Implementation of adaptive mesh refinement into an Eulerian hydrocode

In this paper the implementation of an adaptive mesh refinement scheme into an existing three-dimensional Eulerian hydrocode is described. The adaptive strategy is block-based, which was required in order to leave the existing data structure intact. The focus of this work is on the development of refinement and unrefinement procedures that are conservative and preserve the locations of material interfaces, as well as error indicators suitable for use in an Eulerian hydrocode. Introduction Adaptive mesh refinement refers to a scheme for finite difference and finite element codes wherein the size and distribution of the computational mesh is changed dynamically so that the solution complies with some specific constraint. There are many different types of constraints of interest, depending on the goals of the computation, but a common constraint is that the error be held constant over the entire computational mesh. There are many advantages to adaptive refinement schemes; most importantly is that problems are solved in the most computational time and memory-efficient manner. The benefits of adaptive mesh refinement have already been demonstrated in many application areas (for example, linear elasticity, gas dynamics, and acoustics), but adaptive refinement for shock wave physics codes is still a new area of research. Eulerian codes in particular may benefit significantly from adaptive mesh refinement since typically it is necessary to include a large number of elements in a simulation, even in regions where there may not be any material initially. In this paper the implementation of an adaptive mesh refinement scheme into an existing three-dimensional Eulerian hydrocode is described. The adaptive strategy is block-based, which was required in order to leave the existing data structure intact. The focus of this work is on error indicators suitable for use in an Eulerian hydrocode, as well as refinement and unrefinement procedures. Overview of Adaptive Strategy When implementing adaptive refinement into an existing code, it is very important to consider the organization and data structure of the target application code. In the present case, the data is organized in (/,J,K) logical blocks that correspond to the mesh used in the problem, as is shown in Fig. I. Within a block, the mesh contour lines must remain parallel to the coordinate axes and constrained nodes are not permitted. However, different values of !, J, and/or K are permitted in adjacent blocks. Thus a reasonable approach for implementation of adaptivity, which preserved the existing data structure in the code, was to limit refinement to the block level. Furthermore, in order to simplify the refinement process as well as communication between blocks, the refinement/unrefinement was limited to isotropic 2: I ratios along adjacent blocks. This is illustrated in Fig. 2, where a set of communicating blocks is shown. Ghost cells are incorporated along the edges of the block, and the contents of these cells comes form a cell combine or split as is needed from the adjacent block. In this way, each block sees exactly the information it expects to see. ! : : : : i .L _L_ l t l :"~"'";''':'''':-'T''''':''':'''':''':''''": .: :: :: :: :: : ·........~..L...LJ.....:...i.... ; . : .. : * :···· ... T ..... ·T······r ....T··..·~ ,. 11: 11 ;'.:::::::::1 ~=::I i·····.... . : t........ , ; i j i j ...... .l....... L ....l....... L..... ~ Figure 1. Organization of data in target application code Figure 2. 81ock·adapti~·e stmteg)' applied to target application code. A significant part of this effort is to establish the two-way communication between blocks, as well as to make the scheme work in parallel, which is the subject of an accompanying paper [1]. The focus of this work is on the development of refinement and unrefinement schemes, as well as error indicators, suitable for implementation into a three-dimensional Eulerian shock physics code. Refinement and Un refinement Efficient and accurate schemes for refinement ant unrefinement of the cell variables are a crucial element in any adaptive scheme. The refinement and unrefinement strategies used here take advantage of the 1:2 and 2: 1 ratios between parent and child cells, allowing these processes to be carried out rapidly and in a conservative manner. Unrefinement Procedure The collapse of a set of eight child cells into a single parent cell is a simple process. A method for this was implemented by Crawford [1] and was not initially modified any further. It conserves mass and energy; momentum is also conserved if the velocity profiles are linear across the parent cell. Further details of the procedure may be found in [2]. Refinement Procedure The refinement of a parent cell into eight child cells, on the other hand, is complicated by the fact that there may be material interfaces in the parent cell. The interfaces must reconstructed to properly map the cell variables to the children. There are already algorithms in place in the target application code that perform rezoning that could have been used for this; however, these algorithms rely in one-dimensional advection and were thought to be too dispersive given the large number of times the refinement routines are used during a typical calculation. Instead, we can take advantage of the exact geometric overlaps that exist when the refinement is limited to 2 to 1 ratios, and reconstruct the material interfaces in the child cells exactly. This eliminates dispersion errors. Review of Youngs' Method for Interface Reconstruction A key element in the refinement process is the proper mapping of material interfaces when elements are refined. In order to understand this process, it is useful to review Youngs' algorithm for interface reconstruction [3]. The method is a systematic approach for a unique determination of a planar interface separating two materials in a cell, given the volume fractions of the materials in the cell. Conceptually, the method is simple to understand. Where it provides the greatest benefit is by minimizing the number of possible intersection conditions that must be checked when a plane of arbitrary orientation passes through a cell (there are only five when this method is applied). The basic strategy in the Youngs' algorithm is to determine the outward unit normal vector n separating the material of interest from the other materials, and the distance d from the interface plane to a reference comer, measured along a direction parallel to n. If there are only two materials in the cell and the interface plane is assumed to be planar, these two quantities uniquely define the location of the interface plane. There are five possible intersection conditions. These are given in Figure 3, and include the triangle section, quadrilateral section A, pentagonal section, hexagonal section, and the quadrilateral section B. From this comes the interface geometry as well as a value for d. A summary of the relationships needed to determine d for each of the intersection conditions can be found in [3].