Effective Relaxation for Microstructure Simulations: Algorithms and Applications

For a wide class of problems in continuum mechanics like those involving phase transitions or finite elastoplasticity, the governing potentials tend to be not quasiconvex. This leads to the occurrence of microstructures of in principle arbitrarily small scale, which cannot be resolved by standard discretization schemes. Their effective macroscopic properties, however, can efficiently be recovered with relaxation theory. The paper introduces the variational framework necessary for the implementation of relaxation algorithms with emphasis on problems with internal variables in a time-incremental setting. The methods developed are based on numerical approximations to notions of generalized convexification. The focus is on the thorough analysis of numerical algorithms and their efficiency in applications to benchmark problems. An outlook to time-evolution of microstructures within the framework of relaxation theory concludes the paper.