The case for octree meshing in geomechanics

Anyone working with the more advanced side of 3D Numerical Modelling in mining geomechanics, underground geotechnical analysis or pit slope stability has probably been on the receiving end of a heavy dose of triangulation, even if just in the reports they’ve read. Technically, you’ve been looking at faces of tetrahedral elements. And whilst mostly harmless, the choice of element type impacts the modelling approach concerning two important aspects of meshing for geomechanics projects:

1. Mesh quality. This directly affects results quality and solution time and is often linked to the number of elements (again, driving solution time, but also computer memory and storage)
2. Effort. The significant time to build models from geometry cleanup and preparation and resolving mesh problems is often measured in weeks. Sometimes many weeks.

Now I’m a big fan of automated solutions, a ‘just-press-the-button’ kind of guy. Having worked with Dassault Systèmes, a leader in parametric design and computer-aided-engineering (even on public cloud), developments like Parametric Mine Design, where design is created in geometry objects, sound like the holy grail for full Geotech Analysis and Mine Design integration. We’re just not there yet.

So, why are we discussing tetrahedra in the first place??

The results quality of a tetrahedral element is affected by the shape factor (as one choice of possible quality measures), with 1.0 being the ideal element, 0.1 okay-ish and below 0.01 getting into the problem area.

For explicit solvers (including Flac3D and Abaqus) the stable time increment is proportional to the shortest element edge (with caveats, modern codes have ways to work around this within limitations). Total computer time (or wall clock time) increases inversely proportional to time increment. So short edges become computationally expensive. And depending on the geometry (intersections of geology / faults and mine design) mesh refinement can have side effects:

Below is an example where tetrahedral elements were used to closely represent stope geometries and faults, linking back to "Effort" above ... measured in weeks!

Bricks, Cubes and Octree

Bricks (although tempting, not the ones from LEGO), taking cubes as examples, are more efficient in many ways. First, each cube would need at least five tetrahedra (more if you want them equal in size) to cover the same volume. This does not affect the number of degrees of freedom in a problem (DOF) but it does affect the number of constitutive equations to be solved (each element). Second, practical geometries will lead to tet meshes with short edges. Therefore short stable time increments and higher CPU times. And with the time savings brought on by a regular mesh, you can afford to increase the resolution by using smaller elements (within limitations, again), allowing for better representation of the geometries. Importantly, today's computers can handle tens of millions of elements. No problems there anymore.

For particular applications, such as Caving, using model automation based on a regular grid (single size of brick elements), as described in the Caving 2018 and 2022 conference proceedings, can be a very efficient way to enable the modelling process to be part of the mine design. Here, GEOVIA PCBC and SIMULIA Abaqus are integrated into a fully coupled simulation and an automated model build process. [1,2]

Cavroc StopeX

Putting all the above aside, the mesh build process if we chose to use octree meshing, with all its benefits, can be fully automated with today's technology. The example below is from Cavroc's StopeX website.

Using solids in STL or DXF, with a ... reasonable ... quality, the graphical interface guides users through a step-by-step workflow to arrive at their own model, often within the first day for new users.

Now ... the duck

To demonstrate this, I took an old Rhino3D file I created back in 2004 (yes, I am serious!) following the tutorial (rubber ducky).

To create an octree mesh with StopeX I saved STL versions of the geometry and assigned them to the model. About 10min effort. Then I can use this model in Flac3D or Abaqus for analysis. Now this is progress!

Now we'll let the duck bob in the water :)

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Links

[1]https://doi.org/10.36487/ACG_rep/1815_16_Arndt

[2]https://doi.org/10.36487/ACG_repo/2205_64

[3]https://www.researchgate.net/publication/372165503_Forecasting_Mining_Induced_Seismicity_From_Modelled_Energy_Release_In_High_Stress_Stope_Extraction

Disclosure: I work for Mining One which has a formal collaboration?with Cavroc.