After successful completion of the course, students are able to

• formulate coupled nonlinear partial differential equations for multiphysics problems,
• determine the differential geometric character of the physical fields that is decisive for appropriate finite element discretizations and
• recognize essential invariants of various (multi)physics problems.

Theory

• Selected concepts of differential geometry for the formulation and description of partial differential equations (vectors, differential forms, exteriror derivative, Lie derviative)
• Lokal and global invariants and stationary conditions
• Geometrically consistent discretization of (pyhsical) fields
• Formalisms for thermodynamically consistent nonlinear coupled material equations (coupled material laws)

Applications

• Fundamentals of the geometrically nonlinear theory of elasticity
• Fundamentals of the electro- and magnetostatics
• Selected magnetoelectromechanical coupling mechanisms
• Multiscale modeling and numerical homogenization

Lecture is held on the blackboard / on a tablet PC

The lecture tries to find a balance between purely formal considerations and physical interpretation. For this purpose, the lectures also feature numerical simulations of mathematical and physcial problems, models and effects.

The exercise part fosters the understanding of the theory presented in the lecture by means of derivations and calculations by hand. In addition, after a brief introduction to the software NGSolve, the students have the opportunity to experiment with computer simulations of coupled physical problems.

A preliminary zoom meeting is scheduled for the beginning of the winter term 2021/22 (see schedule). It is required to connect to zoom via an official TU Wien address, e.g., x.y@student.tuwien.ac.at.

The link to the meeting can be found in the lecture schedule as well as in the TUWEL course.

Discussion of the lecture's content, in particular theoretical principles of practical relevance. Application of the lecture's content in various (sub-)domains of continuum mechanics and classical physics.

William L. Burke, 1996: Applied differential geometry; Cambridge: Cambridge Univ. Press.

Theodore Frankel, 2011: The Geometry of Physics: An Introduction; Cambridge University Press.

Matthias Rambausek, 2020: Magneto-electro-elasticity of soft bodies across scales; Dissertation at the Universität of Stuttgart.