Update coordinates for gravitational stresses

[willic3]

Description

We need an approximate method to represent the changes in body forces due to gravity when deformation occurs.

Desired Solution

I believe we can do this by updating the coordinates at each time step prior to computing stresses and body forces.

Alternatives

The alternative is to solve the finite strain problem, which introduces nonlinearity.

Additional context

I will outline a procedure for doing this.

[baagaard-usgs]

Proposed solution

1. Decide precisely what PetscVec objects are needed to implement updating the coordinates at the end of each time step and at each iteration of the solve (with the current guess of the solution).
2. Create problems::TimeDependentUpdateCoords object that derives from problems::TimeDependent. This object will hold the PetscVec objects as private data members.
3. Whenever we make use of the solution values in updating the coordinates, we need to project the solution into the FE space of the coordinates.

Update steps

1. Initialization - store original coordinates
2. Poststep - BEFORE output - update PetscVec storing the displacement field relative to the original coordinates
3. Residual + Jacobian calculations (static methods in TimeDependent.cc) - update coordinates PetscVec with current coordinates = original + displacement(t) + solution_displacement [Make a separate function to do this, so the change to computeRHSResidual(), etc is just adding this one function call.]

poststep()

We want to temporarily update the solution field with the total displacement field, so that the output of the displacement field is relative to the original coordinates AND the Cauchy strain and stress are computed using the total displacement field.

The coordinate field should be the original coordinates, not the updated coordinates.

[willic3]

Vectors needed

1. coordinates (existing PetscVec). This now contains the updated coordinates: coordinates = originalCoordinates + displacementT + solution_displacement.
2. solution_displacement (existing PetscVec). This now contains the increment in displacement relative to the previous time step.
3. originalCoordinates (new PetscVec). This keeps a copy of the original coordinates.
4. displacementT (new PetscVec). This is the total displacement at the end of the previous time step. At the end of each time step it is updated by the current solution (increment). I am a bit confused about how this will work, since our current solution represents a displacement increment, but we want stress and strain calculations to be done using the total displacement. Can we pass displacementT into these functions?

[willic3]

This method of approximating gravitational stresses requires the solution be an increment (displacement increment). This is incompatible with our boundary conditions and interface conditions that rely on the solution field NOT being an increment. Thus, this approach seems like a dead end as it would require rewriting the boundary conditions and interface conditions as well as updating the coordinates and solution.