locally refined grids: Application to analysis of solidification microstructure

Abstract

SUMMARYA new method is presented for the solution of free-boundary problems using Lagrangian finite element approximations defined on locally refined grids. The formulation allows for direct transition from coarse to fine grids without introducing non-conforming basis functions. The calculation of elemental stiffness matrices and residual vectors are unaffected by changes in the refinement level, which are accounted for in the loading of elemental data to the global stiffness matrix and residual vector. This technique for local mesh refinement is combined with recently developed mapping methods and Newton's method to form an efficient algorithm for the solution of free-boundary problems, as demonstrated here by sample calculations of cellular interfacial microstructure during directional solidification of a binary alloy.KEY WORDS Local mesh refinement Free boundary Finite element method Co-ordinate mapping Solidification

On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations

“…The h-method has been suggested by Szabo and Babuska [50] who subdivided the elements on which the measure of the error was larger than a prescribed tolerance. Tsiveriotis and Brown [12], also applied it in a free boundary problem by introducing a transition layer of non-conforming quadrilateral elements, and found that the local refinement technique is essential in cases where elliptic grid generators are used because it relaxes the requirements on the mapping equations and adds more flexibility to the handling of the three important characteristics of the grid mentioned above.…”

“…(11), while their volume is calculated in spherical coordinates via Eq. (12). Along the axis of symmetry, if spherical coordinates are used for the flow problem the symmetry conditions become…”

A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations

“…(22) and (23) are the weak forms of the momentum and continuity equations, respectively. By ðÁ; ÁÞ is denoted the inner product defined in the whole physical domain…”

“…Along the interface, nodes were equally distributed, while at the other boundaries reflective boundary conditions were used, allowing in this way the motion of the boundary nodes according to the interface deformation. To improve the accuracy of the solution on the interface, Tsiveriotis and Brown [22] proposed a two-to one-element splitting scheme, for a transition from a smaller number of elements in the bulk to a larger one close to the interface. However, this splitting resulted in elements with great aspect ratios and the generated grid lacked homogeneity; there were regions with extremely high concentration of elements and others where the grid was rather coarse.…”

Adaptive meshing in a mixed regime hydrologic simulation model

“…Local grid refinement is a popular way to increase grid resolution where needed without uniformly refining the mesh [4,6,7,22,25,[28][29][30]32]. Local grid refinement techniques have been developed in two dimensions for triangle and quadrilaterals ( [2,8,[19][20][21]27] and in three dimensions for tetrahedrals [1,12,13,17,18,31] and for hexahedrals [11,14,15].…”