Conversion local stiffness matrix to global

[JoeH78]

hi,

Regarding the triangular plate(thick) element, the book(FEM Analysis by Mircea RADES) that I'm reading refers to:
"Note that nodal coordinates are usually given in global axes. Calculation of the element stiffness matrix referred to local axes requires the local coordinates of nodes 2 and 3. These can be obtained from their global coordinates using the corresponding transformation matrix.

In order to assemble the global stiffness matrix, the element matrices have
to be first transformed from local to global axes through a matrix triple product which is one of the more time-consuming procedures in finite element analysis. "

It's obvious that conversion of coordinates from global to local one are performed in order to simplify the integration of derivatives and integrands. But to comprise the global matrix(assembly), local stiffness matrix should be converted to global ones.

What I've in hand is the 9x9 local stiffness matrix obtained for each element(triangle), global +local coordinates and connectivity data.

So the problem boils down to how to convert those local matrices to global ones, by the datas that I've in hand?

Your comments will be appreciated,

 

[BAretired]

It has been a few years since I attempted to do this type of transformation, but here is a link which may be helpful:

http://www.cece.ucf.edu/people/catb...EB/CES 6116 FEM Review of Matrix Analysis.pdf

BA

 

[ferrarialberto]

Transformation matrices of each finite element are used both to change the reference system of coordinates (from local to global and vice versa) which during assembly of the global stiffness matrix. Multiplying the transformation matrix for the finite element stiffness matrix (local) element gives the element stiffness matrix in the global reference system (to be added to the contributions of other finite element).

Hello.

ing. FERRARI Alberto -

http://www.ferrarialberto.it

 

[JoeH78]

BAretired

Thanks!
At a glance, it focuses on frame structures but the logic should be the same for piece-wise structures (triangular/quad meshing) as well? I noticed something under the "Coordinate Transformation" title, what you refer to as transformation matrix in your article is actually rotation matrix. While I'll have translation+rotation = transformation matrix in my case.
I think that give me the clue, if I compose the rotation matrix (9x9)along the diagonal with accurate inner products and add up the translation matrix, then it should work. Pseudo code is like below :

[GLOBAL ELEMENT MATRIX]= [Translate][sub]1x9[/sub]+[rotation][sub]9x9[/sub]X[LOCAL MATRIX]X[-rotation][sub]9x9[/sub]-[Translate][sub]1x9[/sub]

Where [Translate][sub]1x9[/sub] will have form of [DeltaX1 DEltaY2 0 ......... DeltaY8 0 ][sup]T[/sup]
Could you comfirm it ? (Pay attention to the minuses , I'm not sure they are visible enough)

I still feel bit of uneasy, that is the correct way of manipulating the local matrix like that.Didn't we violate the symmetry of matrix?

Regards,

 

[JoeH78]

Sorry big msistake here (Don't know how to edit the post)
Translation matrix should be 9x9 diagonally furnished, as well.

It's a proof that's enough for today

 

[IDS]

I recommend "Programming the Finite Element Method" by Smith and Griffiths. It goes into the nuts and bolts of FEA methods, including a brief coverage of the theory, and a detailed description of how it works in practice, including full Fortran code for working examples.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[JoeH78]

One of the superb books, it really well describes the frame elements in 1D, 2D, 3D, coupled, uncoupled problems, eigenvalue problems etc..

But what a pity that it doesn't cover shell/plate elements in depth other than in-plane problems, plate bending elements has been covered only in 4.7 Section with simple rectangular elements. Where my achievement is to find the solution to the plate/shell elements.