Hexahedral mesh generation in real time

In a famous talk, Pr. Thomas Hughes estimated to be of the order of a million analyses a day performed in engineering design offices throughout the world. Engineering designs are encapsulated in Computer Aided Design (CAD) systems. Up to manufacturing tolerances, CAD systems exactly represent the geometry of designs. The analysis process begins with CAD geometry but the predominate method of analysis, finite elements, requires a mesh which is an alternative (discrete) representation of a geometry.

Even though efficient 2D mesh generation techniques were already available in the early 1970’s, the first automatic unstructured mesh generation system for general 3D domains was proposed in the early 1990’s with Paul-Louis George’s seminal work on 3D constrained Delaunay triangulation. It is interesting to note that today’s most widely used 3D mesh generation algorithm is still the one developed at that time. Three-dimensional mesh generation is a problem that is extraordinary complicated. Only half a dozen research teams in the world have the technology to build tetrahedral meshes for general domains in an automatic manner. Pr. Jean-Francois Remacle belongs to this short list with Gmsh, the only open source complete mesh generator available today.

In his talk, Pr. Hughes stated: "Mesh generation accounts for more than 80% of overall analysis time and is the major bottleneck". Every simulation engineer is aware that finite elements come with the very high price of mesh generation. Reducing the hassle of mesh generation in finite element analysis has been a major domain of research in the last 20 years. Yet, most of this effort has been devoted to design some alternative simulation methods that would not require a mesh or that would significantly simplify meshing issues. Up to now, none of those methods can be considered as a true alternative to the usual mesh based analysis.

In HEXTREME, the point of view is different. We believe that time has come for pushing innovation in mesh generation. HEXTREME aims at creating two breakthroughs in the art of mesh generation that will be directly beneficial to the finite element community at large: all the developments will be carried out in the open software platform Gmsh (http://gmsh.info) which is actively developed.

1) Hexahedral meshing

Hexahedral meshes in 3D and quadrilateral meshes in 2D are considered to be superior to triangular meshes in the finite element community. There exist nowadays no method for generating hex meshes. The first objective of HEXTREME is to create this breakthrough. and propose a fully automatic and reliable way of generating conforming hexahedral meshes.

2) Real time meshing

In the last decades, the size of the meshes that are used in industry has grown considerably due to the availability of massively parallel computers. It is not uncommon now to generate meshes that
have over 100 million of tetrahedra. In a user point of view, generating a finite element mesh of a complex domain usually involves the generation of some intermediary meshes that are progressively enhanced in order to fulfill some adequate design requirements. Today’s best 3D meshing algorithms are able to generate about five million tetrahedra per minute on one single core. Each iteration in the meshing process takes long minutes and users spend most of their time waiting for the mesh generator to provide outputs. Practitioners consider mesh generation procedures to be too slow: we will fasten the process by at least one order of magnitude.

The second objective of HEXTREME, "Real time meshing" has been addressed. We are able to generate meshes of hundered of millions of tetrahedra in one minute on a standard laptop. The first and most important objective of HEXTRELME, all hex meshing is way more ambitious. A great step has been done recently by the team. The Scheiders'open problem that was unsolved for more than 10 years is now solved. We have now set up an algorithm that is able to find hexahedrizations of small cavities. This algorithm will play a central role in the next months of the project. A first version of an all-hex mesher will be developped before the end of year 2019.

We have essentially broken two world records :
-) We are able to generate over 3 billion tetrahedra per minute on a multicore machine, which is more that 10 times faster than everyting that has been tested up to now.
-) We tackled the challenging task of finding hexahedral meshes with a fixed quadrangulation as their boundary. This constrained problem, while a
common case of the mesh generation problem, had no practical solutions. Even for quadrangulations containing only a small number of
quadrangles, as encountered in templates used in a number of algorithms, hexahedral meshes need to be constructed by hand due to the complexity of
existing methods. While the theoretical conditions under which a solution exists are known, the only known upper bound for the number of hexahedra
needed to mesh a ball-shaped domain was 5396 times the number of input quadrangles on the boundary! We lowered this constant to 78

We are halfway through the project and I was not excpecting such great advances, especially on finding hexahedrizations for small
quadrangulations of the sphere. It is quite probable that a all-hex algorithm will be available in Gmsh during HEXTREME's time, which was the goal of the project.

We are now investingating another route for hex meshing, an even more complicated one. We want to use a frame field to build a global 3D parametrization
of general 3D domains (multibloc decomposition, polycubes) and generate structured meshes. The demand of such meshes in industry is even more critical:
new technologies like isogeometric analysis will not be used without that capability. For my opinion, the fruit is not yet ripe. It could take some more years, maybe 10,
to have all the prerequisites that are necessary for giving a final answer to that problem. There many theoretical obstructions that have to be solved.
We organize a workshop with some of the teams that are interrested in the subject in summer 2019 (www.hextreme.eu/frames2019).

Periodic Reporting for period 4 - HEXTREME (Hexahedral mesh generation in real time)

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