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May 3, 2001, 13:27 
FEM/Linear algebra question

#1 
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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? 

May 3, 2001, 14:29 
Re: FEM/Linear algebra question

#2 
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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.


May 3, 2001, 17:05 
Re: FEM/Linear algebra question

#3 
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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.


May 3, 2001, 17:16 
Re: FEM/Linear algebra question

#4 
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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.


May 3, 2001, 20:53 
Re: FEM/Linear algebra question

#5 
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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 

May 4, 2001, 23:33 
Re: FEM/Linear algebra question

#6 
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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.


May 4, 2001, 23:34 
Re: FEM/Linear algebra question

#7 
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can you explain or give references?


May 4, 2001, 23:47 
Re: FEM/Linear algebra question

#8 
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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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