The vibration reduction index of heavy junctions is predicted by means of a model based on spectral finite elements. This is equivalent to a finite element method but faster and with smaller computational costs. This advantage is used in order to perform a parametric analysis of the vibration reduction index for several junction types: T-shaped, L-shaped and +-shaped. The influence of several parameters such as: damping, junction dimensions or the mass ratio on the vibration reduction index is observed. The study is focussed to provide data and guidelines oriented to the EN-12354 design method for flanking transmission in buildings.
A numerical technique for solving scattering problems is presented. It is
based on a boundary integral equation idea, so the unknowns are localized on the contour (in 2D case) or the surface (in 3D case) of the scattering object. Two major difficulties of traditional boundary integral methods (the appearance of spurious resonances and the necessity to perform numerical integration of singular functions) are overcome by studying the problem in an approximate discrete formulation from the very beginning. The space is filled by cubic blocks, and the shape of the scatterer is formed by a set of blocks removed from the space. Thus, the formulation of the problem is discrete and the continuous Green’s function is substituted by a discrete mesh Green’s function. An analogue of combined field boundary integral equation (CFIE) is developed for this formulation.
An automatic methodology for identifying SEA (statistical energy analysis) subsystems within a vibroacoustic system is presented. It consists in dividing the system into cells and grouping them into subsystems via a hierarchical cluster analysis based on the problem eigenmodes. The subsystem distribution corresponds to the optimal grouping of the cells, which is defined in terms of the correlation distance between them. The main advantages of this methodology are its automatic performance and its applicability both to vibratory and vibroacoustic systems. Moreover, the method allows the definition of more than one subsystem in the same geometrical region when required. This is the case of eigenmodes with a very different mechanical response (e.g. out-of-plane or in-plane vibration in shells).
A new numerical method for solving wave diffraction problems is given. The method is based on the concept of boundary elements; i.e., the unknown values are the field values on the surface of the scatterer. An analog of a boundary element method rather than a numerical approximation of the initial (continuous) problem is constructed for an approximate statement of the problem on the discrete lattice. Although it reduces the accuracy of the method, it helps to simplify the implementation significantly since the Green functions of the problem are no longer singular. In order to ensure the solution to the diffraction problem is unique (i.e., to suppress fictitious resonances), a new method is constructed similarly to the CFIE approach developed for the classical boundary element method.
The final publication is available at Springer via http://dx.doi.org/10.1134/S2070048214020082
La modelización de problemas vibroacústicos en el ámbito de la edificación supone un desafío debido al gran tamaño de los dominios y al amplio rango frecuencial requerido por las normativas. Las técnicas numéricas estándares, como por ejemplo el método de los elementos finitos (MEF), fallan al tratar de alcanzar las frecuencias más altas, puesto que el tamaño de elemento requerido es muy inferior a las dimensiones del problema y el coste computacional asociado es excesivo para tratarse de un cálculo tan cotidiano.El análisis estadístico de energía (SEA) es un marco de análisis de problemas vibroacústicos basado en el comportamiento de las ondas a altas frecuencias. Trata directamente con magnitudes promediadas, tal y como requieren las normativas, y su coste computacional es muy bajo. Sin embargo, presenta numerosas limitaciones a la hora de analizar estructuras reales. Habitualmente la definición del modelo SEA necesita ser complementada con experimentos u otros datos añadidos.Esta tesis se centra en la modelización de problemas de acústica de la edificación con un coste computacional razonable. En ese sentido se han seguido dos líneas fundamentales de investigación.En la primera parte de la tesis se analiza el potencial uso de simulaciones numéricas para extender la aplicabilidad del SEA. En particular, se tratan tres aspectos diferentes: en primer lugar, se desarrolla una metodología sistemática para la estimación de factores de acoplamiento a partir de simulaciones numéricas. Estos factores se estiman a partir de pequeñas simulaciones deterministas y posteriormente se aplican para la resolución de problemas mayores con SEA. En segundo lugar, se presenta un modelo basado en el SEA para acoplamientos no conservativos, así como una estrategia para obtener los factores de acoplamiento conservativos y no conservativos a partir de simulaciones numéricas. Finalmente, se propone una metodología para la identificación de subsistemas SEA con análisis modal. Esta