Here I am referring to Kasparov's KK-theory, a bivariant functor on the category of separable C* algebras. It is well known that $KK(A, \mathbb{C})$ is K-homology and $KK(\mathbb{C}, B)$ is K-theory, both of which can be calculated for a huge collection of C* algebras (often by topological methods).

I am wondering if anybody knows how to actually calculate $KK(A,B)$ when it is not simply isomorphic to K-theory or K-homology (so I guess I also have to exclude $C_0(\mathbb{R})$. It seems that most of the time the interest in KK groups is not actually calculating them but in constructing specific elements (such as the Dirac / dual Dirac elements in proofs of the Baum-Connes conjecture).

It occurred to me that I have never actually seen anybody explicitly calculate $KK(A, B)$ in nontrivial cases. I am left wondering if this is because nobody knows how to do it or if it just isn't particularly useful to do so. Explanations or references are both appreciated.