Add ADR integration

[JBludau]

This is based upon the formulation in https://www.sciencedirect.com/science/article/pii/S0020768317304833 but without the rotational stiffness.

One important difference is: in (75) we use (L_u(t-1)/Delta_ii-L_u(t)/Delta_ii). This corresponds to a -1*(75). Otherwise we get negative damping coefficients due to the approximated stiffness term being negative (the force difference points opposite to the velocity).

The implementation tries to be independent of CabanaPD's particle interface. This allows for easier testing (testcase is just a single mass) and potential reuse.

• Add adapters for particle interface
• Add testcase with particles
• Add interface in Solver class Not sure if that is a good idea ... we need to discuss this probably. In general the interface is still a bit ugly. The points are marked with TODO in comments
• Add convenience functions for creation of an ADRIntegrator
• Convert Dogbone case to use this method
• MPI support/testing (this probably revolves around bounding the iteration to exclude the frozen particles and end at the offset)
• Think about how to guess c for non PMB materials

Related to issue #207

[JBludau]

This is where it differs from the paper

[JBludau]

hmm currently this does not support mpi ... maybe I should remove it

[JBludau]

Hmm...right now the integrator cycles over all particles ... that is probably false and should be corrected somehow ... I guess the best would be to create proxy objects for the slices

[JBludau]

This is the reason why we need a middle step. We calculate the velocity in the middle step but need to overwrite it with the bc if the bc sets a velocity. After that is applied we can go and integrate the displacement

[yaowwwwwww]

"Think about how to guess c for non PMB materials". This is a big question for the application. People cannot calibrate the c for each material, since the material behavior is different.

I am trying to define a bond-based JC strain rate-dependent model, and i define the equivalent plastic strain as the weighted root-mean-square norm of norm of bond plastic stretches in PMB model. The theoretical consistency needs further study.

Maybe use a reasonable mapping method of bond deformation to the material point strain, it will provide a easy way for defining different classical material in the future.

[JBludau]

"Think about how to guess c for non PMB materials". This is a big question for the application. People cannot calibrate the c for each material, since the material behavior is different.

I am trying to define a bond-based JC strain rate-dependent model, and i define the equivalent plastic strain as the weighted root-mean-square norm of norm of bond plastic stretches in PMB model. The theoretical consistency needs further study.

Maybe use a reasonable mapping method of bond deformation to the material point strain, it will provide a easy way for defining different classical material in the future.

I guess we would be fine with guessing a minimal value for c. Due to the ADR using c in the numerator for the fictitious masses, having a higher c than what the force uses increases the fictitious masses and makes the ADR converge slower (The implementation has a safety-factor of 5 anyways). I mainly expect numerical problems if c is far too low. In this case the ADR uses particles that are too light and thus they are underdamped, allowing them to oscillate. If that gets severe, it might lead to large errors in the time integration and thus to wrong solutions which might be hard to detect.

Maybe guessing c for the ADR as we do for PMB via 6E/(pihorizon^4(1-2nu)) is a good approximation for many cases. Basically as long as we guess the maximal stiffness we might be ok (and we still have the safety factor). Nevertheless, if we want to tune for performance that still might be tricky. But other factors might get us there faster (e.g. using something similar to a multi-scale simulation or even just using reduced precision for the ADR)