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+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +7476 +10.21105/joss.07476 + +HOHQMesh: An All Quadrilateral/Hexahedral Unstructured +Mesh Generator for High Order Elements + + + +https://orcid.org/0000-0002-8076-0856 + +Kopriva +David A. + + + + + +https://orcid.org/0000-0002-5902-1522 + +Winters +Andrew R. + + +* + + +https://orcid.org/0000-0002-3195-2536 + +Schlottke-Lakemper +Michael + + + + +https://orcid.org/0000-0001-5650-7095 + +Schoonover +Joseph A. + + + + +https://orcid.org/0000-0002-3456-2277 + +Ranocha +Hendrik + + + + + +The Florida State University, Tallahassee, FL, United +States of America + + + + +San Diego State University, San Diego, CA, United States of +America + + + + +Department of Mathematics; Applied Mathematics, Linköping +University, Sweden + + + + +High-Performance Scientific Computing & Centre for +Advanced Analytics and Predictive Sciences, University of Augsburg, +Germany + + + + +Fluid Numerics, Hickory, NC, United States of +America + + + + +Institute of Mathematics, Johannes Gutenberg University +Mainz, Germany + + + + +* E-mail: + + +1 +11 +2024 + +9 +104 +7476 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2024 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +Mesh Generation +Spectral Element Methods +High Order +Unstructured Quadrilateral and Hexahedral Meshes + + + + + + Summary +

HOHQMesh + (David + A. Kopriva, Winters, Schlottke-Lakemper, Schoonover, et al., + 2024) generates unstructured all-quadrilateral and hexahedral + meshes with high order boundary information for use with spectral + element solvers. Model input by the user requires only an optional + outer boundary curve plus any number of inner boundary curves that are + built as chains of simple geometric entities (lines and circles), user + defined equations, and cubic splines. Inner boundary curves can be + designated as interface boundaries to force element edges along them. + Quadrilateral meshes are generated automatically with the mesh sizes + guided by a background grid and the model, without additional input by + the user. Hexahedral meshes are generated by extrusions of a + quadrilateral mesh, including sweeping along a curve, and can follow + bottom topography. The mesh files that HOHQMesh generates include high + order polynomial interpolation points of arbitrary order.

+ + + Statement of Need +

Spectral element methods (SEMs) use multiple degrees of freedom + within elements to achieve high order accuracy and can be applied to + complex geometries. Details of SEMs can be found in the books by + Deville, Fischer and Mund + (Deville + et al., 2002), Karniadakis and Sherwin + (Karniadakis + & Sherwin, 2005), Hesthaven and Warburton + (Hesthaven + & Warburton, 2008), and Kopriva + (David + A. Kopriva, 2009).

+

Open source spectral element packages now exist to compute + solutions of a wide range of equations such as the compressible and + incompressible Navier-Stokes, ideal and visco-resistive + magnetohydrodynamics, Euler gas dynamics, and shallow water equations, + and include Nektar++ + (Cantwell + et al., 2015), SemTex + (Blackburn + et al., 2019), Sem2dPack + (Ampuero, + 2012), SPECFEM + (Martire + et al., 2021), Nek5000 + (Fischer + et al., 2008), HORSES3D + (Ferrer + et al., 2023), FLEXI + (Krais + et al., 2021), FLUXO + (Rueda-Ramirez + et al., 2017), Trixi.jl + (Ranocha + et al., 2022; + Schlottke-Lakemper + et al., 2021), and NUMA + (Giraldo + et al., 2013).

+

The features of SEMs are now well-established. Like low order + finite element methods, they can be applied to general geometries, but + have exponential convergence in the polynomial approximation order. + Discontinuous Galerkin (DGSEM) versions applied to hyperbolic problems + have exponentially convergent dissipation and dispersion errors + (Ainsworth, + 2004), making them well suited for wave propagation problems. + Discontinuous Galerkin SEMs are also especially suitable when material + discontinuities are present. Approximations exist for high order + quadrilateral/hexahedral and triangle/tetrahedral elements.

+

What some are now calling “classical” spectral element methods use + tensor product bases on quadrilateral or hexahedral meshes. These + bases lead to very efficient implementations and have high order + quadratures that can be used to approximate the integrals found in + weak forms of the equations. Of the widely available spectral element + packages, SemTex, Sem2dPack, Nek5000, FLEXI, FLUXO, Trixi.jl, and + HORSES3D primarily or exclusively use quadrilateral and hexahedral + meshes.

+

Unfortunately, unstructured meshes for quad/hex elements are + difficult to generate even for low order finite elements + (Bommes + et al., 2013). The advantages not withstanding, a major + impediment to the application of SEMs has been the availability of + appropriate general purpose mesh generation software that can generate + elements of arbitrary order, especially in open-source form. In 2002, + Sherwin & Peiró + (2002) + wrote: “The development of robust unstructured high-order methods is + currently limited by the inability to consistently generate valid + computational meshes for complex geometries without user + intervention.” This has remained true particularly for quadrilateral + and hexahedral meshes. For these reasons, HOHQMesh was developed to + generate all-quadrilateral and extruded hexahedral meshes suitable for + use with spectral element methods. HOHQMesh is a direct quadrilateral + mesher, which generates quadrilateral elements by the subdivision + method of Schneiders + (Schneiders, + 2000) rather than indirectly from a triangular mesh or by + curving a low order mesh. It also adjusts the size and curvature of + the elements based on the length scales in the model, rather than + attempting to modify an existing low-order mesh.

