We prove that the solution of any linear mechanical system can be expressed as a linear combination of signal transmission paths. This is done in the framework of the Global Transfer Direct Transfer (GTDT) formulation for vibroacoustic problems. Transmission paths are expressed as powers of the transfer matrix. The key idea of the proof is to generalise the Neumann series of the transfer matrix --which is convergent only if its spectral radius is smaller than one-- into a modified Neumann series that is convergent regardless of the eigenvalues of the transfer matrix. The modification consists in choosing the appropriate combination coefficients for the powers of the transfer matrix in the series. A recursive formula for the computation of these factors is derived. The theoretical results are illustrated by means of numerical examples. Finally, we show that the generalised Neumann series can be understood as an acceleration (i.e. convergence speedup) of the Jacobi iterative method.
Real mechanical systems are usually complex. Their modelling with discretisation techniques involve a large number of degrees of freedom/unknowns. This leads to high computational costs especially at high frequencies. An alternative, the statistical methods, are sometimes limited by strong behavioural requirements. So, sub-structuring can be considered very often as a good alternative. The present research studies the coupling between subsystems from the point of view of transmission path analysis. The interest is focused on the signal transmission rather than the energy distribution. A subsystem identification method is proposed. It is based on the expression of the solution in terms of the powers of the transfer matrix. This is related with the description of high-order paths which are more affected by damping and system properties. Consequently, the identification of subsystems can be more easily done. The method provides a quantification of the degree of coupling between subsystems and of the error caused by the detachment of a system part from the global system in the modelling phase. The output of the presented technique can be a sub-system definition valid for the SEA method or relevant information to be used in the design of measurement procedures.
Macroscopic deformations in embryonic soft tissues are due to the intra-cellular remodelling
and cell intercalation. We here present a computational approach that can handle the
two types of deformations, and also take into account the active cell response. The model resorts
to cell-centred techniques, where particles represent cell nuclei, and to vertex models, where the
vertices represent cell boundaries. This hybrid approach allows to consider separately intracellular
and inter-cellular forces, and at the same time impose cell incompressibility.
In the proposed model, the cell boundaries (defined by vertices) and cell nuclei (or cellcentres)
networks are coupled through an interpolation scheme, which is eventually relaxed in
order to smooth the cell boundaries. We show that this coupling between the two networks
modifies the equilibrium equations and stabilises the vertex network. Incompressibility is implemented
through a penalty method. The resulting model can be implemented in two- and
three-dimensions, and is complemented with active rheological models.
We apply the model to simulate the stretching and relaxation of cell monolayers, and to
simulate wound healing process in the wing disc of Drosophila fly embryo. We show that the
numerical results agree with the experimental measurements.
We have recently proposed a continuous-discontinuous model of material failure . The continuous
model consists of a non-local elastic-damage model. Non-locality is introduced by
working with two displacement fields: the classical field of local displacements and a non-local,
gradient-enriched displacement field. This choice does regularise softening and has a number
of advantages over standard non-local models based on a gradient-enriched equivalent strain
(clear physical meaning of boundary conditions on the non-local variable and correct damage
initiation at the crack tip, among others).
The continuous model can be used to simulate damage inception and progressive material failure.
In many applications, however, an explicit representation of cracks (i.e. discontinuities) is
required. This is the case, for instance, in the simulation of fracking or structural damage due
to leakage of an aggressive agent, where the actual geometry of the crack plays a relevant role
in the fluid-structure interaction.
Penalty methods are a common technique to impose constraints in finite element (FE) analysis. In statics,
penalisation consists in adding virtual stiffness to the FE system. However, this approach is not suitable for dynamics,
because the increase in the stiffness leads to a decrease in the critical time step of conditionally stable
explicit time-integration schemes. To avoid this, we advocate the use of bipenalty methods, i.e. the simultaneous
use of stiffness and mass penalties. By choosing appropriately the stiffness and mass penalty parameters,
the maximum eigenfrequency of the FE system (and, hence, the critical time step) is not changed. This is
shown by analysing two generalised eigenvalue problems: the unpenalised problem prior to the imposition of
constraints and the bipenalised problem. We provide expressions of the CPR (critical penalty ratio) for different
types of finite elements (bars, beams, plane strain/stress quadrilaterals…). To illustrate the generality and
versatility of the bipenalty method, we show a number of numerical examples involving one or several onepoint
or multi-point constraints in a variety of applications: restrained displacements in structural dynamics, a
one-dimensional contact-impact formulation and a two-dimensional crack propagation analysis.
