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UNIVERSIDADE ESTADUAL DE CAMPINASFaculdade de Engenharia Mecânica
Rayston Werner Oliveira Sousa
Vibroacoustic Modeling of PeriodicCylindrical Shells with Internal Fluid via
Wave Finite Element Method
Modelagem Vibro acústica de CascasCilíndricas Periódicas com Fluido Internoutilizando Método dos Elementos Finitos
de Onda
CAMPINAS2018
Rayston Werner Oliveira Sousa
Vibroacoustic Modeling of PeriodicCylindrical Shells with Internal Fluid via
Wave Finite Element MethodModelagem Vibro acústica de Cascas
Cilíndricas Periódicas com Fluido Internoutilizando Método dos Elementos Finitos
de OndaDissertation presented to the School of MechanicalEngineering of the University of Campinas in partialfulfillment of the requirements for the degree of Mas-ter in Mechanical Engineering, in the area of SolidMechanics and Mechanical Design.
Dissertação de Mestrado apresentada à Faculdadede Engenharia Mecânica da Universidade Estadualde Campinas como parte dos requisitos exigidospara obtenção do título de Mestre em EngenhariaMecânica, na Área de Mecânica dos Sólidos e Pro-jeto Mecânico.
Orientador: Prof. Dr. José Maria Campos dos Santos
ESTE EXEMPLAR CORRESPONDE À VER-SÃO FINAL DA DISSERTAÇÃO DEFENDIDAPELO ALUNO RAYSTON WERNER OLIVEIRASOUSA, E ORIENTADO PELO PROF. DR JOSÉMARIA CAMPOS DOS SANTOS.
CAMPINAS2018
Agência(s) de fomento e nº(s) de processo(s): FAPEMA, BM-05024/15ORCID: https://orcid.org/0000-0001-6579-560
Ficha catalográficaUniversidade Estadual de Campinas
Biblioteca da Área de Engenharia e ArquiteturaLuciana Pietrosanto Milla - CRB 8/8129
Sousa, Rayston Werner Oliveira, 1994- So85v SouVibroacoustic modeling of periodic cylindrical shells with internal fluid via
wave finite element method / Rayston Werner Oliveira Sousa. – Campinas, SP :[s.n.], 2018.
SouOrientador: José Maria Campos dos Santos. SouDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade
de Engenharia Mecânica.
Sou1. Cascas (Engenharia). 2. Interação fluido-estrutura. 3. Cristais fonônicos.
4. Propagação de ondas. 5. Método dos elementos finitos. I. Santos, JoséMaria dos, 1953-. II. Universidade Estadual de Campinas. Faculdade deEngenharia Mecânica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Modelagem vibro acústica de cascas cilíndricas periódicas comfluido interno utilizando método dos elementos finitos de ondaPalavras-chave em inglês:Shells (Engineering)Fluid-structure interactionPhononic crystalsWave propagationFinite element methodÁrea de concentração: Mecânica dos Sólidos e Projeto MecânicoTitulação: Mestre em Engenharia MecânicaBanca examinadora:José Maria Campos dos Santos [Orientador]Renato PavanelloAdriano Todorovic FabroData de defesa: 23-03-2018Programa de Pós-Graduação: Engenharia Mecânica
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UNIVERSIDADE ESTADUAL DE CAMPINAS
FACULDADE DE ENGENHARIA MECÂNICA
COMISSÃO DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA
DEPARTAMENTO DE MECÂNICA COMPUTACIONAL
DISSERTAÇÃO DE MESTRADO
Vibroacoustic Modeling of PeriodicCylindrical Shells with Internal Fluid via
Wave Finite Element Method
Modelagem Vibro acústica de CascasCilíndricas Periódicas com Fluido Internoutilizando Método dos Elementos Finitos
de OndaAutor: Rayston Werner Oliveira Sousa
Orientador: José Maria Campos dos Santos
A Banca Examinadora composta pelos membros abaixo aprovou esta Dissertação:
Prof. Dr. José Maria Campos dos Santos, PresidenteDMC-Faculdade de Engenharia Mecânica - UNICAMP
Prof. Dr. Renato PavanelloDMC-Faculdade de Engenharia Mecânica - UNICAMP
Prof. Dr. Adriano Todorovic FabroENM-Faculdade de Tecnologia - UnB
A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vidaacadêmica do aluno.
Campinas, 23 de março de 2018.
Dedication
I dedicate this work to God and my parents.
Eu dedico este trabalho a Deus e aos meus pais.
Acknowledgements
First and foremost, I thank God for my life, for my dreams and for giving me the health andstrength necessary to complete this dissertation.
To all my family for the unconditional hope, especially my parents, Antônio and Rízia, for theirrelentless support at all moments.
To my supervisor, Prof. Dr. José Maria Campos dos Santos, my sincere thanks for opportunity,constant support and commitment to make possible the development of this work, for the valu-able teaching and counselling, for the patience and friendship during this dissertation. I alsowish to express my sincere thanks to Prof. Dr. Jean-Mathieu Mencik at INSA Centre Val deLoire, for his support during this research. His continuous motivation, interest and scientificacumen were essential to complete this project.
To the external members of the jury for the availability, suggestions and participation in mydefense. I would like also to thank to Prof. Dr. Renato Pavanello for valuable discussions, sug-gestions and all the support provided, which made the development of this work viable.
To my labmates at Vibroacoustic Laboratory-DMC (Fernando Ortolano, Danilo Beli, JoshLabaki, Helder Daiha, Lucas Egídio, Guilherme "Jack Sparrow", Otávio "Kamikaze" Tovo,Luis "Didi" Felipe, Daniela Damasceno, Lucas "Vei", Vinícius Lima, Brenno "Pherbes" Cam-pos, George "Yoji" de Assis, Fernando "Hulk", José "Inspetor" Ilmar, Raimundo Lucena, Hélio"PM" Cantanhêde e Edson Miranda Jr.) for all the moments we spent together that I will notforget. To all G.R.E.L.O members and to Danilo "Barney" whose friendship I will treasure formy entire life. I am thankful to fellow of SIFEM for all the fun we had in the time I spent there.I would like also to thank to my housemates, in especial, to Mavd Ribeiro for valuable collab-orations and discussions. I am also grateful, in special, to the Max William for all the supportand companionship in the various moments along this journey.
To FAPEMA, for providing me the financial support necessary to the development of this dis-sertation.
Finally, Iwould like to thank all who in oneway or another contributed to the completion of thiswork.
One can never consent to creep when onefeels an impulse to soar.
Helen Keller
Nunca se pode concordar em rastejar,quando se sente ímpeto de voar.
Helen Keller
Resumo
O método dos elementos finitos de onda é proposto para modelar cristais fonônicos decascas cilíndricas preenchidas com fluido. Essa abordagem numérica é também aplicada paraanalisar a propagação da onda guiada e o comportamento vibro acústico em cascas cilíndricasperiódicas. Neste estudo, assume-se que as cascas cilíndricas são lineares elásticas e modeladascom elementos de casca plana 2D, enquanto os fluidos internos são considerados acústicose modelados com elementos lineares 3D. É utilizada uma suposição periódica de malhas deelementos finitos, o que permite a descrição de uma casca cilíndrica inteira preenchida comfluido em termos de idênticos subsistemas, que são compostos por partes estruturais e partesfluidas e que são montados em conjunto ao longo de uma direção reta. Ao considerar o modelode elementos finitos de um subsistema, pode-se derivar uma relação de matriz de transferênciaque liga as quantidades cinemáticas/mecânicas entre as seções transversais direita e esquerda.Os autovalores e autovetores da matriz de transferência fornecem os chamados modos deonda, isto é, os parâmetros de ondas/números de onda e as formas de onda. Além disso, ométodo dos elementos finitos de onda é aplicado para calcular as respostas vibro acústicasde cascas cilíndricas cheias de fluido de comprimento finito e cujas extremidades esquerda edireita estão sujeitas a vetores prescritos de deslocamentos/pressões e forças elásticas/acústicas.Ademais, o método dos elementos finitos ondulatórios é usado para calcular bandas proibidasem cristais fonônicos elásticos de placas e cascas cilíndricas com distribuição periódica dediferentes propriedades elásticas. Bandas proibidas geradas pelo efeito de espalhamento deBragg são calculadas com o método dos elementos finitos de onda através de diferentes casosde teste. Os resultados são apresentados na forma de diagramas de dispersão e funções deresposta de frequência. A relevância do método dos elementos finitos ondulatórios é claramentedemonstrada em comparação com diferentes soluções analíticas e numéricas. Finalmente, osefeitos das bandas proibidas em PC de cascas cilíndricas com fluido interior são investigados.Portanto, é destacado o potencial do método dos elementos finitos de onda para projetarsistemas periódicos que podem ser usados para atenuação de vibração e ruído.
Palavras-chave: Cascas, Interação fluido-estrutura, Cristais fonônicos, Propagação de ondas,Método dos elementos finitos.
Abstract
The wave finite element (WFE) method is proposed for modeling fluid-filled phononiccrystal (PC) cylindrical shells. This numerical approach also is applied to analyze the guidedwave propagation and vibroacoustic behavior in periodic cylindrical shells. In this study, thecylindrical shells are assumed to be linear elastic and modeled with 2D flat shell elements,while the inner fluids are assumed to be acoustic and modeled with 3D linear elements. Aperiodic finite element (FE) mesh assumption is used which enables the description of awhole fluid-filled cylindrical shell in terms of identical subsystems which are composed ofstructure parts and fluid parts, and which are assembled together along a straight direction. Byconsidering the FE model of a subsystem, a transfer matrix relation can be derived which linksthe kinematic/mechanical quantities between the right and left cross-sections. Eigenvalues andeigenvectors of the transfer matrix provide the so-called wave modes, i.e., the wave parame-ters/wave numbers and the wave shapes. Besides, the WFE method is applied to compute thevibroacoustic responses of fluid-filled cylindrical shells of finite length and whose left and rightends are subject to prescribed vectors of displacements/pressures and elastic/acoustic forces.In addition, the WFE method is used to calculate band gaps in elastic phononic crystal platesand cylindrical shells with a periodic distribution of different elastic properties. Band gapsgenerated by Bragg scattering effect are calculated with the WFE method through differenttest cases. Results are presented in the form of dispersion diagrams and frequency responsefunctions. The relevance of the WFE method is clearly demonstrated in comparison withdifferent analytical and numerical solutions. Finally, band gap effects in PC cylindrical shellswith internal fluid are investigated. Therefore, the potential of WFE method to design periodicsystems that can be used to vibration and noise attenuation is highlight.
Keywords: Shells, Fluid-structure interaction, Phononic crystals, Wave propagation, Finite ele-ment method.
List of Figures
1.1 Examples of real applications . (𝑎) Acoustic muffler in exhaust system; (𝑏) Muf-fler parts that can be modeled by elastic-acoustic models involving PC shellelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 FE model of an elasto-acoustic subsystem. . . . . . . . . . . . . . . . . . . . . 252.2 Flat shell element with six degrees of freedom at a node. . . . . . . . . . . . . 262.3 Cylindrical shell as an assembly of flat elements: local and global coordinates. . 292.4 Structure with internal fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Illustration of state vectors of two consecutive subsystems. . . . . . . . . . . . 342.6 Dispersion curves: (𝑎) without tracking of wave modes; (𝑏) with tracking of
wave modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7 Sketch of a full waveguide connected by identical subsystems and subject to
prescribed acoustic excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8 Flowchart illustrating the computational basic steps in the WFE method. . . . . 403.1 FE mesh of two periodic structures having different elastic properties periodi-
cally distributed along their length. (𝑎) periodic plate; (𝑏) periodic cylindricalshell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Schematics of the FE model of a fluid-filled PC system subject to elastic/acous-tic forces; wave amplitudes related to left- and right-going waves; FE model ofa typically subsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 FE mesh of a cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 FE mesh of a cylindrical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 FE mesh of a fluid-filled cylindrical shell . . . . . . . . . . . . . . . . . . . . . 514.4 FE models of the plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 Dispersion curves for a Levy plate: WFE solutions (blue and green solid lines);
SE solutions (black and red circles). . . . . . . . . . . . . . . . . . . . . . . . 524.6 FE models of the cylindrical shell. . . . . . . . . . . . . . . . . . . . . . . . . 534.7 Dispersion curves for a homogeneous cylindrical shell: WFE solutions ( );
analytical solutions for 𝑠 = 0 (. . . ); analytical solutions for 𝑠 = 1 (∘ ∘ ∘). . . . . 534.8 FE models of the elasto-acoustic subsystem . . . . . . . . . . . . . . . . . . . 544.9 Dispersion curves: WFE solutions ( ); analytical solutions for 𝑠 = 0 (. . . );
analytical solutions for 𝑠 = 1 (∘ ∘ ∘). . . . . . . . . . . . . . . . . . . . . . . . 554.10 FE model of fluid-filled cylindrical shell subject to prescribed pressure. . . . . 564.11 Pressure FRF of the fluid-filled cylindrical shell: FE method (. . . ); WFE method
( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.12 The periodic structure with 𝑁 = 4 substructures (left), and FE mesh of a sub-
structure (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.13 FRF of the PC plate (left); dispersion curves for the bending mode (Right): realpart (blue) and imaginary part (red) of the SE solution; real part (magenta) andimaginary part (green) of the WFE solution; Bragg limit (black line). . . . . . . 59
4.14 Periodic structure with 20 substructures (left), and FE mesh of a substructure(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.15 Frequency response of periodic PC cylindrical shell (left), and dispersion curvesfor two modes in the PC cylindrical shell (right). Real part ( ) and imaginarypart ( ) of the WFE solution. . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.16 Periodic PC system with 20 substructures and subject to the prescribed pressurefield (𝑎) ; FE mesh of a subsystem (𝑏) . . . . . . . . . . . . . . . . . . . . . . . 62
4.17 Dispersion curves of the waves in fluid-filled PC cylindrical shell. Real part of𝛽𝑗𝑅
F ( ) and imaginary part of 𝛽𝑗𝑅F ( ) of the WFE solution. . . . . . . . 63
4.18 Frequency response of periodic fluid-filled cylindrical shell: Homogeneous sys-tem ( ) and PC system ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.19 Periodic PC system with 20 substructures and subject to prescribed forces. . . . 644.20 Frequency response of PC cylindrical shell "in vacuo" ( ) and PC cylindrical
shell with internal fluid ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
List of Tables
4.1 Comparison of values of the natural frequencies for a cylindrical shell . . . . . 494.2 Comparison of values of the natural frequencies for an acoustic cavity . . . . . 504.3 Comparison of values of the natural frequencies for a fluid-filled cylindrical shell 514.4 Total number of DOFs and CPU times involved for computing the FRF . . . . . 574.5 Material properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
List of Abbreviations and Acronyms
Matrices and Vectors
𝒜 - Matrix of size 2𝑛× 2𝑛
B - Strain matrixC - Fluid-structure coupling matrixD - Dynamic stiffness matrixℱ - Vector of excitationsF𝐼 - Distribution of pressures on the coupling surfaceJ - Unit symplectic matrixJ0 - Jacobian matrixI - Identity matrixK - Stiffness matrixL,N - Matrices of size 2𝑛 × 2𝑛 used to express (N,L) eigen-
problemL′,N′ - Matrices of size 2𝑛× 2𝑛 used to express S + S−1 eigen-
problemL𝑒 - Transformation matrix between the local and global co-
ordinate systemsM - Mass matrixn - unit normal vectorp - Vectors of nodal pressuresP𝐼 - Interface termℛ - Diagonal symmetry transformation matrix of size 𝑛× 𝑛
S - Symplectic transfer matrix𝒯 - Diagonal symmetry transformation matrix of size 2𝑛×2𝑛
U - Vectors of nodal displacements/rotationsΨ - Vector of velocity potentialsΦ - Matrix of right-going wave mode shapesΦ⋆ - Matrix of left-going wave mode shapesn - unit normal vector
Matrices and Vectors
F - Vector of external loadsq - Vector of displacements/rotations/velocity potentialsq0 - Vector of prescribed displacements/rotations/velocity po-
tentialsQ - Vector of wave amplitudes𝒬 - Vector of wave amplitudesu - State vectorw′ - Eigenvector of matrix pencils (N′,L′ )𝜇 - Diagonal matrix of right-going eigenvalues of S)𝜑 - Vector of wave mode shapes𝜆 - Matrix of direction cosines
Superscripts
b - relative to a bending motionF - Relative to fluid part(𝑘) - Subsystem indexm - relative to a membrane motionS - Relative to structural part𝑇 - Vector/matrix transpose
Subscripts
𝑒 - relative to structural element𝑒′ - relative to fluid elementFS - relative to free surface boundary conditionI - relative to internal degrees of freedomL - relative to left degrees of freedomR - relative to right degrees of freedomRW - relative to rigid wall boundary condition
Latin Letters
𝑑 - substructure length𝑐0 - speed of sound𝑓𝑣 - volume or surface forcesℎ - thicknessi - imaginary unitK,M - approximate coefficient for rotation degrees of freedom𝑛 - number of degrees of freedom which discretizes the left-
/right cross-section of a subsystem𝑁 - number of substructures within a periodic system𝑢,𝑣,𝑤 - displacement directions𝑥,𝑦,𝑧 - cartesian axes in global system𝑥,𝑦,𝑧 - cartesian axes in local system𝑊𝑖 - weighting function
Greek Letters
𝜔 - angular frequencyΩ - domain𝛽 - wavenumber∆ - space between two consecutive impedance mismatches𝜂 - structural damping loss factor𝜇 - propagation constant or eigenvalue of S𝜆 - eigenvalue of matrix pencils (N′,L′ )𝜌 - mass density𝜈 - Poisson’s ratioℬ𝑓 - frequency range of analysisΓ - boundary interface𝜃𝑗 - rotation degrees of freedom
Table of Contents
1 INTRODUCTION 181.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 THE WAVE FINITE ELEMENT METHOD FOR FLUID-FILLED CYLIN-DRICAL SHELLS 252.1 Subsystem modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Structure part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.2 Fluid part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.3 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Wave mode computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3 Forced response computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Wave expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 WAVE PROPAGATION IN 1D PHONONIC CRYSTAL SYSTEMS 413.1 Phononic Crystal modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 WFE solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 SE method for periodic Levy plate . . . . . . . . . . . . . . . . . . . . . . . . 45
4 NUMERICAL RESULTS 484.1 FE model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.2 Fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.3 Coupled fluid-structure model . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Dispersion curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.1 Levy-type plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.2 Cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.3 Fluid-filled cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Forced response of fluid-filled cylindrical shell . . . . . . . . . . . . . . . . . . 564.4 Application in Phononic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 Phononic crystal plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.2 Phononic crystal cylindrical shell . . . . . . . . . . . . . . . . . . . . . 604.4.3 Fluid-filled phononic crystal cylindrical shell . . . . . . . . . . . . . . 61
5 CONCLUSION 66
5.1 Futures works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.1 Papers in Conferences Proceedings . . . . . . . . . . . . . . . . . . . 68
REFERENCES 69
ANNEX 75
A – Stiffness and mass matrices for a Kirchhoff-Love plate element 75
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1 INTRODUCTION
1.1 Motivation
Cylindrical shells are frequently encountered in many applications, such as gas pipelines,exhaust pipes, aircraft fuselage, submarine hulls, rockets and nanotubes. Furthermore, thesemodels can be used in bioengineering, for instance, in the modeling of the blood vessels of thehuman being. In most real engineering problems, these structures are filled with fluid, whichmakes it all the more important the development of numerical tools capable of predicting effi-ciently the vibroacoustic behavior of these systems.
