The two-scale-transformation method

Abstract

We present the two-scale-transformation method which allows rigorous homogenisation of problems defined in locally periodic domains. This method transforms such problems into periodic domains in order to facilitate the passage to the limit. The idea of transforming problems into periodic domains originates from the homogenisation of problems defined in evolving microstructure and has been applied in several works. However, only the homogenisation of the periodic substitute problems was proven, whereas the method itself was just postulated (i.e. the equivalence to the homogenisation of the actual problem had to be assumed). In this work, we develop this idea further and formulate a rigorous two-scale convergence concept for microscopic transformation. Thus, we can prove that the homogenisation of the periodic substitute problem is equivalent to the homogenisation of the actual problem. Moreover, we show a new two-scale transformation rule for gradients which enables the derivation of new limit problems that are now transformationally independent.