Mosafa, P.; Asadipour, N.; Millán, D.; Rodriguez-Ferran, A.; Muñoz, J.J. International Conference on Computational and Mathematical Biomedical Engineering p. 536-539 Data de presentació: 2015-06-02 Presentació treball a congrés
Macroscopic deformations in embryonic soft tissues are due to the intra-cellular remodelling and cell intercalation. We here present a computational approach that can handle the two types of deformations, and also take into account the active cell response. The model resorts to cell centred techniques, where particles represent cell nuclei, and to vertex models, where the vertices represent cell boundaries. This hybrid approach allows to consider separately intra-cellular and inter-cellular forces, and at the same time impose cell incompressibility. The model is applied to simulate the active stretching of epithelium.
Askes, H.; Rodriguez-Ferran, A.; Hetherington, J. International journal for numerical methods in engineering Vol. 101, num. 11, p. 809-824 DOI: 10.1002/nme.4819 Data de publicació: 2015-03-16 Article en revista
In this paper, the effects of element shape on the critical time step are investigated. The common rule-of-thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four-noded bilinear quadrilaterals and three-noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements. Copyright (c) 2014 John Wiley & Sons, Ltd.
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 continuous-discontinuous strategy for the simulation of failure is presented. The continuous bulk is regularised by means of a gradient-enhanced damage model, where non-locality is introduced at the level of displacements. As soon as the damage parameter is close or equal to 1, a traction-free crack is introduced. To determine the direction of crack growth, a new criterion is proposed. In contrast to traditional techniques, where mechanical criteria are used to define the crack path, here, a geometrical approach is used. More specifically, given a regularised damage field D(x), we propose to propagate the discontinuity following the direction dictated by the medial axis of the isoline (or isosurface in 3D) D(x) = D*. The proposed approach is tested on different two-dimensional and three-dimensional examples that illustrate that this combined methodology is able to deal with damage growth and material separation. Copyright (c) 2014 John Wiley & Sons, Ltd.
This paper presents a cell-centred model for the simulation of planar and curved multicellular soft tissues. We propose a computational model that includes stress relaxation due to cell reorganisation (intercellular connectivity changes) and cytoskeleton remodelling (intracellular changes). Cells are represented by their cell centres, and their mechanical interaction is modelled through active non-linear elastic laws with a dynamically changing resting length. Special attention is paid to the handling of connectivity changes between cells, and the relaxation that the tissues exhibit under these topological changes. Cell--cell connectivity is computed by resorting to a Delaunay triangulation, which is combined with a mapping technique in order to obtain triangulations on curved manifolds. Our numerical results show that even a linear elastic cell--cell interaction model may induce a global non-linear response due to the reorganisation of the cell connectivity. This plastic-like behaviour is combined with a non-linear rheological law where the resting length depends on the elastic strain, mimicking the global visco-elastic response of tissues. The model is applied to simulate the elongation of planar and curved monolayers.
Perez, L.; Campos, A.; Lascano, S.; Oller, S.; Rodriguez-Ferran, A. Mathematical problems in engineering Vol. 2014, p. 782079-1-782079-12 DOI: 10.1155/2014/782079 Data de publicació: 2014-12 Article en revista
The softening elastoplastic models present an unsuitable behavior after reaching the yield strength: unbounded strain localization. Because of the material instability, which is reflected in the loss of ellipticity of the governing partial differential equations, the solution depends on the discretization. The present work proposes to solve this dependency using the meshless Finite Points Method. This meshfree spatial discretization technique allows enriching the governing equations using gradient's plasticity and introducing an internal length scale parameter at the material model in order to objectify the solution.
