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The governing equation of the transient heat transfer problem is

$$C \frac{dT}{dt}+K T = Q$$

$C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $Q$ is the heat flux vector.

I know that it is possible to use a lumped capacity matrix in thermal finite element analysis where the capacity is shifted to the nodes of every single element. Is it also possible to use lumped conductivity matrix? What will be the benefit and use case if this is possible?

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No, you can't lump the $K$ matrix: that would not be a consistent approximation to the second-order differential operator it is supposed to represent.

But if you're trying to be a bit more formal, just write out what that lumped mass matrix would actually be: Most rows of the matrix (corresponding to nodes not next to the boundary) would simply add up to zero. So you would have a matrix that is mostly zero, and you can't expect that to represent the effects of diffusion!

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