-Ringraziamenti
Ringrazio mia madre e mio padre per avere permesso assieme a me tutto questo.
Questa tesi e questa laurea sono il coronamento di un percorso durato cinque anni. Nei primi tre anni ho studiato la matematica e imparato i modi e il linguaggio della scienza piu’ astratta. La sera tornavo col treno da Milano e mi dedicavo alla musica. Tutti intorno a me sanno che mi interesso di questo principalmente: il suono, il suo significato, come manipolarlo, la musica. Cosi’ un giorno mio fratello, anche lui laureato al Politecnico, mi ha informato dell’esistenza di un percorso di studi magistrali dedicato a queste cose. Ringrazio mio fratello per questo: è stato un ottimo consiglio (quasi come quando mi ha consigliato di ascoltare Yamanadu Costa (quasi)).
Ringrazio Raffaele per aver seguito questo progetto nel modo in cui lo ha fatto. Ne ha limato la forma finale, ha sopperito alla naturale pigrizia della mia precisione con la attitudine solida di chi forse diventerà uno scienziato. Lui e il professor Corradi hanno più volte ridirezionato questo progetto con delicatezza e assennatezza e li ringrazio per questo.
Ringrazio alcune persone che hanno frequentemente studiato con me nei due anni precedenti alla tesi. Stefano, che ancora aspetta i suoi stickers.
Nicola, che l’anno scorso ha iniziato a suonare la tromba con successo. Bertan, a cui auguro di trovare una casa, tra la Turchia e l’Italia. Antony, take me to the magic of the moments on a glory night.
Luca, Matteo, Francesco, Carlo, Luca. . . ci siamo tutti aiutati! Grazie. Ringrazio Chiara e lei sa perché (non è perché è la mia ragazza).
Se faccio un altro nome scende una lacrima di nostalgia ed è meglio non far traboccare questo vaso.
Preferirei appoggiarlo e guardarlo per bene, ma la mia mente a quanto pare non ha scaffali liberi per lui. Lo terró in mano per un giorno e una notte, in attesa di consegnare la versione definitiva di questo lavoro, poi probabilmente sparparglieró l’acqua a terra: cosi’ faccio sempre.
E’ un contenitore piu pesante di quello che ho lanciato in aria gioiosamente alla fine del liceo. E’ un grosso vaso, arancio come il tramonto e ruvido come la voce di Don Henley:
“’Cause there is no more new frontier, we have got to make it here”
Abstract (English)
A comparison of different models for the simulation of sound radiated from a thin vibrating structure is presented. A rectangular aluminium plate vibrating in the midst of an infinite rigid baffle is studied as a benchmark case. In this work two models are developed for this situation using FEM and BEM. This models have great flexibility and in fact any shell like structure could be threated in this way. For example many musical instruments feature a shell structure which is vibrating when the instrument is played (the soundboard of a piano is a case) . Its properties influence greatly for the way sound diffuses in space and for the reproduced sound itself. The plate structure could also be a small part of a bigger model, for example of an industrial machine emitting a noise. The FE models here implemented simulate both the plate and the acoustic region in different ways. The plate was modeled as a Shell (thus approximated as a 2D structure) in one model, and was instead represented as the real 3D object in the other. FEM is adopted in both cases for the solid element analysis. In the 3D plate case the plate FEM is coupled with the BEM for the infinite acoustic region and results for the two are found togheter. When modeled as a shell the plate is instead passing data on its surface velocity to a near field acoustic region which is again modeled with FEM. In this model it is this region to be coupled with BEM for the infinite acoustic domain. Also a third completely different model is developed starting from analytical knowledge of the modal shapes and eigenfrequencies of the plate. The model uses modal summation on the plate and Rayleigh’s Integral for the calculation of the radiated sound field. This model is specifically referred to the simple rectangular plate case, which is one of the few cases in which this analytic information is available. The three models are compared. The focus here is not establishing which model is the most efficient, but to describe similarities and differencies in their behaviour in a situation fairly close to convergence. It was found that at lower frequencies mismatch between the models depends on the estimation of the intensity of vibration of the plate. A nearly constant distance between the results is obtained in this frequencies. Going towards the 500-1000Hz range this discrepancy is attenuated but the shapes of velocity and pressure fields becomes harder to approximate. As a consequence the distance (between the models) peaks become higher and more localized. The FE models are built in COMSOL. In the last ten years there was a great development in the area of softwares for engineering simulation; COMSOL is the one used in this work and it is devoted to Finite Element Analysis. It was born in 1998, the Acoustic Module was introduced in 2006, but some very important updates were added just recently: for example coupling of BEM and FEM for the acoustic case was introduced in December of 2017 (release 5.3a).
Abstract (Italiano)
Un confronto di diversi modelli per la simulazione del suono irradiato da una struttura sottile vibrante e’ presentato. Un piatto rettangolare di alluminio che vibra nel mezzo di un piano rigido e infinito e’ studiato come caso benchmark. Nella presente tesi due modelli sono sviluppati per questa situazione usando il FEM e il BEM. Questi modelli sono molto flessibili: qualunque struttura “shell” potrebbe essere trattata in questo modo. Per esempio molti strumenti musicali posseggono una struttura “shell” che vibra quando lo strumento e’ suonato (la tavola armonica di un pianoforte e’ un caso). Le sue proprieta’ influenzano fortemente il modo in cui il suono si diffonde nello spazio e il suono riprodotto stesso. La struttura del piatto potrebbe essere una piccola parte di un modello piu’ grande, per esempio il modello di una macchina industriale che emette un rumore. I modelli FE implementati nella presente tesi simulano diversamente sia il piatto che la regione acustica. Il piatto e’ modellato come uno “shell” (quindi approssimato come un struttura 2D) in uno dei modelli, mentre nell’altro e’ rappresentato come l’oggetto 3D reale. In entrambi i casi la struttura solida e’ analizzata con il FEM. Nel caso del piatto 3D, il FEM sul piatto e’ accoppiato a BEM per la regione acustica infinita e i risultati per questi due sono ottenuti contemporaneamente. Il piatto modellato come “shell” invece comunica i dati sulla sua velocita’ di superficie a una vicina piccola regione acustica modellata anch’essa utilizzando FEM. In questo modello e’ questa regione ad essere accoppiata con il BEM per il dominio acustico infinito. Anche un terzo modello completamente diverso e’ sviluppato partendo da informazioni ottenute analiticamente sulle autofrequenze e forme modali del piatto. Esso utilizza la sovrapposizione di modi sul piatto e l’integrale di Rayleigh per il calcolo del campo acustico radiato. Questo modello e’ specificatamente riferito al caso del piatto rettangolare, che e’ uno dei pochi casi in cui questa informazione analitica e’ disponibile. I tre modelli sono confrontati. La priorita’ nel presente lavoro non e’ stabilire qual’e’ il modello piu’ efficiente, ma evidenziare somiglianze e differenze tra i risultati in una situazione vicina a convergenza. Sono state trovate alcune discrepanze; per le frequenze piu’ basse esse dipendono da differenze nella stima dell’intensita’ di vibrazione del piatto. Una differenza quasi costante tra i risultati e’ osservata in questo caso. Muovendosi nel range 500-1000Hz questa discrepanza e’ attenuata, ma le distribuzioni di velocita’ sul piatto e di suono nell’aria diventano piu’ complesse e difficili da approssimare. Di conseguenza si osservano picchi di distanza (tra i modelli) piu’ alti e localizzati. I modelli FE sono costruiti in COMSOL. Negli ultimi dieci anni c’e’ stato un grande sviluppo nell’ambito di softwares per la simulazione ingegneristica; COMSOL e’ quello usato nella presente tesi ed e’ dedicato alla Analisi ad Elementi Finiti. E’ nato nel 1998, il modulo acustico e’ stato introdotto nel 2006, ma alcuni aggiornamenti molto importanti sono stati fatti solo recentemente: per esempio la possibilita’ di accoppiare il BEM e il FEM per il caso acustico e’ stata introdotta nel Dicembre 2017 (versione 5.3a).
