Hugo Scurto
email: hugo.scurto@ensad.fr

Ludmila Postel
email: ludmilapostel@gmail.com

Soundwalking Deep Latent Spaces

Keywords: AI, Soundwalk, Generative Deep Learning, Virtual World

This paper relates an early art-research collaboration between two practitioners in machine learning and virtual worlds toward new embodied musical experiences of Artificial Intelligence (AI). Instead of a digital music instrument or a music-generating agent, we propose to craft a soundwalk experience where a human person moves through a three-dimensional virtual world to explore a latent sound space generated by deep learning. We report on the diffractive prototyping and iterative crafting of three such soundwalks through/out deep latent spaces, using nn∼ and New Atlantis as computational platforms for AI audio processing and virtual world experimentation. We share critical perspectives emerging from our latent soundwalking practice, with the hope that they contribute to ongoing communitywide reflections toward new AI for musical expression.

INTRODUCTION

Deep generative models are high-dimensional parametric representations of large datasets computed by deep learning. In the sound and music domain, they typically enable to generate unheard-of sounds, by exploring a so-called latent parametric space, which expressively interpolates sounds contained in the original dataset. The versatility of deep generative models enabled the creation of a wide range of new interfaces for musical expression, including gestural music systems [START_REF] Martin | Understanding musical predictions with an embodied interface for musical machine learning[END_REF][START_REF] Erdem | Cavi: A coadaptive audiovisual instrument-composition[END_REF], autonomous generative agents [START_REF] Vigliensoni | Live latent space drum rhythm generation from minimal-size datasets[END_REF], or stream-based sound installations [START_REF] Scurto | ciel[END_REF]. Such musical interfaces in turn contributed to the modern hype surrounding music Artificial Intelligence (AI) [START_REF] Knotts | A survey on the uptake of music ai software[END_REF], despite more nuanced reflections from NIME practitioners [START_REF] Tahiroglu | Ai-terity 2.0: An autonomous nime featuring ganspacesynth deep learning model[END_REF][START_REF] Murray-Browne | Latent mappings: Generating open-ended expressive mappings using variational autoencoders[END_REF].

In response to this modern hype, several academics expressed the need to develop critical practices and representations of AI, both within and beyond the NIME community [START_REF] Morreale | Where does the buck stop? ethical and political issues with ai in music creation[END_REF][START_REF] Caramiaux | explorers of unknown planets" practices and politics of artificial intelligence in visual arts[END_REF]. Sound and music researchers elicited ethical issues raised by corporate scraping of music datasets used for deep generative models, which are often hidden behind commercial discourses on creative potentials of music AI [START_REF] Sterne | Tuning sound for infrastructures: artificial intelligence, automation, and the cultural politics of audio mastering[END_REF]. Scientists shed light on the vast energy consumption and greenhouse gas emission of deep generative models, whose unprecedented performances often imply high computational costs [START_REF] Douwes | Energy consumption of deep generative audio models[END_REF]. Writers have revealed the planetary computational infrastructures that support modern AI applications, and how they perpetuate an extractivist agenda that transcends bodies, spaces and societies [START_REF] Crawford | The atlas of AI: Power, politics, and the planetary costs of artificial intelligence[END_REF].

As a duo of artists-researchers, we are interested in exploring soundwalking as a joint musical interface for deep generative models and critical practice of AI. Soundwalking consists in an individual or collective walk with a focus on listening to the environment [START_REF] Westerkamp | Soundwalking. Sound Heritage[END_REF]. It has joint pedagogical, political and artistic implications, including educating persons to listen to uncanny sounds, giving voice to places and bodies that have been muted throughout history, or articulating musical performances throughout such situated soundscapes [START_REF] Shaw | Ambulation: Exploring listening technologies for an extended sound walking practice[END_REF]. We believe that soundwalking could help reveal social, political and planetary issues raised by modern AI applications, while suggesting novel practices of deep generative models within and beyond NIME [START_REF] Scurto | Deeply listening through/out the deepscape[END_REF].