In last decades, a large number of analytical formulations have been developed whichanalyze the guided elasto-acoustic wave propagation along fluid-filled cylindrical shells (Sinhaet al., 1992; Fuller and Fahy, 1982). Despite this, those analytical theories are fairly limited tothe analysis of homogeneous systems with simple geometry (Mencik and Ichchou, 2007). Tocircumvent this problem, the Finite Element (FE) method can be used. While the FE method iswell suited to model elasto-acoustic waveguides with arbitrary-shaped cross-sections, it requiresdense mesh discretization to accurately describe the system behavior, which penalizes the com-putational efficiency of the method. On the other hand, the Wave Finite Element (WFE) methodcan be considered, especially for periodic systems with many degrees of freedom (DOFs) likethose involving shell elements. The WFE method constitutes a fast and accurate numerical toolfor computing the wave modes in periodic structures, and further their vibroacoustic responses.The WFE method is not constrained to simple models as opposed to the analytical approachesand provides the same level of accuracy as the FE method. However, this wave-based approachis limited in scope to the analysis of constant waveguides whose wave propagation occurs inone direction.
Within the WFE framework, the FE model of a small slice of a whole system is consideredand a transfer matrix relation is expressed to link the kinematic/mechanical quantities betweenthe right and left boundaries of this slice. By considering the Bloch’s theorem, the so-calledwave modes of the periodic system can be computed, which involves assessing the eigenvaluesand eigenvectors of the transfer matrix of the slice (Mencik and Ichchou, 2005). The proposedstrategy reduces considerably the computational time when compared to the FE method.
In addition, an important topic in the study of the wave propagation along these systemsis the development of vibration filtering devices and systems that could be used to insulatenoise and vibration. In recent years, there have been numerous researches in this area, mainly inthe field of Phononic Crystals (PCs) and acoustic metamaterials. Analyzing and understandingthese systems from the computational (modeling) and experimental points of view remains anopen challenge, especially in the engineering field (Hussein et al., 2014). Phononic crystals(PCs) can be seen as composite materials, i.e., structures having different elastic propertieswhich are periodically distributed along their length. One interesting characteristic of PCs (dueto their periodic nature) is to produce band gaps, i.e., stop bands or forbidden bands where
19
there are no propagating waves. These are mostly induced by Bragg scattering effect, i.e., whenimpedance mismatches — i.e., changes in the elastic properties — periodically occur along astructure. This interesting feature enables one to propose efficient solutions for the vibrationand sound insulation of structures.
Most of the studies about PCs have focused on investigating bulk waves. Recently, somestudies have been conducted on elastic waves in plates. Most of these work by consideringthe plane wave expansion (PWE) method along with the Kirchhoff-Love thin plate theory orthe Reissner-Mindlin thick plate theory (Hsu and Wu, 2006; Wu et al., 2011). The main issuewhen using the PWE method is that the structures are assumed to be infinite, i.e., it cannot beapplied to analyze the dynamic behavior of systems such as plates and cylindrical shells of fi-nite dimensions, that is to say, structures which are frequently encountered in the engineeringindustry. Nowadays, the Spectral Element (SE) method have been used to study the band gapproperties in PCs plates (Wu et al., 2013a; Wu et al., 2013b; Wu et al., 2014). This approachpresents an exact solution of the model in the frequency domain. Another technique to eval-uate band gaps in plates and shells has been proposed by Sorokin and Ershova (Sorokin andErshova, 2004; Sorokin and Ershova, 2006) . This provides a means to analyze the formationof band gaps in plates and shells with or without fluid loading using the Floquet theory and theboundary integral equation methodology. However, these methods are limited to existing ana-lytical formulations. This makes it more difficult to analyze the wave propagation, and furtherthe dynamic response of complex acoustic metamaterials with arbitrary-shaped periodicity pat-terns (e.g., stiffened plates with holes). These limitations encourage the use of numerical tools,such as FE and WFE methods.
In this work, the WFE method is applied to calculate the wave propagation and vibroa-coustic response of cylindrical shells filled with an internal acoustic fluid. Although much re-search has been done using WFE method in elasto-acoustic systems (Mencik and Ichchou, 2007;Manconi et al., 2009; Bhuddi et al., 2015), only a few of these focus on systems whose elasticpart is modeled with shell elements. Here lies one motivation of the present work. Further-more, the WFE is applied to compute band gaps in simply-supported plates, cylindrical shellsand fluid-filled cylindrical shells whose elastic properties are distributed periodically. Hence,another motivation behind this work is to demonstrate that the WFE method is accurate andefficient for modeling and simulating these kinds of PCs. This enables to highlight the potentialof the use of this numerical tool in the design of periodic ducts which work as vibration andnoise filters within specific frequency bands such as exhaust mufflers (Figure 1.1). Another ap-plication for this work refers to any shell-like periodic structure in which one needs to reducevibration and noise, such as in aircraft fuselage and rockets. Also, this method can be usedto analyze complex periodic structures with arbitrary-shaped periodicity patterns, and as such,it can be applied to systems which cannot be handled with the analytical and semi-analyticalapproaches, such as SE and PWE methods. This leads the way to interesting prospects for com-puting band gaps in periodic structures whose shapes can be designed so as to magnify band
20
gap effects.
(a)(b)
Figure 1.1: Examples of real applications . (𝑎) Acoustic muffler in exhaust system 1; (𝑏) Mufflerparts that can be modeled by elastic-acoustic models involving PC shell elements 2.
1.2 Literature review
In this section, it is presented a brief review of the works found in the literature in lastyears, which are considered to be more relevant to this research area and to the development ofthe work.
Wave propagation in fluid-filled cylindrical shells has been the subject of many studies.These works propose mainly to evaluate the elasto-acoustic wave mode propagation. The ex-isting literature on analytical theories to evaluate the dispersion relation – i.e., the frequencyevolution of the wavenumbers – is vast. In this context, one of the main works was developedby Fuller and Fahy (1982), who studied the dispersion behavior and energy distributions of freewaves in thin-walled elastic pipes filled with fluid at rest. For this task, these researchers havedeveloped an analytical model using the Donnell–Mushtari equations of motion for cylindricalshells coupled to a fluid loading. Based on energy flow, Pavic (1990) developed an analyticalmethodology to measure the vibrational energy flow in fluid-filled pipes. Beyond that, Pavicpresented a better numerical solution to compute the fluid-loading factor described by Fullerand Fahy (1982).
Regarding the problem of fluid–structure interaction, the finite element methodology hasbeen investigated by a great number of researchers (Sandberg and Ohayon, 2009). One im-portant work that was used as a basis here, was developed by Romero (2007), who studiedthe vibroacoustic behavior of a structural-acoustic coupled model of stiffened aircraft panelsand the sensitivity analysis of the acoustic and structural response of this model under localstructural defects. Shells and beams structural elements for 2D and 3D models coupled withan internal fluid are presented. The flat shell model employed the MacNeal element, which is
1available on http://www.autoanything.com/exhausts-mufflers/77A3570A3075616.aspx on March 1, 20182available on https://br.comsol.com/blogs/multiphysics-simulation-provides-accurate-muffler-designs/ on
March 1, 2018
21
based on the Reissner-Mindlin theory.Ruzzene and collaborators focused on the development of finite element models to study
the dynamic analysis of fluid-filled cylindrical shells and to investigate active/passive controlstrategies in these systems. One of the earliest papers (Ruzzene and Baz, 2000) presented a nu-merical study on the active/passive control of sound radiation and power flow from fluid-loadedcylindrical shells using multiple stiffeners with active/passive constrained damping treatment.After that, Ruzzene and Baz (2001) applied controlled actuators that were distributed period-ically along fluid-loaded shells. The phenomena of wave stop and pass bands as well as theattenuation characteristics were then investigated. In this context, Oh et al. (2002) presenteda numerical and experimental study on the controlled vibration and noise radiation from sub-merged cylindrical shells by passive control. The effect of stiffening of the shell in the atten-uation was demonstrated using a FE model, a FE commercial software package and validatedexperimentally. The results of these works showed that unwanted noises and vibrations can beeliminated or reduced by use of suitable advanced materials or complex geometric shapes inoptimized shells.
One particular technology that has been topic of several studies refers to the use of mech-anisms that allow band gap phenomena in periodic structures. This is mostly induced by Braggscattering effect which occurs when geometric or material discontinuities are introduced in pe-riodic structures. This causes dispersion and destructive interference between the waves (elasticor acoustic), which creates zones of frequency where waves can not propagate (stop-bands orband gaps) (Deymier, 2013). A large number of studies have since appeared in this area.
One the first theoretical basis about Phononic Crystals (PCs) was probably presented bySigalas and Economou (1992). They studied the behavior of elastic and acoustic waves in pe-riodic structures composed by identical spheres distributed periodically within a matrix of ho-mogeneous material. In this case, the band gap phenomena was clearly highlighted.
Hussein et al. (2006) studied the wave propagation and dynamic response in finite andinfinite 1D PCs with layered materials. In this connection, the unit cell was composed of threestrips of alternating materials. The phenomena of pass and stop bands was analyzed concerningthe number of cells that composed a finite structure. The attenuation characteristics in thesestructures were clearly shown in comparison to a equivalent homogeneous structure.
More recently, Silva et al. (2011) studied the longitudinal wave propagation in a periodicrod constituted by a sequence of unit cells, each cell consisting of two layers of alternatingdifferent materials by the same reasoning of Hussein et al. (2006). They confirmed the existenceof longitudinal wave band gaps both numerically and experimentally.
Wu and collaborators have used the spectral element (SE) method to evaluate bandgaps on Kirchhoff-Love thin plates (Wu et al., 2013a) and Reissner-Mindlin thick plate (Wuet al., 2014). In both works, the band gap properties in the periodic plates which have differentmaterial are analyzed. The band gap effect was highlighted with an increase in the unit cellnumber. Furthermore, it is demonstrated that the vibration attenuation can be improved vary-
22
ing the materials quantities in the cell. In addition, a considerable number of researchers haveapplied the band gap theory in shell-based models.
In the field of the PCs, one of the first theoretical works dedicated to cylindrical shellswas presented by Sorokin and Ershova, 2004, who studied the formation of band gaps in platesand shells with or without fluid loading using Floquet-Bloch theory and the boundary integralequation methodology (Sorokin and Ershova, 2004; Sorokin and Ershova, 2006). Also, Shenet al., 2013 has investigated band gaps in fluid-filled cylindrical shells using active and passivecontrol by Finite Element (FE) analysis. In this study, the band gap effects were reinforced byactive control strategies. The formation mechanisms of stop-bands was identified in the attenu-ation curves, acoustic pressures level and frequency response functions.
The most important characteristic of a PC is its periodic structure. In last years, newmethods have since emerged which use periodicity modeling together with numerical modelingto analyze complex structures at a low computational cost. Among these, the WFE method isparticularly important in this context, owing the possibility of this being used in conjunctionwith commercial FE packages. This method has been used as research tools in many studies inVibroacoustic Laboratory of the DMC-FEM-UNICAMP (Nascimento, 2009; Silva, 2015; No-brega, 2015).
In the 70’s, Mead started working on the propagation of waves along 1D periodic struc-tures, i.e., structures made up of identical substructures along a certain straight or curved direc-tion (Mead, 1970). This was the initial impetus for the development of the WFE method. Morerecently, Mencik and Ichchou (2005) proposed a hybrid approach to study multi-mode wavepropagation through waveguide based on this method. They formulated coupling between mul-tiple waveguides in a joint. The diffusion matrix, relating incident and reflected wavemodes,is validated through numerical simulations. This work was based on the Zhong’s method tonumerically describe the wave propagation along one-dimensional periodic elastic structures(Zhong and Williams, 1995). These studies laid the basis for development of many works.
Meanwhile, a considerable number of researchers have used the WFE to analysis theforced response of periodic structures (Waki et al., 2009a; Waki et al., 2009b; Renno andMace, 2010). Mencik (2010) described the formulation the eigenproblem in one-dimensionalfinite structures and made use of Bloch’s theorem and Neumann-to-Dirichlet problems to com-pute forced response in periodic structures. The importance of definition of a reduced wave ba-sis is clearly highlight in the forced response of a beam-like structure and a Reissner–Mindlinplate, as well as the forced response of two waveguides coupled through an elastic junction incomparison with the FE analysis.