Comparative analysis of three alternative modelling approaches to the
simulation of sound radiation from a baffled plate
April 5, 2020
Contents
1 Introduction 8
2 Background 10
2.1 The Galerkin Finite Element Method . . . 13
2.1.1 FEM in Acoustics . . . 15
2.1.2 Discretization of a Solid Structure . . . 18
2.1.3 Discretization of Shells . . . 20
2.2 The Boundary Element Method in Acoustics: fundamentals . . . 24
2.3 FEM and BEM coupling: Costabel’s method . . . 27
2.4 Solvers . . . 30
2.5 Modal superposition and Rayleigh Elements: an anaytical model . . . 31
2.5.1 Modal superposition analysis of the plate . . . 31
2.5.2 Rayleigh Integral . . . 32
3 Implemented models 33 3.1 Model 1 : decoupled analytical model . . . 34
3.2 Model 2: decoupled FEM + FEM/BEM model . . . 35
3.3 Model 3: coupled FEM/BEM model . . . 38
3.4 Error and distance . . . 40
3.5 Comparison of the meshes . . . 40
4 Results 43 4.1 Modal analysis of the plate . . . 43
4.2 Frequency response of the plate . . . 50
4.3 Radiated sound field: modal analysis . . . 57
4.4 Radiated sound field: frequency response . . . 64
4.5 Computational Time . . . 77
5 Conclusions 78
6 Appendix : tensorial notation 80
7 Bibliography 82
-List of Figures
2.1 Example of 3D Mesh . . . 14
2.2 Parametrization of a shell structure, . . . 20
2.3 Visualization of a generic scattering problem . . . 28
3.1 Code for Model 1, . . . 35
3.2 Model 2: MID Mesh for the acoustic area . . . 36
3.3 Model 2: MID Mesh for the acoustic area . . . 36
3.4 Model 2: SUPER Mesh for the plate . . . 37
3.5 Model 3: MID mesh for the 3D plate . . . 38
3.6 Model 3: detail of MID mesh for the 3D plate . . . 39
3.7 Comparison of the meshes used in Models 2 and 3 . . . 41
4.1 Eigenfrequencies 1-26 found by the three models . . . 44
4.2 Eigenfrequencies 25-26 found by the three models) . . . 45
4.3 MAC between couple of models using MID and HIGH/SUPER meshes (High Scale) . . . 46
4.4 MAC between Model 1/2 and Model 1/3 using SUPER/HIGH meshes (Super High Scale) . . . 47
4.7 Comparison of Modal Shapes on the plate: Modes 17 and 20 (MID meshes) . . . 47
4.5 MAC between the three couple of models using MID and HIGH/SUPER meshes (Low Scale) . . . . 48
4.6 MAC between Model 1 and 2 using MID Meshes (MID Low Scale) . . . 49
4.8 Comparison of mean velocities on the plate for ⌘ = 0.01, 0.1 at frequencies 10:10:1000 . . . 50
4.9 Velocity magnitude and Relative Distance on the plate: colormaps for the plate excited at eigenfre-quencies 1 and 4 (MID meshes - ⌘ = 0.01) . . . 51
4.10 Velocity magnitude and Relative Distance on the plate: colormaps for the plate excited at eigenfre-quencies 17 and 20 (MID meshes - ⌘ = 0.01) . . . 52
4.11 Velocity magnitude and Relative Distance on the plate: colormaps for the plate excited at eigenfre-quencies 17 and 20 (HIGH Meshes) . . . 54
4.12 NMSD on the plate for different meshes . . . 55
4.13 Velocity magnitude and Relative Distance on the plate: colormaps for the plate excited at eigenfre-quencies 6 and 25 (MID meshes - ⌘ = 0.01) . . . 56
4.14 Planes on which sound pressure is calculated . . . 57
4.15 Normalized sound pressure magnitude and relative distance on plane XY3 when the plate is assuming modal shapes 1 and 4 (MID Meshes) . . . 58
4.16 Normalized sound pressure magnitude and relative distance on plane XY3 when the plate is assuming modal shapes 8 and 15 (MID Meshes) . . . 59
4.17 Normalized sound pressure magnitude and relative distance on plane XY3 when the plate is assuming modal shapes 20 and 26 (MID Meshes) . . . 61
4.18 NMSD on plane XY3 . . . 62
4.19 Normalized sound pressure magnitude and relative distance on plane XY3 when the plate is assuming modal shape 5 (MID Meshes) . . . 62
4.20 Normalized sound pressure magnitude and relative distance on plane XY1.5 when the plate is as-suming modal shapes 20 and 26 (MID Meshes) . . . 63
4.21 Comparison of sound pressure magnitudes calculated at point (0,0,3)[m] for ⌘ = 0.01 at frequencies 10:10:1000 . . . 64
4.22 Comparison of sound pressure magnitudes calculated at point (0,0,3)[m] for ⌘ = 0.01 at frequencies 10:10:1000 . . . 65
4.23 Comparison of sound sound pressure magnitudes calculated at point (0,0,6)[m] for ⌘ = 0.1 at fre-quencies 10:10:1000 . . . 65
4.24 Pressure magnitude and relative distance on plane XY3 when the plate is assuming excited at eigen-frequencies 1 and 4 (MID meshes - ⌘ = 0.01) . . . 67
4.25 Pressure magnitude and relative distance on plane XY3 when the plate is assuming excited at eigen-frequencies 17 and 20 (MID meshes - ⌘ = 0.01) . . . 68
4.26 Sound pressure magnitude and relative distance on plane XY3 when the plate is assuming excited at
eigenfrequencies 25 and 26 (MID meshes - ⌘ = 0.01) . . . 69
4.27 Sound pressure magnitude and relative distance on plane XY1.5 when the plate is assuming excited at eigenfrequencies 1 and 4 (MID meshes - ⌘ = 0.01) . . . 70
4.28 Sound pressure magnitude and relative distance on plane XY1.5 when the plate is assuming excited at eigenfrequencies 17 and 20 (MID meshes - ⌘ = 0.01) . . . 71
4.29 Sound pressure magnitude and relative distance on plane XY1.5 when the plate is assuming excited at eigenfrequencies 25 and 26 (MID meshes - ⌘ = 0.01) . . . 72
4.30 Pressure magnitude and relative distance on plane ZY when the plate is assuming excited at eigen-frequencies 17 and 20 (MID meshes - ⌘ = 0.01) . . . 73
4.31 NMSD for sound pressure magnitude colormaps on XY3 (MID and HIGH meshes) . . . 74
4.32 NMSD for sound pressure magnitude colormaps on XY1.5 and ZY (MID meshes) . . . 75
List of Tables
1 Fixed parameters . . . 33 2 Summary of the models . . . 33 3 Eigenfrequencies from the three models . . . 44
1 Introduction
In the present work the study of acoustic radiation from thin vibrating structures is tackled, with a focus on the comparison of different methods for performing this analysis. This type of structures are in engineering literature referred to as shells: 3D objects with a dimension much smaller than the other two. For an efficient analysis these objects are often considered as 2D: they are absorbed in a surface, which is often the surface at the middle between the exterior wide surfaces. When excited by a harmonic force these objects start to vibrate at the same frequency of the force itself. Particles of air in contact with their surface move with the same frequency and communicate this harmonic displacement to adjacent particles. In other words a sound wave is radiated. Many musical instruments feature a shell structure which is vibrating when the instrument is played. Its properties are fundamental for the way sound diffuses in space and for the reproduced sound itself. An example could be the soundboard of a piano or a surface of the harmonic box of a guitar. The plate structure could also be a small part of a bigger model, for example of an industrial machine emitting a noise. The way this noise radiates and ways to reduce its intensity could be the object of the analysis [33]. In reality the movement of the air, caused by the movement of the shell, is influencing the shell vibration itself; if the feedback of the air is considered a structure-acoustic coupling problem arises [20]. In the case of radiation in an infinite air medium (exterior radiation) an accurate analysis can be per-formed neglecting this effect, and this is what is done in the present. Sound radiation is analyzed starting from its source: a vibrating object.
