Fixed compositional field (and temperature) boundary conditions on deforming boundaries

[anne-glerum]

By default, fixed temperature boundary conditions are prescribed on both the inflow and outflow parts of the user-selected boundaries. For compositional fields, the default is to only prescribed values on inflow parts. To determine what is an inflow or outflow part of the boundary, the velocity at the boundary is queried in the replace_outflow_boundary_ids() function in helper_functions.cc. This can be done by asking the prescribed velocity plugin for a value, or, in case of a deforming boundary, by extracting the velocity at the location in question from the current_linearization_point.

However, for deforming boundaries, the velocity at the boundary moving up or down does not necessarily correspond to actual outflow or inflow. For example, a free surface moves up and down according to (a projection of) the velocity at the boundary, without any in/outflow. A boundary that is deformed by FastScape, is moved by uplift and by erosion or deposition. In this case, an uplift ("outflow") could coincide with deposition of material, for which a boundary condition would be appropriate. Uplift plus erosion would I think not require a boundary condition to be prescribed. Depending on the relative magnitudes of uplift and erosion, the new surface height could be larger or smaller than the previous one, but the value on the boundary should reflect the material that was eroded above it, not a prescribed value.

In the advection equations, the mesh velocity is subtracted from the material velocity first. @gassmoeller suggested to do a similar thing to determine in/outflow on deforming boundaries. For a free surface using the normal projection, I think this would work (depending on the projection used). It would lead to a zero velocity at the surface, and thus no outflow boundaries. For other mesh deformation methods, I can't fully wrap my head around it yet, but I think it could work. For example, when an uplifted area is eroded more than it is uplifted, the mesh velocity would be negative and the boundary would be considered an outflow boundary (as I think it should). When uplift and deposition occur, the boundary would be considered an inflow boundary, which I think is also correct. Any thoughts on this solution would be appreciated!