Modelling Generalized Continua with Homological Algebra

Abstract: Structure-preserving discretization within the frameworks of Finite Element Exterior Calculus (FEEC) and Finite Element Tensor Calculus (FETC) motivates the investigation of the differential structures underlying physical models, namely, the spaces in which variables reside and the differential operators that connect them. These structures are encoded in differential complexes and their associated cohomology. In computational electromagnetism, discretizing the entire differential complex avoids spurious solutions and yields many desirable properties.

Recent efforts to extend this perspective to continuum mechanics have also led to progress in modelling. In elasticity, strain and stress tensors fit naturally into the elasticity (Calabi, Kröner) complex, which can be viewed as a special case of the Bernstein–Gelfand–Gelfand (BGG) construction. The intermediate steps in the BGG construction turn out to correspond to continuum microstructures, such as the Cosserat model. From this viewpoint, the BGG construction can be interpreted as a cohomology-preserving elimination of microstructural degrees of freedom.

In this talk, we explore mechanical models, such as classical elasticity, the Cosserat model, various plate models (dimension reduction), continuum defects, and mixed-dimensional models, through the lens of differential complexes, the BGG machinery, and the Čech complex.

Computer scienceMathematics

Audience: researchers in the topic

Modelling of materials - theory, model reduction and efficient numerical methods (UNCE MathMAC)