+

Examples of meshes generated by HOHQMesh have been published in + Winters & Kopriva + (2014), + David A. Kopriva & Gassner + (2016), + Acosta-Minoli et al. + (2020), + Manzanero et al. + (2020), + Ersing & Winters + (2024), + Ranocha et al. + (2024), + Marbona et al. + (2024), + plus Wintermeyer + (2018) + and Eriksson + (2024).

+ + + Features +

HOHQMesh is designed to require minimal input from the user through + the use of a control file. The model defines the geometry in terms of + an outer and one or more inner boundary curves.

+

HOHQMesh features include:

+ + +

Unstructured all-quadrilateral or hexahedral meshes

+ + +

Isoparametric polynomial boundary approximations of arbitrary + order

+ + +

Automatic geometry-guided refinement

+ + +

Optional user specified local refinement

+ + +

Interior boundaries to separate regions of different + properties

+ + +

Symmetric mesh generation

+ + +

Hexahedral meshes from extrusion, rotation, and sweeping of a + quadrilateral mesh, with or without scaling

+ + +

Bottom topography variations, defined through functional form + or input topography data, for extruded hexahedral meshes with + automatic resolution of topographic features

+ + +

HOHQMesh is available as an open-source software package under the + MIT license and runs on Linux, macOS, and Windows + (David + A. Kopriva, Winters, Schlottke-Lakemper, Schoonover, et al., + 2024).

+ + + Examples +

In 1959 the Malpasset dam in France failed and flooded the Reyran + river valley down to the Mediterranean sea + (Hervouet + & Petitjean, + 1999),(Goutal, + 1999). Fig + [fig:Total] shows a + mesh of the valley and a portion of the Mediterranean with 2392 fourth + order elements generated by HOHQMesh in 0.44s on an Apple MacBook Pro + with a 2.3 GHz Quad-Core Intel i7. A zoom of the western portion of + the mesh is shown in Fig. + [fig:zoom]. The + geometry model consists only of an outer boundary, which was specified + as a cubic spline, and no inner boundaries. A control file to generate + this mesh can be found in the + Examples/2D/Malpasset + directory. Fig. + [fig:WaterHeight] + shows a spectral element computation of the water heights using the + mesh of Fig. + [fig:Total] in the + package TrixiShallowWater.jl + (Ersing + et al., 2023), which is part of the Trixi.jl + (Schlottke-Lakemper + et al., 2021) ecosystem.

+ +

Spectral element mesh for the Reyran river valley + including a portion of the Mediterranean + Sea

+ + + +

Western portion of the Reyran valley + mesh

+ + + +

Spectral element computation of the water heights at + 1985s after the break of the Malpasset + dam

+ + +

Further examples can be seen in the + gallery. + Control files for 29 examples that collectively include all the + capabilities of the mesher are available in the + Examples + directory, a table of which is presented in the documentation under + Pre-Made + Examples. Finally, 21 meshes are generated and validated as + part of the automated test option.

+ + + Related Software +

Special purpose quad/hex spectral element grid generators for + simple geometries are openly available as part of some spectral + element solvers. The preprocessor for FLEXI, HOPR, for instance, will + generate Cartesian boxes and meshes built from combinations of + Cartesian boxes, cylinders and spheres.

+

Spectral element solvers that currently can read meshes generated + by HOHQMesh include

+ + +

FLUXO + (Rueda-Ramirez + et al., 2017)

+ + +

Trixi.jl + (Schlottke-Lakemper + et al., 2021)

+ + +

HORSES3D + (Ferrer + et al., 2023)

+ + +

The preprocessor HOPR + (Hindenlang + et al., 2015) can also read and modify quad meshes generated by + HOHQMesh.

+

HOHQMesh can be used with the graphical front end HOHQMesh.jl + (David + A. Kopriva, Winters, Schlottke-Lakemper, & Ranocha, 2024). + It is a wrapper package that augments HOHQMesh with interactive + functionality giving a user the ability to create and visualize the + meshes without the need to compile from source.

+ + + Acknowledgements +

This work was supported by a grant from the Simons Foundation + (#426393, #961988, David Kopriva). Andrew Winters was funded through + Vetenskapsrådet, Sweden grant agreement 2020-03642 VR. Michael + Schlottke-Lakemper received funding through the DFG (Deutsch + Forschungsgemeinschaft, German Research Foundation) research unit + FOR-5409 (project number 463312734). Michael Schlottke-Lakemper and + Hendrik Ranocha were supported by a DFG individual grant (project + number 528753982). Hendrik Ranocha was supported by the Daimler und + Benz Stiftung (Daimler and Benz foundation, project number + 32-10/22).

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