Overview
The study introduces a novel computational method, the Quadratic Moment-Matched Sigmoid (QMMS) approximation, designed to enhance the efficiency of the tangent of hyperbola for interface capturing (THINC) method when utilizing quadratic surface representation and Gaussian quadrature (THINC/QQ) on unstructured grids. THINC/QQ, in its original formulation, necessitates topology-dependent numerical quadrature for computing cell and face integrals and employs Newton iteration for determining the surface constant. The QMMS approximation addresses these computational demands by evaluating these quantities through the mean and variance of the quadratic field, while concurrently preserving the hyperbolic-tangent reconstruction profile and the boundary variation diminishing (BVD) formulation inherent to the THINC approach.
Research Context
The THINC method, specifically its THINC/QQ variant, is employed for interface capturing on unstructured computational grids. A core challenge within this method involves the computation of cell and face integrals, which traditionally relies on numerical quadrature whose specifics are dependent on the topological characteristics of the grid cells. Additionally, determining the surface constant for the quadratic representation involves an iterative process, specifically Newton iteration. These requirements contribute to the computational overhead during the reconstruction stage of numerical simulations.
Approach
The QMMS approximation is developed to provide closed-form solutions for quantities that traditionally require iterative or topology-dependent numerical methods. This is achieved through the integration of a slope-matched Gaussian cumulative distribution function and a moment-matched Gaussian model. These models are utilized to derive approximations for both the surface constant and the face averages. The application of QMMS fundamentally alters the evaluation process for these quantities, eliminating the reliance on Newton iteration for the surface constant and obviating the need for topology-dependent runtime quadrature for integral evaluations. The method maintains the core hyperbolic-tangent reconstruction profile and adheres to the BVD formulation, which are key characteristics of the THINC method.
Findings
- The QMMS approximation successfully evaluates cell and face integrals and the surface constant from the mean and variance of the quadratic field.
- The approach retains the hyperbolic-tangent reconstruction profile and the boundary variation diminishing (BVD) formulation.
- A slope-matched Gaussian cumulative distribution function and a moment-matched Gaussian model provide closed-form approximations, thereby eliminating Newton iteration for the surface constant and topology-dependent runtime integral evaluation for face averages.
- Assessment across single-cell tests and nine two- and three-dimensional benchmarks indicated that the principal transported and shock structures remained closely aligned with those obtained using THINC/QQ.
- Local differences between QMMS and THINC/QQ results were observed, concentrated primarily near discontinuities, contact regions, and developed shear layers.
- Reconstruction-stage profiling demonstrated a reduction in computational cost with QMMS. Mean THINC/QQ-to-QMMS time ratios were measured at 2.12 in two dimensions and 1.85 in three dimensions.
- QMMS effectively replaces topology-dependent runtime integral evaluation with a common moment-based formulation, reducing reconstruction-stage cost.
Why This Matters
The introduction of the QMMS approximation offers a computational efficiency gain for the THINC/QQ method on unstructured grids. By replacing iterative and topology-dependent calculations with closed-form moment-based approximations, the method reduces the runtime overhead associated with integral evaluations and surface constant determination. This directly impacts the computational cost of the reconstruction stage in simulations utilizing interface capturing techniques.