Mencik (2013) demonstrated the efficiency of wave-based matrix formulations to com-pute the frequency response functions (FRFs) of various structures involving rectangular flatshells, including assemblies of waveguides connected in arbitrary ways. The coupling elementsare modeled by using the FE method. The coupling conditions at the joint interfaces are ex-pressed through Lagrange multipliers. The results show a good agreement with FE full models.
23
The reduction of CPU time is evident by employed of the WFE method.Recently, Mencik and Duhamel (2015) have proposed a model reduction technique to
describe the dynamic behavior of periodic composed of arbitrary-shaped substructures. Theyproposed a generalized eigenproblem based on S + S−1 transformation technique, which pre-serves the symplectic structure of the problem. A hybrid technique based on the ComponentMode Synthesis (CMS) is introduced for analyzing 2D and 3D large-sized models. Also, it wasshown that the computation of the eigensolutions drastically sped up with the Lanczos methodto compute a reduced number of wave modes only.
One interesting work was performed by Silva et al. (2016), who presented a strategy forpassive vibration control involving adding periodic arrays of locally resonant devices in a 1Drods and a 3D aircraft fuselage. The dispersion curves of the periodic structure are assessedby means of the WFE method. The vibration decrease and band gap phenomena are clearlydemonstrated through spatial distribution of the total displacement of the periodic structures.
Nobrega et al. (2016) studied band gaps in elastic metamaterial rods containing an arrayof local resonators modeled by spring-mass resonators of multi-degree-of-freedom (M-DOF).Numerical experiments experiments could be carried out by the use of WFE and WSE meth-ods. Miranda Jr. and Dos Santos (2018) analyzed the band gap effect in 1D phononic crystalbeam made of different materials. The forced response is obtained by the WFE, WSE and PWEmethods, and validated experimentally. In addition, they studied the influence of the percentageof polyethylene and steel on the attenuation characteristics of the unit cell.
Moreover, several WFE strategies have been proposed to calculate the wave propagationand vibroacoustic response of elasto-acoustic systems. One of the main works was carried outby Mencik and Ichchou (2007) in which a symmetric matrix formulation is considered to assessthe wave propagation in fluid-filled periodic structures. Also, Manconi et al. (2009) applied theWFE method to axisymmetric fluid-filled pipes. In both works, the FE model of the elementarycell is made of three-dimensional elements. More recently, the WFE method was used to inves-tigate the vibroacoustic behavior and acoustic radiation of axisymmetric elastic pipes (Bhuddiet al., 2015) and elastic pipes of arbitrary cross-sections (Gobert and Mencik, 2016) interactingwith internal and external acoustic fluid.
1.3 Objectives
The main objective of this work is to develop the modeling of fluid-filled elastic cylin-drical shells with spatial periodic distribution as well as analyze the wave propagation andvibroacoustic behavior in these systems via Wave Finite Element (WFE) method.
The specific objectives of this work are listed in the following:∘ Implement a FE model of the fluid-filled cylindrical shell.∘ Implement SE method and analytical solutions for Levy plates, cylindrical shells and
fluid-filled cylindrical shells and validate by means of dispersion curves the WFE method.
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∘ Verify the relevance of the WFE method in terms of accuracy and CPU time savings tocompute the vibroacoustic response of a fluid-filled cylindrical shell, in comparison withthe FE method.
∘ Propose the WFE method to modeling phononic crystal cylindrical shells with internalfluid and demonstrate the accurate and efficient of this numerical approach for simulatingcoupled acoustic-structural phononic crystal systems.
∘ Compute and investigate band gaps and attenuation effects in phononic crystal plates andcylindrical shells with and without internal acoustic fluid via WFE method.
1.4 Outline of the dissertation
This dissertation is organized into five chapters and one annex. In this chapter is presenteda brief literature review, which describes the previous work in the areas related to this study.
The rest of this text is organized as follows. The basics of the WFE method for fluid-filledcylindrical shell systems are presented in Chapter 2. The FE modeling of a periodic subsystemis shown first. This FE model is based on the symmetric matrix formulation which consists inusing the displacements and velocity potentials as field variables. Also, the wave-based strategyfor computing the dispersion curves and the frequency response functions (FRFs) of periodicsystems are presented.
Chapter 3 presents the WFE-based procedure for analysis of wave propagation in periodicPC systems with 1D elastic properties distribution. Here, the so-called S + S−1 transformationtechnique is considered to compute the waves modes with accurate precision. This alternativeeigenvalue problem is used when the substructures presents complex geometry, i.e., arbitrary-shaped. Also, the WFE strategy to compute the forced response of periodic structures is recalled.The SE method for homogeneous and PC Levy-plates is reviewed.
Numerical experiments are brought in Chapter 4. Dispersion curves are assessed and com-pared with analytical theories. The proposed WFE-based approach is validated by analyzing thevibroacoustic response of a fluid-filled cylindrical shell of finite length and subject to harmonicacoustic loads in comparasion with FE analysis. Comparisons are made with the SE methodfor plates, the Flügge's theory for shells and Donnell's for fluid-filled cylindrical shell whichhighlight the accuracy of the numerical predictions. In addition, the relevance of the proposedmethod to identify the location and width of band gaps in the dispersion curves and frequencyresponse functions is discussed.
Finally, a brief summary of the conclusions and suggestions for future works are outlinedin Chapter 5. At the end, a list of the publications that resulted from progress of this work ispresented.
In Annex A, it is presented the mass and stiffness matrices of a thin rectangular isotropicplate bending element, which have been used in FE formulation for shell element presented inSection 2.1.1.
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2 THE WAVE FINITE ELEMENT METHOD FOR FLUID-FILLEDCYLINDRICAL SHELLS
This chapter presents the Wave Finite Element (WFE) method to describe the elasto-acoustic wave propagation and vibroacoustic response in fluid-filled cylindrical shells. Here,only one-dimensional (1D) periodic systems are considered. In Section 2.1 it is presented theformulation and strategies used in the Finite Element (FE) model associated with the method.The numerical procedure to calculate the elasto-acoustic wave modes is recalled in Section 2.2.In Section 2.3 it is presented the strategy used to compute the harmonic response in a pipe withinternal fluid by means WFE method. Finally, the computational implementation of the methodused in this work is discussed. The concepts and theory presented here are based on Mencikand Ichchou (2007), Romero (2007), Bhuddi et al. (2015), Mencik and Ichchou (2005).
2.1 Subsystem modeling
The WFE method starts by considering the FE model of a slice/subsystem of a periodicsystem. The system is periodic in the sense that it is composed of identical subsystems, of length𝑑, along a certain straight direction (say 𝑧-axis) (Mencik and Ichchou, 2005). The subsystemsunder concern are made of a structure part which is supposed to be elastic and dissipative, andan inner fluid part consisting in an acoustic fluid which is supposed to be inviscid, compressibleand at rest (Morand and Ohayon, 1992).
Figure 2.1: FE model of an elasto-acoustic subsystem.
Consider a subsystem 𝑘 (𝑘 = 1,...,𝑁 ) as shown in Figure ( 2.1). The left and right cross-sections (boundaries) of the subsystem are meshed using the same number 𝑛 of DOFs. Here,the total number of DOFs on the boundaries is expressed as 𝑛 = 𝑛S + 𝑛F, where 𝑛S and 𝑛F arethe numbers of DOFs on the boundaries of the structure part and fluid part, respectively. In thepresent study, the FE mesh of the subsystem (cf. Figure 1) involves 2D flat shell elements forthe structure part (cylindrical shell) and 3D acoustic elements for the fluid part (internal fluid).
26
In the formulation used to model the fluid-structure problem, the Lagrangian descriptionis used to express the structural domain while the Eulerian description is used to express thefluid domain. All the details about this formulation are based on the reference works of Morandand Ohayon (1995), Romero (2007), Sandberg and Ohayon (2009) and Zienkiewicz and Taylor(2000).
2.1.1 Structure part
A shell element combines a bending behavior with an “in-plane” (membrane) motion. Asthe governing equations of a curved shell are quite complex due to the curvature of the middlesurface, many alternative strategies have been used as a means of overcoming this problem. Onepractical approach is the development of flat shell elements that include both in-plane and bend-ing motions by combining a membrane element together with a Kirchhoff-Love plate element(Zienkiewicz and Taylor, 2000; Quek and Liu, 2003). This provides an effective procedure foranalysis of shells of arbitrary shape, especially due the simplicity of its formulation comparedto the more complex curved shell elements.
In this framework, a cylindrical shell is meshed with 2D rectangular thin flat shell ele-ments with four nodes and six degrees of freedom (DOFs) per node (i.e., three displacementsand three rotations) that incorporate both bending and membrane actions with drilling DOFs,as shown in Figure 2.2 (Zienkiewicz and Taylor, 2000; Bathe, 2015). The membrane-bendingcoupling appears when the element matrices are assembled into the global matrix. In this case,the curvature of the shell is done by changing the orientation of the elements in space, i.e., whenflat elements are assembled at different angles. Therefore, if the curvature of the shell is verylarge, a fine mesh of elements has to be used.
Figure 2.2: Flat shell element with six degrees of freedom at a node.
To represent the membrane effects of the element is used a 2D solid rectangular element,which corresponds to DOFs of 𝑢 and 𝑣. This element has two degrees of freedom per node for a
27
total of eight degrees of freedom per element. More details about the formulation of the stiffnessKm
𝑒 and mass Mm𝑒 matrices for the quadrilateral membrane element can be found in Cook et al.
(2001) and Romero (2007).To develop the quadrilateral plane element, it must be defined in the isoparametric coor-
dinate system (𝜉, 𝜁). The displacement is expressed within the element interpolation of nodaldisplacements. For four-node rectangular plane elements, the displacements can be interpolatedas
𝑢 = Nm𝑒u , 𝑣 = Nm
𝑒v (2.1)
where Nm𝑒 is the shape function for the four-node 2D solid element. Once the shape function and
nodal variables have been defined, element stiffness and mass matrices can then be expressedas follows (Romero, 2007):
Km𝑒 =
∫ 1
−1
∫ 1
−1
ℎ(Bm𝑒)
𝑇EBm𝑒det(J0)𝑑𝜉𝑑𝜁 , Mm
𝑒 = 𝜌
∫ 1
−1
∫ 1
−1
𝑡(Nm𝑒)
𝑇Nm𝑒det(J0)𝑑𝜉𝑑𝜁,
(2.2)where Bm
𝑒 J0 are termed the strain matrix and Jacobian matrix, respectively, which can be ob-tained by the following equations
Bm𝑒 =
⎡⎢⎣𝜕𝑁1
𝜕𝑥0 𝜕𝑁2
𝜕𝑥0 𝜕𝑁3
𝜕𝑥0 𝜕𝑁4
𝜕𝑥0
0 𝜕𝑁1
𝜕𝑦0 𝜕𝑁2
𝜕𝑦0 𝜕𝑁3
𝜕𝑦0 𝜕𝑁4
𝜕𝑦𝜕𝑁1
𝜕𝑦𝜕𝑁1
𝜕𝑥𝜕𝑁2
𝜕𝑦𝜕𝑁2
𝜕𝑥𝜕𝑁3
𝜕𝑦𝜕𝑁3
𝜕𝑥𝜕𝑁4
𝜕𝑦𝜕𝑁4
𝜕𝑥
⎤⎥⎦ (2.3)
J0 =
[ ∑𝜕𝑁𝑖
𝜕𝜉𝑥𝑖
∑𝜕𝑁𝑖
𝜕𝜉𝑦𝑖∑
𝜕𝑁𝑖
𝜕𝜁𝑥𝑖
∑𝜕𝑁𝑖
𝜕𝜁𝑦𝑖
], 𝑖 = 1,2,3,4. (2.4)
Also, E is the material constant matrix for the plane stress problems (Cook et al., 2001). Notethat ℎ and 𝜌 are the thickness and density of the elastic element, respectively.
To represent the bending effects of the flat shell element is used a rectangular plate el-ement, corresponding to DOFs of 𝑤, 𝜃𝑥 and 𝜃𝑦. In this study is considered a thin plate withsmall deformations based on the Kirchhoff theory. The basic assumptions for this classical the-ory closely parallels the Euller-Bernoulli beam theory, i.e., the mid-surface of the plate remainsunstretched during deformations; the lateral dimension of the plate is at least 10 times its thick-ness; and transverse shear stresses are small compared to normal stresses and hence can beneglected (Szilard, 2004). The development of the stiffness Kb
𝑒 and mass Mb𝑒 matrices for the
quadrilateral thin plate element based on Kirchhoff theory can be found in Szilard (2004).As it turns out, the element stiffness matrix (24× 24) and element mass matrix (24× 24)
of a shell element are expressed as follows (Romero, 2007):
KS𝑒 =
⎡⎢⎣ Km𝑒 0 0
0 Kb𝑒 0
0 0 KI
⎤⎥⎦ , MS𝑒 =
⎡⎢⎣ Mm𝑒 0 0
0 Mb𝑒 0
0 0 MI
⎤⎥⎦ , (2.5)
28
where Km𝑒 and Mm
𝑒 are, respectively, the 8×8 element stiffness and mass matrices correspondingto the membrane motion; Kb
𝑒 and Mb𝑒 are, respectively, the 12 × 12 element stiffness and mass
matrices corresponding to the bending motion (see Annex A); finally, K is an approximatecoefficient defined as one-thousandth of the smallest diagonal value of the element stiffnessmatrix (Bathe, 2015). Regarding mass matrix, in this work it is assumed the same strategy forcoefficient M. This procedure was adopted by Romero (2007) and it leads to accurate results.These factors reflect the rotational DOFs 𝜃𝑧 in Equation (2.5). The addition of such terms mayavoid presence of singular issues (Zienkiewicz and Taylor, 2000).
As a result, the global stiffness and mass matrices of the structural part are expressed as(Zienkiewicz and Taylor, 2000; Romero, 2007):
KS =∑
𝑒(LS
𝑒)𝑇KS
𝑒LS𝑒 , MS =
∑𝑒(LS
𝑒)𝑇MS
𝑒LS𝑒, (2.6)
where∑
represents the assembly procedure of the global system and LS𝑒 is the transformation
matrix between the local and global coordinate systems of the rectangular shell element that hasthe following expression:
LS𝑒 =
⎡⎢⎢⎢⎢⎣Λ 0 0 0
0 Λ 0 0
0 0 Λ 0
0 0 0 Λ
⎤⎥⎥⎥⎥⎦ , (2.7)
where
Λ =
[𝜆 0
0 𝜆
], (2.8)
with 𝜆 being the matrix of direction cosines between the two sets of reference axes, whichexpress as
𝜆 =
⎡⎢⎣ cos(𝑥,𝑥) cos(𝑥,𝑦) cos(𝑥,𝑧)
cos(𝑦,𝑥) cos(𝑦,𝑦) cos(𝑦,𝑧)
cos(𝑧,𝑥) cos(𝑧,𝑦) cos(𝑧,𝑧)
⎤⎥⎦ , (2.9)
in which the local system is represented by (𝑥,𝑦,𝑧) and the global system is represented by(𝑥,𝑦,𝑧). Figure 2.3 shows the global axis and the transformed local axis.
Finally, the dynamic equilibrium equation for the structural part can be written as:
− 𝜔2MSU + KSU = FS, (2.10)
where U is the vector of nodal displacements/rotations and FS is the nodal equivalent forcesvector.
29
Figure 2.3: Cylindrical shell as an assembly of flat elements: local and global coordinates.