A new continuous-discontinuous strategy to describe failure of quasi-brittle materials is presented. For the early stages of the failure process, a gradient-enhanced model based on smoothed displacements is employed. As soon as the damage parameter exceeds a critical value Dcrit<1, a cohesive crack is introduced. A new criterion to estimate the energy not yet dissipated by the bulk when switching models-from continuous to continuous-discontinuous-is proposed. Then, this energy is transferred to the cohesive crack thus ensuring that the continuous and the continuous-discontinuous strategies are energetically equivalent. Compared to other existing techniques, this new strategy accounts for the different unloading branches of damage models and thus, a more accurate estimation of the energy that has to be transferred is obtained. The performance of this technique is illustrated with one- and two-dimensional examples.
En aquesta tesi, presentem una nova estratègia per tal de descriure el procés de fallida de materials quasi-fràgils, com ara el formigó. Típicament la simulació numèrica d'aquest procés s'ha dut a terme mitjançant models de dany o models de fractura. Els primers -models continus- descriuen la fractura com un procés de localització de deformacions on el dany creix i es propaga. Els models de fractura, en canvi, són models discontinus que introdueixen de manera explícita discontinuïtats en el camp de desplaçaments. Recentment s'han proposat estratègies que combinen aquestes dues teories clàssiques. Tot i que aquestes formulacions alternatives permeten simular millor el procés de fallida, encara queden alguns aspectes per aclarir, especialment pel que fa al canvi de models -de l'estratègia contínua a la discontínua.En aquesta tesi es presenta una nova estratègia contínua-discontínua. El nostre principal objectiu és proposar nous mètodes per tal de resoldre tres de les dificultats que presenten aquests models combinats: (1) solucionar la dependència patológica de la malla d'elements finits que presenten els models locals amb reblaniment; (2) determinar la trajectòria de la fissura i (3) assegurar-se que el canvi de models -delcontinu al discontinu- es fa de manera que les dues estratègies siguin energèticament equivalents.En primer lloc, ampliem l'ús -per tal de poder simular problemes dos i tres dimensionals- d'una estratègia alternativa que regularitza el reblaniment de les lleis de tensió-deformació. Aquí la no-localitat s'introdueix a nivell del camp de desplaçaments i no a través d'una variable interna com succeeix en les formulacions estàndards. Per aquest motiu, proposem noves condicions de contorn combinades per l'equació de regularització (pel camp de desplaçaments suavitzat). Tal com s'observa en diferents exemples dos i tres dimensionals, aquestes condicions permeten simular de manera físicament realista les primeres etapes del procés de fallida.En segon lloc, presentem una nova formulació combinada on les fissures es propaguen a través del medi regularitzat. Per tal de definir la trajectòria d'aquestes fissures, utilitzem un criteri geomètric, a diferència dels criteris mecànics clàssics. En particular, sigui D(x) un camp regularitzat de dany, les discontinuïtats es propaguen seguint la direcció marcada per l'eix mitjà de la isolínia (o isosuperfície mitjana en 3D) D(x) = D*. És a dir, utilitzem aquí aquesta eina geomètrica -molt emprada en d'altres aplicacions com ara l'anàlisi d'imatges, la visió artificial o la generació de malles- per tal de propagar les fissures. En aquest cas, donem també exemples dos i tres dimensionals.Finalment, proposem un nou criteri per tal d'estimar l'energia que l'estructura encara no ha dissipat en el moment en que canviem de model, per tal que pugui ser transferida a la fissura cohesiva. D'aquesta manera, s'assegura que l'estratègia contínua i la contínua-discontínua siguin energèticament equivalents. En comparació amb d'altres tècniques, aquesta estratègia té en compte les diferents branques de descàrrega dels models de dany i permet estimar de manera més precisa l'energia que cal transmetre. Per tal de mostrar aquest balanç energètic es duen a terme diferents exemples en una i dues dimensions.
