Adaptive mesh refinement in CTH : Implementation of block - adaptive multi - material refinement and advection algorithms

An adaptive mesh refmement (AMR) version of CTH is currently under development. This project is being conducted jointly by researchers at the University of Texas and at Sandia National Laboratories. The AMR version of CTH represents a significant milestone in the ten-year development of this legacy code. CTH is a multi-material wave propagation code used by many analysts in the DoD user community to simulate large deformations, large strain rates, and strong shocks in solid, liquids and gases. The numerical procedure is based on a finite volume formulation of very general forms of the continuum equations. As such, it can be applied to a wide variety of problems. The computational mesh used in CTH is Eulerian; materials and material interfaces are permitted to flow through the mesh as the calculation proceeds. The incorporation of new algorithms into the AMR version of CTH is described. A block refinement algorithm that preserves the location of material interfaces has been implemented into a working version of the code. This algorithm uses advanced interface tracking to map materials; this minimizes the dispersion normally associated with the process. A multimaterial advection algorithm, which is basically a generalization of Youngs' interface reconstruction method to cells with mismatched faces, is described. Results from three-dimensional examples problems are shown that effectively illustrates the improvements afforded by these new algorithms. 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. Here, the application code is CTH (McGlaun et al. 1990), a threedimensional multi-material Eulerian wave propagation code designed for modeling very large deformations and strong shocks. The data in CTH is organized in (l,J,K) logical blocks that correspond to the mesh used in the problem. Within a block, the mesh contours are constrained to remain parallel with the coordinate axes, and the introduction of hanging, or constrained, nodes is not permitted. However, adjacent blocks are permitted to have different values of l, J and K. Thus, a reasonable approach for the implementation of adaptivity, which preserves the original data structure used in CTH, is to limit refinement/unrefinement to the block level. Furthermore, in order to simplify the algorithms for communication between blocks, the refinement/unrefmement was limited to isotropic 2: I ratios between adjacent blocks. This process is illustrated in Fig. I, where a set of communicating blocks is shown. The contents of the ghost cells along the periphery of the blocks are provided by information coming from adjacent blocks. Since these adjacent blocks may be at a different resolution, calculating the contents of the ghost cells may involve a cell split/combine process. r-r-,..··T ·r0.: 'TT.'·''''.lTTT ..'::I i ''''"";",,),, " ....... ! ,~It "1 i·i······ ;-"i I"" ,J , ' 1.1 i ~1 '-i ~ :! tl.······.. ,.· ... · ·,······!··· ;I~VC! ,····'·i:: ' iIi .. , , , , , r,~· dm ::,. 'i l ! : : ;......... , .1 i........1 j .. : J I .. ... . = : i L:::::l:::::.L:J::::::c:J.::::J ....... ·; Figure 1. Block-adaptive strategy applied to target application code. A significant part of this effort was to establish the two-way communication between blocks, as well as to make the scheme work in parallel, which is the subject of another paper (Crawford 1999). The focus of this work is on the development of refinement schemes, as well as error indicators, suitable for implementation into a three-dimensional Eulerian shock physics code. REFINEMENTIUNREFINEMENT The collapse of 8 child cells (in three dimensions) into a single parent cell is a simple process, which for brevity will not be completely described here. For example, intensive quantities (such as specific internal energy) in the parent cell are mass averages of the intensive quantities in the child cells, masses and volumes are simply sums of the values from the child cells, and material volume fractions are volume averages from the child cells. The refinement process, on the other hand, requires a parent cell to be split into 8 child cells. This process is complicated by the fact that material interfaces exist within the cell; thus the location of these interfaces must be preserved when the refinement is done. To accomplish this, it is useful to review the algorithms used for interface tracking in CTH. Review of Interface Tracki ng In CTH, the location of material interfaces within a cell is interpreted using Youngs' algorithm (Youngs 1987). Youngs' algorithm is basically a systematic procedure for determining the position and orientation of the interface plane separating two materials in a computational cell, given the volume fractions of materials in the cell as well as the surrounding cells. The basic procedure used in Youngs' algorithm is to determine the outward unit