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Electromagnetic modeling of carbon-fiber reinforced composite materials using the wave digital concept

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An Erratum to this article was published on 15 February 2017

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Abstract

This paper deals with the modeling of the carbon fiber composite thin plate exposed to an external electromagnetic field. The underlying anisotropic partial differential equations are solved using the wave digital concept. After an appropriate change of the dependent variables and a coordinate transformation, a multidimensional passive reference circuit is derived along with its wave digital implementation. Numerical results, predicted using the multidimensional wave digital filtering approach, are compared to those obtained using the finite element method. Some stability related aspects are discussed. Besides, an electrostatic application of the presented wave digital model is outlined.

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  • 15 February 2017

    An erratum to this article has been published.

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Acknowledgements

The author is very grateful to H. Menana for many fruitful discussions on the subject of this paper as well as to the anonymous reviewers for suggesting several improvements to the original manuscript.

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Correspondence to Slimane Rezgui.

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This paper is dedicated to the memory of Professor Alfred Fettweis.

An erratum to this article is available at https://doi.org/10.1007/s11045-017-0473-0.

Appendices

Appendix 1: N-port series adaptor

The N-port series adaptor shown in Fig. 26 is the wave digital equivalent of the series connections in the MDKC of Fig. 2. The reflected power wave at port \(\nu \) of an N-port series adaptor is given as (Bilbao 2004):

$$\begin{aligned} d_\nu =c_\nu -\frac{2\sqrt{R_\nu }}{\mathop {\varvec{\sum }}\limits _1^N {R_j } }\mathop {\varvec{\sum }}\limits _1^N {\sqrt{R_j }} c_j \end{aligned}$$
(33)

Likewise, the symmetric Jaumann structure along with its wave digital symbol is displayed in Fig. 27.

Fig. 26
figure 26

N-port series adaptor along with its scattered waves and its port resistances

Fig. 27
figure 27

a Four-port symmetric Jaumann structure used in Fig. 2; b Its corresponding differential adaptor

For the special choice of port resistances shown in Fig. 27, the scattered waves from the differential adaptor are given by (Basu and Fettweis 2015):

$$\begin{aligned} d_1= & {} \frac{1}{\sqrt{2}}\left( {c_4 +c_3 } \right) , \end{aligned}$$
(34)
$$\begin{aligned} d_2= & {} \frac{1}{\sqrt{2}}\left( {c_4 -c_3 } \right) , \end{aligned}$$
(35)
$$\begin{aligned} d_3= & {} \frac{1}{\sqrt{2}}\left( {c_1 -c_2 } \right) , \end{aligned}$$
(36)
$$\begin{aligned} d_4= & {} \frac{1}{\sqrt{2}}\left( {c_1 +c_2 } \right) . \end{aligned}$$
(37)

It should be noted that the above scattering operations can be carried out simultaneously for all spatial grid points if sufficient computational resources are available.

Appendix 2: Anisotropic diffusion equation of the magnetic field

Neglecting the first terms on the right hand sides of Eqs. (4) and (5), i.e., adopting the magneto-quasistatic hypothesis for validation purposes, and combining the ensued relations with Eq. (3) one comes up with the following anisotropic diffusion equation of the magnetic field inside the ply:

$$\begin{aligned}&-\frac{\partial }{\partial x}\left( {\frac{\sigma _{xx} }{\sigma _{xx} \sigma _{yy} -\sigma _{xy} ^{2}}\frac{\partial H_z }{\partial x}+\frac{\sigma _{xy} }{\sigma _{xx} \sigma _{yy} -\sigma _{xy} ^{2}}\frac{\partial H_z }{\partial y}} \right) \nonumber \\&-\frac{\partial }{\partial y}\left( {\frac{\sigma _{xy} }{\sigma _{xx} \sigma _{yy} -\sigma _{xy} ^{2}}\frac{\partial H_z }{\partial x}+\frac{\sigma _{yy} }{\sigma _{xx} \sigma _{yy} -\sigma _{xy} ^{2}}\frac{\partial H_z }{\partial y}} \right) . \nonumber \\&+\mu \frac{\partial H_z }{\partial t}=0 \end{aligned}$$
(38)

This equation is solved with the FEM method (Silvester and Ferrari 1996) using the MATLAB PDE Toolbox (The MathWorks Inc. 2016) with the external magnetic field as boundary condition.

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Rezgui, S., Mohellebi, H. & Féliachi, M. Electromagnetic modeling of carbon-fiber reinforced composite materials using the wave digital concept. Multidim Syst Sign Process 29, 405–430 (2018). https://doi.org/10.1007/s11045-016-0467-3

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