2.1.2 Fluid part
The governing equation of the fluid domain is given by the Helmholtz equation (Morandand Ohayon, 1992):
∇2𝑝 +𝜔2
𝑐20𝑝 = 0 in ΩF, (2.11)
where 𝑝 is the acoustic pressure of the fluid, ∇2 is the Laplacian differential operator; 𝜔 and𝑐0 stand for the pulsation and the speed of sound, respectively. The boundary conditions ofrigid wall ΓRW, free surface ΓFS and fluid-structure interface Γ (see Figure 2.4) are as follows(Romero, 2007; Morand and Ohayon, 1992):
𝑝 = 0 on ΓFS, (2.12)
𝜕𝑝
𝜕𝑛= 0 on ΓRW, (2.13)
𝜕𝑝
𝜕𝑛= 𝜔2𝜌0un
F𝑒′ on Γ, (2.14)
where nF𝑒 is the outward unit normal vector at the boundary of the fluid domain and u is the
prescribed displacement field on Γ. The FE model of the fluid part can be obtained by consid-ering the method of Weighted Residuals (Bathe, 2015). The essence of this method is to forcethe residual to zero over the whole domain ΩF. To do so, the residual 𝑅 is multiplied by aweighting function 𝑊𝑖 and force the integral of the weighted residual to zero over this domain(Galli, 1995): ∫
ΩF
𝑊𝑖𝑅𝑑Ω =
∫ΩF
𝑊𝑖
(∇2𝑝− 1
𝑐20𝑝 + 𝑓𝑣
)𝑑Ω = 0, (2.15)
in which 𝑓𝑣 represents the volume or surface forces.Integrating by parts over such domains by means of the Green theorem and applying
the boundary conditions defined in Equations (2.12), (2.13), (2.14) has the weak form of the
30
equilibrium equation∫ΩF
∇𝑊𝑖∇𝑝𝑑Ω +1
𝑐20
∫ΩF
𝑊𝑖𝑝𝑑Ω =
∫Γ
𝑊𝑖𝜕𝑝
𝜕𝑛𝑑Γ +
∫ΩF
𝑊𝑖𝑓𝑣𝑑Ω. (2.16)
In this case, the Galerkin method is used and it is assumed that the weighting function is𝑊𝑖 = 𝛿𝑝.
The discretization, in the context of Finite Element method, is made taking into accounta finite basis by means of the classical polynomial approximation for each element, it is yields:
𝑝 = NF𝑒′p, (2.17)
𝑝 = NF𝑒′p, (2.18)
𝛿𝑝 = NF𝑒′𝛿p, (2.19)
where NF𝑒′ is the matrix of shape functions for a 3D linear element and p is the vector of nodal
pressures. Substituting the Equations (2.14), (2.17), (2.18), (2.14) in Equation (2.16) leads to
∑𝑒′
∫ΩF
(∇NF
𝑒′
)𝑇∇NF𝑒′𝑑Ωp+
1
𝑐20
∫ΩF
(NF
𝑒′
)𝑇NF
𝑒′𝑑Ωp = −𝜌0
∫Γ
(NF
𝑒′
)𝑇nF𝑒′N
S𝑒𝑑ΓU+
∫ΩF
NF𝑒′𝑓𝑣𝑑Ω
.
(2.20)Thus, the system of equations for an acoustic fluid domain can be expressed as:
MFp + KFp = P𝐼 + FF, (2.21)
Hence, the global stiffness and mass matrices of the fluid part are obtained as follows:
KF =∑
𝑒′
∫ΩF
(BF𝑒′)
𝑇BF𝑒′𝑑Ω , MF =
∑𝑒′
1
𝑐20
∫ΩF
(NF𝑒′)
𝑇NF𝑒′𝑑Ω, (2.22)
P𝐼 =∑
𝑒′−𝜌0
∫Γ
(NF
𝑒′
)𝑇nF𝑒′N
S𝑒𝑑ΓU , FF =
∑𝑒′
∫ΩF
NF𝑒′𝑓𝑣𝑑Ω, (2.23)
where BF𝑒′ is formed by the partial derivatives of the shape functions NF
𝑒′ and U is the vectorof structural acceleration. Moreover, the contribution of the interface and external forces termscan be expressed by P𝐼 and FF, respectively. In this case, 3D isoparametric elements with eightnodes and one DOF per node (pressure) are considered.
31
2.1.3 Fluid-structure interaction
The elasto-acoustic problem is schematically sketched in Figure ( 2.4). In this formula-tion, the boundary conditions and cinematic continuity between the structural domain ΩS andfluid domain ΩF are ensured by coupling conditions, which are defined in the interface Γ (Sand-berg and Ohayon, 2009).
Figure 2.4: Structure with internal fluid.
Consider the normal vector as n = nF𝑒′ = −nS
𝑒. On Γ, the fluid is felt by the structure as asurface force field. Therefore, the effect of the distribution of pressures on the coupling surfaceΓ can be expressed as (Romero, 2007; Galli, 1995):
F𝐼 =
∫Γ
(NS
𝑒
)𝑇𝑝𝑑Γ. (2.24)
By applying the Equation (2.17), yields:
F𝐼 =∫
Γ
(NS
𝑒
)𝑇nNF
𝑒′𝑑Γp. (2.25)
Thus, the coupling conditions between the elements 𝑒 and 𝑒′ in the structural and fluid parts aremanaged by considering a coupling matrix C :
C =∑
𝑒𝑒′
∫Γ
(NS
𝑒
)𝑇nNF
𝑒′𝑑Γ, (2.26)
In this case, the interface is considered as a third element, independent of the structuraland fluid elements as proposed by Romero (2007). The coupling is performed by linear inter-polation for the fluid element and cubic interpolation for the structural element (displacementand rotations are considered).
Moreover, by considering the coupling matrix (Equation 2.26), note that the interface termP𝐼 can be written as:
P𝐼 = −𝜌0C𝑇 U. (2.27)
32
In this way, the Equation (2.21) is rewritten as:
MFp + KFp = −𝜌0C𝑇 U + FF, (2.28)
For the structure part, the coupling is introduced by the distribution of pressures F𝐼 asfollows:
MSU + KSU = Cp + FS, (2.29)
Assuming time harmonic motion, the time derivatives of the elastic and acoustic variablescan be written as 𝑑2/𝑑𝑡2 = 𝜕2/𝜕𝑡2 = −𝜔2. By considering the Equations (2.28) and (2.29),the following unsymmetric formulation (U,p) can be obtained (Romero, 2007;Morand andOhayon, 1992):
− 𝜔2
[MS 0
𝜌0C𝑇 MF
](U
p
)+
[KS −C
0 KF
](U
p
)=
(FS
FF
), (2.30)
where U and p are the vectors of nodal displacements and nodal pressures, respectively; FS andFF are the structural and acoustic force vectors, respectively.
Mencik and Ichchou (2007) proposed the use of a symmetric matrix formulation basedon displacements and velocity potentials as field variables, with a view to expressing a transfermatrix which is symplectic (this provides the usual properties that waves should occur in pairs,i.e., right-going and left-going wave modes). This symmetric matrix formulation is expressedas follows:
Dq = F, (2.31)
where q and F represent the vectors of kinematic variables:
q =
(U
Ψ
), F =
(FS
1i𝜔FF
), (2.32)
where Ψ is the vector of velocity potentials defined as p = −i𝜔𝜌0Ψ. Also, D represents thedynamic stiffness matrix of the subsystem (Morand and Ohayon, 1992):
D = −𝜔2
[MS 0
0 −𝜌0MF
]+ i𝜔
[0 𝜌0C
𝜌0C𝑇 0
]+
[KS 0
0 −𝜌0KF
]. (2.33)
Notice that the structure part is supposed to be damped with a loss factor 𝜂, i.e., thestiffness matrix of the structure part is complex and expressed as KS = ℜKS(1 + i𝜂).
After that, Equation (2.31) is condensed on the left and right boundaries (2 × 𝑛):[D*
LL D*LR
D*RL D*
RR
](qL
qR
)=
(FL
FR
), (2.34)
33
in which the subscripts L and R refer to the left and right boundaries of the subsystem, and D*
is the condensed dynamic stiffness matrix of the subsystem, expressed by
D* = DBB −DBID−1II DIB, (2.35)
where the subscripts B and I refer to the boundary DOFs (i.e., those on the left and right cross-sections) and the internal DOFs, respectively. In case of no internal DOFs D* = D. It is impor-tant to remember that vectors of nodal displacements/rotations and velocity potentials (qL andqR) and forces (FL and FR) are partitioned into elastic and acoustic variables.
Therefore, Equation (2.31) can be reformulated in state vector form on the left and rightboundaries of the subsystem, as follows (Mencik and Ichchou, 2005):
uR = SuL, (2.36)
where
uR =
(qR
FR
), uL =
(qL
−FL
). (2.37)
Here, S is the transfer matrix of the subsystem (size of 2𝑛× 2𝑛) which is expressed as
S =
[−D*−1
LR D*LL −D*−1
LR
D*RL −D*
RRD*−1LR D*
LL −D*RRD
*−1LR
], (2.38)
It should be emphasized that the matrix S is symplectic, which means that S𝑇JS = J (Zhongand Williams, 1995), where
J =
[0 I𝑛
−I𝑛 0
]. (2.39)
By considering two consecutive subsystems 𝑘 − 1 and 𝑘, as shown in Figure (2.5), theconditions of continuity of the displacements/rotations and velocity potentials, and the action-reaction raw for the forces at the coupling interface must be satisfied as:
q(𝑘−1)R = q
(𝑘)L , F
(𝑘−1)R = −F
(𝑘)L . (2.40)
Then, substituting these relations in Equation ( 2.36), it yields the expression that allowslinking the kinetic quantities on the left boundaries of elasto-acoustic subsystems 𝑘 − 1 and 𝑘:
uL(𝑘) = SuL
(𝑘−1). (2.41)
As a result of Bloch’s theorem, the state vector uL(𝑘) can be expanded as (Mencik, 2014):
uL(𝑘) =
∑𝑗𝜑𝑗𝑄
(𝑘)𝑗 , (2.42)
34
Figure 2.5: Illustration of state vectors of two consecutive subsystems.
in which 𝜑𝑗 are the wave shapes and 𝑄(𝑘)𝑗 are the wave amplitudes. In the case of a periodic
system the wave amplitudes between two consecutive subsystem 𝑘 and 𝑘 − 1 are linked as:
𝑄(𝑘)𝑗 = 𝜇𝑗𝑄
(𝑘−1)𝑗 , (2.43)
where 𝜇𝑗 are the eigenvalues S. This allows to express the WFE eigenproblem, whose solutionsrefer to the wave modes traveling along the periodic structure.
2.2 Wave mode computation
Following the WFE framework (Mencik and Ichchou, 2005; Zhong and Williams, 1995),the elasto-acoustic wave modes (𝜇𝑗,𝜑𝑗) are obtained by solving an eigenproblem of the form
S𝜑𝑗 = 𝜇𝑗𝜑𝑗, (2.44)
where 𝜇𝑗 and 𝜑𝑗 are the eigenvalues and eigenvectors of S. In accordance with Bloch’s theoremthe eigenvalues 𝜇𝑗 have the meaning of propagation constants, expressed as 𝜇𝑗 = exp(−i𝛽𝑗𝑑)
where 𝑑 is the subsystem length and 𝛽𝑗 designates the wavenumbers. Also, the eigenvectors 𝜑𝑗
have the meaning of wave shapes defined as 𝜑𝑗 = [(𝜑q)𝑗𝑇 (𝜑F)𝑗
𝑇 ]𝑇 where in this case
𝜑q =
(𝜑S
q
𝜑Fq
), 𝜑F =
(𝜑S
F
𝜑FF
). (2.45)
Here, 𝜑Sq and 𝜑S
F are associated with the displacements and forces on the boundaries of the struc-ture part, while 𝜑F
q and 𝜑FF are associated to the velocity potentials and acoustic forces on the
35
boundaries of the fluid part. Due to the symplectic property of the matrix S, the eigensolutionsoccur in pairs as (𝜇𝑗,1/𝜇𝑗), i.e., there exist 𝑛 right-going wave modes
(𝜇𝑗,𝜑𝑗)
𝑗=1,...,𝑛
and 𝑛
left-going wave modes
(𝜇⋆𝑗 ,𝜑
⋆𝑗)𝑗=1,...,𝑛
, where 𝜇𝑗 with |𝜇𝑗| < 1 and 𝜇⋆𝑗 = 1/𝜇𝑗 with
𝜇⋆𝑗
> 1.
In matrix form, those wave modes are written as:
𝜇 = (𝜇⋆)−1 = diag 𝜇j𝑗=1,...,𝑛, (2.46)
Φ =
[Φq
ΦF
]=
[(𝜑q)1...(𝜑q)𝑛
(𝜑F)1...(𝜑F)𝑛
], Φ⋆ =
[Φ⋆
q
Φ⋆F
]=
[(𝜑⋆
q)1...(𝜑⋆q)𝑛
(𝜑⋆F)1...(𝜑
⋆F)𝑛
], (2.47)
in which Φq, Φ⋆q, ΦF and Φ⋆
F are 𝑛×𝑛 matrices partitioned into structure and fluid components.The direct computation of the eigensolutions of the matrix S in Equation (2.44) may be
subject to numerical ill-conditioning on the inversion of submatrix DLR, especially when thenumber of boundary DOFs is large. This is due especially to the fact that the eigenvectors areexpressed in terms of displacement/rotation (and velocity potential) and force/moment compo-nents, whose values can be largely discrepant (Mencik, 2010). To overcome this issue, Zhongand Williams (1995) proposed a alternative generalized eigenvalue problem by making use ofthe symplectic property. In this alternative formulation the eigenvalue problem takes into ac-count the vectors of displacements/rotations (and velocity potential) only, in order to avoid theconditioning problems related to the force/moment components. The following (N,L) eigen-problem method can be considered (Zhong and Williams, 1995; Mencik, 2014):
Nw𝑗 = 𝜇𝑗Lw𝑗, (2.48)
where
L =
[I𝑛 0
−D*LL −D*
LR
], N =
[0 I𝑛
D*RL D*
RR
]. (2.49)
In Equation (2.48), w𝑗 are the eigenvectors which are expressed in terms of displacementand velocity potential components, only. The original eigenvectors 𝜑𝑗 are retrieved as 𝜑𝑗 =
Lw𝑗 . Notice that, when the subsystems are symmetric with respect to their mid-plane, the right-going and left-going wave modes are linked as follows:
Φ⋆ = 𝒯 Φ, (2.50)
where
𝒯 =
[ℛ 0
0 −ℛ
], (2.51)
is a block diagonal matrix that includes the 𝑛 × 𝑛 diagonal symmetry transformation matrixℛ with −1 or 1 components. Here, the matrix ℛ is expressed taking into account the elasticand acoustic variables. Equation (2.50) provides an analytical means to enforce the coherencebetween the right-going and left-going wave modes, i.e., the fact that they occur in pairs with
36
the same velocities (Mencik, 2010). Without that, the WFE method may suffer from numericalerrors which may be dramatic especially for computing the vibroacoustic responses of periodicsystems.
Another important step within the WFE framework is tracking of wave modes. In a peri-odic systems with many degrees of freedom (DOFs), many wave modes are calculated at eachdiscretized frequency step. In this process, the eigenvalues and associated eigenvectors can bedisordered in different discrete frequencies. To circumvent this issue, a usual procedure to checkthe correlation of the resulting modes has been defined in Mencik (2010). Consider two wavemodes 𝑟 and 𝑠, where 𝜇𝑟 = 1/𝜇𝑠 at the frequency 𝜔𝑖 and a sufficiently small step ∆𝜔. The wavemode 𝑠 at the frequency 𝜔𝑖 + ∆𝜔 is chosen so that:
Φ𝑟(𝜔𝑖)𝑇
‖Φ𝑟(𝜔𝑖)‖J
Φ𝑠(𝜔𝑖 + ∆𝜔)
‖Φ𝑠(𝜔𝑖 + ∆𝜔)‖
= max𝑘
Φ𝑟(𝜔𝑖)
𝑇
‖Φ𝑟(𝜔𝑖)‖J
Φ𝑘(𝜔𝑖 + ∆𝜔)
‖Φ𝑘(𝜔𝑖 + ∆𝜔)‖
. (2.52)
This tracking criterion is based on sympletic orthogonality properties of the wave modes (Zhongand Williams, 1995). The result of the application of a wave mode tracking procedure is clearlyhighlighted in the figure below, that shows the dispersion curves relative to non-tracked (Fig-ure 2.6 (a)) and tracked (Figure 2.6 (b)) wave modes, i.e., this criterion allows that wave modesare clearly identified over the frequency domain.
