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?
INTELLIGENT WORK FORUMS
FOR ENGINEERING PROFESSIONALS Come Join Us!Are you an
Engineering professional? Join EngTips Forums!
*EngTips's functionality depends on members receiving email. By joining you are opting in to receive email. Posting Guidelines 

Join your peers on the Internet's largest technical engineering professional community.
It's easy to join and it's free.
Here's Why Members Love EngTips Forums:
Register now while it's still free!
Already a member? Close this window and log in.
RE: question about finite element method formulation
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
raj
RE: question about finite element method formulation
RE: question about finite element method formulation
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 FEProgram system for nonlinear problems in structural mechanics.
regards
RE: question about finite element method formulation
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.
Thanks for your hint.
regards
RE: question about finite element method formulation
models of for aerodynamic systems which are non selfadjoint
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.
RE: question about finite element method formulation
[El][Er]=
RE: question about finite element method formulation
RE: question about finite element method formulation
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 email. thanks again.
regards
RE: question about finite element method formulation
RE: question about finite element method formulation
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
RE: question about finite element method formulation
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 FEMethode (in german) explained this phenomenon in the same manner.
regards