The project delivered on all three proposed algorithmic fronts and expanded far beyond the original scope.
A complete computational geometry toolkit was developed for high-order numerical methods on polygonal and polyhedral meshes: robust polygon clipping, combinatorial partitioning for efficient numerical integration, novel quadrature rules achieving near-minimal node counts on arbitrary shapes, and reconstruction strategies for unstructured grids.
Semi-orthogonalized polynomial basis functions were developed for arbitrary convex polygons, achieving efficient hardware vectorization through fully contiguous memory access at quadrature points. The quadrature work was extended to three-dimensional polyhedra, where a moment-fitting approach achieved node counts surpassing the best published results in the literature, including some results for standard hexahedral cells.
The project's central achievement was a new quaternion-based mathematical formulation of the unified model. The standard formulation evolves a matrix field that simultaneously encodes rotation, shear, and volume changes in a material. Standard numerical interpolation of this matrix destroys its rotation content, making the model unusable for turbulent flows. The quaternion formulation decomposes the field into separate rotation and deformation components, with the rotation represented as a unit quaternion, enabling geometrically correct numerical treatment. This reduces computational cost by approximately four orders of magnitude.
A new Riemann solver for low-speed flows, a novel piecewise-quadratic reconstruction technique, and novel boundary conditions and exact analytical solutions for the quaternion system were developed. The simulation code was optimized at every level, from instruction-level vectorization to parallel input/output on computing clusters, achieving 40% of flops-throughput-adjusted theoretical peak on production runs. These innovations combined to enable Taylor-Green Vortex simulations (a standard turbulence benchmark) of the 15-equation quaternion system at Reynolds number 3200: a single-octant simulation at 2304^3 cells (equivalent full-domain resolution 4608^3, approximately 180 billion evolved unknowns) at Mach 0.03 on 48 nodes (9216 cores), and a full-domain simulation at 3072^3 cells (approximately 430 billion evolved unknowns) at Mach 0.1 on 96 nodes (18,432 cores), both on the University of Cologne's RAMSES cluster, a university-scale facility more than an order of magnitude smaller than the national supercomputers typically employed for computations at this scale.