0.5 1 1.5 2 2.5 3
104
0
200
400
()
[m- 1
]
0.5 1 1.5 2 2.5 3
Frequency [Hz] 104
-400
-200
0
()
[m- 1
]
(a)
0.5 1 1.5 2 2.5 3
104
0
200
400
()
[m-1
]
0.5 1 1.5 2 2.5 3
Frequency [Hz] 104
-400
-200
0
()
[m-1
]
(b)
Figure 2.6: Dispersion curves: (𝑎) without tracking of wave modes; (𝑏) with tracking of wavemodes.
2.3 Forced response computation
2.3.1 Wave expansion
Consider a periodic system composed of 𝑁 subsystems (see Figure 2.7) subject to arbi-trary boundary conditions on its left and right boundaries. The strategy used here consists in
37
expanding the vectors of displacements/rotations (and velocity potentials) and forces/moments,on the left boundary of a subsystem 𝑘, using wave bases
𝜑𝑗
𝑗∪𝜑𝑗
⋆𝑗. The vectors of dis-
placements/rotations and velocity potentials (qS and qF) over the subsystem boundaries areachieved by considering the following wave mode expansions (Mencik, 2010):
q(𝑘)L = ΦqQ
(𝑘) + Φ⋆qQ
⋆(𝑘) , q(𝑘)R = ΦqQ
(𝑘+1) + Φ⋆qQ
⋆(𝑘+1), 𝑘 = 1, . . . ,𝑁, (2.53)
where
Φq =
[ΦS
q
ΦFq
], Φ⋆
q =
[ΦS⋆
q
ΦF⋆q
], (2.54)
in which ΦSq and ΦS⋆
q are 𝑛S × 𝑛 matrices that reflect structural components, while ΦFq and ΦF⋆
q
are 𝑛F × 𝑛 matrices that reflect acoustic components. Also, Q and Q⋆ are 𝑛 × 1 vectors ofwave amplitudes defined at the left and right boundaries of the excited whole periodic system,respectively. As convention, the wave amplitudes may be expressed as (Mencik, 2014):
Q(𝑘) = 𝜇𝑘−1Q , Q⋆(𝑘) = 𝜇𝑁−𝑘+1Q⋆, (2.55)
whereQ = Q(1) , Q⋆ = Q⋆(𝑁+1), (2.56)
Substituting Equation ( 2.55) into Equation ( 2.53), it yields:
q(𝑘)L = Φq𝜇
𝑘−1Q+Φ⋆q𝜇
𝑁+1−𝑘Q⋆ , q(𝑘)R = Φq𝜇
𝑘Q+Φ⋆q𝜇
𝑁−𝑘Q⋆, 𝑘 = 1, . . . ,𝑁, (2.57)
Assume, for the sake of clarity, that those left and right boundaries are subject to two pre-scribed vectors of displacements/velocity potentials (q0 = q
(1)L and q′
0 = q(𝑁)R ). More precisely,
the vectors q0 and q′0 are to be expressed as follows:
q0 =
(qS0
qF0
), q′
0 =
(qS0′
qF0′
). (2.58)
Consider a elasto-acoustic system with arbitrary boundary conditions at the interfaces (𝑁) and(1), as shown in Figure 2.7. Thus Equation ( 2.57) can be rewritten as:
q0 = ΦqQ + Φ⋆q𝜇
𝑁Q⋆ , q′0 = Φq𝜇
𝑁Q + Φ⋆qQ
⋆. (2.59)
Hence, by considering Equation (2.59), the matrix equation may be expressed as:[Φq Φ⋆
q𝜇𝑁
Φq𝜇𝑁 Φ⋆
q
](Q
Q⋆
)=
(q0
q′0
). (2.60)
However, when inverting the matrix can occur ill-conditioning (Mencik, 2010). To cir-
38
Figure 2.7: Sketch of a full waveguide connected by identical subsystems and subject to pre-scribed acoustic excitations
cumvent this issue, the following well-conditioned matrix equation can be established (Men-cik, 2014): [
I𝑛 Φ−1q Φ⋆
q𝜇𝑁
Φ⋆−1q Φq𝜇
𝑁 I𝑛
](Q
Q⋆
)=
(Φ−1
q q0
Φ⋆−1q q′
0
). (2.61)
This matrix formulation is of the form 𝒜𝒬 = ℱ , where 𝒬 = [Q𝑇Q⋆𝑇 ]𝑇 is the vectorof wave amplitudes for the right-going and left-going wave modes, while ℱ is the vector ofexcitations. The computation of the vectors of wave amplitudes follows as 𝒬 = 𝒜−1ℱ ; also,the vectors of displacements/velocity potentials and structural/acoustic forces at the boundariesof any subsystem (𝑘) can be simply retrieved from Equation (2.57).
2.4 Implementation
In this section are described the main computational tasks involving the WFE methodin the scope of this work. The preprocessing was carried out using the commercial soft-ware ANSYS® in APDL programming, while in processing and post processing was usedMATLAB®. The numerical procedure is schematically sketched in Figure (2.8).
The first step is to build the FE mesh of the subsystem. Due to needing to analyze elasto-acoustic systems with complex geometries, the ANSYS® was used to generate the mesh data.The processing is started with a MATLAB® code for reading the finite element mesh data file(coordinates, incidence, boundary conditions, geometrical and material properties). The FE ma-trices were computed in a computational code based on the project MEFLAB (originally writtenin FORTRAN), a program under development in the Department of Computational Mechanicsof the Faculty of Engineering Mechanics of the UNICAMP. The simulation program moduleshave all been updated and reimplemented in MATLAB®, a convenient interactive language
39
which allows direct graphical output and practicality in matrix manipulation.Thenceforth the WFE procedure is performed at each discrete frequency within a band
of interest. The dynamic stiffness matrix (Equation 2.33) is then condensed on the left andright boundaries of the subsystem (Equation 2.34). This allows to expressing the N and L
matrices (Equation 2.49). The wave modes (𝜇𝑗,𝜑𝑗) are calculated by solving the WFE eigen-problem (Equation 2.48). Additionally, the symmetric relation between wave modes is appliednumerically (Equation 2.50). After that, the tracking criterion is applied to sort the wave modes(Equation 2.52).
The implementation of the forced response following the steps listed in Section 2.3. Thelength of the finite system must be defined, together with the number of subsystems 𝑁 thatmake up the excited whole structure. The excitation is hence set, according to the DOFs of thesubsystem. Once the symmetry between wave modes is performed, the boundary condition aredefined (Equation 2.59) and the wave amplitudes are calculated by means of the matrix equation𝒜𝒬 = ℱ (Equation 2.61). At the end, the vectors of nodal displacements/velocity potentials (orforces) on the coupling interfaces, between the subsystems are evaluated at the response point(Equation 2.61).
Finally, the post processing is carried out in MATLAB® which allows the graphical anal-ysis of the numerical results, especially by means dispersion curves and FRFs.
40
Figure 2.8: Flowchart illustrating the computational basic steps in the WFE method.
41
3 WAVE PROPAGATION IN 1D PHONONIC CRYSTAL SYSTEMS
This chapter presents the methodology used in this work to compute band gaps inPhononic Crystal (PC) periodic systems, whose elastic properties periodically vary as shownin Figure 3.1. For this task, the strategy used for passive vibration control is to model multi-layered structures, i.e., made of periodic strips with two materials. This provides a means toshow band gaps generated by Bragg scattering effects, i.e., when impedance mismatch zoneare introduced in a periodic system. In this case, the WFE method is used to asses band gapsin dispersion curves. Moreover, it constitutes an efficient and accurate numerical means for ob-serving attenuation generated by band gaps through the analysis of frequency response functions(FRFs). In order to evaluate the potential of use of the proposed approach, numerical experi-ments are carried out on a PC plate and on a PC cylindrical shell with and without internal fluid.A comparative study is proposed between the WFE method and the SE method for plates. TheSE method uses an exact analytical solution issued from the Levy-plate theory, while the WFEmethod uses FE models.
(a)(b)
Figure 3.1: FE mesh of two periodic structures having different elastic properties periodicallydistributed along their length. (𝑎) periodic plate; (𝑏) periodic cylindrical shell.
In this case, the band gaps are mostly induced by Bragg scattering effect, i.e., whenimpedance mismatches (changes in the elastic properties) periodically occur along a structure.In fact, this phenomena generally appear around frequencies governed by the Bragg condition,i.e., ∆ = 𝑛′′(𝜆/2) (𝑛′′ = 1,2,...) with 𝜆 the wavelength and ∆ the space between two consecu-tive impedance mismatches.
The contribution of this chapter is the first to demonstrate that the WFE method is accurateand efficient for modeling and simulating of periodic shell-based systems exhibiting band gapsphenomena. In addition, it is proposed to use S + S−1 transformation technique (Mencik andDuhamel, 2015) to compute the waves modes in periodic PC systems with complex periodicitypatterns (material distributions).
42
The chapter is organized as follows. The basics of the WFE method applied to analyze PCsystems are carried out in Section 3.1. Here, the so-called S + S−1 transformation technique isconsidered to compute the waves in periodic structures with accurate precision. Also, the WFEstrategy to compute the forced response of periodic structures is recalled. In Section 3.2, the SEmethod for homogeneous and PC Levy-plates is reviewed.
3.1 Phononic Crystal modeling
3.1.1 WFE solution
Consider a periodic structure composed of identical substructures which are made up ofstrips with different material properties as shown in Figure 3.1. Within FE framework, it isassumed that the substructures are meshed in the same way with the same number 𝑛 of degreesof freedom (DOFs) on their left and right boundaries. Hence, the dynamic stiffness matrix of agiven substructure can be expressed by
D = −𝜔2M + (1 + i𝜂)K, (3.1)
where M and K are the mass matrix and stiffness matrix of the substructure, respectively.Also, 𝜔 and 𝜂 are the angular frequency and the loss factor, respectively. The key idea here is tomodel a substructure (unit-cell) made of strips of different materials (e.g., Mat1−Mat2−Mat1).For elasto-acoustic case, consider the dynamic stiffness matrix defined in Equation (2.33). Thestructural part of periodic fluid-filled shell system is made of multi-layered structures that arealternated periodically along the 𝑧-direction, as shown in Figure 3.2. Attention should to be paidabout the size of the FE mesh, i.e., it has to be fine enough to accurately capture the wavelengthsof the wave motion which physically occur, e.g., in an equivalent infinite structure. As a rule ofthumb, at least six elements per wavelength should be considered (Manconi et al., 2009).
As a result, the dynamic equilibrium equation of the substructure can be expressed as⎡⎢⎣ DLL DLI DLR
DIL DII DIR
DRL DRI DRR
⎤⎥⎦⎛⎜⎝ qL
qI
qR
⎞⎟⎠ =
⎛⎜⎝ FL
0
FR
⎞⎟⎠ , (3.2)
and by condensing the matrix D on the left and right substructure boundaries, result:[D*
LL D*LR
D*RL D*
RR
](qL
qR
)=
(FL
FR
), (3.3)
where D* is the dynamic stiffness matrix of the substructure condensed, which is expressed byEquation (2.35).
The PC modelling can be involve complex structures with arbitrary-shaped periodicity
43
patterns. For this cases, the (N,L) eigenproblem may be prone to numerical issues, except whenthe substructures are symmetric with respect to their mid-plane. Moreover, in order to avoidnumerical issues in computation of the wave modes (𝜇𝑗,𝜑𝑗), an alternative well-conditionedgeneralized eigenproblem based on the so-called S+S−1 transformation technique can be con-sidered (Mencik and Duhamel, 2015). First, it involves a well-posed eigenproblem based onskew-symmetric matrices to compute the eigensolutions of the transfer matrix of the substruc-tures. The feature of the proposed algorithm is that it preserves the symplectic nature of thewaves, i.e., the fact that they occur in pairs and that each pair of waves travel in opposite direc-tions with the same speed. In this sense, numerical dispersion issues can be overcome. For thesake of clarity, the key steps of this procedure are recalled hereafter.
The formulation described here can be found in the reference work of Mencik andDuhamel (2015). Consider the following generalized eigenproblem express by:
N′w′𝑗 = 𝜇𝑗L
′w′𝑗, (3.4)
where
L′ =
[0 I𝑛
D*LR 0
], N′ =
[D*
RL 0
−(D*LL + D*
RR) −I𝑛
], (3.5)
and w′𝑗 are the eigenvectors, which are partitioned into displacement/rotation (and velocity
potential) and force/moment components. In this case, numerical errors may occasionally oc-cur due to ill-conditioning of the eigenvalue problem. The key idea, to overcome this issue,is to write the eigenvalue problem as a function of the vectors of displacements/rotations (andvelocity potential) only, as mentioned in Section 2.2.
As such, within the framework of the S + S−1 transformation technique, the followinggeneralized eigenproblem is considered with double eigenvalues 𝜆𝑗 of the form 𝜆𝑗 = 𝜇𝑗 + 1/𝜇𝑗
(Mencik and Duhamel, 2015):((N′JL
′𝑇 + L′JN′𝑇 ) − 𝜆𝑗L
′JL′𝑇)z𝑗 = 0, (3.6)
where
N′JL′𝑇 + L′JN
′𝑇 =
[D*
RL −D*LR (D*
LL + D*RR)
−(D*LL + D*
RR) D*RL −D*
LR
], (3.7)
and
L′JL′𝑇 = N′JN
′𝑇 =
[0 −D*
RL
D*LR 0
]. (3.8)
The nice feature of the eigenproblem (3.6) is that it involves skew-symmetric matrices(N′JL
′𝑇 +L′JN′𝑇 ) and L′JL
′𝑇 . In this sense, it can be shown that the symplectic nature of theeigenproblem is preserved, i.e., the computed eigenvalues 𝜆𝑗 = 𝜇𝑗 + 1/𝜇𝑗 and 𝜆⋆
𝑗 = 𝜇⋆𝑗 + 1/𝜇⋆
𝑗
for the right-going and left-going wave modes will be identical. Once the eigenvalues 𝜆𝑗 in
44
Equation (3.6) are found, the double eigenvalues (𝜇𝑗,𝜇⋆𝑗) can be found analytically by solving a
quadratic equation of the form𝑥2 − 𝜆𝑗𝑥 + 1 = 0, (3.9)
whose solutions are
𝜇𝑗, 𝜇⋆𝑗 =
𝜆𝑗 ±√
𝜆2𝑗 − 4
2. (3.10)
Finally, the wave shapes (𝜑𝑗,𝜑⋆𝑗) can be retrieved as follows
𝜑𝑗 =
[I𝑛 0
D*RR I𝑛
]w′
𝑗 , 𝜑⋆𝑗 =
[I𝑛 0
D*RR I𝑛
]w
′⋆𝑗 , (3.11)
wherew′
𝑗 = J(L′𝑇 − 𝜇⋆
𝑗N′𝑇 )z𝑗 , w
′⋆𝑗 = J(L
′𝑇 − 𝜇𝑗N′𝑇 )z𝑗, (3.12)
where z𝑗 is the eigenvector (eigenproblem (3.6)), which are expressed in term of displace-ments/rotations (and velocity potential) components.