A new strategy to describe failure of quasi-brittle materials -concrete, for example- is presented. Traditionally, numerical simulation of quasi-brittle failure has been tackled from two different points of view: damage mechanics and fracture mechanics. The former, which belongs to the family of continuous models, describes fracture as a process of strain localisation and damage growth. The latter, which falls in the family of discontinuous models, explicitly introduces displacement discontinuities. Recently, some new approaches that merge these two classical theories have been devised. Although these combined approaches allow a better characterisation of the whole failure process, there are still some issues that need to be addressed, specially regarding the model switching -from the continuous to the continuous-discontinuous strategy.
The goal of this thesis is to present a new contribution in this direction. Our main concern is to properly account for the three main difficulties that emerge when dealing with combined strategies: (1) the pathological mesh-dependence exhibited by local softening models needs to be corrected; (2) the crack-path location has to be determined and (3) the switching from the continuous to the continuous-discontinuous strategy should be done in such a way that the two approaches are energetically equivalent.
First, we extend the applicability to a two- and three-dimensional setting of an alternative approach to regularise strain-softening -where non-locality is introduced at the level of displacements rather than some internal variable. To this end, we propose new combined boundary conditions for the regularisation equation (for the smoothed displacement field). As illustrated with different two- and three-dimensional examples, these boundary conditions allow to obtain physical realistic results for the first stages of the failure process.
Second, we present a new combined formulation that allows the propagation of cracks through a regularised bulk. To define the crack-path, instead of the classical mechanical criteria, we propose to use a geometrical criterion. More specifically, given a regularised damage field D(x), the discontinuity propagates following the direction dictated by the medial axis of the isoline (or isosurface in 3D) D(x) = D*. That is, a geometric tool widely used for image analysis, computer vision applications or mesh generation purposes is used here to locate cracks. We illustrate the capabilities of this new approach by carrying out different two- and three-dimensional numerical tests.
Last, we propose a new criterion to estimate the energy not yet dissipated by the bulk when switching models, so it can be transferred to the cohesive crack. This ensures that the continuous and the continuous-discontinuous strategies are energetically equivalent. Compared to other existing techniques, we present a strategy that accounts for the different unloading branches of damage models thus better estimating the energy that has to be transferred. We illustrate the performance of this technique with one- and two-dimensional examples.
En aquesta tesi, presentem una nova estratègia per tal de descriure el procés de fallida de materials quasi-fràgils, com ara el formigó. Típicament la simulació numèrica d'aquest procés s'ha dut a terme mitjançant models de dany o models de fractura. Els primers |models continus| descriuen la fractura com un procés de localització de deformacions on el dany creix i es propaga. Els models de fractura, en canvi, són models discontinus que introdueixen de manera explícita discontinuïtats en el camp de desplaçaments.
Recentment s'han proposat estratègies que combinen aquestes dues teories clàssiques. Tot i que aquestes formulacions alternatives permeten simular millor el procés de fallida, encara queden alguns aspectes per aclarir, especialment pel que fa al canvi de models |de l’estratègia contínua a la discontínua.
En aquesta tesi es presenta una nova estratègia contínua-discontínua. El nostre principal objectiu és proposar nous mètodes per tal de resoldre tres de les dificultats que presenten aquests models combinats: (1) solucionar la dependència patològica de la malla d'elements finits que presenten els models locals amb reblaniment; (2) determinar la trajectòria de la fissura i (3) assegurar-se que el canvi de models del continu al discontinu| es fa de manera que les dues estratègies siguin energèticament equivalents.
En primer lloc, ampliem l’ús |per tal de poder simular problemes dos i tres dimensionals d'una estratègia alternativa que regularitza el reblaniment de les lleis de tensió-deformació. Aquí la no-localitat s'introdueix a nivell del camp de desplaçaments i no a través d'una variable interna com succeeix en les formulacions estàndards. Per aquest motiu, proposem noves condicions de contorn combinades per l’equació de regularització (pel camp de desplaçaments suavitzat). Tal com s'observa en diferents exemples dos i tres dimensionals, aquestes condicions permeten simular de manera físicament realista les primeres etapes del procés de fallida.
En segon lloc, presentem una nova formulació combinada on les fissures es propaguen a través del medi regularitzat. Per tal de definir la trajectòria d'aquestes fissures, utilitzem un criteri geomètric, a diferència dels criteris mecànics clàssics. En particular, sigui D(x) un camp regularitzat de dany, les discontinuats es propaguen seguint la direcció marcada per l'eix mitjà de la isolínia (o isosuperfície mitjana en 3D) D(x) = D_. _Es a dir, utilitzem aquí aquesta eina geomètrica, molt emprada en d'altres aplicacions com ara l’anàlisi d'imatges, la visió artificial o la generació de malles| per tal de propagar les fissures. En aquest cas, donem també exemples dos i tres dimensionals.
Finalment, proposem un nou criteri per tal d'estimar l'energia que l'estructura encara no ha dissipat en el moment en que canviem de model, per tal que pugui ser transferida a la fissura cohesiva. D'aquesta manera, s'assegura que l’estratègia contínua i la contínua-discontínua siguin energèticament equivalents. En comparació amb d'altres tècniques, aquesta estratègia té en compte les diferents branques de descàrrega dels models de dany i permet estimar de manera més precisa l'energia que cal transmetre. Per tal de mostrar aquest balanç energètic es duen a terme diferents exemples en una i dues dimensions.
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.
A new combined strategy to describe failure of quasi-brittle materials is presented thus allowing the complete description of the process, from initiation of damage to crack propagation. For the early stages of the process, and in order to overcome the well-known problems characterising local descriptions of damage (e.g. mesh-dependence), a gradient-enhanced model based on smoothed displacements is employed. In order to deal with material separation, this continuous description is coupled to a cohesive crack when damage parameter exceeds a critical value. Some difficulties may arise when dealing with the transition from regularised damage models to evolving cracks: crack initiation, crack-path direction, energetic equivalence... In this work, a discrete cohesive crack is introduced when the damage parameter exceeds a critical value. On the one hand, and to determine the crack-path direction, the medial axis of the already damaged profile is computed. That is, a geometric tool widely used in the computer graphics field is used here to track the crack surface. Since this technique is exclusively based on the shape of the regularised damage profile, no mesh sensitivity is observed when determining the crack direction. On the other hand, and to define the cohesive law, an energy balance is imposed thus ensuring that the fracture energy not yet dissipated in the damage zone is transferred to the crack.
Wave propagation through solids is strongly influenced by th
ture, especially if this microstructure is periodic. This u
sually leads to the emergence of
band gaps, i.e. frequency ranges that cannot propagate thro
ugh the material . These
phenomena can be taken into account with nonlocal continuum
models. They outperform
the classical elastic models in the sense that they can captu
re dispersive behaviours and
band gaps within a continuous medium . The addition of a hi
gher order inertia term,
characterised by a certain intrinsic length, leads to the di
spersion of the waves propagated
through the solid medium. Microperforated plates (MPP) are
an example of solid ele-
ments with periodic microstructure. In this case the micros
tructure consists of a set of
holes distributed throughout the plate. In this work, the in
fluence of the microstructural
properties of the MPP (hole size, inter-hole distance and ho
le pattern) on the emergence
of band gaps is studied. Moreover, an equivalent nonlocal co
ntinuum model for the MPP
is developed, relating the plate microstructure with the ch
aracteristic length of the higher
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.
Hetherington, J.; Rodriguez-Ferran, A.; Askes, H. International journal for numerical methods in engineering Vol. 93, num. 5, p. 465-482 DOI: 10.1002/nme.4389 Data de publicació: 2013-02-03 Article en revista
In finite element (FE) analysis, traditional penalty methods impose constraints by adding virtual stiffness to the FE system. In dynamics, this can decrease the critical time step of the system when conditionally stable time integration schemes are used by introducing spurious modes with high eigenfrequencies. Recent studies have shown that using mass penalties alongside traditional stiffness penalties can mitigate this effect for systems with a one single-point constraint. In the present work, we extend this finding to include systems with an arbitrary set of multipoint constraints. By analysing the generalised eigenvalue problem, we show that the values of spurious eigenfrequencies may be controlled by the choice of stiffness and mass penalty parameters. The method is demonstrated using numerical examples, including a one-dimensional contact–impact formulation and a two-dimensional crack propagation analysis. The results show that constraint imposition using the bipenalty method can be employed such that the critical time step of an analysis is unaffected, whereas also displaying superiority over the mass penalty method in terms of accuracy and versatility.
