Science targets

[brendanjmeade]

@mallickrishg

Short-term:

1. Accurate stored strain energy calculations. Revisit Feldl and Bilham (2006) without the infinities. How would earthquakes off of the main structures drain energy from the system? Great 2D figures with real topography!
   • Requires: BEM with topography.
2. Kinematic postseismic slip to quasi-dynamic postseismic afterslip for the 2015 Ghorka earthquake. Short- and longer-term afterslip on non-planar geometry.
   • Requires: BEM with topography and RS-like friction evolution.
3. Earthquake sequences at/near the HRF on non-planar geometry with multiple splays.
   • Requires: BEM with topography and RS-like friction evolution. Ben Thompson has a beautifully written DAE solver for this: https://tbenthompson.com/book/c1qbx/part6_qd.html

Longer-term:

1. Quasi-static and quasi-dynamic with gravity, off-fault deformation, power-law VE.
   • Kelvin kernel to represent body forces.
2. Quasi-static and quasi-dynamic with material property variations, especially near surface.
   • Requires: BEM with topography and material property contrasts.
3. One thing that we haven't discussed but has been discussed by many others is failure/deformation induced by gravitational and surface loads. We could do this very accurately even for very rough topography.
   • Requires: BEM with topography.

[mallickrishg]

The 3 short-term projects look do-able with the current code. Some development work needs to be done for the ODE stuff to deal with the extra constraints on overlapping nodes. I also need to check how well we do when it comes to intersecting faults. This is something we will encounter for the half-space/topographic free surface problem when our source intersects the free surface. Looking into this today.

3. One thing that we haven't discussed but has been discussed by many others is failure/deformation induced by gravitational and surface loads. We could do this very accurately even for very rough topography.

   • Requires: BEM with topography.

Is this related to some of Dave Pollard's group's work on jointing and fracturing near the surface? Or are you thinking about glacial/monsoonal loads?

[mallickrishg]

I realize we actually need to first write a short follow up article to your bem2d.jl paper showing a number of mathematically important but underappreciated properties of the new solutions. Perhaps this is something we can target for the near future?

1. Linear basis functions cannot get rid of the stress singularities for typical slip distributions.
2. The addition of continuity and smoothness constraints is necessary for accurate stress calculations in the near field, and as you put it previously - a bridge between the bem world and fracture mechanics
3. Current ideas about slip vectors on non planar faults need updating, whether it is fractal rough surfaces or smooth long wavelength topography. Slip in a purely shear sense can occur only under very limiting conditions. We may want to show how different the stress or traction kernels are for the above cases for constant slip vs 3qn.

[brendanjmeade]

@mallickrishg

Is this related to some of Dave Pollard's group's work on jointing and fracturing near the surface? Or are you thinking about glacial/monsoonal loads?

I'm thinking about the earthquake and tectonic loading questions considered in papers like:

• https://www.science.org/doi/10.1126/science.aab2210
• https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2003JF000097
• https://www.nature.com/articles/nature03562

There's a ton of interesting stuff here!

[brendanjmeade]

@mallickrishg

I realize we actually need to first write a short follow up article to your bem2d.jl paper showing a number of mathematically important but underappreciated properties of the new solutions. Perhaps this is something we can target for the near future?

I think this is reasonable...otherwise the first application paper is going to have a 5 page methods section, although that could work too. It might be worth considering putting it in an applications paper as I think applications papers get more eyes on them.

1. Linear basis functions cannot get rid of the stress singularities for typical slip distributions.

I think we also have to make clear that the use of on-fault stresses from constant elements gives incorrect stresses even for the case of uniformly spaced elements. This was a surprise to me and it's generally tacitly assumed to just be ok.

2. The addition of continuity and smoothness constraints is necessary for accurate stress calculations in the near field, and as you put it previously - a bridge between the bem world and fracture mechanics

We should write about this carefully in the context of Quadrature by expansion and Isogeometric approaches. These change the game a bit and are more mainstream BEM work outside of earth science. They bring additional constraints about the far-field continuity of slip and C2 meshes. There are also ongoing efforts to use super-high order integration to do this directly with point sources. These other methods are great in theory, but there really isn't code for this and they are not at the just go ahead and use it stage.

I think the real thing that we have here is that it's essentially an analytic solution to the problem that solves the primary challenge with C0 meshes. The only numeric part is the inverse, and in theory this could be done entirely by back substitution.

3. Current ideas about slip vectors on non planar faults need updating, whether it is fractal rough surfaces or smooth long wavelength topography. Slip in a purely shear sense can occur only under very limiting conditions. We may want to show how different the stress or traction kernels are for the above cases for constant slip vs 3qn.

Strong agree!

[brendanjmeade]

One other type of toy science calculation that might interesting, not sure, is a single Coulomb failure stress calculation. Sure we can only do it in 2D but hear me out. Most of the 3D calculations are done with Okada or triangular dislocation elements on with no-planar surfaces. There are stress singularities all over the place for these. There might be a philosophical point to be made about how different it would look if a nearly equivalent about of slip were realized on a non-planar surface where slip and it's gradients are assumed to be continuous. It would be a much less interesting pattern but that's the point.

[mallickrishg]

I sort of get what you are saying but I could not follow exactly what calculations you are proposing we do. Could you give me some more details about how we would set this problem up?

[brendanjmeade]

Consider this figure from modeling of the 1992 Landers event. There are a bunch of small "flare-like" red regions that are likely not real but rather discretization of effects associated with slip gradients and non-planar slip surfaces. Near fault tips there are likely to be real things but along strike there shouldn't be much other than CFS decreases...I think!

[mallickrishg]

Ahhh I get it!