To forced response computation, for the sake of clarity, consider a periodic structure com-posed of 𝑁 substructures like the one displayed in Figure 3.4, whose left side is subjected toa force vector F0 and whose right side is subjected to prescribed displacements (vector q0).Within the WFE framework, the displacement vector and force vector on the left and rightboundaries of a substructure 𝑘 (𝑘 = 1, . . . ,𝑁 ) are expressed in terms of wave mode shapes, asfollows (Mencik, 2014):
q(𝑘)L = Φq𝜇
𝑘−1Q+Φ⋆q𝜇
𝑁−𝑘+1Q⋆ , q(𝑘)R = Φq𝜇
𝑘Q+Φ⋆q𝜇
𝑁−𝑘Q⋆, 𝑘 = 1, . . . ,𝑁, (3.13)
− F(𝑘)L = ΦF𝜇
𝑘−1Q + Φ⋆F𝜇
𝑁−𝑘+1Q⋆ , F(𝑘)R = ΦF𝜇
𝑘Q + Φ⋆F𝜇
𝑁−𝑘Q⋆, 𝑘 = 1, . . . ,𝑁,
(3.14)where Φq, Φ⋆
q, ΦF and Φ⋆F are 𝑛× 𝑛 matrices of wave shapes, expressed by
Φq =[𝜑q1 · · ·𝜑q𝑛
], Φq
⋆ =[𝜑⋆
q1 · · ·𝜑⋆q𝑛
], (3.15)
ΦF = [𝜑F1 · · ·𝜑F𝑛] , ΦF⋆ = [𝜑⋆
F1 · · ·𝜑⋆F𝑛] . (3.16)
For elasto-acoustic case, these eigenvectors are represented by the Equation (2.45). Also, inEquations. (3.13) and (3.14), 𝜇 is the 𝑛 × 𝑛 diagonal matrix of wave parameters 𝜇𝑗 for theright-going wave modes, i.e., 𝜇 = diag𝜇𝑗𝑗=1,...,𝑛 where |𝜇𝑗| < 1 and ‖𝜇‖2 < 1. Finally,Q = [𝑄1 · · ·𝑄𝑛]𝑇 and Q⋆ = [𝑄⋆
1 · · ·𝑄⋆𝑛]𝑇 are 𝑛 × 1 vectors of wave amplitudes defined at
the left and right sides of the whole periodic structure, respectively. As a result, by consideringEquations. (3.13) and (3.14) as well as the boundary conditions at the left and right sides of the
45
Figure 3.2: Schematics of the FE model of a fluid-filled PC system subject to elastic/acousticforces; wave amplitudes related to left- and right-going waves; FE model of a typically subsys-tem.
whole structure, a well-conditioned wave-based matrix equation can be expressed as follows:[I𝑛 Φ−1
F Φ⋆F𝜇
𝑁
Φ⋆−1q Φq𝜇
𝑁 I𝑛
](Q
Q⋆
)=
(−Φ−1
F F0
Φ⋆−1q q0
). (3.17)
The matrix formulation (3.17) is of the form 𝒜𝒬 = ℱ , were 𝒬 = [Q𝑇 Q⋆𝑇 ]𝑇 and ℱstands for the vector of excitations. Solving the matrix equation yields the vectors of waveamplitudes as 𝒬 = 𝒜−1ℱ . The displacement/force vectors at any substructure boundary followfrom Equations (3.13) and (3.14).
3.2 SE method for periodic Levy plate
The SE method for a Levy-type plate may be derived in analogy with the Kirchhoff-Loveplate ( Lee and Lee, 1999; Doyle, 1997; Campos and Dos Santos, 2015). In the frequencydomain, the governing equation of a Kirchhoff-Love plate is given by (Arruda et al., 2004):
𝐷∇2∇2𝑤(𝑥,𝑦) − 𝜔2𝜌ℎ𝑤(𝑥,𝑦) = 𝐹 (𝑥,𝑦), (3.18)
where 𝐷 = 𝐸ℎ3/12(1 − 𝜈2) is the bending rigidity, ℎ is the thickness, 𝜈 is the Poisson ratio, 𝜌is the density, 𝑤(𝑥,𝑦) is the transverse displacement and 𝐹 (𝑥,𝑦) is the applied surface load. Byconsidering the Levy-plate theory, the transverse displacement of a flat plate which is simply
46
supported along two parallel edges (𝑥−direction) is expressed as follows:
𝑤(𝑥,𝑦) =∞∑𝑛=1
[A𝑛𝑒
−i𝛽1𝑛𝑥 + B𝑛𝑒i𝛽1𝑛𝑥 + C𝑛𝑒
−𝛽2𝑛𝑥 + D𝑛𝑒𝛽2𝑛𝑥]sin(𝛽𝑦𝑛𝑦), (3.19)
where 𝛽1𝑛 and 𝛽2𝑛 are wavenumbers:
𝛽1𝑛 =√𝛽2𝑝 − 𝛽2
𝑦𝑛 , 𝛽2𝑛 =√𝛽2𝑝 + 𝛽2
𝑦𝑛, (3.20)
where𝛽𝑝 = (𝜔2𝜌ℎ/𝐷)1/4 , 𝛽𝑦𝑛 =
𝑛𝜋
𝐿𝑦
. (3.21)
The coefficients A𝑛, B𝑛, C𝑛 and D𝑛 in Equation (3.19) are determined from the boundaryconditions at the left and right ends of the plate (Arruda et al., 2004). Consider for instance asimply-supported strip of length 𝑑s, it becomes possible to link the shearing forces and bendingmoments on the sides 𝑥 = 0 and 𝑥 = 𝑑s to the displacements and slopes on the same sides. Thisyields the dynamic stiffness matrix of the strip (Arruda et al., 2004;Lee and Lee, 1999):
DSE = [B] [A ]−1 , (3.22)
where
B = 𝐷
⎡⎢⎢⎢⎢⎣𝛼1 𝛼1 𝛼1 𝛼1
𝛾1 𝛾1 𝛾2 𝛾2
−𝛼1𝑒−i𝛽1𝑑s 𝛼1𝑒
i𝛽1𝑑s −𝛼2𝑒−𝛽2𝑑s 𝛼2𝑒
𝛽2𝑑s
−𝛾1𝑒−i𝛽1𝑑s 𝛾1𝑒
i𝛽1𝑑s −𝛾2𝑒−𝛽2𝑑s 𝛾2𝑒
𝛽2𝑑s
⎤⎥⎥⎥⎥⎦ , (3.23)
A =
⎡⎢⎢⎢⎢⎣1 1 1 1
−i𝛽1 i𝛽1 −𝛽2 𝛽2
𝑒−i𝛽1𝑑s 𝑒i𝛽1𝑑s 𝑒−𝛽2𝑑s 𝑒𝛽2𝑑s
−i𝛽1𝑒−i𝛽1𝑑s i𝛽1𝑒
i𝛽1𝑑s −𝛽2𝑒−𝛽2𝑑s 𝛽2𝑒
𝛽2𝑑s
⎤⎥⎥⎥⎥⎦ . (3.24)
Here, 𝛼1, 𝛼2, 𝛾1 and 𝛾2 are four coefficients given by:
𝛼1 = i𝛽31 +i𝛽2
𝑦(2− 𝜈)𝛽1, 𝛼2 = −𝛽32 +𝛽2
𝑦(2− 𝜈)𝛽2, 𝛾1 = 𝛽21 + 𝜈𝛽2
𝑦 , 𝛾2 = −𝛽22 + 𝜈𝛽2
𝑦 .
(3.25)By considering the dynamic stiffness matrix DSE (Equation (3.22)), the transfer matrix S
of the strip can be derived (Goldstein et al., 2010):
S =
[−(DSE
LR)−1DSE
LL (DSELR)
−1
−DSERL + DSE
RR(DSELR)
−1DSELL −DSE
RR(DSELR)
−1
]. (3.26)
The transfer matrix of a substructure, e.g., which is made up of three strips with different mate-
47
rial properties (Mat1− Mat2− Mat1), follows as:
S = SMat1SMat2SMat1. (3.27)
In the same way as for the WFE method, the eigensolutions of the transfer matrix S of thesubstructure (three strips) — say, 𝜇SE
𝑗 — are calculated, which provides the wavenumbers 𝛽SE𝑗 :
𝛽SE𝑗 = −
ln(𝜇SE𝑗 )
i𝑑, (3.28)
where 𝑑 = 𝑑s1 + 𝑑s2 + 𝑑s3 is the substructure length. It is worth to mention that in this case, it isnot necessary to use of S + S−1 transformation technique because the symplectic property ofthe eigenvalue problem is inherent to the WFE method.
A code implemented in MATLAB® for the numerical analysis of wave propagation inLevy plate by means of the SE method was based on Arruda et al. (2004). This script wasadapted for phononic crystal case.
48
4 NUMERICAL RESULTS
In this chapter, numerical experiments are carried out to highlight the relevance of theWFE method in terms of accuracy and efficiency, in comparison with conventional methods.The proposed WFE-based approach is validated by analyzing the vibroacoustic and dynamicresponse of periodic systems of finite length and subject to harmonic mechanic/acoustic loads.Dispersion curves are assessed and compared with different analytical theories. Also, band gapsgenerated by Bragg scattering effect are calculated with the WFE method through different testcases.
4.1 FE model validation
The first test case refers to validation of the FE models implemented in this works. Theyare described in Section ( 2.1). The results were compared with commercial software ANSYS®
by mean of the modal analysis through natural frequencies.
4.1.1 Structural model
To check the accuracy of the FE computational code, which was implemented inMATLAB®, several simulations were performed in the form of static and dynamic analysis.Firstly, the analysis of the Kirchhoff-Love plate with different boundary conditions was per-formed and compared with analytical solutions (Blevins, 2001). These results are not presentedin this section. Here, the validation of this element is represented in the cylindrical shell model,which uses the plate model as the basis for its formulation.
Consider a cylindrical shell modeled by means of 2D rectangular thin flat shell elementsand that is meshed with 240 elements as shown in Figure ( 4.1). In the ANSYS® model theSHELL63 element was used. The geometrical properties of the model are: length 𝑑 = 0.15 m; aradius of the cylindrical shell 𝑅S = 0.15 m and thickness ℎ = 0.0025 m. The structure is madeof steel with Young's modulus 𝐸S = 2.1 × 1011 Pa, density 𝜌S = 7850 kg/m3 and Poisson’sratio 𝜈S = 0.3. Two boundary conditions were applied: free-free and clamped-clamped. For thefree-free case the rigid body frequencies are not presented.
Table 4.1 shows the comparison of first seven natural frequencies of the shell obtained bythe ANSYS® and the present FE implementation. The differences in the frequencies betweenthe two programs are very small. They are less than 1.5 %. The results are reliable and accurate.The convergence of the solutions validates the cylindrical shell model by the assembly of flatelements.
49
Figure 4.1: FE mesh of a cylindrical shell
Table 4.1: Comparison of values of the natural frequencies for a cylindrical shell
Natural Frequency [Hz]Free-Free Clamped-Clamped
MATLAB® ANSYS® Relative Error [%] MATLAB® ANSYS® Relative Error [%]73.91 73.82 0.081 2100.69 2107.5 0.323
133.89 134.48 0.436 2111.08 2117.8 0.317209.18 208.94 0.115 2164.77 2174.9 0.466333.46 336.42 0.876 2218.69 2222.1 0.153401.35 400.69 0.165 2372.16 2385.3 0.551562.79 569.50 1.178 2378.98 2391.9 0.540649.76 648.23 0.236 2515.14 2515.5 0.014
4.1.2 Fluid model
Consider now a cylindrical cavity modeled by means of 3D hexahedral acoustic elementsand that is meshed with 792 elements as shown in Figure ( 4.2). In the ANSYS® model theFLUID30 element was used. The geometrical properties of the model are: length 𝑑 = 0.15 m;radius of the cavity 𝑅F = 0.15 m. The fluid is air with a density of 𝜌F = 1.21 kg/m3 and a speedof sound of 𝑐F = 343 m/s.
Table 4.2 shows the comparison of first seven natural frequencies of the acoustic cavityobtained by the ANSYS® and the present FE implementation. The differences in the frequenciesbetween the two programs are very small. They are of the order of 10−3 %. The comparisonsshow that the implementation is correct and its results are reliable and accurate.
50
Figure 4.2: FE mesh of a cylindrical cavity
Table 4.2: Comparison of values of the natural frequencies for an acoustic cavity
Natural Frequency [Hz]MATLAB® ANSYS® Relative Error [%]
675.05 675.05 ~01124.00 1124.00 ~01128.70 1128.70 ~01156.43 1156.40 0.0031339.04 1339.00 0.0031417.93 1417.90 0.0021565.48 1565.50 0.001
4.1.3 Coupled fluid-structure model
Figure (4.3) shows a coupled model of fluid-filled cylindrical shell implemented. InANSYS® model was used SHELL63 and FLUID30 elements with the application of the FSI
command. The shell surface is made of steel and it is meshed with 96 elements. The shell isfully free at its boundaries. This shell is filled with air that is meshed with 720 elements. Herethere are still 96 interface elements. Coupled natural frequencies are obtained. The rigid bodyfrequencies are not presented.
Table 4.3 shows the comparison of first seven natural frequencies of the elasto-acousticsystem obtained by the ANSYS® and the present FE implementation. The differences in theresults by the two programs are within 2 %. From this table, it is evident that the results agreevery well with each other.
51
Figure 4.3: FE mesh of a fluid-filled cylindrical shell
Table 4.3: Comparison of values of the natural frequencies for a fluid-filled cylindrical shell
Natural Frequency [Hz]MATLAB® ANSYS® Relative Error [%]
1.67 1.65 1.2313.03 3.05 0.7454.74 4.70 0.9127.54 7.65 1.3969.11 9.05 0.737
12.75 13.01 2.03714.77 14.67 0.664
4.2 Dispersion curves
4.2.1 Levy-type plate
Consider a simply-supported plate made up of identical substructures which are meshedby means of 2D rectangular plate elements with four nodes and three DOFs per node (onedisplacement and two rotations) as shown in Figure (4.4). This example is proposed to comparethe WFE method with the SE method. The geometrical properties of the related substructureare: length of 𝑑 = 0.01 m; width of 𝐿𝑦 = 0.2 m and thickness of ℎ = 0.001 m. The materialis considered with Young’s modulus 𝐸 = 193 GPa, density 𝜌 = 8030 kg/m3, Poisson’s ratio𝜈 = 0.27 and damping loss factor 𝜂 = 0.001.
The dispersion curves — i.e., the frequency evolution of the wavenumbers 𝛽𝑗 — for thewaves in the homogeneous plate are shown in Figure 4.5. The WFE solutions are plotted (solidlines) along with the SE solutions (circles). In the frequency range analyzed, 2 wave modes cut-on. The imaginary values of the wavenumbers are represented as negative values (Doyle, 1997).It can be seen that the WFE solutions rigorously match the reference SE results, hence givingcredit to the WFE method. To achieve the WFE modeling, attention has been paid on discretiz-ing the substructure with a sufficient number of elements to accurately capture the analytical
52
x
zy
Substructure
d
Wave Propagation
Figure 4.4: FE models of the plate.
waves occurring in an equivalent infinite Kirchhoff-Love plate. In this sense, the WFE methodcan be proven highly accurate to compute the waves in homogeneous plates, and more generallyin periodic structures.
Frequency [Hz]
0 50 100 150 200 250 300
Wavenum
ber β
[m
-1]
-60
-40
-20
0
20
40
Figure 4.5: Dispersion curves for a Levy plate: WFE solutions (blue and green solid lines); SEsolutions (black and red circles).
4.2.2 Cylindrical shell
Consider now a homogeneous cylindrical shell composed of identical substructures whichare meshed with 2D rectangular thin flat shell elements as shown in Figure (4.6). A homoge-neous cylindrical shell is analyzed which is made of stainless steel. The geometrical propertiesof a substructure are: length of 𝑑 = 0.015 m, radius of 𝑅 = 0.05 m and thickness of ℎ = 0.0025
m. The material is considered with Young’s modulus 𝐸 = 193 GPa, density 𝜌 = 8030 kg/m3,Poisson’s ratio 𝜈 = 0.27 and damping loss factor 𝜂 = 0.001.
53
Substructure
Wave Propagation
zy
x
d
Figure 4.6: FE models of the cylindrical shell.
The dispersion curves of the structure are computed with the WFE method and comparedto the Flügge shell theory (Liu et al., 2009) (see Figure 4.7). Here, the parameters 𝛽𝑗𝑅 areplotted as functions of the non-dimensional frequency Ω = 𝜔𝑅/𝑐𝐿 where 𝑐𝐿 is the longitudi-nal wave speed (Liu et al., 2009). The WFE solutions (solid lines) are plotted along with theanalytical solutions (dots and circles) that concern the “breathing modes” (zeroth order Besselfunction for fluid, 𝑠 = 0) and “beam modes” (first-order Bessel function, 𝑠 = 1). In this case,again, it is seen that the WFE method is rigorously similar to the analytical solution.