A test case is adopted: the case of a rectangular aluminium plate, simply supported, vibrating in the midst of an infinite rigid baffle. Three different models are developed for the study of sound radiation in this case. A model is developed starting from analytical knowledge of the modal shapes and eigenfrequencies of the plate. This model is specifically referred to the simple rectangular plate case, which is one of the few cases in which information about eigenfrequencies and modal shapes is obtainable in an analytical fashion. Whenever modal information is available it is possible to apply the modal summation technique. This technique has been used for an anlogous benchmark case in [39],[44] and [34]. In [31] modal summation is used for the analysis of the vibration of a violin, while in [21] it is used to perform high frequency analysis of seismographic waves. When performing high frequency analysis, modal summation is a particulary advantageous approach because its computational cost stays nearly constant. It requires anyway information about many higher modes. The modal summation approach is a quite empirical ap-proach, based on a formula for the calculation of contribution of modes, which are then superposed. When studying a dynamic system a more physical approach could be adopted and a differential equation can be found describing the dynamic system at hand. The standard procedure utilizes the principle of virtual work to obtain an equation which is then discretized using Finite Element Methods. FE (Finite Element) models have great versatility and require information only about geometries and materials involved. They however tend to become very costly in the mid and high frequency range and it has been said that they are mainly suitable for a low frequency analysis. A FEM approach was adopted for the study of a violin in [35], where detailed information about the geometry of the harmonic box was retrieved using a Computed Tomography scanner. FEM was applied again to study the violin in [45] with a focus on the shell structure constituting the harmonic box and considering acoustic-structure interaction. In the present two FE models are developed in COMSOL.
The second step of the analysis is concerning sound radiation from the plate. As in [39], [34] and [44] the modal summation approach is coupled to Rayleigh’s Integral [36]. Rayliegh’s formula is a semplification of the Represen-tation Formula which requires as input only the velocity on the plate. The FEM procedure is instead coupled to the BEM procedure using a classical coupling technique devised by Costabel [15]. With this method the two variables to input in the Representation formula are found, and in a final step this formula is used to obtain radiated pressure values. Coupling of FEM and BEM is a standard approach for the study of radiation of waves, because it allows to have a solution far from the source without increasing the dimension of the model. The only information required is on the boundary of the source or on the boundary of a small near-field area around the source (See [22] for examples of application).
Coupling the modal summation procedure with Rayleigh’s method a very quick process is obtained to analyze sound radiation from a structure. On the contrary FEM-BEM coupling delivers a linear system with a dense matrix, very costly to solve. Increasing the frequency and/or the complexity of the considered geometry the computational cost could easily become prohibitive. Indeed a light and a heavy procedure, obtained following completely different prin-ciples, are compared. If the same results were obtained by the two, with no doubt the Modal Summation/Rayleigh
Integral procedure would be the choice (in a situation were needed modal information is available). Unfortunately a substantial mismatch will be uncovered by the present work. Choosing one method instead of the other, noticeably different results are found. It is thus important to gain knowledge about the particular behaviour of the single algorithms. This information is increasing consciousness on the meaning of results obtained choosing one method instead of the other. This is the purpose of the present work.
Programs like COMSOL allow in a flexible and relatively simple way a state of the art implementation (from mesher to solver) of the FEM-BEM technique. Models developed for this purpose have great flexibility and in fact any shell like structure could be threated in this way. In the last ten years there was a great development in the area of softwares for Engineering Simulation; COMSOL was born in 1998 but the chronology of the releases shows how its expansion in different application areas quickened in the last years. The Acoustic Module was introduced in 2006, but some very important updates were added just recently: for example coupling of BEM and FEM for the acoustic case, heavily used in this thesis, was introduced in December of 2017 (release 5.3a).
Efficiency of the software itself is not the focus here, neither it is establishing which model is the most efficient. The purpose of this work is describing similarities and differencies in the results in a situation fairly close to convergence. The frequency range 10-1000Hz is selected (a low to MID frequency range). The models are run for this frequencies with a focus on the eigenfrequencies.
In chapter 2 a theoretical background is given. Not all the arguments necessary to the developement of a similar model starting from zero are described: for example mesh theory is not touched, and the solvers used are super-ficially described. Instead the focus was posed on physical models allowing a discrete solution starting from the governing equation. Arguments in this chapter allow for a complete understanding of the physical modeling process underlying the COMSOL models. Also a simpler Model 1 is described: the Mode Summation procedure is described starting from the bending plate equation and Rayleigh’s Method is introduced.
In chapter 3 the case of study and the construction of the models are detailed, along with consideration about the meaning of forecoming results and details about the meshes used.
2 Background
The argument of FEM and BEM coupling plays an important role when analysing radiation patterns; tipically, in these problems, waves of some sort (acoustic, seismic, electromagnetic for example) travel away from a zone modeled in detail into a zone approximated as an infinite homogeneous domain. The representation formula allows the calculation of a solution in a finite set of points of the infinite domain, requiring just the discretization of the boundary enclosing the finite interior domain. Since the 1980s much has been said on the coupling of the two methods. The objective is the creation of a linear system delivering a solution in the finite domain, but also the values required to set up the representation formula (namely the restriction to the boundary of the main variable and the corresponding flux in the finite domain). The Boundary Element Method is a discretization technique which, knowing one of the two variables, allows to know the other through the solution of a dense linear system (for example see [9]). Having a FEM solution related to the interior domain means knowing also its restriction to the boundary with the infinite domain: if proper transmission conditions are set on the boundary between the two, this shared variable can be exploited to generate a unique linear system. This is what Nedelec achieved in 1980 through a variational formulation (a weak formulation) of the problem and using a Sobolev Space framework [26] . In the MID eighties, Martin Costabel [16] modified Nedelec’s method and provided a formulation requiring a simpler computation, able to reflect the symmetry of the original problem, if present. His method is the one referred to in COMSOL’s manuals and is described in paragraph 2.3. A particular type of coupling is obtained if the problem on the boundary is expressed in integral form through a particular Dirichlet condition: in this case, a FEM-Dirichlet Boundary Condition Iteration (FEM-DCBI) coupling problem is obtained. This formulation has been compared to the more classical FEM-BEM in [2] and proved to be less accurate but also less heavy, in terms of computational time. In the same paper also numerical schemes for solving the linear system obtained are described: the system is divided into a first part related to FEM, which is usually sparse and a denser one related to BEM. Then a Conjugate Gradient scheme is applied to the FEM part while GMRES is applied to the BEM part, trying to solve a reduced system instead of using the complete dense matrix.
When dealing with infinite domains, FEM-BEM is nowadays one of the alternatives. The problem could be tackled entirely using a “domain” scheme like FEM, but introducing particular elements at the boundary called infinite elements. These elements use a “radial” coordinate able to simulate the stretching of the elements towards infinity and the behaviour of wave-like solutions if these are subject to appropriate radiation conditions. Infinite Elements’ performance and cost depend greatly on the number of points used to discretize this radial direction. In [5] the three most important formulations are compared: (i) the unconjugated Burnett formulation, (ii) the conjugated Burnett formulation and [10], (iii) the unconjugated Astley-Leis formulation [4]; formulations differ only in the choice of the radial funtions. In the paper the unconjugated Burnett formulation is confirmed as the best performer for near-field computation while the Astley-Leis formulation proves to be the most effective in the far-field.
When the behaviour of the solution in the far field is not of interest, truncation of the infinite domain is performed, usually with a Perfectly Matched Layer scheme; it was invented by Berenger in the nineties for electromagnetic problems [7]. In the PML layer (constituted by elements of the FE scheme), a stretching of the coordinates in a direction radial to the finite domain is performed; in this way the behaviour of an absorbing layer is simulated with almost no reflections detected. The method has been validated for acoustic problems [23] and also a computationally lighter variant (a simplification of the model) called Nearly Perfectly Matched Layer (NPML) was developed for this case [24]. The technique has not been considered in the present because sound fields in the far field are of maximum interest. Actually, in the present work only techniques making use of the representation formula are considered. The BEM technique was already mentioned: in the present it is extensively used because of the simplicity of the models used; however this is not granted: the BEM delivers dense matrices which require O(N3) operations for
inversion. In the last two decades techniques were devised to extend BEM to large scale problems. In [38] a fast multipole boundary element method (FMBEM) for 3D acoustic problems is presented in which the iterative solver GMRES and a multipole expansion of the kernels are used. Also another very well known issue about the non uniqueness of BEM solutions at certain high frequency resonances is addressed: a combination of CBIE and HBIE was applied to overcome this problem. These techniques overdetermine the system of equations to reduce the number of solutions to one. A presentation of this kind of practices can be found in [11].
On the other side, FEM is also a discretization technique for differential problems but delivering sparse matrices. In this case direct sparse solvers can be used: memory is saved expoloiting sparsity and using out of core memory.
Refined solvers exist like PARDISO ([17] and [41]) (Parallel Sparse Direct Multi-Recursive Iterative Solver) or MUMPS (MUltifrontal Massively Parallel sparse direct Solver [3]). For this reason and for its versatility FEM is one of the most used discretization technique in the last thirty years. Its convergence (depending on the element dimension) has been discussed in [25] and specifically for COMSOL’s implementation in [8]. It is found that theoretical convergence rate of O(hp), where p is the polynomial interpolation degree, is often not reached (especially
when non smoothness is present in the considered geometry). In the present work the classical Galerkin formulation will be presented as in [1] starting from the weak formulation of the differential problem at hand. A very basic scheme was applied for this simple case, but nowadays more complex and versatile formulation exist like the Discontinuous Galerkin method (see [13] for a generic introduction and [32] for the acoustic case). In this formulation discontinuities in the solution at the boundaries between elements are allowed: a stabler behaviour is obtained also for very complex and non smooth structures. The FEM model developed here for the simple test case of a rectangular plate doesn’t require any preliminary knowledge apart from material and geometry; it can be thus easily adapted to any sufficiently thin vibrating structure. In [14] the FE analysis is applied to subsequent steps of the construction of a piano soundboard. It is an example of how this type of analysis is of great interest for the design of musical instruments.
Arguments cited until here are related to the two FE models developed in COMSOL, described in paragraphs 3.2 and 3.3. A much simpler and much lighter model was developed in MATLAB, exploiting some assumptions of the specific study case. Costly BEM schemes are dropped for this procedure and the link between the solid and the acoustic domain is constituted by the Rayleigh Integral [40]. The formula can be seen as a semplification of the representation formula for the plate case and requires the presence of an infinite baffle. A quick calculation technique of this integral using the FFT has been given by Wallace in [43] for planar structures of any shape: similarities between formulas for the FFT and the function to be evaluated in the Integral are exploited. Also the FEM procedure was replaced by the extremely light mode summation method. This method requires knowledge of the modal shapes and eigenfrequencies, which is retrievable analytically only in some cases like circular or rectangular plates; in this situation it relies on the Kirchoff-Love plate model, and it is thus flawed by its assumptions. These are
1. Every straight line of the plate which was originally perpendicular to the plane bounding surfaces remains straight under deformations, and perpendicular to the surfaces which were originally parallel to the plane bounding surfaces;
2. All the elements of the middle surface remain unstretched.
This conditions are largely met for the low frequency range but are less precise for shapes changing very quickly in space (i.e. higher frequency range). Fixing the frequency, a ready to use equation can be found to assess modal shapes and eigefrequencies. Much has been said on the subject using K-L model: a recent discussion on the effect of boundary conditions in the radiating properties of a plate has been given in [39]. The quantity tackled in the paper was the radiation efficiency which relates the actually radiated power of the plate to the power radiated by a piston having the same size and material of the plate, and vibrating altogheter at the selected frequency. Results for the radiated power were averaged over all possible excitation positions to obtain a parameter describing the plate properties in a very general way. In [34] results for different excitation points are showed and the single point radiation efficiency variation around the average one is analyzed. In the present results are taken for a single point because a similar type of average would be extremely costly and potentially imprecise on FEM models; a new Frequency Sweep would have to be done for every point and results would have to be averaged on a finite set of points, while in Squicciarini’s paper a continuous average is defined, using a continuous theoretical integration. This type of calculations are made possible by the presence of Model 1. FE models can’t rely on such information, but once their behaviour is assessed for the considered test case their use can easily be extended to more complex thin structures. The analytical knowledge of modal shapes for the rectangular case has been used in [42] to derive closed forms for the plate radiation efficiency at small wavenumbers. The analytical tools discussed here (knowledge of the plate and Rayleigh Integral) were instead used from Maidnaik [29] to obtain closed formulae estimating the radiated acoustic field for single modes of a rectangular plate in a baffle. His formulae are the basis for many other models. Here such formulae are not required since once the integral is set up, classical numerical methods for integration are used to carry out the calculation. In the present a trapezoidal calculation proved to be sufficient.
In the following the methods just discussed are introduced and described to the extent needed for the understanding of the models. In chapter 2.1 FEM is detailed: the Galerkin Formulation is derived and then examples are given. These examples will serve to clarify the Galerkin procedure, but will also be directly involved in the explanation of the models; they represent the two cases met in our experiments: the 3D solid case, the shell case and the acoustic case. In chapter 2.2 BEM is explained at its fundamentals for the acoustic case; also possibility to extend the algorithm to semi infinite regions is discussed. In chapter 2.3 Costabel’s method is described and concepts from chapter 2.1 and 2.2 will be used to describe the equation solved by COMSOL. Derivation of the linear system is also briefly detailed. All these methods result in linear systems, thus a discussion on solvers for this types of system is added. Iterative Krylov solvers are the focus as they are the choice for dense matrices like the ones resulting from BEM. Finally chapter 2.5 is about theory underlying Model 1 for the radiation of the plate: Kirchoff-Love plate theory and Rayleigh elements are introduced.
2.1 The Galerkin Finite Element Method
The Galerkin method is a procedure for the discretization of differential equations. It’s a very versatile way of finding a discrete set of parameters describing the desired solution. In the following these parameters represent the value of the solution at nodal points; a very similar approach exists making a different choice: in the Wave Based Method [18] weights are searched for functions solving the equation but not satisfying the boundary conditions. A post processing step is required in that case. In the present the classical method is derived starting from the most general premises; the concept of weak formulations is presented and the integral equation to be discretized is derived for a general differential problem. In the following an example of derivation is given for the acoustic case. Also the structural case is presented but starting from the Lagrange equation. Both cases are directly referred to in the description of the models which are the focus of this thesis. Weak formulations are also derived in the case of shells, briefly describing an alternative model adoptable in the case of thin structures. In this approach the presence of a dimension much smaller than the others is exploited and a 2D model is developed for a 3D object.
Weak Formulations. Posing the problem to be solved in the most general terms, an unknown function u is sought
such that it satisfies a certain differential equation set A(u) = 8 < : A1(u) A2(u) ... 9 = ;= 0 (2.1)
within a certain domain, togheter with certain boundary conditions B(u) = 8 < : B1(u) B2(u) ... 9 = ;= 0 (2.2)
From the previous it follows that Z ⌦ vTA(u)d⌦⌘ Z ⌦ [v1A1(u) + v2A2(u) + ...] d⌦⌘ 0 (2.3) where v = 8 < : v1 v2 ... 9 = ;
is a set of arbitrary functions equal in number to the number of equations, called test functions. A more powerful statement can be derived: if eq 2.3 is satisfied for all possible v then the differential equations 2.1 must be satisfied
at all the points of the domain. This is obvious if an instance of u such that A(u) 6= 0 at any point is considered:
then immediatly a test function can be found such that 2.3 is not satisfied. Similarly the weak formulation of the boundary conditions can be written as 2.2
Z ¯
vTB(u)d⌦⌘ Z
[¯v1B1(u) + ¯v2B2(u) + ...] d⌦⌘ 0 (2.4)
the integral statement Z
⌦
vTA(u)d⌦ + Z
¯
vTB(u)d⌦ = 0 (2.5)
Is equivalent to the satisfaction of the differential equations 2.1 and of the boundary conditions 2.2. In the above it is assumed that integrals are capable of being evaluated: this restricts in some way v, u to functions which don’t result in any term in the integral becoming infinite.
Fig. 2.1:
Mesh for Model 2. Here elements are 3D tetrahedral and shape function will have to vary in 3 coordinates.
Discretization and Galerkin Formulation. The Finite Element Method is a discretization technique for differential
problems like the one in the previous, defined over a domain ⌦ which could be one, two or three dimensional. Instead of searching for a continuous function solving the given problem, the values of a finite set of parameters aidescribing
the solution is wanted. A continuous solution is then recovered using a to weight a set of functions, called shape functions. In many applications (and in the ones of this thesis) a represents the value of u in a set of points called
nodal points. They define the edges of the elements (see FIG 2.1) in which the problem domain is partitioned.
In this case shape functions will have the following property Ni(N odalP ointj) =
(
1 i = j
0 i6= j (2.6)
therefore interpolating the known values.
In other words, an approximation of the function u is sought depending on a finite set of values, and the approximate solution is given by ˆ u = n X i=1 Niai= Na (2.7)
Simirarly in place of the test functions v = n X j=1 wj aj ¯v = n X j=1 ¯ wj aj (2.8)
where aj is arbitrary. Inserting the above in 2.5 the following equation is obtained
Z ⌦ wjTA(Na)d⌦ + Z ¯ wjTB(Na)d⌦ = 0 (2.9)
in which the aj terms have been already factorized and neglected. A(Na) represents the residual or in other
words the error obtained by substitution of the approximation into the differential equation (and
Different methods can now be derived choosing different test functions wj: for example if wj = Nj is chosen
(the shape functions as test functions) the Galerkin method is obtained. The Galerkin method, because of the choice of test functions, often leads to symmetric matrices1. In the next it will be seen how this formulation can
lead to a linear system of the form
Xa + f = 0 (2.10)
through the example of the FEM for Acoustics.
2.1.1 FEM in Acoustics
Let’s start from the Helmholtz equation governing the steady-state acoustic pressure at any location (x,y,z) in a bounded fluid domain ⌦, enclosed by a boundary surface and generated by a time-harmonic external source distribution q at frequency ! = 2⇡f.
O2p(x, y, z) + k2p(x, y, z) = j⇢
0!q(x, y, z) (2.11)
where k = !/c = 2⇡f/c is the acoustic wavenumber, c is the speed of sound and ⇢0 is the ambient fluid mass
density. The boundary is partitioned into three parts, accounting for three possible types of boundary condition: ⌦ = ⌦p[ ⌦v[ ⌦z and
• p = ˜p on p, e.g imposed pressure;
• vn=⇢0j⌦@p@n˜ = ˜vn on v, e.g imposed normal velocity;
• p = ˜Zvn on z, e.g imposed normal impedance.
The weighted residual concept provides an equivalent integral formulation of the Helmholtz Equation. The concept defines a steady-state acoustic pressure field in a bounded fluid domain ⌦ as a pressure field, for which the integral equation
Z
⌦
¯
p O2p + k2p + j⇢0!q dV = 0 (2.12)
is satisfied for any weighting function ¯p, that is bounded and uniquely defined within volume ⌦ and on its boundary surface . Equation 2.12 can be rewritten (integrating the first term by parts) as
Z ⌦ @ @x ✓ ¯ p@p @x ◆ + @ @y ✓ ¯ p@p @y ◆ + @ @z ✓ ¯ p@p @z ◆ dV Z ⌦ ✓ @ ¯p @x @p @x+ @ ¯p @y @p @y + @ ¯p @z @p @z ◆ dV (2.13) + Z ⌦ k2ppdV +¯ Z ⌦ j⇢0! ¯pqdV = 0 or Z ⌦ (O¯pOp) dV Z ⌦ k2ppdV =¯ Z ⌦ j⇢0! ¯pqdV + Z ⌦O (¯pOp) dV (2.14) The divergence theorem is now applied Z
⌦ @Fi @xi dV = Z FinidS (2.15)
where Fi is a vector field to the last term of equation 2.14, obtaining
Z ⌦O (¯pOp) dV = Z ✓ ¯ p@p @n ◆ d⌦ = Z (j⇢0! ¯pvn) d⌦ (2.16)
The ’weak form’ of the weighted residual formulation of the Helmholtz equation is obtained Z ⌦ (O¯pOp) dV Z ⌦ k2ppdV =¯ Z ⌦ j⇢0! ¯pqdV Z (j⇢0! ¯pvn) d⌦ (2.17)
• R⌦(O¯pOp) dV
R
⌦k2ppdV¯ being the bilinear form integrated in ⌦;
• R (j⇢0⌦¯pvn)representing some of the boundary conditions rewritten in Costabel’s theory as trace operators.
In the next, it is shown how the weighted residual formulation can lead to a discrete linear system explicitly for the acoustic case.
Acoustic Stiffness Matrix. This matrix is derived from the first term in 2.17. Let B = @N
and ˆp be the vector of nodal values for the solution, then Z ⌦O¯pOˆp = {¯p i} Z ⌦ BTB dV{ˆpi} = {¯pi}[K]{ˆpi} (2.18)
where [K] is a {nf⇥ nf} matrix called “stiffness matrix”; it can be seen as an inverse mass or mobility matrix for
the volume of air, relating the pressure to an acceleration. An expression for the single matrix terms is then Kij = Z ⌦ @Ni @x @Nj @x + @Ni @y @Nj @y + @Ni @z @Nj @z dV (2.19)
Notice that shape functions Ni and Nj are non zero only in the elements where the nodal points i and j belong.
Also the summation property of integrals can be used to calculate the integral over the whole domain by summing the ones in the single elements; this allows to limit the integration over those elements in which the shape functions are non zero, meaning
Kij= mij X e=1 Z ⌦e ✓ @Ne i @x @Ne j @x + @Ne i @y @Ne j @y + @Ne i @z @Ne j @z ◆ dV (2.20)
where mij is the number of elements to which both node i and node j belong, and ⌦e is the considered element.
Since each node shares elements just with a few adjacents nodes the matrix [K] will be sparsely populated. Often the calculation is carried out in a two step procedure: first stiffness matrices for each element are calculated as
Kije = Z ⌦e ✓ @Ne i @x @Ne j @x + @Ne i @y @Ne j @y + @Ne i @z @Ne j @z ◆ dV (2.21)
and then the general matrix is assembled by simple addition of the corresponding entries. The procedure here detailed for the acoustic case is general to every FEM application.
Acoustic Mass Matrix. In a completely similar way, starting from 2.17 it is possible to rewrite the second term
in the discrete form
k2Z ⌦ ¯ pˆpdV = !2 {˜pi} ✓Z ⌦ 1 c2[N T][N ]dV ◆ {ˆpi} = !2{˜pi}[M]{ˆpi} (2.22)
where [M] is a {nf ⇥ nf} matrix called “mass matrix”; really it’s a compressibility matrix for the air volume,
relating the pressure to a displacement. A formula for each of its elements can be easily found Mij = mij X e=1 Z ⌦e ✓ 1 c2N e iNje ◆ dV (2.23)
Acoustic Excitation Vector The first integral term in the right hand side of 2.17 may be expressed as Z ⌦ (j⇢0! ˜pq) dV ={˜pi} Z ⌦ j⇢0![N]Tq dV ={˜pi}T{Qi} (2.24)
where {Qi} is a (nf⇥ 1) acoustic source vector. Let’s suppose now that q is an acoustic point source of strength ¯qi
located at node i, with node i not on the surface, that is
q(x, y, z) = ¯qi (xi, yi, zi) (2.25)
this will result in an acoustic excitation vector which is zero in every position apart from where is positioned the source Qi= j⇢0! ¯qi.
Other terms and final form: The last term in the right hand side of 2.17 allows for the introduction of boundary
conditions. The integration is divided between the three different surfaces defined before, to separate the three boundary condition situations
Z (j⇢0! ¯pvn) = Z v (j⇢0! ¯pvn) dS + Z p (j⇢0! ¯pv.n) dS + Z Z (j⇢0! ¯pA˜p) dS (2.26)
Let’s now see how different conditions can be dealt with:
• The term R v(j⇢0! ¯pvn) dS allows for the definition of an input velocity vector
Vni=
Z
v
(j⇢0! ¯pNvn) d (2.27)
• The term relative to impedance allows for the introduction of a matrix called Damping Matrix which is not discussed here (for details on this and the calculation of the previous see [19])
• To introduce the prescribed pressure condition a different procedure is required, because the termR p(j⇢0! ¯pv.n) dS
is not allowing for the introduction of the known values. The values will be directly assigned to each node located on the boundary ⌦p; it is then possible to delete a corresponding number of equations from the final
system. This is usually done by eliminating each row that expresses the weighted residual formulation, in which the global shape function of a node on the boundary surface p is used as weighting function. These
equations orthogonalize the error on the pressure predictions in the region near the boundary surface with respect to the shape functions in this region. Their elimination is motivated by the fact that this prediction error is smaller than the errors in the other regions of the fluid domain, since the exact pressure values at the nodes of the boundary surface ⌦pare a priori assigned.
Now instead of 2.17 the following linear system can be written
{˜pi}T [K] !2[M] {ˆpi} = {˜pi}T({Qi} + {Vni}) (2.28)
or
[K] !2[M] {ˆpi} = {Fi} (2.29)
where the lines related to known pressure values have been eliminated. Solving this system gives the values of the target function at nodal points, leading to an approximation of the continuous solution through interpolation based on shape functions.
2.1.2 Discretization of a Solid Structure
In this case, the starting equation is the Lagrange equation, defining the motion for the small vibration of bodies. These are described in discretized form as points applying an elastic force on each other. The Lagrange equation can be derived for the elastic case from the Hamilton Principle (for example) like in [30], which is in turn based on the very general fact that the work done by a conservative force can be obtained from the change in potential energy. Denoting as u the displacement function for a generic point in the structure a kinetic energy associated to it is defined
T (u) = 1 2m ˙u
2 (2.30)
and a potential energy or strain energy
U (u) =1 2ku
2 (2.31)
where m, k are respectively mass associated to that point and the stiffness of the corresponding ideal spring. These are the conservative forces discussed above. Their virtual work is equated to that done by non conservative forces
Wnc= (f c ˙u) u (2.32)
Where the term c ˙u is related to viscous Damping: a potential D called dissipation function is associated to it D = 1 2c ˙u 2 (2.33) such that c ˙u = @D @ ˙u (2.34)
The dissipation function represents the instantaneous rate of energy dissipation. Actually in this model this term will be approximated to simplify the introduction of damping in the structure. Lagrange Equation then reads
d dt ✓@T @ ˙u ◆ +@D @ ˙u + @U @u = f (2.35)
Substituting for a single degree of freedom basic one dimensional linearized vibration equation is obtained
m¨u + c ˙u + ku = f (2.36)
from the term related to the kinetic energy the inertia force is obtained while from the elastic potential term the restoring force of the spring is derived. In the case of a multi degrees of freedom system the deformation of the whole body is described by n indipendent displacements q1, ..., qn, then
T = T ( ˙q1, ..., ˙q2) (2.37)
U = U (q1, ..., q2) (2.38)
D = D( ˙q1, ..., ˙q2) (2.39)
and a matrix notation can be devised such that T =1 2{ ˙q} T [M]{ ˙q} and ⇢d dt ✓@T @ ˙q ◆ = [M]{¨q} (2.40) U =1 2{q} T [K]{q} and @U@q = [K]{q} (2.41) With
• [K] = square symmetric matrix of stiffness coefficients.
The vector {Q} = (Q1, ..., Qn)is introduced to represent generalized forces applied to the considered points. From
now on to approximate losses due to damping a (usually small) imaginary part to the Stiffness Matrix will be added. The equation obtained is
[M]{¨q} + (1 + ⌘j) [K] {q} = [Q] (2.42)
Where ⌘ isotropic loss was introduced. Let’s now define the potentials appearing for the case of a three dimensional solid, just like the 3D plate in one of the models discussed in the present. The derivations are based upon the linear theory of elasticity. This means that both the stress-strain and the strain-displacement relations are linear. The state of stress in a three-dimensional elastic body is defined by the stress components (which are referred to Cartesian axes x, y, z) [ ] = 2 4⌧yxx ⌧xyy ⌧⌧xzyz ⌧zx ⌧zy z 3 5 (2.43) with ⌧ij= ⌧jii, j = x, y, z (2.44)
The stresses are usually derived from the strains quantity ["] = 2 4"yxx "xyy xzyz zx zy "y 3 5 (2.45)
Denoting with (u, v, w) the displacement components in the directions of the axes then the strains-displacement relations are "i= @u @i i = x, y, z (2.46) ij = @u @j + @v @i i = x, y, z (2.47)
For the three dimensional isotropic solid, strains derived from 2.46 and 2.47 are used to define the energy potential U as U =1 2 Z V { } T {"} dV (2.48) With { }T =⇥ x y z ⌧xy ⌧xz ⌧yz⇤ (2.49) {"}T =⇥"x "y "z xy xz yz⇤ (2.50)
But a strain-stress relationship is available in matrix form as
{ } = [D] {"} (2.51)
with [D] symmetric matrix. For an anisotropic material, it contains 21 independent material constants, while in the case of an isotropic material it is
[D] = E (1 + v) (1 2v)⇥ 2 6 6 6 6 6 6 4 1 v v v 0 0 0 1 v v 0 0 0 1 v 0 0 0 1 2(1 2v) 0 0 Sym 1 2(1 2v) 0 1 2(1 2v) 3 7 7 7 7 7 7 5 (2.52)
with E Young Modulus and v Poisson’s Ratio depending on the material. Inserting 2.52 in 2.48 the following form for the potential energy is obtained
U = 1 2 Z V {"} T[D] {"} dV (2.53)
Fig. 2.2:
Parametrization of a shell structure (image from [12])
For the kinetic energy simply it is written depending on the three displacement components T = 1
2 Z
V
⇢ ˙u2+ ˙v2+ ˙w2 dV (2.54)
with ⇢ material density. Now terms in 2.42 are defined for the 3D case and the equation is ready to be discretized. The Galerkin method is applied with shape functions and nodal values just as in the previous chapter. A finite linear system can be obtained following the same procedure of the acoustic case.
2.1.3 Discretization of Shells
The Reissner-Mindlin kinematical assumption. Most shell models are based on assuming admissible
displace-ments profiles through the shell thickness, connecting the displacedisplace-ments of points on a line orthogonal to the midsurface of the object. The assumption is realistic in the case of a solid with a dimension much smaller than the others. Specifically for this case it is assumed that any material line in the direction of this dimension remains straight and unstretched during the deformations: the formulation is due to Reissner-Mindlin. In this chapter tensorial notation will be extensively used: it is introduced in Appendix. The limitations introduced by the R-M assumption are expressed by the following
U(⇠1, ⇠2, ⇠3) = u(⇠1, ⇠2) + X
=1,2
⇠3✓ (⇠1, ⇠2)a (⇠1, ⇠2) (2.55) where the material line in the direction of a3at the coordinates (⇠1, ⇠2)is considered (See FIG 2.2 for a representation
of the parameters involved). All greek indeces will range from 1 to 2. In this paragraph the Einstein summation
convention will be used and summation indices will be understood for all indices appearing once as a subscript and
once as a superscript. For example 2.55 would be written as:
u(⇠1, ⇠2) + ⇠3✓ (⇠1, ⇠2)a (⇠1, ⇠2)
The displacement u(⇠1, ⇠2)represents an infinitesimal displacement of the middle line which is causing a
displace-ment of the same amount for all the points on the ortoghonal line. The term ✓ (⇠1, ⇠2)a (⇠1, ⇠2)is due to rotation
of the line dependent on ✓1 and ✓2. A rotation of the midsurface without a component a3 can be defined because
the rotation of an infinetly thin material line is uniquely defined by a rotation vector normal to that line. Now the definition of the Green Lagrange strain tensor depends on the U displacement: precisely it is half the increment of the 3D metric tensor. For a linear analysis only the linear part of this tensor is considered , obtaining (note the “,”
at the subscript means a derivation) : eij =1
2(gi· U,j+ gj· U,i) i, j = 1, 2, 3 (2.56) the covariant components are computed for the linearized strain tensor as functions of u and ✓. The following form is obtained (see [12] for detailed calculation of the derivative of tensors involved)
8 > < > : e↵ = ↵ (u) + ⇠3 ↵ (u, ✓) ⇠3 2↵ (✓) ↵, = 1, 2 e↵3= ⇣↵(u, ✓) ↵ = 1, 2 e33= 0 (2.57) where 8 > > > > < > > > > : ↵ (u) = 12 u↵| + u |↵ b↵ u3 ↵ (u, ✓) = 12 ⇣ ✓↵| + ✓ |↵ b u |↵ b↵u | ⌘ + c↵ u3 ↵ (✓) =12 ⇣ b ✓ |↵+ b↵✓ | ⌘ ⇣↵(u, ✓) = 12 ✓↵+ u3,↵+ b↵u
Where the tensors , and ⇣ are called the membrane strain, bending strain and shear strain tensor respectively.
The “basic shell model” To derive an equation to discretize, a relation between strains and stresses on the plate
has to be established. Citing Bathe-Chapelle ([12]) “Most classical shell models are based on the assumption that
the state of the stresses in the shell corresponds to place stress tangent to the midsurface of the shell, or at least approximately so”. Considering an isotropic loss material, in a general curvilinear coordinate system Hooke’s law
reads: ij = Hijkle kl (2.58) with Hijkl= L 1gijgkl+ L2 gikgjl+ gilgjk (2.59)
and L1, L2Lame’ constants depending on the Young Modulus and Poisson’s coefficient:
L1= E
v
(1 + v) (1 2v) (2.60)
L2= E
2 (1 + v) (2.61)
If the assumption that normal stress 33 is zero holds, the following modified constitutive equation is obtained
(again refer to [12] for detailed calculation) ( ↵ = C↵ µe µ with C↵ µ = H↵ µ H ↵ 33H µ33 H3333 ↵3= 1 2D ↵ e 3 with D↵ = 4H↵3 3 (2.62) the basic shell model as defined by the following variational formulation can now be derived
Z ⌦ ⇥ C↵ µe↵ (U)e µ(V) + D↵ e↵3(U)e 3(V)⇤dV = Z ⌦ F· VdV (2.63)
where V(⇠1, ⇠2, ⇠3) is an arbitrary test function satisfying the Reissner-Mindlin kinematical assumption. The symbol
Fdenotes the external 3D loading applied to the shell structure. The variational formulation (or weak formulation) just given represents a mathematical shell model since the unknowns as well as the test functions are given by a sets of tensors defined on the shell midsurface.
The plate model An equation ready to discretize can be immediately derived from 2.63 for a very generic shell
structure: the “shear membrane bending model”(s-m-b) is the starting point for many other formulations (see [12]). Here simplifications are immediately introduced due to the assumption of dealing with a planar surface that is equal to imposing a3 constant when no force is applied (thus constant rotations ✓1,2) . As a consequence
g↵ = a↵ (2.64)
g↵ = a↵ (2.65)
g = a (2.66)
Also the strain values take the simpler form 8 > > > > > > < > > > > > > : ↵ (u) =12 u↵| + u |↵ ↵ (u) = 12 ✓↵| + ✓ |↵ ¯ ⇢↵ (u) = u3|↵ ↵ (✓) = 0 ⇣↵(u, ✓) = 12(✓↵+ ue,↵) (2.67)
This implies a decoupling between the unknowns (u1, u2)and (✓, a3). As a result the s-m-b model is now described
by two variational problems. Inserting 2.57 and 2.67 in 2.63 it is obtained: Z ! ⇥ tC↵ µ ↵ (u12) µ(v12)⇤dS = Z ⌦ tF12·v12dV (2.68)
with F12= (F1, F2)and u12= (u1, u2) , v12= (v1, v2), and
Z ! t3 12C ↵ µ ↵ (✓) µ(⌘) dS + Z ! ⇥ tD↵ ⇣↵ (u3, ✓)⇣ µ(v3, ⌘)⇤dS = Z ! tF3v3dS (2.69)
where t represents the thickness of the plate. The problem described in 2.68 and 2.69 is a special case of the more general smb model, which represents a well posed problem. More specifically assuming
• F 2 L2(S)(e.g. is integrable in two dimensions);
• The essential boundary conditions enforced in V are such that no rigid body motion is possible;
Equations 2.68 and 2.69 represent a well posed mathematical problem, (i.e. there is a unique solution (u, ✓)).
A proof of this fact can be found in [12].
Discretization of the plate shell equation Equation 2.69 can be rewritten in symbolic form
A(U, V) = F(V) (2.70)
such that a general form for the discretized problem would be denoted as
Ah(Uh, V) = Fh(V) (2.71)
The mesh has to be set just on the midusrface, and a specific type of elements called general shell elements is devised. The following relation between local element coordinates (r, s, z) and position vector x inside an element is defined x = k X i=1 i(r, s) ✓ x(i)+ zt (i) 2 a (i) 3 ◆ (2.72)
where i(r, s) denotes 2D Lagrange Shape functions while x(i),a(i)3 , t(i) are defined on the midsurface and are
position, unit normal vector at nodes and thickness. With this strategy a 3D mesh is obtained with a single layer of elements using the thickness parameter. “Degenerating” a 3D solid finite element into a shell element (using the R-M assumption) allows for the use of general 3D constitutive rules on a 2D model. Let’s now consider the chart (⇠1, ⇠2, ⇠3) describing the midsurface of the shell: starting from the given element an approximate chart
is derived, which is the one used for the discretized problem. Let’s define two types of one to one mapping, one defining interpolation on the midsurface
I( )(⇠1, ⇠2) = k
X
i=1
i(r, s) (⇠(i)1 , ⇠2(i)) (2.73)
and the other one is the equality
⇠3= zt
2 (2.74)
Also values t(i)and a(i)
3 are discretized in an equivalent manner and this interpolation is denoted as I(ta3)(⇠1, ⇠2).
The mapping between (⇠1, ⇠2, ⇠3)and the approximate geometry is thus given by
x =I( ) +z
2I(ta3) =I( ) + ⇠
3I(ta3)
t (2.75)
This gives a characterization of the geometric approximation involved in Ah and Fh. This approximation consists
in using the approximate chart
h(⇠1, ⇠2, ⇠2) =I( )(⇠1, ⇠2) + ⇠3I(ta3)(⇠ 1, ⇠2)
t (2.76)
The internal virtual work can now be directly obtained by using the left side of 2.68 and 2.69 and by substituting the approximate chart for the exact one to derive all geometric coefficients. Components C↵ µ and D↵ depend
on the approximate geometry through the use of approximate components ¯g↵ instead of g↵ with
¯ gi=
@ h
@⇠i (2.77)
Similarly for the strains
¯ eij(U) = 1 2 ✓@U @⇠i¯gj+ @U @⇠jg¯i ◆ (2.78)
Using this procedure (namely procedure P), problems 2.68 and 2.69 have a unique solution. Furthermore assuming the solution of the basic shell model is smooth and writing as uh the approximate solution, the following error
estimate is valid
||u uh||Vu 0 Chmin{p,2} (2.79)
where the norm is defined on the Sobolev space Vu=H1⇥ H1(see [12] for a proof). This natural strategy leads to
suboptimal convergence, limited to the quadratic order. More complex strategies can be used to achieve a better result, but they are not discussed here.
2.2 The Boundary Element Method in Acoustics: fundamentals
The Boundary Element Method (BEM) is a discretization procedure used to limit the discretized area to the surface of a volume, or, as in our case, to the surfaces within an infinite domain. In radiation (and also scattering) problems like the one at hand it is a classic procedure to apply FEM to a finite domain and to use BEM for the infinite region. In the present, the finite domain is an air region in the proximity of the source for the first model and an aluminium plate for the second. In scattering problems, FEM is used for the domain containing the reflecting bodies; outside of this region the domain is approximated as homogeneous and infinite. BEM gives a solution of the equation (for example the Helmholtz equation), which is valid in the finite domain and also in desired locations of the infinite domain. The choice of BEM is here fundamental, otherwise a bigger FE model would be needed everytime the analyzed acoustic region gets more distant. In the following, the standard procedure to derive BEM for a closed domain is described. Then, the derivation is extended to infinite regions and to semi-infinite regions, rquiring the adoption of the method of images.
Starting from the divergence theorem Z
FinidS =
Z
⌦
@Fi
@xidV (2.80)
where Fi is a vector field (in particular, the derivation exploits this reformulation of a volume integral into a
boundary integral) and passing through Green’s Identities the Representation Formula is obtained. It defines a function satisfying a particular equation (in the case at hand the Helmholtz Equation) in the required domain by calculating an integral on a surface. This allows the calculation on infinite domains if values are required only on a limited set of points. A derivation of the method for the case of a generic potential is found in [11]: for the acoutic case, it is sufficient to describe formulae to be plugged in the Green’s Second Identity. The velocity potential U of acoustic waves propagating in a fluid is satisfying the linear wave equation
O2U = 1
c2
@2U
@t2 (2.81)
Supposing that U is time-harmonic, e.g. U(x, t) = (x)ei!t with ! = 2⇡f = kc angular frequency, the Helmholtz
equation is obtained
O2 + k2 = 0 (2.82)
where is a reduced velocity potential; from , values for acoustic pressure and its derivative are immediately derived:
• acoustic pressure p = i⇢! , with ⇢ fluid density; • normal derivative of the pressure @p@n = i⇢!v, with v =
@
@n acoustic velocity.
The equation is the same in 2.11 but now the unknown is the velocity potential of the pressure. Let ⌦ be an acoustic region bounded by a closed surface , on which boundary conditions of this type hold:
• prescribed acoustic pressure: = ¯; • prescribed acoustic velocity: @@n = v = ¯v,
and let’s assume that one of the two is known in every point of . Focusing on the acoustic case and starting from the Green’s Second Identity Z
⌦ ( O2G GO2 )dV = Z ( @G @x G @ @x)dS (2.83)
Here G is a fundamental solution of the Helmholtz equation, thus a solution of
O2G(X0, x) + k2G(X0, x) = (X0, x) (2.84)
where X0 is called source point, x stands for the field point and is the Dirac delta function. G represents the
solution of the Helmholtz equation in the case of a point source in X’ and for the three dimensional case it has the form
G = e
ikr
where r =pX0 x. Substituting equation 2.82 in equation 2.83 Z ⌦ (O2G + kG) dV =Z ( @G @x G @ @x)dS (2.86)
Then, introducing property 2.84 in the above gives the boundary integral representation of the problem (X0) =
Z @ (x) @n G(X
0, x)dS Z (x)@G(X0, x)
@x dS (2.87)
This formula is the starting point for the Boundary Element Method part of the Costabel Coupling technique described in 2.3. The singularity of G and @G
@x when r ! 0 is such that the integral around the singular point is
well defined, thus the source point can be moved to the surface (this passage is represented using x0 instead of X0),
introducing a point in which the function to integrate has a discontinuity. The following equation is obtained c(x0) (x0) =
Z @ (x) @n G(x
0, x)dS Z (x)@G(x0, x)
@x dS (2.88)
where the coefficient c is introduced to take into account source points positioned on non-smooth surfaces. BEM is a numerical method for solving boundary integral equations, based on a discretization procedure. It requires two types of approximation: the first is geometrical, involving a partition of the boundary into Nesmall elements j.
This allows to rewrite 2.87 as
(X0) = Ne X j=1 Z j @ (x) @n G(X 0, x)dS (x)@G(X0, x) @x dS (2.89)
The second approximation is functional: and his derivative are discretized using nodal points and shape functions just as in the previous chapter. Using a piecewise constant approximation, the simplest one, the previous becomes:
(X0) = Ne X j=1 @ (xj) @n Z j G(X0, x)dS Ne X j=1 (xj) Z j @G(X0, x) @x dS (2.90)
Now fixing the nodal points as source points the following set of equations is obtained (xi) = Ne X j=1 @ (xj) @n Z j G(xi, x)dS Ne X j=1 (xj) Z j @G(xi, x) @x dS i = 1, ..., Ne (2.91) Now equation 2.91 can be written as
H = GQ (2.92) With Gij = Z j G(xi, x)dS (2.93) ¯ Hij= Z j @G(xi, x) @x dS Hij = ¯Hij+ ij (2.94) where:
• H,G are the influence matrices defined before, calculated using for example Gauss quadrature; • , Qare vectors containing the nodal values of the potential and of its derivative.
Supposing that either one of the two values is known for each node, the linear system can be solved with every unknown on one side:
AX = F (2.95)
As discussed in section 2.4.5, once the vectors , Q are available, it is possible to use the representation formula for points in ⌦: the acoustic pressure function so defined is satisfying the Helmholtz Equation.
Infinite Regions The theory developed until here is valid for closed domains wrapped into the considered surface.
In order to extend it to the infinite region case, as the one discussed in the present work, a fictous external boundary
1 is considered: a sphere with ray going to infinity and centered in one point of the real boundary . Also the
Sommerfield Condition has to be imposed to the solution, guaranteeing that it represents a wave going from the source to infinity and not vice versa. It reads for three dimensional problems:
lim r!1 r ✓@ @r ik ◆ = 0 (2.96)
Formula 2.88 can be written as c(x0) (x0) = Z @ (x) @n G(x 0, x)dS (x)@G(x0, x) @x dS + Z 1 @ (x) @n G(x 0, x)dS (x)@G(x0, x) @x dS1 (2.97) inserting the Sommerfield condition at infinity as @
@r = @ @n = ik Z 1 @ (x) @n G(x 0, x)dS (x)@G(x0, x) @x dS = Z 1 @G(x0, x) @x ikG(x 0, x) (x)dS 1 (2.98)
In order to show that the integral over the infinite plane disappears, the explicit form of G is considered for three dimensional problems @G(x0, x) @x ikG(x 0, x) = eikr 4⇡r ✓ ik 1 r ◆ @r @n+ ik eikr 4⇡r = 1 r @r @n (2.99) Considering that @r @n ! 1 and 1
r! 0 for large values of r, and that satisfies the sommerfield Condition, the result
is obtained. Thus the representation formula is applicable also to infinite regions.
Semi-Infinite Regions: Finally the case of the models in this thesis is considered, in which the presence of an
infinite rigid baffle is simulated. This can be seen as an additional surface H over which @@n = 0and thus where
total reflection occurs for any incident wave. The following integral equation is therefore valid (X0) = Z @ (x) @n G(X 0, x)dS Z + H (x)@G(X 0, x) @x dS (2.100)
The potential is different from zero on the baffle so it is necessary to modify the Green Function so that @G(X0, x)
@x = 0 on H (2.101)
This is achieved using the method of images: a fictous source is added at a point X00 symmetric to X0 with respect
to the baffle, counting for the reflection from the rigid infinite plane. The modified fundamental solution for the three dimensional Helmhlotz equation has form
G(X0, x0) = e
ikr1
4⇡r1
eikr2
4⇡r2 (2.102)
where r1= d(X0, x), r2= d(X00, x). This function satisfies both the Helmholtz equation and the condition 2.101.
Plugging this modified Green’s Function in the representation formula leads to the starting point for BEM in semi infinite regions.
2.3 FEM and BEM coupling: Costabel’s method
In the Comsol models presented in this work, the problem has been tackled starting from the sound source, solving the vibration equation also for the plate. This allows the algorithm to be very general: the vibration of any thin structure could be (knowing its geometry, material and point of excitation) analysed with a FEM model. The analysis sets up the BEM model automatically for the calculation of the radiated pressure field. If used togheter the two techniques require coupling2; a single equation must be set to recover at the same time:
• the values of the pressure field in the FEM region and on the border between the BEM and FEM regions; • the values of ' which will be defined as the restriction of the conormal of the derivative of the pressure
field to the border between the two regions.
The equation discussed in this chapter is the one set up by COMSOL, and the resulting linear system is the one its solver will deal with; however it is not delivering all the values of interest, but only the unknowns in the finite domain and the values necessary to run the BEM algorithm.
The next section is devoted to the description of this method for the generic case. In the next ones the con-tribution of each part of the model for our case will be detailed.
Costabel’s coupling The theory in this chapter has been revised and rewritten in more abstract ways for example
in [27], in which a matrix formulation is devised for the acoustic boundary problem (this paper will be cited later when describing the elements of the Calderon projector for the Helmholtz problem). The method in [27] is inspired directly by Costabel’s work which is referred to because:
• Costabel is the author cited by the software guide itself;
• The method is in the style of what can be found in the engineering literature of BEM, thus fitting better in the explanation of the analysis procedure here presented;
• Costabel’s paper refers in a more direct way to a mixed FEM-BEM formulation and presents an equation clearly distinguishing the terms related to the FEM from the ones related to the BEM.
In [16], comparing this method with Johnson-Nedelc’s one [26], the state of the art for coupling before Costabel’s, the following advantages are highlighted:
1. It is applicable for strongly elliptic systems of second order or more in any dimension;
2. If the boundary value problem and the discretization procedure are symmetric, symmetric matrices are ob-tained;
3. A convergence proof and asymptotic error estimates are generally available, in the case of second order systems (e.g. linear elasticity theory) even for non-smooth domains with the presence of corners and edges.
Proof and details for these properties can be found in section 4 of [16].
The starting problem. The problem is defined in the domain ⌦ which is decomposed in ⌦j, j = 1, 2: while ⌦1
is bounded, ⌦2 con be an infinite region (See FIG 2.3). In the case discussed here, ⌦1 is either the plate or the
near-field acoustic region. ⌦2 is the infinite acoustic region in both cases; the analysis of scattering in an infinite
domain is a typical situation managed using Costabel’s technique. c is defined as the boundary between the two
domains (@⌦1\ @⌦2) and j, j = 1, 2are the remaining boundaries of the respective domains. For second order
problems like the one at hand, boundaries are only required to be Lipschitz continuous, thus corners and edges (like the ones in the 3D rectangular plate model) are allowed; in the general case smoothness is required. Let’s now define the operators appearing in the formulation of the problem:
• Let P1 and P2, in the specific case of this work, be strongly elliptic differential operators of order 2m with
