Hi,

I have two local stiffness matrices (6x6) calculated in SMath. Now I want to assemble them into a single global stiffness matrix (8x8):



Red numbers represent degrees of freedom. The assemblage is done in such a way that terms from these two local matrices which correspond to the same degree of freedom have to be added. So for example I take the term 5-5 (column numbered as 5 and row numbered as 5) from k1 and add it to the term 5-5 from k2. The problem is that these terms are located in different parts of the matrices. For example aforementioned term 5-5 is actually in the 3rd column and 3rd row of the first matrix, in the 5th column and 5th row of the second matrix and goes to the same place (5th column, 5th row) of the global matrix. Of course I can do it manually but it's a lot of work for larger matrices (even here it takes some time). Is there an automated way to do it in SMath ?

Thanks in advance for your help.

There should be two ways for three methods I could think of.
I won't add algorithm style here, these are enough I guess.
You can also combine method1 and method2 but I wouldn't call this a new way.

Regards

Stiffness matrix_rev.sm (59kb) downloaded 44 time(s).

Edited by user 29 May 2021 23:55:33(UTC)  | Reason: Not specified

Thank you very much. After the assembly, selected rows and columns of the global stiffness matrix have to be removed to account for boundary conditions. For example I would like to delete columns 1-4 and rows 1-4. So far I was just manually rewriting remaining terms to new matrix (K_new) but again it's quite tedious work. What's the best way to do it automatically in SMath ? Column and row numbers for deletion are not always ordered like that. For example it could be 2 an 5-8 as well.

Is this what you are wishing to accomplish?
Specified rows and columns automatically removed from matrix.
There are also matrix extension commands to remove columns and rows.
But I wanted to write those functions without extension.

Regards

PS: If you want to see the results without removal,
assigning rowsdel=[9] and colsdel=[9] shall be enough.
The rowsdel and colsdel should always be a vector.

Stiffness matrix_rev2.sm (46kb) downloaded 37 time(s).

Edited by user 30 May 2021 02:41:22(UTC)  | Reason: Not specified

Better removal for columns and rows added, algorithm rewritten.
And much more conventional way for not removal implemented.
Entering zero for not remove wanted row or col is easier.

Regards

Stiffness matrix_rev3.sm (54kb) downloaded 45 time(s).

I combined this with previous code for matrix assembly and it works great.

Thanks again, you guys are awesome, you really helped me a lot. And thank you Alvaro for sharing your FEM implementations in SMath. The 7th one (plates) is just what I'm working on now.

Great, thanks Alvaro :-) Are these examples from some book ?

This example shows the use of an index vector (f, contains the unconstrained dofs) for reduction of the global system.

The input is via dof and connectivity tables.

Each bar element has 4 dofs defined by a line in the connectivity matrix
Each global dof is a line in the dof matrix with each containing

- initial position
- external nodal load
- constraint flag (1-constrained, 0-free)
- prescribed displacement (only relevant if dof is constrained, can be nonzero)


Fachwerk UE2.sm (81kb) downloaded 52 time(s).

... ½ automated left for clic/clic/DONE gurus.

Here I attach the other elementary stiffness matrices. I didn't tested these in FE programs (always worked with rigid-rigid ends in FE), but when I was student of indeterminate structures (displacement method) I did some demonstrations of stiffness factors for each case, and these are well verified (except for the last page, that I wrote a few minutes ago just to complete the case of unitary rotation about the hinge (not needed for displacement method)).

Having in mind that the k(i,j) element of an elementary stiffness matrix is the reaction value in the j-displacement direction caused by an i-unitary displacement, you could trace each of the values found in my demonstrations.

Hope it works for you and please comment if it did once you have implemented it into your code.

I attach those demonstrations too (in spanish, but structural mechanics is a universal language )

Regards!

Alvaro

Element stiffness - beam.sm (25kb) downloaded 51 time(s). Element stiffness - demonstration.pdf (2,395kb) downloaded 49 time(s).

Edited by user 21 April 2022 12:32:00(UTC)  | Reason: Corrected stiffness matrices