3D cache-oblivious multi-scale traversals of meshes using 8-reptile polyhedra

In the field of numerical simulation, a multi-scale grid and its traversal can have a huge impact on the cache-efficiency of the calculations performed. We present the Haverkort element traversal and refinement scheme which is based on the bisection of 8-reptile tetrahedra. We developed a stack-assignment scheme that allows the Haverkort traversal to use stacks for storing the input data, the output data, and the temporary data of vertices and elements. Our parallel plane approach to assign stacks to vertices gave solution that requires at most 8 stacks to store temporary vertex data for uniform and multi-scale refined grids independent of the refinement level. Furthermore 8 is also the lower bound for the number of stacks for our parallel plane approach. Additionally we show that a general lower bound exists of 5 temporary stacks for storing the vertex data during a traversal. The combination of the Haverkort traversal with our Constant Stack algorithm is cache-oblivious, suitable for multi-level cache hierarchies and maintains its performance for multi-level refined grids and allows for fast and efficient space-filling-curve-based (re)partitioning. The Constant Stack algorithm has a time complexity of O(1) for stackassignment and -access and can achieve a cache-miss ratio of less than 5.2% .For all of these reasons we expect that a constant-number-of-stacks solution can compete with and outperform numerical simulations using cache-optimization techniques based on loop blocking when running on CPUs or GPUs. We apply the approach found for obtaining a constant-number-of-stacks solution for the Haverkort element traversal and a refinement scheme was applied to two other suitable 8-reptile polyhedra. Traversal and refinement schemes are given for an 8-reptile bisected cube and an 8-reptile bisected triangular prism needing 9 and 7 stacks respectively. this page is intentionally left blank