FEM, Matlab and the modes of an element

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Discussion Overview

The discussion revolves around the eigenvalue analysis of a stiffness matrix for a 4-noded finite element in a plane strain problem using Matlab. Participants explore the expected zero-eigenvalue modes, specifically focusing on how to visualize these modes as two pure translations and one rotation, as well as the implications of using only the stiffness matrix without a mass matrix.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant reports obtaining five non-zero eigenvalues and three zero-eigenvalue modes, which they believe should correspond to two translations and one rotation.
  • Another participant suggests checking the orthogonality of the eigenvectors using the condition ##\phi^T M \phi = I##, questioning the handling of isoparametricity in the analysis.
  • There are suggestions to use the 'axis equal' command in Matlab to accurately visualize the shapes of the elements.
  • Concerns are raised about the implications of combining eigenvectors, as this may violate orthogonality and affect the validity of the eigenvalue analysis.
  • One participant asserts that it is possible to compute eigenvalues and eigenvectors using only the stiffness matrix, while others express skepticism about the physical interpretation of such results.
  • There is a discussion about the potential issue with Matlab reporting combinations of eigenvectors instead of the expected pure modes.
  • Participants express uncertainty about the physical meaning of eigenvectors derived solely from the stiffness matrix and seek clarification on how to interpret these results.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of the mass matrix in the eigenvalue problem or the interpretation of eigenvectors derived from the stiffness matrix alone. Multiple competing views remain regarding the expected forms of the eigenvectors and the functionality of Matlab in this context.

Contextual Notes

There are unresolved questions about the assumptions underlying the eigenvalue analysis, particularly regarding the physical interpretation of results obtained without a mass matrix and the handling of isoparametric formulations in finite element analysis.

  • #31
Trying2Learn said:
Matlab reports these eigenvectors and I do not like the way they look. If I am a first time FE coder and I experiment to see the modes, and get those shapes, NOTHING in them informs me that three of the eigenvectors would have been MORE INFORMATIVE had I seen them as pure translations and one pure rotation. How would I have known this in advance.

Well, the zero-valued eigenvalues is a big give-away... They never occur for fully constraint problems.
 
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  • #32
Arjan82 said:
Well, the zero-valued eigenvalues is a big give-away... They never occur for fully constraint problems.
And, oddly, THAT was what I wanted to hear. Those last two sentences, were what I was looking for. And, now that you said them, it is obvious. I should have known that.

Thank you for patience (I know I can be stubborn -- sorry if it came across that way).
 
  • #33
Well, we took some detour, but we got there in the end :) All is ok!
 

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