FEM/Linear algebra question
Hi I have a question:
Suppose I have a Finte element problem which is in the form of a system of linear equations: [K]{u}={F} where [K] is the sparse symmetric stiffness matrix, {u} is the solution vector of DOFs and {F} is the load vector. Is it possible to solve uniquely for [K] if we know what {u} and {F} are? What if I had the node connectivity (ie the grid), would it be possible then? 
Re: FEM/Linear algebra question
Unless K is diagonal you have too many unknowns to uniquely determine the coefficients.....if you have the grid then you could find K just by assembling it in the normal finite element method manner..assumming you have all the other constants that you need.

Re: FEM/Linear algebra question
yeah i thought that'd be the answer. It seems strange though that that should be the case. The way I figured was if you knew the grid then you know which entries in the matrix are nonzero and reduce the problem that way. I was thinking of a sort of inverse problem where you didn't quite know the material properties so you could reconstruct the stiffness matrix via the solution. Oh well.

Re: FEM/Linear algebra question
You could conceivably construct a stiffness matrix using impulse responses/green's function type method..however I don't think it would be worth the effort.

Re: FEM/Linear algebra question
Curious, why you'd want the matrix if you already have the solution? Is the idea: you have "one" solution vector, you want to construct the matrix for possible follow up problems?
Anyway, the answer is already given. Can't do it Adrin Gharakhani 
Re: FEM/Linear algebra question
yeah that's it. Suppose you didn't have the stiffness matrix but needed it for subsequent calculations... My thinking was that the stiffness matrix is essentially a linear transformation so if you know the solution and the force vector then you might be able to recreate the stiffness matrix. But it was probably wishful thinking.

Re: FEM/Linear algebra question
can you explain or give references?

Re: FEM/Linear algebra question
In Meirovitch's Principles of Vibrations there is a decent explanation of construction of a stiffness matrix via Green's function/kernel approach for a linear selfadjoint elastic structure. You may want to do a google search under influence function approach as well.This method of deriving approximate systems of equations from experimental data has been around for a while and was used a lot in aeroelasticity back in the day..

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