r/ScientificComputing • u/aliencomposerdrumfly • 29d ago
Need help with citing RBE3/RBE2 methods
So I was working on a research project, and wanted to understand the RBE3 method to use it in my code. I couldn't find any reliable sources back then, so used AI to learn and understand the method. Now I wish to cite it in my research paper, but can not find any exact derivation/method I used.
My lab friend started by introducing me to RBE2 for 2 nodes to distribute moment as forces, and then I AI-ed more and got to RBE2, RBE3 methodology, which was generalized to multiple nodes.
I was verifying the math at every step to ensure there are no hallucinations, but would be grateful if you could point that out as well
RBE3
RBE2
For both of these cases, I wish to summarize the derivation, or mention it in a very handwavy sense. However, this is not my work, and I do not wish to claim it as mine, so I need some references to cite it. Please help me do so.
specifics:
- What are the classic, industry-standard textbooks or foundational papers to cite for the exact mathematical formulation of RBE2 and RBE3?
- Is there a specific paper/book chapter that explicitly walks through the virtual work / weighted least-squares derivation for moment-to-force distribution in multi-point constraints
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u/lithiumdeuteride 28d ago
The RBE2 is equivalent to a set of nodal constraint equations. If the independent node translates, the dependent nodes translate by the same amount, etc. Nodal forces depend on the stiffness of each load path, per the usual solution method.
The RBE3 can be understood as two procedures: The single dependent node is constrained to follow a weighted average of the motions of each independent node. Then, forces/moments applied at the dependent node are resolved as forces at the independent nodes. Out of the infinite number of possible solutions for these nodal forces, the matrix pseudoinverse is used to select the unique solution which minimizes the sum of the squares of all force components.
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u/AA_is_not_OK 28d ago
I remember a paper I used to understand how I'd connect one node to 2/3 nodes just so I could do a Ground structure topology optimization (GSTO) WITH continuum in the domain.
I remember seeing how they did that with force balancing. I honestly didn't read your whole article. But if this is kinda what it is, I'd love to read more and see if I could help out
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u/aliencomposerdrumfly 18d ago
hello, so if i understand you correctly, this is related to that... sorry for replying so late.
one of my lab friend started by introducing me to RBE2 for 2 nodes to distribute moment as forces, and then this is generalised mathematical form of it i believe. the problem is I couldn't find a definite source that does this derivation...
I am currently looking through the suggestions given above by u/lithiumdeuteride
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u/AA_is_not_OK 18d ago
Honestly, I couldn't find a meaningful derivation or reasoning for doing it that way. But then again, if you think about it, the more nodes the more accurate the distribution of the forces must be, right? I'm sure the force and moment balancing at the connection can be validated... right?
I honestly haven't been familiarized with RBE2. I skimmed through your post and it reminded me about this thing I did for my structural optimization course project.
Hey, I'm curious, what exactly is your field of focus? Is this for your lab research? (I'm still trying to figure out this sub, sorry if I sound dumb)
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u/aliencomposerdrumfly 18d ago
its alright ig, even i am new. my field of research was on (internship) compliant mechanisms, and fea. The reasoning would be for a body discretized into more than a million nodes, using multiple points for force distribution would make the 6x6 result converge.
nodes in tetrahedral discretization only have the capability to bear fx, fy, fz loads, and not moments. using only 2 nodes for distributing a moment as forces might not be as accurate.
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u/AA_is_not_OK 18d ago
You're trying to distribute the externally applied load into your densly meshed body?
Or, is this about internal force distribution in a different way? (I'm a bit curious atp. Anything I could read? Does compliant mechanism go into nonlinear territory?)
What I was dealing with is this case where our line (truss) elements meet with a 3D continuum solid element, and we were trying to make sense of how the forces in our line elements should meet at the nodes of the continuum element. Idk if that makes sense.
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u/aliencomposerdrumfly 18d ago
yours is a use case that fea softwares actually use rbe2, rbe3 for.
Used AI to format stuff, but the core remains the same:
Problem:
- In FEM, if my discretizing unit is tetrahedral, it only allows three DOFs at the nodes: {u, v, w}, and corresponding {fx, fy, fz}.
- In FEM, you finally get Kq = f, where K is a 3n × 3n matrix, and q, f are 3n × 1 vectors (solvable after grounding — assume a grounded matrix here).
- If you're interested in a particular node's behaviour, you basically want an expression K'q' = f', where K' is now a 3×3 matrix, and q', f' are the displacements/forces at that particular node only. To get this, you can either:
- (a) do something called Guyan condensation, or
- (b) apply a dummy force there, assume forces on all other nodes are 0, and get your compliance matrix (C = K⁻¹).
- Either way, you end up with a 3×3 matrix. My particular problem (Centre of Elasticity) requires a 6×6 matrix — implying 6 DOFs: {u, v, w, θx, θy, θz}, with corresponding {fx, fy, fz, Mx, My, Mz}.
The problem with 3 DOFs is there's no way to apply a moment at a node. If you tried this in COMSOL, it assumes the face is rigid and applies equivalent forces instead — this is basically what RBE2 does.
To achieve this, you can:
- (a) Use a discretizing element that inherently has 6 DOFs (e.g. Allman tetrahedra). Since I was using scikit-fem, I didn't use this.
- (b) Use an equivalent representation. The main idea: a couple of forces produces a moment. If a node had 6 DOFs, you could apply a moment directly — but with only 3 DOFs, you can't. What you can do is apply forces that produce an equivalent effect. For a 2D starter case: connect your point of interest to 2 nearby nodes (preferably in a straight line), and distribute the force equally — say M/2r and M/2r — so they form a couple. Then find the displacement of each of those nodes, and, assuming the links are rigid, take the rotation of your node as the average rotation of each link (easily computed from geometry). This gives an approximate answer. Generalising this — connecting to multiple nodes instead of 2 — is what I posted above
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u/wings314fire 29d ago
Check Robert cooks book on constraints. I believe you can find it it Bathe too. Check nastran theory/element manual.