Andr´es Prieto Aneiros PhD Dissertation SOME CONTRIBUTIONS IN TIME-HARMONIC DISSIPATIVE ACOUSTIC PROBLEMS Departamento de Matem´atica Aplicada Facultade de Matem´aticas ¿C´omo me vas a explicar, di, la dicha de esta tarde, si no sabemos porqu´e fue, ni c´omo, ni de qu´e ha sido, si es pura dicha de nada? En nuestros ojos visiones, visiones y no miradas, no percib´ıan tama˜nos, datos, colores, distancias. Pedro Salinas Contents Preface v I Porous materials 1 1 Porous models 3 1.1 Introduction .................................... 4 1.2 Rigid porous models ............................... 6 1.2.1 Darcy’s like model ............................ 7 1.2.2 Allard-Champoux model ......................... 13 1.3 Poroelastic models ................................ 15 1.3.1 Classical Biot’s model .......................... 15 1.3.2 Non-dissipative poroelastic model (closed pores) ............ 18 1.3.3 Non-dissipative poroelastic model (open pores) ............. 21 1.3.4 Dissipative poroelastic model (open pore) . .............. 22 2 Finite element solution of acoustic propagation in rigid porous media 25 2.1 Introduction .................................... 26 2.2 Models for fluid-porous vibrations ........................ 27 2.3 Associated nonlinear eigenvalue problems .................... 30 2.4 Statement of the weak formulation ....................... 34 2.5 Finite element discretization ........................... 35 2.6 Matrix description ................................ 37 2.7 Numerical results ................................. 38 2.8 Conclusions .................................... 41 3 Finite element solution of new displacement/pressure poroelastic models in acoustics 43 3.1 Introduction .................................... 44 3.2 Statement of the problem ............................ 44 3.3 Weak formulation ................................. 47 3.4 Finite element discretization ........................... 48 3.5 Matricial description ............................... 51 i ii Contents 3.6 Numerical solution of cell problems ....................... 53 3.7 Numerical results ................................. 55 3.8 Conclusions .................................... 57 II Perfectly Matched Layers 61 4 A non reflecting porous material: the Perfectly Matched Layers 63 4.1 Introduction .................................... 64 4.2 Wave equation ................................... 66 4.2.1 Time domain equations .......................... 66 4.2.2 Time-harmonic equations ........................ 67 4.3 Cartesian Perfectly Matched Layers ....................... 67 4.3.1 Time-domain equations .......................... 68 4.3.2 A physical interpretation ......................... 68 4.3.3 Time-harmonic equations ........................ 70 4.4 Plane wave analysis of the PMLs ........................ 73 5 An optimal PML in Cartesian coordinates 79 5.1 Introduction .................................... 80 5.2 The time-harmonic acoustic scattering problem ................ 80 5.3 Finite element discretization. ........................... 82 5.4 Determination of the absorbing function .................... 85 5.5 Comparison with classical absorbing functions ................. 89 5.6 Numerical tests .................................. 92 5.7 Conclusions .................................... 94 5.8 Computation of the element matrices ...................... 95 6 An exact bounded PML in radial coordinates 103 6.1 Introduction ....................................104 6.2 Scattering problem ................................105 6.3 Statement of the PML equation .........................106 6.4 PML fundamental solution ............................108 6.5 PML integral representation formula ......................115 6.6 Addition theorem .................................118 6.7 Existence and uniqueness of solutions for the PML equation .........120 6.8 Coupled fluid/PML problem ...........................122 6.9 Discretization and numerical results .......................125 6.10 Appendix .....................................128 6.10.1 Technical results .............................128 6.10.2 Some classical results about the Hankel functions ...........135 Contents iii III Computational applications on dissipative acoustics 139 7 Validation of acoustic dissipative models 141 7.1 Introduction ....................................142 7.2 Statement of the problem. Mathematical modeling ..............144 7.2.1 The Allard-Champoux model ......................144 7.2.2 The wall impedance model ........................145 7.2.3 Computing the wall impedance .....................146 7.3 Planar unbounded wall ..............................148 7.3.1 Plane waves with oblique incidence ...................148 7.3.2 Spherical waves ..............................152 7.4 Curved wall ....................................156 7.4.1 The Perfectly Matched Layer ......................157 7.4.2 Finite-element discretization .......................159 7.4.3 Verification of the numerical methods ..................161 7.4.4 Numerical validation of the wall impedance model for non-planar ge- ometries ..................................164 7.5 Conclusions ....................................166 8 Numerical simulation of locally reacting panels 169 8.1 Introduction ....................................170 8.2 Modelling the panel ................................171 8.2.1 Wall-like impedance ...........................172 8.2.2 Porous veil and micro-perforated plates .................173 8.2.3 Thin porous layer .............................174 8.2.4 Multilayer panel with a rigid back ....................175 8.3 Variational formulation ..............................176 8.3.1 Wall-like impedance ...........................177 8.3.2 Porous veil .................................178 8.3.3 Thin porous layer .............................178 8.3.4 Multilayer panel with a rigid back ....................179 8.4 Finite element discretization ...........................179 8.5 Numerical validation ...............................180 8.6 Numerical results for an absorbing box for reducing noise in rooms ......185 8.7 Conclusions ....................................187 Further research 191 Acknowledgments 192 Resumo en galego 195 Bibliography 203 iv Contents Preface Nowadays the numerical methods have a fundamental role as a tool to reduce the design time and developed costs of new products in fields such as Aerospace, Mechanic, Naval Engineering, etc. From this point of view, sometimes the quick evolution of computers is not enough in all the cases to solve the real-life problems of engineering efficiently and in a practical time. Hence the computational capability of actual computers has to be completed with efficient and renewed numerical methods. One of the problems which has become more relevant from a social point of view is the reduction of acoustic pollution produced by cars, planes, air-conditioned systems, etc., as it is reflected in national and European laws, which are more restrictive in the last years. In this context, it arises the necessity to solve more complex acoustic propagation problems which cannot be tackled with by numerical techniques based on classical methods. Obviously, prototype essays are fundamental to asses the feasibility of the proposed technologies, nevertheless the high cost of prototype production makes necessary that this kind of experiments has to be done in an advanced phase of design, with a product close to the final one. These two factors are the reasons why computational acoustics becomes a scientific field of great importance nowadays and numerical simulation is a relevant tool to do analysis of products and to study innovative systems with comfortable acoustic properties with competitive cost and saving developing time. The sophistication of the acoustic materials related with the real-life problems in the last decades have caused that the mathematical models used to solve acoustic propagation problems have been enriched from a mathematical point of view, and consequently they require the use of new advanced computational and numerical techniques. Among these new models, we focus in this thesis on those derived from the porous materials and, from a computational point of view, on the Perfectly Matched Layer (PML) technique, which allows solving numerically acoustic propagation problems in unbounded domains. In any case, the complexity of the acoustic models and the geometrical configuration of the problems require their resolution to be done by numerical methods as, for instance, the finite element methods. The study presented through this thesis is in the context of the frequency domain, i.e., under the assumption of time-harmonic dependency of the time variable of the acoustic fields. In fact, our attention is focused in acoustic propagation problems in the low range of frequencies, where the discretization by finite element methods is suitable and non excessively expensive from a numerical point of view. v vi Preface In the present dissertation thesis, we distinguish three parts well defined and different but joined by a common topic: the study of acoustic propagation problems in bounded or unbounded domains which involves dissipative material. In some cases the main topic is the acoustic behavior of the porous material, but in other cases, the aim of our study is the use of dissipative materials as a numerical tool to deal with other problems, such as the truncation of unbounded computational domains. So, in the first part of this thesis, we focus our attention on the computation of