One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian–Eulerian methods
Introduction
In the arbitrary Lagrangian–Eulerian (ALE) methods, a fast and accurate remapping method for data transfer of all fluid quantities from the Lagrangian to the rezoned mesh is necessary. Typically, methods based on swept regions are used in practical calculations [25], [21], as these methods are significantly faster than the natural methods based on intersections. Unfortunately in the multi-material ALE, one needs to distribute the fluxes to particular materials, which we believe cannot be done consistently without intersections. The compatibility condition then leads to the requirement of doing the intersections in the entire computational domain. As the intersection-based methods are typically significantly more expensive, this leads to slower numerical computations.
To improve efficiency of the multi-material remapping, the concept of hybrid remap was introduced [5], [16]. It is based on the main idea of performing the expensive intersections only in the vicinity of material interfaces, while in pure material regions the cheap swept region method can be used.
In [5], the basic concepts of the hybrid remapping methods were described and the two step hybrid remapping (TSHR) method was introduced. This method employs the rezoning and remapping phases separately for pure and mixed regions, in two distinct steps. In the first step, only nodes in pure regions are moved and the swept region method is used to remap all fluid quantities from the Lagrangian to this inter-mediate mesh. In the second step, the remaining nodes in mixed regions are moved and the multi-material fluid quantities are remapped using the intersection-based method. This method is applicable for any meshes, however, due to different treatment of pure and mixed nodes, symmetry of the problem can be violated, which can be very important in certain types of problems [24], for example for ICF applications [12], [6]. Moreover, this method requires modifications in the main program routine, which may not be acceptable in the particular production code.
In [16], the combined one-step hybrid remapping (COSHR) method is introduced. This method explicitly combines the swept fluxes in pure material regions with the intersection-based fluxes close to material interfaces. At the buffer regions where both types of fluxes meet, a special treatment is required due to the existence of the corner fluxes in the intersection-based method, while they are missing in the swept region approach. This method keeps the problem symmetry, however, it is only applicable for logically rectangular meshes, its generalization to general polygons is not straightforward.
Both hybrid approaches reduce the computational cost of the remapping step, however, both approaches suffer from particular deficiencies. Here, a novel flux-based swept-region-with-intersections one-step hybrid remapping (SIOSHR) method is introduced, which does not suffer from these disadvantages. In this paper, we only focus on remap of the material masses and material information (volumes and centroids), no additional fluid quantities are considered. In this method, the swept regions are constructed even for multi-material regions of the mesh, and the corresponding multi-material fluxes are constructed by intersecting the swept regions with the pure material polygons of the involved cells. In case of self-overlapping swept region, the swept region is split into triangles which are then intersected with the pure material polygons separately. Therefore, the fluxes in the pure-material regions are consistent with the fluxes in the mixed regions – that is, for each cell there is the same number of fluxes computed by integration over the same swept regions, just the internal structure of the particular swept region can be different. Due to this construction, this method is consistent, conservative, linearity-preserving (even for multi-material cells in case of straight material interface), and more efficient than standard intersection-based method. It does not require modifications of the main routine and preserves symmetry.
This paper is organized as follows. In Section 2, the flux-based remapping framework incorporating geometric exchange integrals is described. In Section 3, the standard single-material remapping methods are reviewed: the remapping method based on cell intersections in Section 3.1, and the swept region remapping method in Section 3.2. In Section 4, the intersections are performed between the original cells and the standard swept regions (which may eventually be decomposed into triangles), and we demonstrate that this approach is equivalent to the standard intersection-based method from Section 3.1. Finally, the new swept-region-with-intersections one-step hybrid remapping method is introduced in Section 5, which combines the standard swept region fluxes from Section 3.2 in pure material regions with the swept region fluxes computed by intersections from Section 4 close to material interfaces. In Section 6, two numerical tests are used to demonstrate properties of the new method and compare it with the standard intersection-based and swept region approaches. This paper is concluded in Section 7.
Section snippets
General considerations
We assume that a two-dimensional domain is fully covered by polygonal cells (a particular cell is denoted by the c symbol in this paper), each cell has its vertices (mesh nodes, a particular mesh node is denoted by the n symbol here). In the context of ALE methods, we consider two different meshes, the original (old, Lagrangian) mesh with cells c, and a new (rezoned) mesh with cells . We assume that both meshes have the same topology and are similar – we assume that every new node remains in
Review of standard single-material remapping methods
In this Section, the standard single-material remapping methods are reviewed. The method based on intersections is described and its flux form (including corner fluxes) is presented. Next, we review the remapping approach based on swept regions, where the particular flux is approximated by integration of the density function from one of its edge neighbors over the whole swept region.
Remap by intersections in swept regions
In this Section, we describe the remapping approach based on intersections of the swept region with the adjacent cells of the original mesh. Similar approach based on classification of all possible swept region shapes is described in [8] for the particular case when one of the meshes is a regular square mesh. In case of self-overlapping swept regions, its decomposition into triangles [15], [14] is used. This method basically combines the swept region approach from Section 3.2 with the
Swept-region-with-intersections one-step hybrid remap (SIOSHR)
Unfortunately, the swept region method in its pure form is not very suitable for multi-material simulations. In the case of the presence of material interfaces, it is not obvious how to approximate various material fluxes. We are aware of several approximations implemented in various codes [2], [3]. For example, it is possible to distribute the total swept mass to the materials according to the cell volume fractions, or approximate the swept region by a single rectangle of the same volume. All
Numerical examples
In this Section, we demonstrate the properties of the SIOSHR method on two selected stand-alone cyclic remapping numerical examples from [16], [5]. We define material volume fractions and material centroids in each cell of the initial computational mesh, and using the MOF material interface method, pure material polygons are defined in each cell. In each material polygon, material density is set by an analytic density function, which is then used for initialization of material masses. At the
Conclusions
In this paper, we have introduced a new swept-region-with-intersections one-step hybrid remapping (SIOSHR) method. We have presented several numerical tests to compare the efficiency and accuracy of the new method with the standard intersection-based method, and the single-material approach based on swept regions. From the numerical tests, we can make several conclusions.
The new method is linearity preserving and second order accurate in both single- and multi-material tests. The numerical
Acknowledgements
This work was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The authors gratefully acknowledge the partial support of the US Department of Energy Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research and the partial support of the US Department of Energy National Nuclear Security Administration Advanced Simulation
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2020, Computers and FluidsCitation Excerpt :In order to combine the advantages of the Lagrangian and Eulerian methods, Hirt et al.[1] proposed an arbitrary Lagrangian-Eulerian (ALE) method in which the grid vertices may move in some arbitrarily specified ways. The indirect ALE approach [2–5] consists of three stages [6]: a Lagrangian stage, in which the solution and the computational mesh are updated; a rezoning stage, in which the grid quality is optimized; and a remapping stage, in which the Lagrangian solution is projected to the rezoned mesh. In the direct ALE schemes [7–10], the grid motion is taken into account directly in the numerical flux computation.