Numerical Algorithms

, Volume 77, Issue 4, pp 1029–1068 | Cite as

Robust intersection of structured hexahedral meshes and degenerate triangle meshes with volume fraction applications

  • Frida Svelander
  • Gustav Kettil
  • Tomas Johnson
  • Andreas Mark
  • Anders Logg
  • Fredrik Edelvik
Open Access
Original Paper


Two methods for calculating the volume and surface area of the intersection between a triangle mesh and a rectangular hexahedron are presented. The main result is an exact method that calculates the polyhedron of intersection and thereafter the volume and surface area of the fraction of the hexahedral cell inside the mesh. The second method is approximate, and estimates the intersection by a least squares plane. While most previous publications focus on non-degenerate triangle meshes, we here extend the methods to handle geometric degeneracies. In particular, we focus on large-scale triangle overlaps, or double surfaces. It is a geometric degeneracy that can be hard to solve with existing mesh repair algorithms. There could also be situations in which it is desirable to keep the original triangle mesh unmodified. Alternative methods that solve the problem without altering the mesh are therefore presented. This is a step towards a method that calculates the solid area and volume fractions of a degenerate triangle mesh including overlapping triangles, overlapping meshes, hanging nodes, and gaps. Such triangle meshes are common in industrial applications. The methods are validated against three industrial test cases. The validation shows that the exact method handles all addressed geometric degeneracies, including double surfaces, small self-intersections, and split hexahedra.


Cut-cell Volume fraction Mesh repair Overlapping triangles Split hexahedra 



This work was supported in part by the Swedish Governmental Agency for Innovation Systems, VINNOVA, through the FFI Sustainable Production Technology program, and in part by the Sustainable Production Initiative and the Production Area of Advance at Chalmers University of Technology. The support is gratefully acknowledged.


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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Fraunhofer-Chalmers Research Centre for Industrial MathematicsGöteborgSweden
  2. 2.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGothenburgSweden

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