[geometry] AutoDiff support for hydroelastic contact

[SeanCurtis-TRI]

Although, much of the hydroelastic contact infrastructure has been coded with a templated scalar, it is not yet ready for computing contact with AutoDiff calculations. The primary reason for this is because the scalars haven't been properly applied.

For example, one long-lived comment discusses the fact that for a given soft mesh, there could be multiple different scalar types: the pose, field value, mesh vertex positions, etc. We currently load double-valued meshes (analogous to all of the "parameterized" geometries). But introducing AutoDiff-valued transforms would produce an AutoDiff-valued ContactSurface.

In addition, there is no value for a bounding volume hierarchy (Bvh) in being anything but double-valued. Other than being used for acceleration of the query, it doesn't contribute to numerical results. But we need to be able to build a Bvh for a T-valued mesh.

There are several possible solutions:

1. Keep everything templated on a single scalar value and convert all properties to the T-values (rather than having double-valued properties in an otherwise AutoDiff-valued system).
2. Revisit the templating so that scalar types that are conceptually independent remain independent.

The exact solution still needs to be determined. The proof that it is done, is that we can perform AutoDiff-valued hydroelastiic contact queries (with, at least, AutoDiff-valued pose data).

[SeanCurtis-TRI]

A solution has been implemented in the "leave meshes double-valued and let derivatives come in through mesh transforms" direction. The full solution will be a PR train:

• [X] #14897 - kill the MeshField base class (polymorphic + template is a bad combination).
• [x] #14904 - Give meshes and fields the ability mix calculation scalar types
• [x] #14951 - Give internal::Plane and internal::PoseHalfSpace the ability to handle mixed scalar types.
• [x] #14943/#14998 - Enable AutoDiffXd for rigid mesh, soft half space.
• [x] #14952 - Enable AutoDiffXd for soft mesh, rigid plane.
• [x] #15012 Enable AutoDiffXd for soft mesh, rigid mesh.
• [x] #15040 Finish wiring up AutoDiffXd-valued ContactSurface all the way up through QueryObject.
• [x] #15219 - Introduce "integration" test to show that the derivative values on the resultant contact forces are what they are expected -- probably create a simple example akin to what was tested in #14951, #14943, or #115012).
  • [x] #14963 Depends on making sure traction calculations respect derivatives

[SeanCurtis-TRI]

@amcastro-tri You've got a checkbox up above for confirming derivatives on contact forces.

[SeanCurtis-TRI]

The initial effort (while still benefitting from some more elaborate tests -- see #15233), is basically complete. We have credible reason for someone to try working with the derivatives in hydroelastic contact.

fyi @hongkai-dai