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.
In order to achieve a better characterisation of a whole failure process, models which
combine damage and fracture mechanics have recently been proposed. Here, a new combined
methodology is presented: in order to describe damage inception and its diffuse propagation,
a gradient-enhanced continuum model based on smoothed displacements is used, which is coupled
to a discontinuous one to describe the final stages of the process.
Special emphasis should be placed on the difficulties concerning the transition between continuous
damage growth and fracture. On the one hand, and in order to conserve the energy
dissipation through the change of models, an appropriate cohesive law must be defined. In this
paper, the proposed technique to define this law is explained. On the other hand, the direction
of the crack path should be determined. Here, a new strategy is proposed: the discontinuity is
propagated following the direction dictated by the medial axis of the damaged domain. That is,
a geometric tool, widely used in the computer graphics field, is used here to locate cracks.
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.
Hetherington, J.; Rodriguez-Ferran, A.; Askes, H. International journal for numerical methods in engineering Vol. 90, num. 3, p. 269-286 DOI: 10.1002/nme.3314 Data de publicació: 2012-04-20 Article en revista
Tamayo, E.; Rodriguez-Ferran, A. Revista internacional de métodos numéricos para cálculo y diseño en ingeniería Vol. 28, num. 3, p. 170-176 DOI: 10.1016/j.rimni.2012.03.006 Data de publicació: 2012 Article en revista
Los modelos de gradiente basados en desplazamientos
suavizados son una alternativa a los modelos
est´andares para simular num´ericamente el proceso
de fallo de materiales. En esta formulaci´on alternativa
coexisten los campos de desplazamientos mec´anicos u
con los desplazamientos suavizados eu, que se obtienen
de solucionar una ecuaci´on de difusi´on-reacci´on. An´alogamente
a los modelos de regularizaci´on est´andares,
prescribir condiciones de contorno en esta formulaci´on
alternativa es un problema abierto. Sin embargo, imponer
estas condiciones para el campo de desplazamientos
(en lugar de la variable de estado interna) presenta ciertas
facilidades. El objetivo de este trabajo es estudiar
la influencia de dichas condiciones: ni las condiciones
de Dirichlet (prescritas en un principio) ni las condiciones
de Neumann homog´eneas (reminiscencia de los
modelos de gradiente est´andares) permiten obtener resultados
num´ericos realistas; mediante las condiciones
de contorno de Neumann no homog´eneas, en cambio, los
resultados son f´ısicamente admisibles. Sin embargo, estas
condiciones no aseguran conservaci´on de volumen,
que es una propiedad interesante en algunos modelos
constitutivos. Por este motivo, se proponen unas nuevas
condiciones de contorno (condiciones combinadas)
que satisfacen las propiedades necesarias para la regu-
larizaci´on: (a) reproducibilidad de orden 1 (u = eu si
u es un campo lineal), (b) desplazamientos suavizados
a lo largo del contorno y (c) conservaci´on de volumen.
En este trabajo se han llevado a cabo varios ensayos
num´ericos bidimensionales con el fin de ilustrar la influencia
de las distintas condiciones de contorno.