Figure 4.7: Dispersion curves for a homogeneous cylindrical shell: WFE solutions ( ); ana-lytical solutions for 𝑠 = 0 (. . . ); analytical solutions for 𝑠 = 1 (∘ ∘ ∘).
Note that the branch 1 corresponds to a complex mode, in which its real and imaginaryparts occurs of the same order in a specific frequency range. For Ω <1, this branch is associatedwith the solution for the ring mode. At Ω ≈ 0.93, it seems that branch 1 bifurcates to twopurely imaginary wavenumbers and for Ω ≥ 1 it becomes a propagating mode. However, somedifferences occur between both solutions at around the bifurcations. These errors around the
54
bifurcation points can be associated, especially, with the approximations in the WFE modeling(e.g., FE mesh and geometrical approximation). This behavior has been highlighted by Waki(2007).
Branch 2 corresponds to the beam type motion. Branch 3 is associated with the longitu-dinal mode for low frequencies, whereas branch 4 rigorously match with the analytical solutionfor the torsional mode. Also, branch 5 cut-on at Ω ≈ 0.57, where from this point it heads forthe torsional motion.
For Ω <1, branch 6 behave in a complex manner until Ω ≈ 1 for which the imaginary partof this complex bifurcates into a pair of evanescent waves with pure imaginary wavenumbersand for Ω >1.43 it is associated with an axisymmetric propagating mode.
4.2.3 Fluid-filled cylindrical shell
In this section, the WFE method is applied to describe the elasto-acoustic wave modepropagation in a fluid-filled cylindrical shell. A subsystem is considered as shown in Figure(4.8), which is meshed with 866 DOFs. The number of DOFs on the left/right cross sectionis 𝑛 = 433, which is split into 𝑛S = 192 DOFs for the structure part and 𝑛F = 241 DOFsfor the fluid part. In this case, the subsystem does not contain internal DOFs. The geometricalproperties of the subsystem are: length 𝑑 = 0.005 m; radius of the cylindrical shell 𝑅S = 0.05
m, thickness of the cylindrical shell ℎ = 0.0025 m where ℎ/𝑅S = 0.05. Regarding the materialproperties, the cylindrical shell is made of steel with the following characteristics: Young'smodulus 𝐸S = 2.1 × 1011 Pa, density 𝜌S = 7850 kg/m3, Poisson’s ratio 𝜈S = 0.3 and dampingloss factor 𝜂S = 0.01. Also, the internal fluid is air, with a density of 𝜌F = 1.21 kg/m3 and aspeed of sound of 𝑐F = 343 m/s.
Figure 4.8: FE models of the elasto-acoustic subsystem
The resulting dispersion curves, i.e., the frequency evolution of the wavenumbers 𝛽𝑗
(Ω ↦→ 𝛽𝑗𝑅F), are presented in Figure (4.9). The results are compared with the analytical so-
lutions issued from the Donnell-Mushtari theory (Fuller and Fahy, 1982). Here, the parameters
55
𝛽𝑗𝑅F are plotted as functions of the non-dimensional frequency Ω = 𝜔𝑅F/𝑐𝐿 where 𝑐𝐿 is
the phase speed of compressional waves in an equivalent Donnell-Mushtari shell (Fuller andFahy, 1982). For the sake of clarity, only 7 modes are displayed over a frequency band ofΩ = [0.1,2]. In this range, these curves illustrate elasto-acoustic propagating modes and modescut-on. It can be noticed that the dispersion characteristics are more complicated when the shellis filled with acoustical fluid, compared with “in vacuo” case. The WFE solutions (solid lines)are plotted along with the analytical solutions (dots and circles) that concern the “breathingmodes” (zeroth order Bessel function for fluid, 𝑠 = 0) and “beam modes” (first-order Besselfunction, 𝑠 = 1).
Figure 4.9: Dispersion curves: WFE solutions ( ); analytical solutions for 𝑠 = 0 (. . . );analytical solutions for 𝑠 = 1 (∘ ∘ ∘).
As it can be seen, the numerical results obtained from the WFE method are in good agree-ment with the analytical theory. The numerical solution accurately reflects the trend of analyticalmodes in whole Ω. The branch associate with torsional elastic mode rigorously matches the an-alytical solution. Numerical solutions for branches that are close to the solution for ring elasticmode cuts off at Ω ≈ 1. It is interesting to note that, in comparison with Figure 4.7, real andimaginary parts of these branches is not so close to the analytical solution. This difference canbe explained by the presence of the effect of fluid loading which is not taken into considera-tion in the analytical model (Mencik and Ichchou, 2007). From Ω ≥ 1, the motion of the shellinteracts strongly with acoustic medium (Fuller and Fahy, 1982).
Beyond that, in the Donnell-Mushtari shell model, the shearing effects and rotary inertiaeffects are not taken into account, which is not the case with the proposed WFE method whereshell elements that incorporate both membrane and bending motions are considered. Hence,
56
several differences occur between the numerical and analytical solutions. These disagreementscan be avoided by decrease frequency step so as to make tracking criterion more efficient (seeSection 2.2). In addition, internal DOFs can be used in the WFE model in order to minimizesuch divergences, especially at high frequencies. However, these tasks increase computationalcost.
4.3 Forced response of fluid-filled cylindrical shell
Numerical simulations are carried out to illustrate the efficiency of the WFE method forpredicting the vibroacoustic response of fluid-filled shells. Here, a fluid-filled cylindrical shellwith a length of 1 m is considered. The material properties and cross-section geometry are thesame as those described in Section 4.2.3. The whole periodic system is modeled by means of𝑁 = 200 identical subsystems of length 𝑑 = 0.005 m, which are connected along the 𝑧-direction(see Section 4.2.3). Regarding FE model, the shell surface is meshed with 6400 elements, whilethe fluid is meshed with 44,800 elements. In fact, WFE method requires a total number ofDOFs smaller than conventional FE method (see Table 4.4). In the present case, the right endof the whole system is clamped (structure part) and subject to zero acoustic pressure (fluidpart); also, the left end is clamped (structure part) and subject to a uniform acoustic pressure(fluid part, p0 = 0.001 Pa ×1𝑛F) (see Figure ). The displacement and pressure solutions arecomputed with the WFE method and compared with the FE method using MATLAB®. Theresults are investigated over a non-dimensional frequency band of Ω = [0.0055,0.55] which isabout ℬ𝑓 = [100,10000] Hz. The frequency response function (FRF) that concerns the pressureat the center node of the circular section (𝑥 = 0, 𝑦 = 0) at the middle of the pipe (𝑧 = 0.5 m) isplotted as shown in Figure 4.11.
Figure 4.10: FE model of fluid-filled cylindrical shell subject to prescribed pressure.
As it can be seen, the WFE solution completely matches the FE one over the whole fre-quency range. To prove the coherence between the methods, the relative error is accessed as:
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0.1 0.2 0.3 0.4 0.5
Non-dimensional frequency,
0
10
20
30
40
50
60
70
80
Pre
ssu
re [
dB
]
Figure 4.11: Pressure FRF of the fluid-filled cylindrical shell: FE method (. . . ); WFE method ().
‖qmeasWFE ‖2 − ‖qmeas
FE ‖2‖qmeas
FE ‖2(4.1)
where qmeas is the measured vector of nodal pressures over the frequency domain and ‖.‖2denotes the 2–norm. Here, it is about 8.76%, which validates the WFE method.
Spurious numerical errors appear at resonances and anti-resonances, which are howeveronly due to frequency sampling and the fact that the fluid is undamped (in other words, high-pressure levels occur at resonance frequencies which are not necessarily well captured due tofrequency sampling).
The CPU times involved in the WFE and FE methods are highlighted in Table 4.4. Inthis case, it takes around 7,500 s with the WFE method to compute the FRF of the systemover the whole frequency band, against 161,990 s with the FE method. Those simulations havebeen realized with the same numerical platform, say using MATLAB® and a processor Intel®
CoreTM i7-3960X. This yields CPU time savings of 95 % in benefit of the WFE method. As itturns out, the WFE method constitutes an efficient/accurate numerical tool for computing thevibroacoustic responses of fluid-filled cylindrical shells.
Table 4.4: Total number of DOFs and CPU times involved for computing the FRF
Approach used DOFs CPU time [s] Reduction [%]FE method 87,033 161,990
WFE method 866 7,500 95
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4.4 Application in Phononic Crystals
Numerical simulations are carried out to illustrate the capability of the WFE method tocalculate band gaps in PC plates, PC cylindrical shells "in vacuo" and fluid-filled PC cylindri-cal shells with 1D periodic elastic properties. The structures under concern are made of stripswith two different material properties. Band gaps generated by Bragg scattering effect are cal-culated with the WFE method through different test cases. Bragg scattering occurs due to theperiodicity of a material. The waves scattered at the interfaces cause destructive interference,which cancelling the propagating waves. The dispersion curves issued from the WFE method.Also, the vibration attenuation in the frequency response functions (FRFs) of the structures isanalyzed.
4.4.1 Phononic crystal plate
Consider a PC plate that consists of a periodic distribution of strips with two differentmaterial properties (see Figure 4.12), i.e., stainless steel and polyacetal. The choice of thesematerials is based on the differences between their elastic properties, which allows the formationof band gaps at low-and mid-frequencies. Those material properties are listed in Table 4.5.
Table 4.5: Material properties.
Material Stainless Steel PolyacetalYoung’s modulus (GPa) 193 3.3
Density (kg/m3) 8030 1418Poisson ratio 0.27 0.35Loss factor 0.001 0.001
The whole periodic structure is made up of 𝑁 = 4 identical substructures with a lengthof 3𝑑 = 0.3𝑚 m as shown in Figure 4.12. The periodic structure is excited on its left sideby means of a uniform distribution of transverse forces (Figure 4.12). Also, the magnitude ofthe transverse displacement is analyzed at one measurement point on the right side as shown(Figure 4.13). The FRF of the structure is computed over a frequency band of ℬ𝑓 = [0.2,300] Hzalong with the dispersion curves of the bending mode (Figure 4.13). Regarding the dispersioncurves, both WFE and SE solutions are highlighted. Also, considering the Bragg condition (seeSection 3) for 1D PC the Bragg limit is displayed which corresponds to the case 𝛽𝑗 = 𝜋/𝑑 =
10.47 m−1.Figure 4.13 clearly highlights the influence of the periodic distribution of two materi-
als on the formation of Bragg-type band gaps. These occur when the imaginary part of thewavenumber becomes negative (evanescent waves). These band gaps are mostly induced byBragg scattering effect, which is due to the impedance mismatches between the strips. Again,the dispersion curves issued by the WFE method rigorously match the SE solutions over the
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0.1 m
0.1 m
0.1 m
0.2 m
Substructure
Measurement
point
x
z
y
Figure 4.12: The periodic structure with 𝑁 = 4 substructures (left), and FE mesh of a substruc-ture (right).
Figure 4.13: FRF of the PC plate (left); dispersion curves for the bending mode (Right): realpart (blue) and imaginary part (red) of the SE solution; real part (magenta) and imaginary part(green) of the WFE solution; Bragg limit (black line).
whole frequency range. Also, the WFE method can be used to obtain the FRF of the periodicstructure (Figure 4.13). Four band gaps can be observed within the whole frequency range (i.e.,59.1–81.5, 99.3–144.1, 186.9–203.8, and 255.4–300 Hz). The overall position of the band gapsin dispersion curve are in good agreement with those of the FRF calculation. One can see thatthe response levels in the gray areas denoted by the band gaps suffered a decrease, which means
60
that in these frequency ranges, the vibration and wave cannot propagate in the periodic struc-ture. From the result, it is seen that the vibration levels are well attenuated when band gapsoccur, as expected.
4.4.2 Phononic crystal cylindrical shell
Consider now a PC cylindrical shell made up of 𝑁 = 20 substructures as shown in Fig-ure 4.14 . Each substructure is composed of two layers of stainless steel of length 0.01 m (redmaterial) which surround one layer of polyacetal of length 0.03 m (yellow material). The wholeperiodic structure is excited by means of two radial point forces of magnitude 𝐹 = 100 N act-ing in opposite directions on the left cross-section, while it is clamped on the right cross-section(see Figure 4.14). The magnitude of the vertical displacement (𝑦- direction) is analyzed at onemeasurement point at the middle of the cylindrical shell as shown in Figure 4.14. The FRF ofthe structure is computed with the WFE method over a frequency band of ℬ𝑓 = [100,20000] Hzalong with the dispersion curves of two significant modes (Figure 4.15).
Figure 4.14: Periodic structure with 20 substructures (left), and FE mesh of a substructure(right).
Figure 4.15 shows the evolution of band gap effect over the whole frequency band. Inthis case, the Bragg limit gives 𝛽𝑗 = 𝜋/𝑑 = 62.83 m−1. It can be seen that the condition ofBragg limit is respected. In FRF there are clear regions of attenuation, indicating the presenceof the formation of Bragg band gaps. The first region of attenuation spans 2380–2950 Hz. Thisbehavior is not so evident in the dispersion curve. This could be attributed to the fact, thatthis band gap may appear in another mode not shown in the Fig. 4.15, since the cylindricalshell can present a large number of modes. The second band gap spans 3500 Hz until 4800
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Figure 4.15: Frequency response of periodic PC cylindrical shell (left), and dispersion curvesfor two modes in the PC cylindrical shell (right). Real part ( ) and imaginary part ( ) ofthe WFE solution.
Hz. Levels of attenuation reach ≈ 83 dB for this band. Here again, several bad gaps can bewell predicted which agree with the frequency regions over which the vibration levels of thestructure are attenuated (i.e., 6500–8000 Hz, 14100–14800 Hz, and 18000–end frequency rangeHz). The locations of these band gaps are in reasonable agreement with those obtained from thedispersion curves calculations. It is important to note that, in this case, the attenuation is lesspronounced level at zones where the imaginary part of one wave coexists with the real part ofanother. Besides that, the direction of excitation significantly influences the attenuation of thewave that can be observed, since many waves can propagate freely along other directions thatare not excited.
4.4.3 Fluid-filled phononic crystal cylindrical shell
In this section, one aims at showing the coupled acoustic-structural band gap effect inphononic crystal systems that involves shell-based models filled with acoustic fluid. The WFEmethod has been utilized to calculate dispersion diagrams and frequency response functions(FRFs).
The periodic system that is considered here, consists of a cylindrical thin shell with a in-ternal fluid. The shell surface is made of materials with different acoustic impedances as shown
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in Figure 4.16(a). The material characteristics of the elastic cylindrical shell are listed in Ta-ble 4.5. Assume that the shell was filled with air (see Section 4.2.3). This PC-like structureis commonly investigated in much of the literature. Hence, beginning the investigation of suchconventional PC system and its related band gap phenomena provides a good basis for under-standing the more complex PC systems that can be examined in future works. In this case, bandgaps are dependent of the length of the unit cell, generally occurring at high frequencies. Thiseffect is due to the fact that the frequency band where this phenomena occurs is inversely pro-portional to wavelength. Thus, it is difficult to achieve a low-frequency Bragg-type band gapfor small size PCs.
Consider a unit-cell made of strips with two different materials as shown in Figure 4.16(b).The length of the subsystem is 𝑑 = 0.05 m. Here, the total number of DOFs used to discretizethe subsystem is 4763, with 𝑛 = 433 DOFs over the left/right boundary and 𝑛I = 3897 internalDOFs.
(a) (b)
Figure 4.16: Periodic PC system with 20 substructures and subject to the prescribed pressurefield (𝑎) ; FE mesh of a subsystem (𝑏) .
