The accurate modelling of mechatronic systems leads to so-called multi-field problems, which are described by a system of non-linear partial differential equations. These systems cannot be solved analytically and thus numerical calculation schemes have to be applied. Thereby, the finite element (FE) method has been established as the standard method for numerically solving the coupled system of partial differential equations describing the physical fields including their couplings.

In detail, the course will teach the physical / mathematical modelling and its FE formulations of the following coupled fields

**Aeroacoustics**

- Sound generation by turbulent flows according to Lighthill's analogy
- Approximation of Lighthill's analogy for low Mach numbers
- FE formulation for the radiated sound
- Approximation of free field conditions by absorbing boundary conditions and the Perfectly Matched Layer (PML) technique
- Non-conforming finite elements

**Electromagnetics-mechanics**

- Maxwell's equations
- Vector potential formulation for magneto-dynamics
- Nonlinear finite elements (Newton method) using edge finite elements
- Coupling mechanism (electromagnetic forces, motional electromotive force)
- FE formulation for the coupled field problem including moving / deforming solid bodies

**Electromagnetics-Heat**

- Multi-harmonic ansatz for the solution of the nonlinear electromagnetic partial differential equations in the frequency domain
- Finite elements of higher order to efficiently resolve eddy currents in electric conductive structures
- Coupling mechanism (Joule's losses due to currents, temperature dependent material parameters)
- FE formulation of the coupled field problem