In this paper, we relate early conceptual and technical work toward soundwalking deep latent spaces. Specifically, we leveraged our respective expertise in machine learning and virtual worlds to craft a soundwalk experience where a human person moves through a three-dimensional virtual world to explore a deep latent sound space. Rather than engineering deep learning to map walking movement with sound [START_REF] Murray-Browne | Latent mappings: Generating open-ended expressive mappings using variational autoencoders[END_REF], we adopted a diffractive approach to machine learning [START_REF] Scurto | Prototyping machine learning through diffractive art practice[END_REF], using existing computational platforms to prototype and iteratively craft such musical experiences while probing them together. Our wish is that such latent soundwalks may help reveal novel representations for AI that better attend to socio-political aspects of music practices.

The next sections successively describe the computational platforms and our musical experiences. We end by sharing critical perspectives raised by our latent soundwalks, with the hope that they contribute to ongoing community-wide reflections toward new AI for musical expression.

COMPUTATIONAL PLATFORMS

This section describes nn∼ and New Atlantis, the two computational platforms that we used to iteratively craft our musical experiences, along with our current working prototype, whose workflow is shown in Figure 1.

nn∼

nn∼ is a Max/MSP and PureData external for AI audio processing 1 . It enables to easily explore deep latent spaces and play with audio streams they generate. The current version works in real-time over standard CPUs, which makes it more affordable than other GPU-based deep learning platforms for music [START_REF] Tahiroglu | Ai-terity 2.0: An autonomous nime featuring ganspacesynth deep learning model[END_REF], while keeping energy costs standard.

Different interaction modalities can be used as input with nn∼. The first is latent parameters, which consists in generating sound in a way similar to standard sound synthesisers, i.e., by exploring different parameter combinations to produce different sounds. The second is live audio: it consists in reconstructing audio coming either from a live stream or an offline audio file by transferring timbres that are contained in the deep generative model. The third one is unconditional generation, which consists in autonomously producing an audio stream based on temporal probabilistic modeling of the deep generative model.

The versatility of nn∼ enabled the creation of diverse music AI experiences, including human-machine live improvisations, infinite audio livestreams over the Web, or samplebased offline music composition. Yet, to the best of our knowledge, it has never been used to create musical experiences based on soundwalking, nor have been other AI platforms or deep generative models for sound and music.

New Atlantis

New Atlantis is a web-based virtual platform dedicated to sound art experimentation2 . It enables to compose sound spaces through a three-dimensional virtual world, and to explore them through individual or collective walking. The latest version consists of a web-based environment, which makes it usable on both standard CPUs and smartphones, without requiring specific commercial software or hardware.

Different interaction modalities can be used to compose and explore soundwalks in New Atlantis. Composing consists in positioning sound samples in the three-dimensional space, and adding virtual boxes to create reverberation and filtering effects, following acoustic rules of the physical world. Exploring typically consists in freely navigating through sound spaces by walking and flying with varying speed, using either keyboard and mouse or a USB gamepad. Virtual objects can also be added as spatial landmarks suggesting specific soundwalks and atmospheres to the audience.

New Atlantis has been used in artistic and pedagogical contexts, ranging from sound art installations and experimental music performances to workshops on computational and sound art. Yet, to the best of our knowledge, it has never been used to soundwalk deep latent spaces, nor have been other three-dimensional platforms or virtual worlds dedicated to artistic creation.

Current Prototype

Our current prototype sought to link New Atlantis with nn∼ to craft soundwalks through/out deep latent spaces. Within New Atlantis, we implemented a Javascript script that get the camera's position (x, y, z) and azimuthal orientation (φ) in the virtual world every 100 ms, and sends it locally through a WebSocket connection. Within our CPUbased machine, we implemented a Python script that receives data from the WebSocket connection, and sends it locally to Max/MSP using the OSC protocol. Future implementation may rely on Node for Max and the jweb object.

