Periodic Reporting for period 1 - MoMeNTUM (Modern high order numerical Methods based on No-compromise moving Voronoi Tessellations: a Unified solver for continuum Mechanics)
Berichtszeitraum: 2024-01-01 bis 2025-12-31
Computer simulations of fluids, solids, and their interactions rely on solving complex mathematical equations on powerful computers. MoMeNTUM developed advanced numerical methods for a comprehensive mathematical framework: a unified model of continuum mechanics UMCM, also known as the Godunov-Peshkov-Romenski GPR model, which describes fluids, elastic and plastic solids, and material failure within a single set of equations. While very general, this model had never been used successfully for turbulent flow simulations due to fundamental computational obstacles.
The project set out to overcome these obstacles through three algorithmic innovations: moving computational meshes; automatic sub-mesh generation for robust shock capturing on general polyhedral grids; and novel basis functions to close the performance gap between flexible unstructured meshes and efficient Cartesian grids. The overarching goal was a high-performance simulation code capable of running the full unified model on modern supercomputers, applied to demanding turbulent flow benchmarks that serve as standard tests for state-of-the-art fluid dynamics solvers.
The expected impact was twofold: demonstrating that a single mathematical model can deliver competitive results for turbulent flows, a domain traditionally requiring specialized solvers, and producing algorithmic tools broadly applicable across computational science and engineering.
The project set out to overcome these obstacles through three algorithmic innovations: moving computational meshes; automatic sub-mesh generation for robust shock capturing on general polyhedral grids; and novel basis functions to close the performance gap between flexible unstructured meshes and efficient Cartesian grids. The overarching goal was a high-performance simulation code capable of running the full unified model on modern supercomputers, applied to demanding turbulent flow benchmarks that serve as standard tests for state-of-the-art fluid dynamics solvers.
The expected impact was twofold: demonstrating that a single mathematical model can deliver competitive results for turbulent flows, a domain traditionally requiring specialized solvers, and producing algorithmic tools broadly applicable across computational science and engineering.
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.
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.
The quaternion reformulation of the unified model of continuum mechanics resolves a long-standing computational barrier. The standard model formulation could not be used for turbulent flow simulation due to systematic numerical corruption of the rotation field. The quaternion approach eliminates this pathology and produces better results at much lower resolution for the Taylor-Green Vortex at Reynolds number 100: the standard formulation with a third-order scheme at 224^3 grid cells was reported to be not sufficiently accurate, while the quaternion formulation with a simpler second-order scheme at just 32^3 cells produces better results. Since the total cost of explicit time-stepping schemes scales with the fourth power of the mesh size (three spatial dimensions plus the time dimension via the CFL condition), this resolution difference alone corresponds to gains of several orders of magnitude. This improvement is entirely due to the reformulation of the mathematical model, not the numerical scheme, which is in fact simpler and cheaper. This opens the model to the broader computational fluid dynamics community: simulations that were previously out of reach even on the largest supercomputers are now feasible on university-scale infrastructure.
The quadrature rules for arbitrary convex polygons and polyhedra achieve near-minimal node counts with guaranteed positive weights and require no numerical optimization. For three-dimensional polyhedra, the moment-fitting approach produces rules with fewer nodes than the best published results, including some results for standard hexahedral cells. These rules have broad applicability in computational geometry, mesh generation, and numerical integration across scientific and engineering disciplines.
The split Riemann solver for low-Mach-number flows offers a simple and efficient alternative to established approaches, applicable to any compressible flow solver. The piecewise-quadratic reconstruction and its formal connection to third-order methods provide a practical tool for reducing numerical dissipation in finite volume schemes.
The Taylor-Green Vortex simulations at Reynolds 3200, carried out with an explicit finite volume code for a 15-equation system on a university-scale computing cluster, demonstrate that the unified model of continuum mechanics, equipped with the quaternion formulation, is not merely a theoretical framework but a practical simulation tool for turbulent flows at scales previously accessible only to purpose-built incompressible solvers.
The quadrature rules for arbitrary convex polygons and polyhedra achieve near-minimal node counts with guaranteed positive weights and require no numerical optimization. For three-dimensional polyhedra, the moment-fitting approach produces rules with fewer nodes than the best published results, including some results for standard hexahedral cells. These rules have broad applicability in computational geometry, mesh generation, and numerical integration across scientific and engineering disciplines.
The split Riemann solver for low-Mach-number flows offers a simple and efficient alternative to established approaches, applicable to any compressible flow solver. The piecewise-quadratic reconstruction and its formal connection to third-order methods provide a practical tool for reducing numerical dissipation in finite volume schemes.
The Taylor-Green Vortex simulations at Reynolds 3200, carried out with an explicit finite volume code for a 15-equation system on a university-scale computing cluster, demonstrate that the unified model of continuum mechanics, equipped with the quaternion formulation, is not merely a theoretical framework but a practical simulation tool for turbulent flows at scales previously accessible only to purpose-built incompressible solvers.