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# question about finite element method formulation

## question about finite element method formulation

(OP)
could someone tell me if all the stiffness are always symmetry in finite element formulation or not?
Replies continue below

### RE: question about finite element method formulation

Hi,
in the stuctural mechanic, the stiffness matrix is usually symetric, but if there is a guide on the structure, then it results to unsymmetric matrix. In the couple field analysis there are many cases, in which the matrix is unsymetric. To my know the matrix is also unsymmetic in the accoustic analysis. Normally you can only implemented the full (direct) solution methode in the case of  unsymeteric matrix, so the modal superposition is not valid. Any suggestion?

regards

### RE: question about finite element method formulation

Even the presence of a guide in strutural mechanics problems desnt affect the symmetry. The symmetry  is attribute to the bettis law or maxwell reciprocal theorem of the system. One has to see where at all this is violated. In formulation where in apart from the field variable as unknown if the derivatives are taken as unknowns in some cases ther is likely hood of higher derivaties not follwing reciprocal law
raj

### RE: question about finite element method formulation

You can use modal superposition if the system is unsymmetric..you just have to find the right and left eigenvectors of the generalized problem.

### RE: question about finite element method formulation

hi,
i think that the stiffness matrix of a structure per se (betti's law) is symmetric, but if guie is presence then there is a so called 'quasi stiffness matrix' due to the displacement dependenci of the load increment, i.e load correction matrix (non linear effect). This quasi stiffness matrix is in general unfortunately unsymmetric. But, there are some specials strategies/algorithmics to handle with such unsymmetric matrix (see : Argyris et al in : A General FE-Program system for nonlinear problems in structural mechanics.

regards

### RE: question about finite element method formulation

Hi all,
thanks FEA_BEM, you are right that the unsimmetry of matrix when guide is precense is due to the non linearity. or do you have another idea about this raj?
pja, i am interested in your way solving the unsymetric matrix with left and right eigenvectors, can you please hint me to some text books, or this is only your idea?
I have actually referred to ANSYS solution capabilities about this matter. I have thought that ANSYS is right, because modalsuperposition presumes that the generalized matrix is diagonal and hence symmetric.

regards

### RE: question about finite element method formulation

Look at papers by Dowell,Hall and Thomas on reduced order
models of for aerodynamic systems which are non self-adjoint
systems. The gist of it is as follows. Say you have
a nonlinear system
{dq/dt}=[Q]+{B(t)} (1)
where the q's are the flow variables and Q is a nonlinear function of the q's and B is the "forcing" term. If you linearize the system about some static solution
[Q(qo1,qo2,...)]+{Bo}={0} (2)
you get a linear system of equations
{dq_pert/dt}=[J]{q_pert}+{B_pert} (3)
where q_pert is the small perturbation solution about the static solution and J is the Jacobian matrix [\partial Q/ \partial q]..however this Jacobian matrix is not symmetric so if one wants to use an eigenvalue expansion to solve one has to solve for the right and left eigenvectors(after setting B_pert to zero in equation 3):

lambda{q_r}=[J]{q_r}    (4)
lambda{q_l}=transpose([J]){q_l} (5)

where the orthogonality is now

transpose({q_l})[J]{q_r}=0 (6)
transpose({q_l}){q_r}=0 (7)

where q_l and q_r are eigenvectors
for two different eigenvalues. So now if one wants to use an eigenvalue solution of equation 3 one expands q_pert as
{q_pert}=[Er]{a_pert} (8)
where Er is the matrix of right eigenvectors and a are the modal coordinates. Putting this into equation 3 and multiplying through by [El],the matrix of the left eigenvectors, and using the orthogonality conditions of equations 6 and 7(along with assuming that the eigenvectors are normalized s.t.
[El][Er]=) (9)
one gets
{da_pert/dt}=[LAMBDA]{a_pert}+[El]{B_pert} (10)
where [LAMBDA] is the diagonal matrix of eigenvalues.

[El][Er]=

### RE: question about finite element method formulation

ok I'm trying to say that [El][Er]= the Identity Matrix..jeesh

### RE: question about finite element method formulation

Hi pja,
many thanks for your post. Actually i do not understand all of your post (may be because of the notation) but i will find the mentioned papers. If you have the papers in PDF format, then it would be great, if youcand sent it to me via e-mail. thanks again.

regards

### RE: question about finite element method formulation

Sorry I don't have them in electronic form only hardcopy...they are in the AIAA journal..starting from about 95 on...they first applied the methods to simple linear vortex lattice aerodynamic models and more recently to more complicated Euler and NS codes..however the modal superposition stuff is just math and can be applied to any type of non-self adjoint problem.

### RE: question about finite element method formulation

Well i do agree that it is a nonlinear case. Provided u dont assume it to be linear static!!?

But then again under nonlinear analysis using incremental techniques(piecewise linear and itrative like newtons method) why does stiffness matix become unsymmetric. I dont understand.

Hope u all make it clear !!
raj

### RE: question about finite element method formulation

The problem I outlined was for a nonlinear *fluid* flow problem..ie the Navier Stokes equations which is unsymmetric in nature..ie my mention of non-self adjoint...however the same methods can be applied to non-self adjoint structural problems as well provided you put them in first order form as I have above.

### RE: question about finite element method formulation

Hi,
as stated by FEM_BEM (sorry, i have call you FEA_BEM before), that 'quasi stiffness matrix' due to the displacement dependenci of the load increment, i think this is related to the so called 'follower force', i.e the force/load conforms with the deformation. So this effect is independent to the solving technique. Btw, i have the book from Argyris also, titeld The FE-Methode (in german) explained this phenomenon in the same manner.

regards

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