What Is The Difference Between A Kinematic (RBE2) And A Distributing (RBE3) Coupling In FEA?
As CAE analysts, one of the most challenging aspects of our job is balancing the tradeoff between simulation accuracy and solver efficiency. What good is a simulation model that runs in five minutes but doesn’t mimic the real-world physics? Conversely, how valuable is a well-correlated methodology that takes weeks or months to solve? Although both of the aforementioned examples can have use cases in a production environment, I think most of us would agree that a simulation workflow that provides adequately accurate predictions within a reasonable time frame is much more valuable to an engineering organization (since it can be used to drive the decision-making process).
One of the ways that FEA analysts combat exorbitant run times is by simplifying some aspects of the analysis. A common example of a simplification that is often made in numerical simulations involves representing 3D geometry by using 2D elements, which was briefly discussed in a recent blog from our Element Selection in Abaqus series. Perhaps the most common, though, involves the use of multipoint constraint elements for modeling connections, loads, and boundary conditions.
What is a Multipoint Constraint?
As the name suggests, a multipoint constraint is a type of Finite Element which allows a relationship to be defined between several nodes simultaneously. Commonly referred to as MPC’s, these elements allow constraints to be imposed between different degrees of freedom of the model and can be used to simulate a wide range of physical phenomena, including nonlinear and non-homogeneous behavior. Most often, though, MPC’s are used to represent more basic interactions: rigid connections and distributed connections, known respectively as RBE2’s and RBE3’s (or Kinematic Couplings and Distributed Couplings).
What are Kinematic (RBE2) and Distributing (RBE3) Couplings?
Despite being the most commonly used types of multipoint constraints, RBE2’s and RBE3’s are often misused by analysts and can lead to egregiously inaccurate predictions. So, let us first start by understanding the key difference between kinematic couplings (RBE2) and distributed couplings (RBE3):
RBE2 elements add infinite stiffness to the nodes being constrained
RBE3 elements provide a distributed connection, which does not influence the local (or global) stiffness of the model.
Digging a little bit deeper, the reason why RBE2’s are infinitely rigid and RBE3’s inherently have zero stiffness boils down to the element definition and the concept of dependent and independent nodes. In the case of an RBE2, the motion observed at the independent node governs the motion that occurs at all of the dependent nodes. For example, if the independent node of an RBE2 moves by 1 mm in the X-direction, then every dependent node will also move by 1 mm in the X-direction. Because there is zero relative motion between the nodes of the RBE2, this type of MPC effectively rigidizes whichever portion of your model is included within the constraint.
Contrary to RBE2’s, whereby the behavior of one node dictates the behavior of many, RBE3’s work by averaging the behavior of many nodes in order to determine the response at a single node. In other words, the motion of the Dependent Node of an RBE3 is calculated by taking the weighted average of the motion of the many Independent Nodes. Because there is no equation governing the relative motion between the Independent Nodes of an RBE3, this type of connection does not contribute any stiffness to the model.
When to Use Kinematic (RBE2) and Distributing (RBE3) Couplings
Now that we understand the inherent differences between these frequently used MPC’s, which should we use? Does it matter? Simply put, absolutely! Though seemingly innocuous, applying an inappropriate constraint can drastically alter your simulation results (which makes sense considering we are literally changing the stiffness matrix). Unfortunately, there are not necessarily any hard and fast rules which govern when to use an RBE2 vs. an RBE3. However, the modeling assumptions being made should correspond to the expected physics of the problem being simulated, so a good place to start is by asking yourself this question: when loaded, do you anticipate the nodes included in the MPC to move together (RBE2) or move independently (RBE3)? Another important consideration is whether rigidizing (or not rigidizing) a portion of the model using MPC’s will change the overall deformation behavior.
A classic example of how the MPC type can drastically change the simulation results can be observed when using RBE2’s or RBE3’s to connect a lumped mass to its mating component. Consider a simulation in which an automobile engine is represented as a point mass and must be connected to the engine mounts: should we use an RBE2 or RBE3? If we use an RBE2 to connect the point mass to the four engine mount locations, this means that all four mounts deform together; if we use an RBE3, though, we will see different displacements at each of the mounts based on the respective local stiffness. Which seems more representative of how this will behave in real life? In this case, an RBE2 is the more appropriate choice since the engine is so stiff. Because it does not deform significantly when loaded, the stiffness of the engine effectively ensures there is no relative motion between the engine mounts; therefore, an RBE2 is more suitable for this scenario since it better mimics the real-world physics at play.
Next, let us consider a situation in which an RBE3 provides a better representation of the physics. Imagine a scenario in which half of a circle is loaded, but the other is not (for instance, a bolt shank loading one side of a bolt hole). Intuitively, we know that the bolt hole will elongate and ovalize as load is applied. Since we expect there to be relative motion between the nodes being constrained, we must select an MPC type that does not add any stiffness to the model (RBE3). In this case, if we were to model this behavior using an RBE2, our local stiffness would be incorrect, which would prevent the hole from ovalizing (it will still elongate, but it will not be able to narrow), and will yield inaccurate local stresses.
Understanding and selecting coupling types can be difficult and confusing, especially for users that are relatively new to FEA. In this article, we've learned the difference between the two main types of couplings, as well as taken a look at a couple of examples where we might want to select them. We hope this helps you in your future analysis efforts and maybe even causes you to think twice next time you add a coupling to your FEA model.
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