MPCs: a quick summary

Many years ago one of my coworkers built a model in Optistruct and was disappointed to see the RBE that he'd built failed to provide the restraint he expected.  He called up the technical support at Altair, and after some back and forth about the RBE element the support representative finally lost his patience: "It is not an element, it is a system of equations!"

So what are these equations, and what do they do?
First of all, they are indeed different from an element.  Although FEA software may refer to them as a type of element, they have a fundamental difference from a normal element. A normal element adds to the global stiffness, mass, and damping matrices. An MPC (multi point constraint) is more like an SPC (single point constraint), in that it modifies instead of adds to the matrices that describe the model in order to describe the special behavior of these MPCs.

So all that sounds interesting, but it hasn't really told you much about what they do. I've thought of different ways to describe them to my coworkers, and the best I've come up with is this analogy. Basically, you're including a group of grids in a type of government, and you're letting them know what kind of leader they have. So what's an RBE2? An RBE2 is a monarchy, where the control grid is king, and tells the other grids what to do. An RBE3, on the other hand, is more like a democracy, where the President does what the people tell it to do. So what works and what doesn't work when we try to make use of these MPCs?

What Works

The above works because the king can tell the dependent nodes what to do.  There are constraints on grids, but not on the same ones as the RBE2.

The above also works because the dependent grid knows what to do because the nodes that vote for it at least know what they want; if the elements beneath them were unconstrained then the whole set of elements would not know what to do.

This one works because the voters still know what they want: the constraints tell them what they want! The RBE2 above the RBE3 also knows what to do because the president is telling it what to do.  Ok, My analogy is getting old.  I'll switch to talking about independent/dependent grids.

As you can see above, I lied, the deputies are out!  Through the use of UM grids the RBE3 is not the dependent element, and can now be told what to do.

The above also works, as the UM grids allow the top RBE3 to not have its dependent grid be the same as the bottom RBE3 dependent grid.

Finally, the above works because the top RBE2 is only connected to independent grids.

You can download the above examples at RBEThingsYouCanDo.bdf

What Doesn't Work

Well this won't work, the RBE2 is trying to tell the dependent grids what to do, and the constraints are doing the same thing.

The above also doesn't work.  There are nodes which are dependent nodes in two separate RBE2's.  

The above also has problems.  The UM grids have been set, making them no longer independent, but those grids also have constraints, so again we have grids where MPCs and constraints are both trying to tell  a grid what to do.

Finally, the above is also bad.  You can't make a grid the dependent grid of more than one RBE3.

You can download the examples of things that don't work at RBEThingsYouCantDo.BDF

There are fun tricks you can do once you're feeling comfortable with RBE3s.  The two most interesting are:
-Applying load without adding stiffness.  If you have a certain total load that needs to be applied across a cross section of a model, you can use an RBE3 to apply it.  This is particularly useful at the ends of a beam.
-Constraining the overall motion of a larger body.  If it's known that something is sitting on a very soft support, such as rubber or water, an RBE3 can be created across the supported region, and through the setting of the UM card you can constrain the former dependent grid to control the overall average motion of the model.

There is, of course, something that analysts commonly do when they need to make a part of their model very stiff but don't have the time or experience to construct MPCs without errors: create elements with very very stiff properties that will behave similarly to MPCs.  The upside to this approach is that it will typically yield a very similar answer, the downside is that it may take a few iterations to find a stiffness that is stiff enough, yet not so stiff the stiffness matrix becomes ill-formed.

So wouldn't it be great if there was an MPC that automatically made a very stiff element that was just stiff enough?  Something that automatically created a perfectly rigid body but then had stiffness that connected it to the dependent grids?  Well there is in fact an element that does this.  This type of element is typically called a Lagrangian element, and it can be selected in certain load cases in Nastran and is the behavior for Kinematic (RBE2 style) and distributing (RBE3 style) elements in Abaqus.  I'll cover this topic in more detail in a later post.

Update 28 Sep 2021: Due to a Google security update the example file links expired, should be working again.