Geometric stiffness term in Stable Constrained Dynamics
Posted: Thu Apr 05, 2018 4:36 am
This is in reference to "Stable Constrained Dynamics": https://hal.inria.fr/hal-01157835/document
Equation (18) / (21)
I'm having trouble understanding how to build this K "geometric stiffness" term.
K = (∂J^T / ∂x) λ
Where J is the constraints jacobian and λ is the constraint force magnitudes.
What I do know - based on its usage in (21), K should be a (3n x 3n) matrix in 3D where n is the number of particles lets say.
What I'm confused about - the jacobian J is a (C x 3n) matrix where C is the number of constraints. λ is (C x 1). This doesn't seem to work out in terms of the matrix dimensions...
What am I missing here?
If I consider only a single constraint, then it does appear to work out in terms of the dimensions - I end up with λ being a scalar and K ultimately being (3n x 3n). However, that leads to the question of how to then build K such that it contains all of the individual constraint K's (one K for each constraint I guess)?
Equation (18) / (21)
I'm having trouble understanding how to build this K "geometric stiffness" term.
K = (∂J^T / ∂x) λ
Where J is the constraints jacobian and λ is the constraint force magnitudes.
What I do know - based on its usage in (21), K should be a (3n x 3n) matrix in 3D where n is the number of particles lets say.
What I'm confused about - the jacobian J is a (C x 3n) matrix where C is the number of constraints. λ is (C x 1). This doesn't seem to work out in terms of the matrix dimensions...
What am I missing here?
If I consider only a single constraint, then it does appear to work out in terms of the dimensions - I end up with λ being a scalar and K ultimately being (3n x 3n). However, that leads to the question of how to then build K such that it contains all of the individual constraint K's (one K for each constraint I guess)?