Abstract
In this chapter, we will study the fundamentals of the FE method, which are important for simulating the behavior of piezoelectric sensors and actuators. The focus lies on linear FE simulations. Section 4.1 deals with the basic steps of the FE method, e.g., Galerkin?s method. Subsequently, the FE method will be applied to electrostatics (see Sect. 4.2), the mechanical field (see Sect. 4.3), and the acoustic field (see Sect. 4.4). At the end, we will discuss the coupling of different physical fields because this represents a decisive step for reliable FE simulations of piezoelectric sensors and actuators. For a better understanding, the chapter also contains several simulation examples.
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Notes
- 1.
Ansatz functions are also called shape, basis, interpolation, or finite functions.
- 2.
For 3-D electromagnetic problems, edge (Nédélec) finite elements are often applied instead of nodal (Lagrangian) elements.
- 3.
Alternatively to the effective mass matrix \(\varvec{\mathsf {M}}^{\star }\), the Newmark scheme can be defined for the effective stiffness matrix \(\varvec{\mathsf {K}}^{\star }\).
- 4.
The argument for the position \(\mathbf {r}\) is mostly omitted in the following.
- 5.
For the sake of clarity, the arguments for both space and time are mostly omitted in the following equations of this chapter.
- 6.
Alternatively to the approach with amplitude and phase, the complex frequency domain can be represented by real and imaginary parts.
- 7.
The sound pressure distribution corresponds to the spatially resolved sound pressure magnitudes.
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Rupitsch, S.J. (2019). Simulation of Piezoelectric Sensor and Actuator Devices. In: Piezoelectric Sensors and Actuators. Topics in Mining, Metallurgy and Materials Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57534-5_4
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