técnica consiste en realizar un análisis cluster basado en los modos propios del problema, y permite la detección de subdivisiones óptimas para dominios complejos, incluso si varios subsistemas coexisten en la misma región geométrica.En la segunda parte de la tesis, se analiza la transmisión sonora a través de paredes dobles desde diferentes puntos de vista, por ser éste un ejemplo paradigmático de las complejidades asociadas a las simulaciones vibroacústicas. En primer lugar, se presenta una compilación de modelos clásicos para este problema. A continuación, se propone la utilización del método de las capas finitas como una nueva manera de discretizar el campo de presiones en la cavidad interior de las paredes dobles, especialmente cuando esta se encuentra parcialmente llena con material absorbente. Este método combina una discretización de tipo MEF en la dirección perpendicular a la pared con funciones trigonométricas en las dos direcciones coplanarias con la misma. El coste computacional de esta técnica es inferior al del MEF, pero también permite la aplicación de las condiciones de continuidad y equilibrio entre capas fluidas. Seguidamente, esta técnica se compara tanto con datos experimentales como con otros modelos predictivos, con objeto de verificar la influencia de distintas simplificaciones habituales en estos modelos.Por último, se presenta la combinación de métodos deterministas y estadísticos como una posible solución para la modelización de problemas vibroacústicos compuestos por paredes dobles y otros elementos. El análsis global se realiza con SEA, pero se utilizan simulaciones numéricas de pequeñas partes del problema para obtener los parámetros necesarios. La combinación de ambas técnicas permite la realización de simulaciones con un coste computacional razonable.
Modelling vibroacoustic problems in the field of building design is a challenging problem due to the large size of the domains and the wide frequency range required by regulations. Standard numerical techniques, for instance finite element methods (FEM), fail when trying to reach the highest frequencies. The required element size is too small compared to the problem dimensions and the computational cost becomes unaffordable for such an everyday calculation.
Statistical energy analysis (SEA) is a framework of analysis for vibroacoustic problems, based on the wave behaviour at high frequencies. It works directly with averaged magnitudes, which is in fact what regulations require, and its computational cost is very low. However, this simplified approach presents several limitations when dealing with real-life structures. Experiments or other complementary data are often required to complete the definition of the SEA model.
This thesis deals with the modelling of building acoustic problems with a reasonable computational cost. In this sense, two main research lines have been followed. In the first part of the thesis, the potential of numerical simulations for extending the SEA applicability is analysed. In particular, three main points are addressed: first, a systematic methodology for the estimation of coupling loss factors from numerical simulations is developed. These factors are estimated from small deterministic simulations, and then applied for solving larger problems with SEA. Then, an SEA-like model for non-conservative couplings is presented, and a strategy for obtaining conservative and non-conservative coupling loss factors from numerical simulations is developed. Finally, a methodology for identifying SEA subsystems with modal analysis is proposed. This technique consists in performing a cluster analysis based on the problem eigenmodes. It allows detecting optimal SEA subdivisions for complex domains, even when two subsystems coexist in the same region of the geometry.
In the second part of the thesis, the sound transmission through double walls is analysed from different points of view, as a representative example of the complexities of vibroacoustic simulations. First, a compilation of classical approaches to this problem is presented. Then, the finite layer method is proposed as a new way of discretising the pressure field in the cavity inside double walls, especially when it is partially filled with an absorbing material. This method combines a FEM-like discretisation in the direction perpendicular to the wall with trigonometric functions in the two in-plane directions. This approach has less computational cost than FEM but allows the enforcement of continuity and equilibrium between fluid layers. It is compared with experimental data and also with other prediction models in order to check the influence of commonly assumed simplifications.
Finally, a combination of deterministic and statistical methods is presented as a possible solution for dealing with vibroacoustic problems consisting of double walls and other elements. The global analysis is performed with SEA, and numerical simulations of small parts of the problem are used to obtain the required parameters. Combining these techniques, a realistic simulation of the vibroacoustic problem can be performed with a reasonable computational cost.
