Fractal and Stochastic Homogenization using Variational Techniques

• Applicant: Dr. Martin Heida
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441154659

Many natural heterogeneous structures exhibit self-similarity. Examples include geology, soil physics or artificial materials such as metal foams. A recently introduced concept for fractal homogenization in geology is to be generalized to a wider class of applications. The method to be developed allows homogenization of partial differential equations with statistically self-similar coefficients. In this case, "homogenization" means that the self-similarity of the coefficients beyond any given iteration level is replaced by a homogeneous structure. This results in a hierarchy of homogenized problems. This method is applied to variational problems of elasticity. Although fractals are quite complex entities, their statistical self-similarity makes it possible to apply various established methods of homogenization and error estimation. More specifically, the goal of this project is to achieve a general homogenization theory for elastic fractal structures, using results of qualitative and quantitative stochastic homogenization, the ergodic theory for fractals by Zähle, and calculations from the field of a posteriori error estimation. The results of the project form the basis for a further study of evolutionary variational problems such as elasto-plasticity or damage.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2256: Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials