After successful completion of the course, students are able to

• formulate coupled nonlinear partial differential equations for multiphysics problems,
• determine the differential geometric character 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

The exercise part fosters the understanding of the theory presented in the lecture. For the theoretical and formal content, the students carry out derivations and prove identities where details have been deliberately omitted in the lecture. Concepts and techniques introduced in the lecture part are applied to practically relevant examples and by that help the students to master their new tools.

After a brief introduction to the software NGSolve, the students perform numerical Simulation that connect coupled partial differential equations with phenomenological models and physical effects.

Active participation in weekly exercises as well as the solution of exercise sheets.