A study on the optimal procedure for obtaining SEA (statistical energy analysis) coupling loss factors (CLF) numerically is presented. The energies of an SEA system with two subsystems (one excited, the other one unexcited) are obtained from deterministic numerical simulations. Three different ways of isolating the CLF are explored: from the power balance of the excited subsystem (first approach) or the unexcited subsystem (second approach) and from the power transmitted through the connection (third approach). An error propagation analysis shows that the first approach is unreliable and that the second approach is the best option. As application examples, the CLF between some typical building structures is computed. These examples illustrate the potential of the estimated CLFs to solve larger problems with SEA and show the influence of the type of excitation on the coupling loss factor estimation. Finally, a simplified technique to account for the effect of studs in double walls with SEA is presented.
The main challenge for models of building acoustics is being able to consider all the geometrical and physical details of real structures with a reasonable computational cost for high frequencies. The SEA (Statistical Energy Analysis) framework is suitable for these frequencies, but presents some difficulties for dealing with complex structural configurations. For instance, modelling absorbing materials with SEA is an open issue, since they are neither reverberant subsystems nor conservative couplings. In this work, a model to account for absorbing materials with a SEA-like approach is performed. It is obtained by analogy with an electrical circuit. This approach is combined with numerical simulations in order to solve vibroacoustic problems in real structural configurations (including complex geometries or dissipative connections) throughout the entire frequency range required by regulations. The proposed technique is applied to modelling the sound insulation of double walls. These walls consist of two leaves of plasterboard connected through metallic studs and filled with a layer of absorbing material. The combination of numerical simulations and SEA arises as a good technique for modelling the acoustic behaviour of real life structures with an affordable computational cost.
In this work an energy model for the acoustic insulation of absorbing ma-
terials is shown. This model is an extension of Statistical Energy Analysis (SEA)  in
order to account for the effect of non-conservative connections [2, 3].
The energy-based approach allows to solve sound insulation problems in large domains
(such as those in building acoustics) in an efficient way for the whole frequency range
required by regulations (50-5000 Hz). In particular, this approach is applied here for the
study of the insulating behaviour of an absorbing layer (mineral wool) filling the cavity of
a double wall. The absorbing layer is considered as a non-conservative connection between
the two leaves of the wall.
This model is combined with detailed numerical computations to obtain the loss factors
associated to the connection. With these parameters, a combined system including the
transmission between rooms and double walls can be stated.
Obtained results show that absorbing layers can be modelled as non-conservative cou-
plings and incorporated in an SEA-like system to compute the sound insulation in buildings
The transmission of sound through slits and openings between cuboid-shaped rooms is analysed. A deterministic model that describes the pressure fields inside the rooms in terms of eigenfunctions and uses the Dirichlet-to-Neumann technique in order to reproduce the slit effect is presented. An efficient formulation of the problem is obtained thanks to the splitting of the original domain into three domains: sending room, slit, receiving room. The geometry and boundary conditions of the problem can be modelled in detail like in an element-based
numerical technique (such as the finite element method) but with smaller computational costs. The model is compared with numerical solutions, existent models and published experimental data. Afterwards it is used to analyse some aspects such as the influence of slit dimensions, opening position, room properties (dimensions and absorption) that cannot be taken into account with the available models. These usually suppose that the slit or opening connects two unbounded acoustic domains.
Modelling absorbing materials with statistical energy analysis (SEA) is an open issue. They are neither
reverberant subsystems nor conservative couplings. The absorbing material layers located inside the
cavities of double walls should be treated as non-conservative couplings between the wall leaves.
However, the standard SEA formulation cannot take into account non-conservative couplings.
In this work, an equivalent circuit analogy is used to deduce how to introduce these couplings in an
SEA-like system. Besides, a technique for obtaining the SEA-like factors associated to a double wall
filled with absorbing material is presented. These factors are computed from numerical simulations of
the vibroacoustic leaf-absorbing material-leaf system and applied for solving larger problems with
Double walls usually consist of two leaves of material connected by steel studs. Aside from improving
the structural performance, studs create a vibration transmission path which connects the two leaves.