On the generative deep learning side, we relied on RAVE3 , a state-of-the-art deep generative model for raw audio waveforms [START_REF] Caillon | Rave: A variational autoencoder for fast and high-quality neural audio synthesis[END_REF]. Two advantages of RAVE over other deep generative models are that audio waveforms are modeled at a 48 kHz sampling frequency, i.e., standard audio quality, and that they are generated in real-time with low latency, i.e., less than one second on a standard CPU. One disadvantage is that audio generation can be perceived as choppy due to RAVE's architecture based on waveform grains [START_REF] Caillon | Rave: A variational autoencoder for fast and high-quality neural audio synthesis[END_REF]. Future work may explore deep generative models for raw spectrogram generation to achieve smoother sound output.

MUSICAL EXPERIENCES

In this section, we describe three musical experiences that we iteratively crafted based on our current prototype. The first and second authors leveraged their respective expertise in machine learning and virtual soundwalks to implement and test these experiences. In each of these experiences, the second author was not told the type of latent sound space at stake: rather, the first author only told them to explore the deep latent space by freely soundwalking the three-dimensional space, using a keyboard and mouse, as long as they wanted to. Extracts of these three musical experiences are shown on a supplementary video material created afterwards by the first author4 .

Walking Into Latent Sounds

The first experience sought to create unexpected encounters with voices and sonic bodies of a deep latent space. The first author used a RAVE deep generative model pre-trained on the VCTK dataset-monophonic English voice data [START_REF] Caillon | Rave: A variational autoencoder for fast and high-quality neural audio synthesis[END_REF], which expressively interpolates between latent noises and human voices. They crafted a one-to-one mapping between the camera's three-dimensional position in New Atlantis to the three first latent parameters of nn∼, leaving remaining parameters at a static null value. The sole visual landmark was one small sphere placed at the origin of the space.

After a few minutes of careful exploration, the second author commented that the experience felt like "a new game for [them]". Specifically, the fact that the camera's movement was mapped to latent parameters created the feeling to be "within a mass of sound". This differs from previous experiences with New Atlantis, which often reproduce acoustic properties of the physical space using located sound sources. Also and interestingly, they did not perceive our current prototype's low latency during the soundwalk.

Despite enthusiasm, the experience destabilised the second author, who felt a bit lost in such an abstract latent space. As a consequence, they sought to link each three dimensions of the virtual space to timbre variations of sound, based on linear displacements relative to the spherical visual landmark. Such an approach was suggested by how space was designed in New Atlantis: "From the moment that there is only one point of reference, I turn around it".

Still, listening alone did seemed to provide the second author with a few landmarks within the latent space. First, they sought to find uncanny voices contained in the latent space: "In fact, there is silence and there are voices, and since we have no other visual clues, we try to find such sounds by ear." Then, they discovered extreme values of the latent space through their corresponding noisy continuums: "when I get to white noise, if I continue to move forward there's no more evolution, so that means it's a bit of an invisible barrier to the world you've built."

Embodying Listening With 3D Landmarks

Based on this feedback, the first author crafted a second experience by adding visual landmarks to the same latent sound space using New Atlantis. Specifically, they added ellipsoid objects of different sizes, colors and relative distances, to create a three-dimensional space that encourages both slow-and fast-paced movements. Their positioning related to latent sound space was deliberately incidental: landmarks could just as easily indicate timbre variations, as be placed in the middle of a static noise continuum.

While remaining abstract, such visual landmarks suggested a way of moving within the latent sound space, which seemed to have cultivated embodied listening along the soundwalk experience: "With the spheres, I understand space very well, so it really allows me to listen much more, and to try to move in relation to what I hear, which is really cool". Interestingly, the second author spent much more time soundwalking this experience, even if the latent sound space and mapping were the same as in the first experience.

From an aesthetic perspective, the experience seemed to relate to an uncanny soundscape that entangle recognisable sonic bodies with noisy continuums: "There are moments of transition which are very beautiful [...]. There are times when you slowly come close to one or several spheres, and you first recognise voice, then it slowly evolves into something else... There's something quite soaring, a little bit like ambient music, something that makes you want to let yourself be carried in the evolution of sound".