The dispersion curves of the PC system are assessed by means of the WFE methodas shown in Figure (4.17). Here are shown elastic and acoustic modes. It should be no-ticed that the added impedance mismatch yields several band gaps in whole frequency bandℬ𝑓 = [2000,20000]. Bragg band gap formation occurs in frequency ranges where there are onlyevanescent waves. In this case, the Bragg limit gives 𝛽𝑗𝑅
F = (𝜋/𝑑)𝑅F ≈ 3. It can be seen thatthe condition of Bragg limit is respected.
To highlight further those structural-acoustic coupling band gaps, the reduction of thesound levels in PC periodic systems can be achieved through the use of the WFE method. Con-sider a whole system composed by 𝑁 = 20 unit cells, and with boundary conditions at ends,i.e., at the right end left ends of the cylindrical shell is clamped (U = 0). The acoustic part of thewaveguide is subjected to uniform pressure fields (p0 = 0.001 Pa ×1𝑛F) on the left end, whilethe right end is subject to zero acoustic pressure as illustrated in Figure Figure 4.16(a). Here, themagnitude of the acoustic pressure is analyzed at one measurement point at the center node ofthe circular section (𝑥 = 0, 𝑦 = 0) at the middle of the pipe (𝑧 = 0.5 m). Hence, the FRF of thePC system is analyzed which is done by considering a frequency band of [100 Hz,10000Hz],say, by considering a sample of 990 discrete frequencies which are equally spaced with a fre-
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Figure 4.17: Dispersion curves of the waves in fluid-filled PC cylindrical shell. Real part of𝛽𝑗𝑅
F ( ) and imaginary part of 𝛽𝑗𝑅F ( ) of the WFE solution.
quency step of 10 Hz.Figure 4.18 compares FRFs obtained of the PC system and of the homogeneous system
presented in Section 4.3. It is possible to see the presence of clear regions of pressure attenuation(grey areas), indicating the presence of the formation of Bragg structural-acoustic coupled bandgaps. The first region of attenuation spans 4120–4480 Hz and the second 5580 Hz until 6210Hz. Beyond that, attenuation zones are also open at 6620–6950 Hz and 7340–7970 Hz.
Figure 4.18: Frequency response of periodic fluid-filled cylindrical shell: Homogeneous system( ) and PC system ( ).
It can be observed that at lower frequencies the structural-acoustic coupling is weak. This
64
is highlighted by reasonable agreement between FRFs until 1200 Hz. As the frequency is in-creased the effect of the shell motion become more accentuated in the acoustic field. This effectallows Bragg band gap to occur in acoustic waves.
To highlight the effect of fluid loading in the band gap behavior, one additional numericalexperiment has been carried out. Consider a PC cylindrical shell interacting with an internalfluid and subjected to vertical point forces of magnitude 𝐹 = 100𝑁 acting in opposite directionson the cross-section as illustrated in Figure 4.19. The FRF of the PC system is displayed inFigure 4.20. Here, the magnitude of the vertical displacement is assessed at a measurement pointlocated 0.5 m far from the excitation sources, along the 𝑧−direction. The frequency response,in this case, is compared to the solution for the case PC shell "in vacuo" (Section 4.4.2).
Figure 4.19: Periodic PC system with 20 substructures and subject to prescribed forces.
Figure 4.20: Frequency response of PC cylindrical shell "in vacuo" ( ) and PC cylindricalshell with internal fluid ( ).
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It can be seen that the presence of acoustic field significantly affects the band gap phe-nomena. The band gap regions at 3510–4730 Hz and 6600–7750 Hz practically disappearedwhen internal fluid is considered. This effect occurs because the propagating acoustic wave ac-companies the elastic wave in each coupled element, i.e., the coupling with the acoustic modeaffects the behavior of the elastic modes in order to reduce the band gap effect. However, thefirst band gap (2380-3000Hz) remains practically the same. This effect is due to the fact that thelongitudinal elastic mode and duct acoustic mode, which govern the wave propagation at low-frequencies, do not contribute strongly to the fluid and the shell motion, respectively (Sorokinand Ershova, 2004).
It is worth pointing out that the main achievements of this work involve a passive vi-bration/noise control of periodic systems through the use of numerical tools based on wavesmodes to model complex structures (e.g., fluid-filled cylindrical shells). In other words, theWFE method besides providing the same level of accuracy as the FE method, also improves thecomputational efficiency by the modeling of just a single substructure. For instance, consideringthe present case, the corresponding FE model of the system would have involved 87,033 DOFs,compared to 4763 DOFs with the WFE method. This fact greatly reduces the computationalmemory in the solution of the problem.
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5 CONCLUSION
This work presented the application of the WFE method to analyze the numerical wavepropagation in fluid-filled phononic crystal cylindrical shells, for the purpose of investigatingthis approach in the design of periodic systems exhibiting band gaps. For this task, a finiteelement code to analyze coupled structural-acoustic systems involving flat shell elements wasimplemented in MATLAB® and validated with the commercial software ANSYS®. The FEmodel was then used together with WFE method to provide elasto-acoustic wave modes andanalyzed the wave description in periodic systems.
Simulated results calculated by WFE are presented and verified with the SE method andanalytical theories where it demonstrates the validation of computational codes developed. Theaccuracy of the WFE method has been clearly demonstrated when compared to these analyticaltheories. The dispersion diagrams for Levy-type plates, cylindrical shells and fluid–filled cylin-drical shells have been computed with accurate precision. These results shown how importantit is to discretizing the substructure with a sufficient number of elements to accurately capturethe waves occurring along periodic systems.
In addition, the WFE method has been applied to compute the vibroacoustic responsesof fluid-filled cylindrical shells of finite length whose left and right boundaries are subject toprescribed pressures. The WFE-based procedure called wave mode expansion was applied tocompute the forced responses in theses systems. This method constitutes an efficient meansto compute the FRFs in periodic systems. The results showed to be accurate compared to FEsolutions. As a second analyze, the related CPU time saving was more than 90 % in comparisonwith the conventional FE analysis. Hence, the WFE method has been proven to be efficient tocompute their vibroacoustic responses, especially for periodic systems with many degrees offreedom (DOFs) like those involving shell elements and fluid-structure interaction.
It was also studied the elastic wave propagation in 1D elastic PC structures, i.e., made ofdifferent material properties which are periodically distributed along their length. Numerical ex-periments have been carried out which clearly highlight the relevance of the proposed approachfor predicting band gaps. The examples, proposed here, were validated given results comparingthem when possible with SE method. The dispersion diagrams for PC Levy-type plates werevalidated in comparison with reference SE solutions. In particular, the WFE method was able toprecisely identify the locations and widths of band gaps which occur in the PC plate. Moreover,the application of the WFE approach in PC cylindrical shells showed that this method can beused in shell-like periodic structures, providing interesting results for passive vibration controlin these systems.
Finally, the WFE method has been proposed for modeling periodic multi-layered cylindri-cal shells with internal fluid. In particular, it could be applied to improve the design of periodicsystems so as to magnify their band gap mechanisms. An investigation of coupled acoustic-structural band gap phenomena in this system has also been carry out. The result showed a re-
67
gion of significant attenuation of acoustic pressure level, especially in higher frequencies wherethe coupling becomes more accentuated. In addition, could be observed that the presence ofa loading fluid significantly affects the band gap characteristics, in certain situations, reducingtheir effects. Hence, the potential of the use of WFE method in the study of passive vibrationand sound control in fluid-filled periodic structures is clearly highlighted. In this way, given thecomplexity of the subject, this work should be interpreted as the first step in a project that, inorder to obtain results that are closer to industrial applications.
5.1 Futures works
In this section suggestions for further research are proposed.
∙ Improve the computational performance of the method proposed in this work by meansof using parallel programming and code optimization.
∙ Investigate a WFE-based model reduction using the dynamic substructuring technique todescribe the vibroacoustic behavior of fluid-filled periodic structures. This is motivatedby the need to reduce even more CPU time involved in the computation of the wave prop-agation along this kind of periodic systems, since for modeling of complex metamaterialsa large number of DOFs is required.
∙ Verify experimentally the results obtained for the fluid-filled phononic crystal cylindricalshell.
∙ Perform a broad investigation on physical phenomena that involve the coupled structural-acoustic band gaps in PC systems.
∙ The use of WFE-based approaches to designing acoustic metamaterials equipped witha local resonators periodically distributed in fluid-filled pipes. This is motivated by thefact in PC systems Bragg’s law implies that is difficult to achieve a Bragg band gap atlow-frequency range.
∙ Expand the WFE-based approach to analyze the wave propagation along waveguides con-nected in arbitrary ways by elastic-acoustic junctions. This is motivated by the need toanalyze industrial piping systems composed of periodic straight pipes and curved pipes(elbows or bent pipes).
∙ Investigate the use of WFE method for structural-acoustic analysis of systems under struc-tural damage. This can be used in industrial applications for damage detection in aircraftfuselages and gas pipelines.
∙ Investigate the use of WFE-based approach in topology optimization of periodic elastic-acoustic systems such as mufflers and duct with a periodic Helmholtz resonators. Thisallows to designing optimal structures that improve the band gap mechanisms.
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5.2 List of Publications
5.2.1 Papers in Conferences Proceedings
∘ R. W. O. Sousa, J.-M. Mencik and J. M. C. Dos Santos, “Modeling Fluid-FilledCylindrical Shells with Wave Finite Elements”, Proceedings of the XXXVIII IberianLatin-American Congress on Computational Methods in Engineering (CILAMCE),November 5-8, Florianópolis, SC, Brazil, 2017.
∘ R. W. O. Sousa, J.-M. Mencik and J. M. C. Dos Santos, “Band gaps in plates and cylindri-cal shells with 1D periodic elastic properties”, 24th International Congress of MechanicalEngineering (COBEM), December 3-8, Curitiba, PR, Brazil, 2017.
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ANEXO A – Stiffness and mass matrices for a Kirchhoff-Loveplate element
A rectangular element with four node is considered. Each node has three degrees of free-dom: displacement normal to the plane of the plate and two rotations in the 𝑥 and 𝑦 directions.More details about formulation can be found in Szilard (2004). Here, 𝐸, 𝜌 and 𝜈 are the Young’smodulus, density and Poisson’s ratio, respectively. Beyond that ℎ is the plate thickness; 𝑙𝑥 is theelement length and 𝑙𝑦 is the element width. In the following 𝑟 = 𝑙𝑥/𝑙𝑦.
The stiffness matrix is:
Kb𝑒 =
𝐸ℎ
180(1 − 𝜈2)
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
𝐹 𝐺 −𝐻 𝐿 −𝑀 −𝑁 𝑂 −𝑃 −𝑄 𝐼 𝐽 −𝐾
𝐺 𝑅 −𝑍 −𝑀 𝑇 0 𝑃 𝑈 0 −𝐽 𝑆 0
−𝐻 −𝑍 𝑉 𝑁 0 𝑋 𝑄 0 𝑌 −𝐾 0 𝑊
𝐿 −𝑀 𝑁 𝐹 𝐺 𝐻 𝐼 𝐽 𝐾 𝑂 −𝑃 𝑄
−𝑀 𝑇 0 𝐺 𝑅 𝑍 −𝐽 𝑆 0 𝑃 𝑈 0
−𝑁 0 𝑋 𝐻 𝑍 𝑉 𝐾 0 𝑊 −𝑄 0 𝑌
𝑂 𝑃 𝑄 𝐼 −𝐽 𝐾 𝐹 −𝐺 𝐻 𝐿 𝑀 𝑁
−𝑃 𝑈 0 𝐽 𝑆 0 −𝐺 𝑅 −𝑍 𝑀 𝑇 0
−𝑄 0 𝑌 𝐾 0 𝑊 𝐻 −𝑍 𝑉 −𝑁 0 𝑋
𝐼 −𝐽 −𝐾 𝑂 𝑃 −𝑄 𝐿 𝑀 −𝑁 𝐹 −𝐺 −𝐻
𝐽 𝑆 0 −𝑃 𝑈 0 𝑀 𝑇 0 −𝐺 𝑅 𝑍
−𝐾 0 𝑊 𝑄 0 𝑌 𝑁 0 𝑋 −𝐻 𝑍 𝑉
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(A.1)
where,
𝐹 = (42 − 12𝜈 + 60𝑟2 + 60𝑟−2)ℎ2/(𝑙𝑥𝑙𝑦) , 𝑄 = (15𝑟−1 − 3(1 − 𝜈)𝑟)ℎ2/𝑙𝑥, (A.2)
𝐺 = (30 + 3𝑟−1 + 12𝜈𝑟−1)ℎ2/𝑙𝑦 , 𝑅 = (20𝑟 + 4(1 − 𝜈)𝑟−1)ℎ2, (A.3)
𝐻 = (30𝑟−1 + 3𝑟 + 12𝜈𝑟)ℎ2/𝑙𝑥 , 𝑆 = (10𝑟 − (1 − 𝜈)𝑟−1)ℎ2, (A.4)
𝐼 = (−42 + 12𝜈 − 60𝑟2 + 30𝑟−2)ℎ2/𝑙𝑥𝑙𝑦 , 𝑇 = (10𝑟 − 4(1 − 𝜈)𝑟−1)ℎ2, (A.5)
𝐽 = (30𝑟 + 3(1 − 𝜈)𝑟−1)ℎ2/𝑙𝑦 , 𝑈 = (5𝑟 + (1 − 𝜈)𝑟−1)ℎ2, (A.6)
𝐾 = (15𝑟−1 − 3𝑟 − 12𝜈𝑟)ℎ2/𝑙𝑥 , 𝑉 = (20𝑟−1 + 4(1 − 𝜈)𝑟)ℎ2, (A.7)
𝐿 = (−42 + 12𝜈 − 60𝑟−2 + 30𝑟2)ℎ2/𝑙𝑥𝑙𝑦 , 𝑊 = (10𝑟−1 − 4(1 − 𝜈)𝑟)ℎ2, (A.8)
𝑀 = (−15𝑟 + 3𝑟−1 + 12𝜈𝑟−1)ℎ2/𝑙𝑦 , 𝑋 = (10𝑟−1 − (1 − 𝜈)𝑟)ℎ2, (A.9)
𝑁 = (30𝑟−1 + 3(1 − 𝜈)𝑟)ℎ2/𝑙𝑥 , 𝑌 = (5𝑟−1 + (1 − 𝜈)𝑟)ℎ2, (A.10)
𝑂 = (42 − 12𝜈 − 30𝑟2 − 30𝑟−2)ℎ2/𝑙𝑥𝑙𝑦 , 𝑍 = (15𝜈)ℎ2, (A.11)
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𝑃 = (−15𝑟 + 3(1 − 𝜈)𝑟−1)ℎ2/𝑙𝑦. (A.12)
The mass matrix is:Mb
𝑒 = QMQ (A.13)
where
Q =
⎡⎢⎢⎢⎢⎣𝛽 0 0 0
0 𝛽 0 0
0 0 𝛽 0
0 0 0 𝛽
⎤⎥⎥⎥⎥⎦ , 𝛽 =
⎡⎢⎣ 1 0 0
0 𝑙𝑥 0
0 0 𝑙𝑦
⎤⎥⎦ , (A.14)
and
M =𝜌ℎ𝑙𝑥𝑙𝑦25200
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
3454 461 −461 1226 199 274 394 −116 116 1226 −274 −199
461 80 −63 199 40 42 116 −30 28 274 −60 −42
−461 −63 −80 −274 −42 −60 −116 28 −30 −199 42 40
1226 199 −274 3454 461 1226 −274 199 394 −116 −116 −116
199 40 42 461 80 63 274 −60 42 116 −30 −28
−274 42 −60 461 63 80 199 −42 40 116 −28 −30
394 116 −116 1226 274 199 3454 −461 461 1226 −199 −274
−116 −30 28 −274 −60 −42 −461 80 −63 −199 40 42
116 28 −30 199 42 40 461 −63 80 274 −42 −60
1226 274 −199 394 116 116 1226 −199 274 3454 −461 −461
−274 −60 42 −116 −30 −28 −199 40 −42 −461 80 63
−199 −42 40 −116 −28 −30 −274 42 −60 −461 63 80
.
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(A.15)