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Inverse of the 4th rank tensor
Thu, 2012-10-25 17:03 - Mubeen
Hi all,
I am looking for an algorithm to get the inverse of a 4th rank tensor (e.g. the compliance tensor S_(ijkl) from elastic stiffness tensor C_(ijkl)) S_(ijkl)=C_(ijkl)^(-1)
I am programming in FORTRAN, and for this purpose I wasn't able to find neither any algorithm nor any existing subroutine.
If anyone at this forum has any idea about this inversion, kindly guide me.
Best regards,
Mubeen.
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alternate method for similar problem
Though I have not found any direct way to get the inverse of 4th rank tensor, I have a different idea related to similar problem for the case.
I had to solve the equation
A:B=C
for B (where A=4th rank tensor, B,C = 2nd rank tensors), and A & C have already known values.
Instead of searching for inverse of A to get B, I expanded A:B , which resulted in 9 expressions A_ijkl * B_kl (i,j=1,2,3), and these 9 expressions were equal to the 9 entries in 2nd rank tensor C_ij (i,j=1,2,3).
With this result, I setup a system of 9 linear algebraic equations (LAEs) in which B_kl (k,l=1,2,3) were the only unknowns.
e.g.
A_11kl * B_kl = C_11
A_12kl * B_kl = C_12
...
A_33kl * B_kl=C_33 (for k,l=1,2,3)
These LAEs can be solved by one of the many existing methods e.g. Gaussian elimination.
Hence using this route, the inverse tensor wasn't required anymore.