Sensing Deep Learning Spatialities

The first author crafted a third experience based on this embodied listening feedback. They used a stereophonic RAVE deep generative model [START_REF] Scurto | Deeply listening through/out the deepscape[END_REF], trained on a custom soundscape dataset collected over the planet [START_REF] Sinclair | Locus stream open microphone project[END_REF]. In addition to previous position mapping, they implemented a one-to-one mapping between the camera's azimuthal orientation and the stereophonic speaker position, using the Spat library. They removed all visual landmarks but the spherical origin to experiment with this stereophonic latent sound space.

The stereophonic aspect of this deep latent space was subtle, yet perceivable by the second author: "There are much more things that take place in the sound space itself, before I even move. And there are moments when I arrive at places where I can really hear a right-left". This sonic depth helped the second author locate themself within the latent space: "My feeling is that there are hinge points that are not visual like the spheres, but that are sonorous, which gives the impression to pass from one place to another, contrary to previous experiences, which felt like one single place".

Interestingly, spatial properties of this latent sound space seemed to differ from those of the physical space: "When you move towards a sound, you don't actually get closer [...] but it does change the stereo". While very preliminary, this uncanny attribute of stereophonic latent spaces may pave the way for music AI practices that focus on spatialities, as the second author suggested: "there are really moments, or places-it's funny that I just said moments to express how very written it feels, in fact-there are places where something particular happens".

CRITICAL PERSPECTIVES

In this section, we share critical perspectives emerging from our early art-research collaboration, looking at latent soundwalk against music AI, and at deep latent spaces against virtual worlds, with the hope that it contributes to communitywide reflections toward new AI for musical expression.

Music AI as Latent Soundwalk

The one-to-one mappings crafted for our musical experiences might seem quite limited from a technical perspective. Yet, we believe they enabled to reveal subtle differences between soundwalking and musical gestures as movementsound relations. While musical gestures typically seek to expressively control sound within relatively short time scales of music performance, soundwalking seeks to reveal sounds of our socio-material environments within larger scales of time and space, thus implying different embodied listenings.

While these two practices are not mutually exclusive, we suggest that future work might delve more into soundwalkinspired mappings between movement and latent sound spaces. For example, latent parameters mapped to camera position could change over different time scales to simulate soundscape evolution over climatic cycles of the planet. Or, displacements in specific parts of the space could trigger abrupt changes in latent parameter mappings to simulate alterations of the sonic environment based on human actions.

On a conceptual level, we suggest that latent soundwalks could inspire how we may practice music AI. Rather than engineering AI to autonomously perform human musical tasks [START_REF] Sterne | Tuning sound for infrastructures: artificial intelligence, automation, and the cultural politics of audio mastering[END_REF], practitioners might design interfaces that support encountering uncanny sounds through embodied exploration of deep latent spaces [START_REF] Tahiroglu | Ai-terity 2.0: An autonomous nime featuring ganspacesynth deep learning model[END_REF]. We argue that such embodied interfaces might foster new AI practices that do not impinge on existing musical labor [START_REF] Morreale | Where does the buck stop? ethical and political issues with ai in music creation[END_REF], similar to how soundwalk fosters ethical awareness of our environments [START_REF] Westerkamp | Soundwalking. Sound Heritage[END_REF]. The computational platforms used to craft our latent soundwalks enabled to experiment with spatial landmarks as guides for music AI. Specifically, we found that visual landmarks seemed to deepen embodied listening throughout latent sound spaces in relation to walking in a three-dimensional space. While such insights may be common for virtual world and soundwalk practitioners, we believe they may be novel for deep generative models practitioners.

Future work may craft three-dimensional worlds in greater detail to create narratives through latent soundwalks, for example using concrete objects, spaces and textures. Moreover, collective modes of soundwalking could also give shape to such latent virtual worlds, relying on avatars to walk through space, and providing social landmarks along soundwalks. We will build on New Atlantis' web standards to craft such collective musical experiences, by implementing online, live audio streaming from deep generative models.

