Constraint Matrix Role in Global Matrix Assembly

[Ronankeating]

Dear All,

I'm very familiar with the composition of finite element local stiffness, mass matrices as per any arbitrary element rod, beam, plate, shell and integration of it into a global stiffness matrix. But I find it but of an obscure on how to integrate the constraint matrix into the global stiffness and mass matrices. It's an obvious that some DOFs are constrained, expressed in the form of another DOFs and stored in that constraint matrix but it is not clear for me which row/column operation is performed based on constraint matrix input in the global matrices.

Probably it resembles very much to the Guyan reduction, since its a simple form of reduction, but I couldn't find any example on net which shows the explicit way of implementing it, I will be very pleased if someone sheds some light on that.

Regards,

 

[SteamKing]

I recommend the book, 'Applied Finite Element Analysis', 2. ed., by Larry Segerlind.

Here is a link to a pdf copy:

ftp://161.53.116.242/Predavanja_vjezbe_programi_rokovi/Metoda%20konacnih%20elemenata/Finite%20Element%20Method%20%28FEM%29%20Collection/Applied%20Finite%20Element%20Analysis%20-%20Larry%20J.%20Segerlind.pdf

If you look at Appendix 3, pages 417-420, the author shows how the stiffness matrix is modified to accommodate the constraints imposed.

 

[Ronankeating]

Thank you in advance SteamKing,

That's another book worth reading which was not in my FEM library.
After reading the suggested Appendix couple of questions comes in mind. The example given is fairly straightforward and easy to understand for the targeted displacements, it gives clear solution to the symmetric matrices which is general in engineering problems. But,
1.) Is that technique valid for non-symmetric matrices ?

2.) Even having the Fortran flexibilities on matrix row/column manipulation, technique requires manipulation per inner-product of matrix which I believe is a lot of work on computer side (e.g. many nested "for" loops). Right?

3.) How about if we are trying to solve problem where Phi_1, Phi_2 and Phi_4(regarding the given example in the book) are constrained relatively to each other (i.e. Phi_1=Phi_2=Phi_4). That is another way of expressing the rigid diaphragms where you can assume in practice that those DOFs are similar and can be represented in identical fashion. Which path do I have to follow then ?

Regards,

 

[SteamKing]

1. I'm not sure how to answer this question. AFAIK, FEM always generates banded symmetric stiffness matrices. However, Boundary Element Methods (BEM) generate non-banded stiffness matrices which are fully populated (i.e., there are no zeroes).

2. The Segerlind book gives many example routines. You should be able to examine these and judge for yourself how computationally intensive applying constraints to the stiffness matrix is. Formation of the stiffness matrix and application of the constraints takes only a relatively small amount of time. The solution of the stiffness equations is where the major computational effort takes place.

3. I really can't help you here. It has been many years since I went thru the Segerlind book when I taught myself about implementing FEM. There are many other books which delve into the guts of FEM.
Consult the references here:
http://en.wikipedia.org/wiki/Finite_element_method

The books by Zienkiewitz and Bathe delve into many different applications of FEM; they may discuss its application to your particular problem. There are many sites on the net, and there are many other books which take an approach similar to Segerlind in looking at how to program the FEM for various problems.

 

[Ronankeating]

Thank you in advance,