In this lecture the current state of numerical simulation using the Finite-Element (FE) method for coupled field problems, which appear within the design process of modern mechatronic systems (e.g., sensors and actuators), is presented.  In combination to the theory of magnetic, mechanical and acoustic fields with all their couplings, we will discuss practical applications occuring during the design and optimization of modern mechatronic systems (numerical simulation of electromagnetic actuators, of  piezoelectric positioning drives, vibrational induced sound generated by machines and automobiles, etc.)

Mechatronic systems are based on the mutual interaction of physical fields, e.g., the mechanical field with the electromagnetic field. In most cases, the fabrication of prototypes within the design process is a lengthy and costly task. Furthermore, not all parameters of interest can be measured (e.g., magnetic field or mechanical stresses inside a solid body) and the measurement setup may influence the (dynamical) behavior of the prototype. Since for the development of mechatronic systems all the different coupling mechanisms of the involved physical fields have to be considered, the design process is a very complex task. Therefore, an increasing need for reliable and usable computer modeling tools capable of precisely simulating the multi-field interactions arises. Such appropriate computer-aided engineering (CAE) tools offer many possibilities to the design engineer. Arbitrary modification of sensor geometry and selective variation of material parameters are easily performed and the influence on the mechatronic device can be studied immediately. Thus, a CAE-based design can tremendously reduce the number of necessary prototypes within the design process.

 The accurate modeling of such transducers 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. Thus, a static, a transient, a time-harmonic as well as eigen-frequency analysis including nonlinearities (e.g., material nonlinearities, geometric nonlinearities, etc.) can be performed very efficiently.

The lecture starts with a detailed discussion of the Finite-Element (FE) method including the main aspects for computer implementation. Thereby, a python program will be available, which contains the main routines for computation of mechanical problems. In a next step,  the current state of numerical simulation using the Finite-Element (FE) method for coupled field problems, which appear within the design process of modern sensors and actuators, is presented. In combination to the theory of magnetic, mechanical and acoustic fields with all their couplings, we will discuss practical applications occurring during the design and optimization of modern mechatronic systems (numerical simulation of electromagnetic actuators, of piezoelectric positioning drives, vibrational induced sound generated by machines and automobiles, etc.).

Numerical Simulation of Mechatronic Sensors and Actuators, Manfred Kaltenbacher, 3rd ed. Springer, 2015.

Basics of the Finite Element Method

Baiscs of mechanical, electromagnetic and acoustic fields