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I am writing a 3d linear finite element code to solve the standard linear elasticity equation on a tetrahedron mesh of a gearbox. Notice that, the two rectangular plates above the gearbox are fixed, which means their displacement are zeros.

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The standard linear elasticity model is $$ -\mu\Delta \bf u - (\lambda + \mu)\nabla \text{div} \bf u = \bf f $$

where $\bf u$ is the displacement, $\mu$ and $\lambda$ are the Lamé parameters. Then I get the weak formulation as follows:

$$ \int_\Omega \sigma(u):\varepsilon \rm(v) dx = \int_\Omega fv\rm d x $$

where $\sigma$ is the stress and $\varepsilon$ is the strain. I use linear finite element to discrete above weak formulation and get

$$ KU=F $$

where $K$ is the stiffness matrix, $U$ is the displacement vector, and $F$ is the loads.

At last, I want to use RBE2 element to add some constraints to the above finite element model. I add an extra node $A$ to the tetrahedron mesh, and connect it to a boundary node sets $\{B_i\}_{i=1}^N$ of the tetrahedron mesh. Then I apply a load on node $A$, include a force $F$ and a moment $M$.

I have tried my best to search a document which can explain the details of the mathematical and physical of RBE2 element clearly but got nothing. So I need your suggestion and help, thanks very much.

One can take a look at this document, which explain RBE3 element very clearly but the RBE2 element is still not clear for my case.

Edit: For RBE2 element, the $A$ is an independent node, which displacements (transition and rotation) are unknown. Furthermore, the displacements of $\{B_i\}_{i=1}^N$ are also unknown. I just know the concentrated load $F$ and moment $M$ on $A$, so my question is how to apply the $F$ and $M$ into $KU=F$?

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    $\begingroup$ What is the constraint that you want to impose? $\endgroup$
    – nicoguaro
    Jun 20 '18 at 23:21
  • $\begingroup$ I want to apply a force $F$ and a moment $M$ on the node $A$ to the fem model by RBE2 element. $\endgroup$
    – Huayi Wei
    Jun 21 '18 at 0:56
  • $\begingroup$ Is your question: how are RBE2 (Rigid Body Elements) implemented? Or do you want to ask how to use a force and moment on your model? $\endgroup$
    – nicoguaro
    Jun 21 '18 at 2:57
  • $\begingroup$ @nicoguaro, I want to ask how to use a force and moment on my fem model by RBE2 element. $\endgroup$
    – Huayi Wei
    Jun 21 '18 at 3:15
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    $\begingroup$ It is unclear exactly what you are asking but if you want to understand better how to implement such a rigid element, this question might be of some help: scicomp.stackexchange.com/questions/29721/… $\endgroup$ Jun 21 '18 at 20:09

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