There is interest in reliable models of the acoustic performance of these structures, for the frequency
range required in regulations. Statistical energy analysis allows reaching high frequencies with a low
computational cost. However, the best SEA approach for modelling double walls is not clear in the
literature. The cavity may be considered as a subsystem or treated as a connecting device between the
two leaves. The effect of the cavity is also often neglected compared to the coupling provided by the
studs. In this work, numerical techniques are used to evaluate these approaches and to define a
combined deterministic–statistical approach that accounts for all the transmission phenomena.
The finite layer method (FLM) is presented as a discretisation technique for the computation of noise transmission through double walls. It combines a finite element method (FEM) discretisation in the direction perpendicular to the wall with trigonometric functions in the two in-plane directions. It is used for solving the Helmholtz equation at the cavity inside the double wall, while the wall leaves are modelled with the thin plate equation and solved with modal analysis. Other approaches to this problem are described here (and adapted where needed) in order to compare them with the FLM. They range from impedance models of the double wall behaviour to different numerical methods for solving the Helmholtz equation in the cavity. For the examples simulated in this work (impact noise and airborne sound transmission), the former are less accurate than the latter at low frequencies. The main advantage of FLM over the other discretisation techniques is the possibility of extending it to multilayered structures without changing the interpolation functions and with an affordable computational cost. This potential is illustrated with a calculation of the noise transmission through a multilayered structure: a double wall partially filled with absorbing material.
Double walls are increasingly used in construction. Due to this, there is interest in reliable models of their sound insulation for the frequency range reguired in regulations (50-5000Hz). These models can be either statistical or deterministic. In this work, the finite layer method (FLM) is presented as a numerical technique for solving the problem in a deterministic way. it is used for discretising the Helmholtz equation in the cavity and combines a finite element method (FEM) discretisation in the direction perpendicular to the wall with trigonometric functions in the two in plane directions. The FLM exploits the simple geometry of the double wall and accounts for all its boundary and interface conditions with a reasonable computational cost. The statistical energy analysis (SEA) is a more suitable framework of analysis for vibroacoustic problems in large domains such as buildings. However, the best SEA approach for modelling double walls is not clear in the literature. The cavity is considered as a subsystem or treated as a connecting device between the two leaves depending on the autor. The finite layer method is a used to evaluate the performance of these two approaches, concluding that both considerations have to be taken into account together to reproduce the real behaviour. Finally, the FLM is used to define a combined deterministic energy based approach to deal with this kind of problems.
The finite strip method, widely employed in structural mechanics, is extended to solve acoustic and vibroacoustic problems. The acoustic part of the formulation, including how to handle the most typical acoustic boundary conditions and the fluid structure interaction, is presented. Several realistic problems where the three-dimensional domain of interest has extrusion symmetry are solved. These examples illustrate the advantages of the method: it has smaller computational costs than the finite element method and consequently the analyzed frequency range can be increased.
Nouri, N.; Ziaei-Rad, N.; Díaz-Cereceda, C.; Poblet-Puig, J.; Rodriguez-Ferran, A. Congress on Numerical Methods in Engineering p. 1-12 Presentation's date: 2011-06-15 Presentation of work at congresses
Díaz-Cereceda, C.; Hetherington, J.; Poblet-Puig, J.; Rodriguez-Ferran, A. Journal of sound and vibration Vol. 330, num. 12, p. 2801-2817 DOI: 10.1016/j.jsv.2010.12.019 Date of publication: 2011-01-17 Journal article
Vibration transmission through structural connections is modelled in a deterministic way by means of modal analysis. This model is used first to study the effect of elastic joints across the floor in the transmission of impact noise. They are an effective means of reducing impact noise propagation, and can almost eliminate it for small values of the joint stiffness. The method is also used to study the acoustic relevance of studs in lightweight floor transmission. Different ways of modelling the studs are presented and compared. For the examples developed, the best option is to use springs for modelling the studs rather than more complex models involving springs and beams. Also the different behaviour of point and line connections is verified, as well as the influence of the position of the studs.
