Second-order accurate normal reconstruction from volume fractions on unstructured meshes with arbitrary polyhedral cells

TL;DR

This paper presents a novel second-order accurate method for reconstructing normal fields from volume fractions on unstructured polyhedral meshes, improving accuracy and robustness in complex 3D geometries.

Contribution

The paper introduces a new least squares fitting approach for normal reconstruction that accounts for volume conservation and simplifies computations on unstructured meshes.

Findings

Achieves second-order convergence in normal alignment

Demonstrates robustness on convex and non-convex hypersurfaces

Provides new insights into the minimization process

Abstract

This paper introduces a novel method for the efficient second-order accurate computation of normal fields from volume fractions on unstructured polyhedral meshes. Locally, i.e. in each mesh cell, an averaged normal is reconstructed by fitting a plane in a least squares sense to the volume fraction data of neighboring cells while implicitly accounting for volume conservation in the cell at hand. The resulting minimization problem is solved approximately by employing a Newton-type method. Moreover, applying the Reynolds transport theorem allows to assess the regularity of the derivatives. Since the divergence theorem implies that the volume fraction can be cast as a sum of face-based quantities, our method considerably simplifies the numerical procedure for applications in three spatial dimensions while demonstrating an inherent ability to robustly deal with unstructured meshes. We discuss the theoretical foundations, regularity and appropriate error measures, along with the details of the numerical algorithm. Finally, numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes are presented, where we obtain second-order convergence for the normal alignment and symmetric volume difference. Moreover, the findings are substantiated by completely new deep insights into the minimization procedure.