<p>This paper uses the scaled boundary finite element method (SBFEM) to study the wave propagation in a phononic crystal (PC). The SBFEM is a general semi-analytical method where a problem domain is divided into subdomains satisfying the scaling requirement. It offers the advantages of the finite element method (FEM) and the boundary element method (BEM), avoiding some drawbacks and making it very attractive for PC applications. In this paper, the SBFEM is formulated using the Bloch–Floquet theory to model periodic PC unit cells. This is an unprecedented modeling, since it is the first paper to use this methodology. The combined use of SBFEM with the Bloch–Floquet theorem provides a robust and efficient framework to accurately design and study PCs, enabling applications such as vibration control and acoustic insulation. The interest in elastic metamaterials (EMs) and PCs started in many engineering applications as vibration and noise control devices around a decade ago. PCs consist of two or more different materials periodically distributed, producing stop band or band gaps characteristic, where no elastic/acoustic waves propagate. The effect of Bragg scattering is analyzed through the dynamic responses obtained for different cases. The results are computed in the form of elastic band structure, forced response, and wave mode shapes. The SBFEM results are compared with those obtained by the FEM and plane wave expansion (PWE) method. Analyses were performed for various frequency ranges, such as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40430_2025_5814_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$f=0-250$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mn>250</mn> </mrow> </math></EquationSource> </InlineEquation> Hz, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40430_2025_5814_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$f = 0-300$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mn>300</mn> </mrow> </math></EquationSource> </InlineEquation> Hz, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40430_2025_5814_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$f = 0-10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> kHz, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40430_2025_5814_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$f = 0-15$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mn>15</mn> </mrow> </math></EquationSource> </InlineEquation> kHz. The analysis range depends on the geometric properties and the frequency range in which vibration attenuation through band gaps is desired. The relative errors of the natural frequencies calculated using SBFEM and FEM were computed for two cases: SBFEM I and SBFEM II. It was observed that for the SBFEM II case, the errors remained within <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40430_2025_5814_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$0.42\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.42</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>. A systematic error analysis was conducted, along with a systematic mesh convergence study, and a simple sensitivity analysis of the filling fraction with respect to the width of the generated band gaps was also performed. For the analyzed cases, comparing computational times shows that SBFEM is considerably more efficient than FEM. In all analyses performed for PCs, it was demonstrated that the SBFEM exhibits higher efficiency and better performance compared to the FEM, establishing itself as a highly effective method for the design and analysis of PCs and EMs.</p>

Dynamic analysis of 2-D phononic crystals by scaled boundary finite element method

期刊论文
发表日期：

2025年8月28日

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H. V. Cantanhêde,
E. J. P. de Miranda Jr.,
J. M. C. Dos Santos

摘要

This paper uses the scaled boundary finite element method (SBFEM) to study the wave propagation in a phononic crystal (PC). The SBFEM is a general semi-analytical method where a problem domain is divided into subdomains satisfying the scaling requirement. It offers the advantages of the finite element method (FEM) and the boundary element method (BEM), avoiding some drawbacks and making it very attractive for PC applications. In this paper, the SBFEM is formulated using the Bloch–Floquet theory to model periodic PC unit cells. This is an unprecedented modeling, since it is the first paper to use this methodology. The combined use of SBFEM with the Bloch–Floquet theorem provides a robust and efficient framework to accurately design and study PCs, enabling applications such as vibration control and acoustic insulation. The interest in elastic metamaterials (EMs) and PCs started in many engineering applications as vibration and noise control devices around a decade ago. PCs consist of two or more different materials periodically distributed, producing stop band or band gaps characteristic, where no elastic/acoustic waves propagate. The effect of Bragg scattering is analyzed through the dynamic responses obtained for different cases. The results are computed in the form of elastic band structure, forced response, and wave mode shapes. The SBFEM results are compared with those obtained by the FEM and plane wave expansion (PWE) method. Analyses were performed for various frequency ranges, such as $f=0-250$ $f = 0 - 250$ Hz, $f = 0-300$ $f = 0 - 300$ Hz, $f = 0-10$ $f = 0 - 10$ kHz, and $f = 0-15$ $f = 0 - 15$ kHz. The analysis range depends on the geometric properties and the frequency range in which vibration attenuation through band gaps is desired. The relative errors of the natural frequencies calculated using SBFEM and FEM were computed for two cases: SBFEM I and SBFEM II. It was observed that for the SBFEM II case, the errors remained within $0.42\%$ $0.42 %$ . A systematic error analysis was conducted, along with a systematic mesh convergence study, and a simple sensitivity analysis of the filling fraction with respect to the width of the generated band gaps was also performed. For the analyzed cases, comparing computational times shows that SBFEM is considerably more efficient than FEM. In all analyses performed for PCs, it was demonstrated that the SBFEM exhibits higher efficiency and better performance compared to the FEM, establishing itself as a highly effective method for the design and analysis of PCs and EMs.