The vibroacoustic equations can be solved by means of the finite element method. A
discretisation of the structure and the acoustic domains is required and highly influences the quality of the numerical solution. There exist meshing criteria (a priori error estimators) for the case of the Helmholtz equation but these studies have not focused their attention in the case of the vibroacoustic problem. The fluid structure interaction represents a new source of numerical errors and meshes in the interaction zone should be designed by not only taking into account the physical properties of the acoustic medium but also the mechanical properties of the structure. The goal of the work is to obtain an a priori error estimation criterion for the vibroacoustic problem and Illustrate its efficiency by means of numerical experiments.
Sound transmission through partitions can be modeled as an acoustic fluid–elastic structure interaction problem. The block Gauss–Seidel iterative method is used in order to solve the finite element linear system of equations. The blocks are defined, respecting the fluid and structural domains. The convergence criterion is analyzed and interpreted in physical terms by means of simple one-dimensional problems. This analysis highlights the negative influence on the convergence of a strong degree of coupling between the acoustic domains and the structure. A selective coupling strategy has been developed and applied to problems with strong coupling (e.g. double walls).
Steel studs are used in double walls to provide structural stability. This creates a vibration transmission path between leaves that can often be more critical than the airborne path through the cavity. Some of the existing models for sound transmission consider the studs as elastic springs. The spring stiff ness may be taken as the cross-section elastic stiff ness of the stud, but this leads to an underestimation of the vibration transmission. A procedure to obtain more accurate parameters to be used in vibration and sound insulation models is presented. The results show that they must be obtained from dynamic models and/or experiments.
The radiation efficiency of a structural element is required by some models in order to predict its sound insulation. A common assumption is that the radiation on both sides of the element is the same. This is not true for asymmetrical structural elements like lightweight floors consisting of a beam-supported flat board. The radiation efficiency is larger on the beam side, because the beams act as exciters and increase the pressure level in the room. These different radiation efficiencies are calculated here for a two-dimensional cross-section by using finite elements and boundary elements. The obtained preliminary results illustrate that considering a single radiation efficiency can be a source of errors and that further investigation is required in order to improve predictions.
Numerical models for the calculation of sound transmission in double walls are presented. The finite element method (FEM) or analytical solutions (for rectangular domains) are used for the acoustic part of the problem while the wall vibration is solved by means of structural finite elements. The vibroacoustic problem is formulated in the frequency range: the acoustic domains (rooms) are described by the Helmholtz equation, the absorbent materials as an equivalent fluid and the structures by means of dynamic linear elasticity. The acoustic and structural parts of the problem are coupled. The influence of the stiffness and spacing of the studs on the performance of lightweight walls is studied. The studs have been modelled with beam finite elements or by means of mechanical constraints. The effect of the boundary conditions of the walls and its dimensions are also analyzed.
In this paper, the characterization of metallic studs used to mount lightweight double wall systems is studied both experimentally and numerically. The metallic studs are usually considered by introducing translational and rotational springs to couple the plasterboards composing the double wall. Therefore, the characterization involves determining these spring characteristics. The performance of this type of lightweight double wall in terms of sound transmission is presented in a companion paper. Different experimental setups have been investigated to determine the equivalent translational and rotational spring values. These experimental setups are described and involve the measurement of an input mobility. A finite element model of the laboratory tests has been developed. Shell and massive finite elements are employed in order to reproduce the experimental setups. A comparison of the measured and numerical results is shown. The FEM modelling is intended to help in developing new type of studs for double walls in order to obtain better sound transmission performance.
Two applications of numerical techniques to the study of sound transmission are shown.
In the first one the sound reduction index of a double wall in the low frequency range is computed solving a vibro acoustic problem by means of the finite element method. It is a typical situation where the uncoupling of the problem is not possible. In the other, the structural spectral element method is used in order to compute the vibration transmission factor. It is a required parameter to compute the indirect transmissions of sound by means of the standard EN-12354.
The main numerical techniques actually employed in the modelling of sound transmission are rapidly reviewed. The most important restricting aspects for the application of the finite element method to vibro acoustic problems are presented. They are the cause of the restriction of the use of FEM to the low frequency range. However, several new numerical techniques (most of them concentrated in the invention of new methods for the Helmholtz equation in the
mid frequency range) which use a priori known information about the solution are being developed. The new characteristics of them are mentioned and a small example of a structural calculation shown.