Ways of speeding up solving for contact constraints?

What do you mean by, "refining"? Are you talking about mesh (ie increasing element density)?  There are a couple ways that the type of contact can effect your model:

1. Allow Penetration is the most basic, and is usually referred to as a Free interface definition. With this, geometry from one part/surface can pass through geometry from another part/surface. For example, if you have two 1" cube blocks of steel that are butted up against one another and you define an Allow Penetration interface, then you'll be able to move one block relative to the other, even if you try to push one block into the other (it will just pass through it). This does not have an impact on solution time, and is typically easier for the meshing engine when meshing.
   
   
2. Bonded comes next, and there are a couple ways SW can handle this.
   
   The most basic way is done with node sharing, which causes nodes at the interface to be merged together. For example, if you have two 1" cube blocks of steel that are butted up against one another and you define a Bonded interface, then the model will behave (and be meshed) as one 1"x1"x2" block of steel. If you look at where the interface is, you'll notice the element edges from one block will go to nodes that fit with element edges from the other block. This method does not impact the solution time, but is a little more difficult for the meshing engine to do, as it has to ensure nodal compatibility at the interface.
   
   Another method involves using something called multipoint constraint equations to "link" the two surfaces together. If you look at where the interface is, you'll notice the element edges from one block will go to nodes that most likely won't fit with element edges from the other block. This method does not require the nodes at the interface to line up, but the downside is that stress results at the interface are inherently inaccurate, and that the multipoint constraint equations are computationally "expensive" for the solver.
   
   
3. Finally, there is No Penetration, which is the most complex. With this, geometry from one part/surface can't pass through geometry from another part/surface. For example, if you have two 1" cube blocks of steel that are butted up against one another and you define a No Penetration interface, then you'll be able to move one block relative to the other except for if you try to push one block into the other (it will not pass through it). This type of interface has serious impacts on the solution time, as this time of interface results in a nonlinear analysis (which requires an iterative solution scheme).