Abstract: The Langlands-Kottwitz method seeks to express Frobenius-Hecke traces on the cohomology of Shimura varieties in terms of objects in local harmonic analysis, namely orbital and twisted orbital integrals. We present a local analogue of that, relating the cohomology of some local Shimura varieties with twisted orbital integrals. The discrete summation over Kottwitz triples in the global case is replaced by an integral over semi-simple conjugacy classes. As an application, we explain a new point of view towards Rapoport's conjecture on the vanishing of certain twisted orbital integrals, which is a key ingredient in the global Langlands-Kottwitz method for a non-quasi-split prime. This is joint work with Rong Zhou.