Modeling of loudspeaker using hp -adaptive methods
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Modeling of loudspeaker using hp-adaptive methods Lukáš Koudela · Pavel Karban · Oldˇrich Tureˇcek · Ladislav Zuzjak
Received: 20 November 2012 / Accepted: 22 January 2013 / Published online: 31 January 2013 © Springer-Verlag Wien 2013
Abstract Numerical modeling of harmonic acoustic field produced by a common commercial loudspeaker is carried out. The goal is to obtain the directional characteristic of the device. So far, these characteristics could only be obtained by measurements, which was very complicated and also expensive. Another benefit of the model consists in obtaining a reference mathematical model for other acoustic devices, such as diffusors or resonators. The continuous mathematical model is derived from the physical laws and is considered for the ideal fluids with properties of a perfect continuum. The model of the loudspeaker in the axial symmetry (2D) is described by the Helmholtz differential equation for harmonic acoustic field and for the correct solution the appropriate boundary conditions and material properties are also defined. The task is solved numerically by a fully adaptive higher-order finite element method developed by the hp-FEM group. The convergence of results is discussed for the different adaptive techniques and the advantages of the use of hp-adaptivity are demonstrated. Selected results of the computations are then compared with experiments performed in an anechoic chamber at the University of West Bohemia and based on the standard CSN EN 60268-5. Keywords Helmholtz equation · Wave equation · Sound pressure level · Higher-order finite element method · Numerical analysis · hp-adaptivity L. Koudela (B) · P. Karban · O. Tureˇcek · L. Zuzjak Faculty of Electrical Engineering, University of West Bohemia, Pilsen, Czech Republic e-mail: [email protected] P. Karban e-mail: [email protected] O. Tureˇcek e-mail: [email protected] L. Zuzjak e-mail: [email protected]
123
S474
Mathematics Subject Classification
L. Koudela et al.
65M60 · 68N01
1 Introduction Loudspeakers (called also speakers for short) are electroacoustic transducers intended for reproduction of sound in response to an electrical signal [2]. The electric signal is lead through the loudspeaker wiring into the coil. The coil forming a part of the electromagnet generates magnetic field that has either the same or opposite direction (which depends on the actual direction of the current) to the field produced by the permanent magnet. When both fields are of the same direction, the electromagnet is attracted to the permanent magnet, in case of opposite directions the electromagnet is repelled. In this way, the electromagnet fastened to the diaphragm (made of paper, plastics or fabric) moves fast forwards and backwards, which makes the diaphragm vibrate. And these vibrations produce pressure waves propagating in the ambient medium, which is perceived as noise. Depending on the frequency range, the design of loudspeakers is very different. By standard, they are divided into three groups: low-frequency, middle-frequency and
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