Multi-material swept face remapping on polyhedral meshes

Highlights

• Conservative interpolation of multi-material fields free of mesh-mesh intersections.

• Second-order accurate remap at pure material regions and first order accurate at interfaces.

• Computationally effective remapping of multi-material fields for Arbitrary Lagrangian-Eulerian hydrodynamics.

• Comparison of an intersection remap to a flux remap for immiscible fluids.

• Open-source implementation of interface reconstruction and remap on polygonal/polyhedral meshes.

Abstract

Remapping is a conservative interpolation of a discretized intensive quantity between two meshes. In this article, we propose a novel multi-material flux remapping method that avoids the geometric computation of mesh-mesh intersections needed for an accurate intersection based remap. The flux remap is applicable to scalar quantities such as material density describing the multi-material flow between meshes with the same connectivity but small mesh displacements. The method is described for two- and three-dimensional polygonal/polyhedral meshes as it is implemented in Portage.¹ Another open source library, Tangram,² is used to calculate material interfaces in cells containing more than one material. Performance and accuracy of the flux remap are discussed with respect to Arbitrary Lagrangian-Eulerian simulations and compared to an accurate intersection based remap. In particular, cyclic remapping shows that the accuracy of the flux remap is limited to first order on material boundaries while maintaining second order accuracy in pure material regions.

Introduction

Hydrodynamic simulations of multi-material flows are important in many areas of research. For example, they are crucial for the design of Inertial Confinement Fusion (ICF) targets. Outer shells of these targets consist of several layers of different materials with different physical properties. In many cases, it is practical to assume that these materials are immiscible, meaning that each material occupies its own spatial region which is a subset of the entire computational domain. These regions are naturally moving along with the flow which makes it suitable for a Lagrangian framework, where an underlying, interface-aligned computational grid evolves with the flow. However, vorticial components of the flow can result in an unacceptable tangling of the grid.

In Arbitrary Lagrangian-Eulerian (ALE) methods [1], mesh untangling and smoothing called rezoning is employed to restore the quality of the computational grid to acceptable levels. Rezoning requires simulation fields to be accurately and conservatively transferred from the original Lagrangian grid to the new rezoned grid. A complication is that some cells of the rezoned mesh now overlap cells containing different materials from the Lagrangian mesh, creating new multi-material cells, even if the original Lagrangian mesh cells were all single material. Rather than subdivide the rezoned mesh along material boundaries which could potentially lead to poor quality cells again, one can choose to represent affected cells as multi-material or mixed cells and only track the volume fractions³ in those cells for carrying forward the Lagrangian simulation. However, when it is time to perform another rezone/remap step, one must contend with the scenario where a cell of the rezoned mesh overlaps one or more multi-material cells. Simply transferring quantities into this rezoned cell in proportion to volume fractions in the overlapped Lagrangian cells is inaccurate and diffusive resulting in smearing of material boundaries. A more accurate approach is to reconstruct the pure material sub-cells of a multi-material cell using a material interface reconstruction algorithm [3], [2], [4] and computing the overlap of the rezoned cell with these sub-cells rather than the full cell. Compared to single-material methods, multi-material remapping requires conservation of material volumes in addition to the standard requirements of mass, momentum, total energy, etc. conservation. All aspects of ALE methods are discussed in the seminal article [5], e.g.

In this paper, we focus only on a particular aspect of an ALE method: conservative interpolation or remapping of a single scalar quantity, such as the material density, in a multi-material flow simulation. More specifically, we assume that this quantity is adequately defined by its mean density and the volume which it occupies inside a polytopal computational cell. This is a common representation in both cell-centered [6] and staggered [7] ALE methods. Accurate multi-material remapping methods [8] exist and are based on the computation of exact intersections between reconstructed material polytopes on a Lagrangian mesh with the cells of the rezoned mesh. However, geometric calculation of these intersections is computationally expensive and this cost can easily dominate the whole ALE method. Therefore, many codes [9], [10], [11], [12] rely on simplified algorithms usually formulated using fluxes through faces of neighboring cells, and are aptly named flux-based, swept-face, or just swept remap. The flux formulation requires the displacement between the meshes to be limited. In particular for the presented method, the volume defined by a face displacement between the two meshes has to be smaller that the volume of a particular cell-face pyramid (3) in the contributing multi-material cell on the Lagrangian mesh, see (31) for details.

A second order, single material swept remap using piece-wise linear density reconstruction was introduced in [13]. Compared to a second-order intersection remap, the single material swept remap generally shows comparable errors while it can be more sensitive to some pathological mesh displacements (like an hour-glass distortion) and to the mesh orientation with respect to the underlying density field [14]. Possible combination of a swept remap with an intersection remap is investigated in [15]. For multi-material problems, hybrid algorithms [16], [17], [18] combine an intersection remap around material interfaces with a swept remap in pure material regions. The history of advection-like remapping methods for ALE simulations is summarized in [9]. One approximate advection based multi-material method [10] relies on additional corrections to ensure that computed material volumes remain positive and that they sum up to the bulk volume. Another method [19] is based on a simple assumption about a layered material layout oriented (either parallel or perpendicular to a face) in order to determine which materials should be fluxed through the face. Advection multi-material remap can be also implemented [11] in an operator split fashion dimension-by-dimension.

A multi-material swept-face remap using a Piece-wise Linear Interface Calculation (PLIC) [3] is used in the LANL code FLAG [20], [21]. Here we present the algorithm in 2D/3D and compare it to an accurate intersection remap. In 1D, swept-face remap is identical to intersection remap. Both remapping methods are implemented in an open source library called Portage [22].

This article is organized as follows: mesh notation, field description (for single and multi-material cases), and field reconstruction methods are provided in Section 2. In Sections 3 and 4, we provide a detailed description of the intersection and swept face remapping methods. In Section 5 we provide a brief overview of the various software libraries implementing the methods described and used for the numerical experiments. Finally, in Section 6, we present results for various comparisons between these two methods followed by discussions and conclusions in Section 7.