Bennett, T.; Rodriguez-Ferran, A.; Askes, H. European journal of mechanics. A, Solids Vol. 31, num. 1, p. 131-138 DOI: 10.1016/j.euromechsol.2011.08.005 Data de publicació: 2012-01-01 Article en revista
A gradient enhanced model is formulated which simulates micro-structurally induced wave dispersion and is capable of achieving mesh objective results when modelling strain-softening materials. Length scale parameters are introduced to incorporate both micro-inertial effects and strain smoothing. The model is formulated such that all the gradient enhancement terms are contained within the kinetic energy functional, whilst all non-linear terms are linked to the strain energy functional. In addition, discretisation of the governing equations is performed in a manner such that only C0C0-continuity is required and is symmetric in the elastic range. Dispersion analysis is performed to show the correct format that a constitutive model must conform to in order that regularisation of strain-softening material models will occur. This dispersion analysis is then used to explore how the two length scales may influence the width of the zone in which strain localisation takes place. The efficacy of the model in simulating strain-softening behaviour in a mesh objective manner and the effects of wave dispersion on damage initiation and accumulation are demonstrated in numerical examples.
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.
Tamayo, E.; Rodriguez-Ferran, A. International Conference on Computational Modeling of Fracture and Failure of Materials and Structures p. 294 Data de presentació: 2011-06-06 Presentació treball a congrés
Rodriguez-Ferran, A.; Bennet, T.; Askes, H.; Tamayo, E. International journal of solids and structures Vol. 48, num. 9, p. 1382-1394 DOI: 10.1016/j.ijsolstr.2011.01.022 Data de publicació: 2011-05-01 Article en revista
A general non-local approach to regularise strain-softening continua is presented.
The key idea is to introduce the gradient-type enrichment at the level of displacements (rather than some internal variable), so the model is formulated
with two distinct displacement fields. In fact, gradient models based on two displacement fields are usual in non-local elasticity, where the goal is to avoid the shortcomings of classical (local) elasticity (i.e. strain singularities in statics, non-dispersive behaviour in dynamics). We show that such a gradient elasticity backbone model can be combined with any standard nonlinear constitutive driver to render a regularised model for softening inelasticity.
To illustrate the generality of the approach, two prototype models (isotropic
damage and von Mises plasticity) are discussed. The numerical examples show that the regularised models exhibit all of the desired features: mesh insensitivity, imperfection size insensitivity and description of size effects.
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 Data de publicació: 2011-01-17 Article en revista
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.
Askes, H.; Caramés-Saddler, M.; Rodriguez-Ferran, A. International Conference on Computational Structures Technology p. 1-9 DOI: 10.4203/ccp.93.339 Data de presentació: 2010-09-14 Presentació treball a congrés
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.
A continuous-discontinuous model to simulate numerically an entire failure process is presented.
Crack inception and its propagation is modelled by means of a gradient non-local model based on non-local displacements. To simulate properly the final stages of the process, a discrete crack approach (X-FEM) is used, where both local and non-local displacements are modelled as discontinuous fields. In this paper, this new combined approach is studied in detail and one- and two-dimensional examples are carried out to validate it.
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).
Askes, H.; Caramés Saddler, M.; Rodriguez-Ferran, A. Proceedings of the Royal Society of London. Series A, mathematical and physical sciences Vol. 88, num. 7-8, p. 1389-1408 DOI: 10.1098/rspa.2009.0350 Data de publicació: 2009-12-11 Article en revista
Penalty functions can be used to add constraints to systems of equations. In computational mechanics, stiffness-type penalties are the common choice. However, in dynamic applications stiffness penalties have the disadvantage that they tend to decrease the critical time step in conditionally stable time integration schemes, leading to increased CPU times for simulations. In contrast, inertia penalties increase the critical time step. In this paper, we suggest the simultaneous use of stiffness penalties and inertia penalties, which is denoted as the bipenalty method . We demonstrate that the accuracy of the bipenalty method is at least as good as (and usually better than) using either stiffness penalties or inertia penalties. Furthermore, for a number of finite elements (bar elements, beam elements and square plane stress/plane strain elements) we have derived ratios of the two penalty parameters such that their combined effect on the critical time step is neutral. The bipenalty method is thus superior to using stiffness penalties, because the decrease in critical time step can be avoided. The bipenalty method is also superior to using inertia penalties, since the constraints are realized with higher accuracy.