The two-scale transformation method

TL;DR

This paper rigorously proves the two-scale transformation method for homogenisation on locally periodic domains, establishing a solid mathematical foundation and new transformation rules for gradients to derive independent limit problems.

Contribution

It develops a rigorous two-scale convergence framework and transformation rule, validating the two-scale transformation method for homogenisation.

Findings

Established a rigorous two-scale convergence concept.

Proved the equivalence of the transformation method to homogenisation.

Derived new limit problems independent of transformations.

Abstract

We prove the two-scale transformation method which allows rigorous homogenisation of problems defined on locally periodic domains by transformation on periodic domains. The idea to consider periodic substitute problems was originally proposed by M. A. Peter for the homogenisation on evolving microstructure and is 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 to prove this method. Moreover, we show a new two-scale transformation rule for gradients which allows to derive new limit problems that are now transformationally independent.