More conceptually, we suggest that virtual worlds could be used as a metaphor for deep latent spaces. Indeed, deep latent spaces have deeper social and aesthetic implications than standard synthesisers or timbre spaces, through the culturally-situated datasets and human work they build on [START_REF] Crawford | The atlas of AI: Power, politics, and the planetary costs of artificial intelligence[END_REF]. Yet, commercial discourses that anthropomorphise AI tend to hide this complexity [START_REF] Sterne | Tuning sound for infrastructures: artificial intelligence, automation, and the cultural politics of audio mastering[END_REF], along with its ecological implications [START_REF] Douwes | Energy consumption of deep generative audio models[END_REF]. We believe that describing deep latent spaces as virtual worlds may better highlight the spatiality of AI-its planetary corporate infrastructures and sociomaterial resources [START_REF] Scurto | Deeply listening through/out the deepscape[END_REF]-, and how these may transform the spaces where musicking and listening take place.

Toward New AI for Musical Expression

Our art-research collaboration had us find compromises between technical engineering of deep generative models and musical experience design. For example, the first author did not perform dimensionality reduction of deep latent spaces [START_REF] Tahiroglu | Ai-terity 2.0: An autonomous nime featuring ganspacesynth deep learning model[END_REF], which tends to produce parameters that are interpretable from a sound perception perspective-e.g., spectral centroid evolving linearly along one parameter. Rather, they opted for elementary mappings between latent parameters and New Atlantis to let the second author experience the raw sonic materiality of deep latent spaces through their virtual soundwalk practice, and interpret latent parameters from a bodily, sensitive perspective.

Along with other artists-researchers [START_REF] Scurto | Prototyping machine learning through diffractive art practice[END_REF][START_REF] Tahiroglu | Ai-terity 2.0: An autonomous nime featuring ganspacesynth deep learning model[END_REF][START_REF] Caramiaux | explorers of unknown planets" practices and politics of artificial intelligence in visual arts[END_REF], we believe that diffracting AI throughout musical expression-e.g., attending to one's subjectivity and responding to more-thanhuman materialities of musical interfaces-may produce situated knowledge on deep generative models that complements that produced by technical engineering [START_REF] Scurto | Deeply listening through/out the deepscape[END_REF]. This is in line with ongoing community-wide reflections toward ethical and epistemological turns for AI [START_REF] Morreale | Where does the buck stop? ethical and political issues with ai in music creation[END_REF]. In future work, we will further entangle NIMEs, soundwalk and AI, by leading latent soundwalks within physical public spaces, using smartphones as joint localisation and microphone devices for a responsive, hybrid listening experience [START_REF] Shaw | Ambulation: Exploring listening technologies for an extended sound walking practice[END_REF]. We believe that such collective practices of deep generative models will be key to reveal AI actions over our environments and ethically reconfigure our representations of AI.

CONCLUSION

This paper related early art-research toward soundwalking deep latent spaces. We look forward to critical perspectives from other NIME practitioners to reconfigure deep generative models in more depth and entangle a plurality of subjectivities within music AI.

Figure 1 :

 1 Figure 1: Our current prototype workflow for soundwalking deep latent spaces.

https://github.com/acids-ircam/nn tilde

https://jonlab.github.io/NewAtlantisWeb/index.html

https://github.com/acids-ircam/RAVE

https://vimeo.com/814921401

ACKNOWLEDGMENTS

We thank Axel Chemla-Romeu-Santos, Antoine Caillon, Peter Sinclair, Jonathan Tanant, Emanuele Quinz, Samuel Bianchini, as well as reviewers for their enriching feedback.

ETHICAL STANDARDS

The speech and soundscape datasets used for training are open-source [1,14] and were fully anonymised in terms of person and place identities. We know of no way to regenerate one's voice or soundscape from the trained models.


Leena Tahkamo

Manuel Bazzana

Pierre Ravel

Francis Grannec

Christophe Martinsons

Georges Zissis

Life cycle assessment of light-emitting diode downlight luminaire: a case study

Keywords: life cycle assessment, light-emitting diode, luminaire, lighting, environmental impact, light source

Introduction

It is estimated that lighting consumes approximately 19 % of the global electricity production (IEA 2006). Globally, the electrical lighting is provided mainly (62 %) by fluorescent lighting, including fluorescent and compact fluorescent lamps (IEA 2006). In contrast, the share of light-emitting diode (LED) in general lighting is low; LED lighting was estimated to cover 6.2 % of the general lighting in the European Union in 2010 [START_REF] Ec | Environmental management -Life cycle assessment -Principles and framework ISO 14044 (2006) Environmental management -Life cycle assessment -Requirements[END_REF]. However, the market penetration of LED technology is rapidly increasing not only in the decorative lighting of coloured, usually low-power applications, but also in general lighting.

