ReALE: A Reconnection Arbitrary-Lagrangian–Eulerian method in cylindrical geometry

Abstract

This paper deals with the extension to the cylindrical geometry of the recently introduced Reconnection algorithm for Arbitrary-Lagrangian-Eulerian (ReALE) framework. The main elements in standard ALE methods are an explicit Lagrangian phase, a rezoning phase, and a remapping phase. Usually the new mesh provided by the rezone phase is obtained by moving grid nodes without changing connectivity of the underlying mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In ReALE we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and permits to follow Lagrangian features much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tesselation machinery. In this work we focus on the extension of each phase of ReALE to cylindrical geometry. The Lagrangian, rezone with reconnection and remap phases are revamped to take into account the cylindrical geometry. We demonstrate the efficiency of our ReALE in cylindrical geometry on series of numerical examples.

A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian–Eulerian computations with nodal finite elements

“…Phase-field models have been successfully applied to interfacial problems, solidification dynamics, viscous fingering, fracture dynamics, vesicle dynamics, and could represent an interesting path forward in multi-material shock hydrodynamics. So far, the use of staggered and cell-centered discretizations [20,28,42,43,[47][48][49][50], have prevented the use of such models, as in general the reconstruction of the gradient fields of thermodynamic variables is not easy in these cases. Nodal-based finite elements are appealing in this context, as the gradient fields are always naturally well defined on element interiors.…”

A fully discrete ALE method over untwisted time–space control volumes

“…In this context, the family of arbitrary Lagrangian-Eulerian (ALE) methods was developed in [1][2][3][4][5][6][7] to solve this problem. Fluid phenomena including strong shocks, contact discontinuities, instabilities of material interfaces, and mixing processes are complicated, and therefore, developing numerically accurate and computationally efficient algorithms for such simulations remains challenging.…”

Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

“…Equivalence conditions are used to devise a class of FV schemes, in which all grid-dependent quantities are defined in terms of FE integrals. In the framework of Lagrangian or ALE solvers, the explosion and implosion problems in two spatial dimensions were solved by Cheng and Shu using a cell-centred Lagrangian scheme [14]; Loubère and others used the reconnection algorithm for ALE framework in cylindrical coordinates [15].The present approach moves from the mixed FV/finite element (FE) approach [16], which has been already successfully applied to the multidimensional Cartesian case [17,18] and the cylindrical case with axial symmetry [19], within the throughflow approximation [20] and more recently, in a unified fashion form for orthogonal coordinate systems [21]. The two-dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived.…”

“…More recently, Maire proposed a cell-centred Lagrangian formulation to capture axisymmetrical, that is, spherical, imploding shock waves [11]; Illenseer and Duschl generalized the two-dimensional central upwind to curvilinear grids [12]; Clain and collaborators extended a multi-slope MUSCL scheme to cylindrical coordinates [13]. In the framework of Lagrangian or ALE solvers, the explosion and implosion problems in two spatial dimensions were solved by Cheng and Shu using a cell-centred Lagrangian scheme [14]; Loubère and others used the reconnection algorithm for ALE framework in cylindrical coordinates [15].…”

“…More recently, Maire proposed a cell-centred Lagrangian formulation to capture axisymmetrical, that is, spherical, imploding shock waves [11]; Illenseer and Duschl generalized the two-dimensional central upwind to curvilinear grids [12]; Clain and collaborators extended a multi-slope MUSCL scheme to cylindrical coordinates [13]. In the framework of Lagrangian or ALE solvers, the explosion and implosion problems in two spatial dimensions were solved by Cheng and Shu using a cell-centred Lagrangian scheme [14]; Loubère and others used the reconnection algorithm for ALE framework in cylindrical coordinates [15].The present approach moves from the mixed FV/finite element (FE) approach [16], which has been already successfully applied to the multidimensional Cartesian case [17,18] and the cylindrical case with axial symmetry [19], within the throughflow approximation [20] and more recently, in a unified fashion form for orthogonal coordinate systems [21]. To link these two different numerical schemes, it is necessary to introduce suitable equivalence conditions that relate the FV metrics, that is, cell volumes and integrated normals, to the FE integrals.…”