The LED light sources represent a relatively new technology. In addition to the indicator and decorative lighting, there are LED lamps and luminaires on the market that possess sufficient luminous flux to be used in general lighting and even in high-power outdoor lighting. The challenges in the heat transfer from the chip through the heat sink have hindered the wider use of the LED technology. Efficient heat transfer is essential in order to slow depreciation of the luminous flux in LED light sources and thus realize long life.

The environmental aspects of LED products have not been studied exhaustively, yet some studies exist. Lim et al. (2011) studied the material contents of indicator LED components of different colours by leachability tests. However, it concerns only the LED components of through-hole technology, and not the LED lamp or luminaire as a whole. [START_REF] Dale | Preliminary Comparative Life-Cycle Impacts of Streetlight Technology[END_REF] studied four different streetlight technologies, one of which was an LED luminaire. The study compared one-to-one replacement of high pressure sodium lamp luminaire to the metal halide, induction or LED lamp luminaire. It did not take into account the luminous flux or the standards and recommendations for street lighting. In addition, the ballasts and drivers were estimated to be similar and thus excluded from the assessment. The LED luminaire was found out to have the lowest environmental impacts that were similar with the ones of the induction lamp luminaire. Navigant consulting (2009) examined several energy-efficient light sources including an integrally-ballasted LED lamp (12 W, 720 lm) and an LED luminaire (18 W, 1170 lm). The life-cycle environmental impacts of an LED lamp have also been compared to incandescent and compact fluorescent lamps (CFL) (Osram 2009, Quirk 2009). The conclusion of the assessments are generally that LED and CFL lamps possess roughly the environmental impacts of the same scale and that the future LED lamp will be more environmentally friendly than the CFL, since no major technological development are expected in the CFL technology.

The environmental performance of the light source is dependent on the luminous efficacy if the analysis uses a proper functional unit taking into account the amount of light produced. The luminous efficacy of incandescent lamp is low, typically 10-15 lm/W, while the one of CFL ranges approximately between 40 and 70 lm/W including the ballast, and fluorescent lamps (T5) up to approximately 100 lm/W including the ballast. The luminous efficacy of LED lamps has a wide range, approximately between 20-120 lm/W (Aalto University 2010). The LED chip luminous efficacy has well exceeded 200 lm/W in laboratories. In contrast to the LED technology, no major improvements are expected in the development of the conventional light source technologies such as incandescent or fluorescent lamp technology. In addition, the lamp technologies of low energy efficiency are being banned in many regions, e.g., the inefficient incandescent lamps and high pressure mercury lamps in the European Union.

Currently, there are LED lamps and luminaires of varying properties available on the market. Notable differences are found in the properties among LED light sources concerning, e.g., the luminous flux, electrical properties such as harmonic distortion, colour temperature, colour rendering, colour shift and lumen maintenance.

This study introduces the life cycle assessment (LCA) of an LED luminaire. It is a case study of one type of recessed downlight luminaire and does not represent the LED technology as a whole. Yet, it is an exemplary case. The use-stage energy consumption was found out to be the most significant environmental aspect in the life cycle of the LED luminaire. The manufacturing stage is dominated by the environmental impacts of the manufacturing of the driver, LED array and aluminium parts. The end-of-life (EoL) scenario has an insignificant impact on the LCA. The analysis is somewhat sensitive to the life estimate of the product. More LCAs of LED products are needed in order to study profoundly the environmental impacts of the technology and the wide product range.

Life cycle assessment of light-emitting diode luminaire

The environmental impacts of an LED luminaire were estimated on the basis of an LCA. The LCA was conducted for an LED downlight luminaire in Figure 1. The LED luminaire in question is a high-quality luminaire used in commercial and retail applications, in which it is intended to replace downlight luminaire equipped with two CFLs. The nominal electrical power of the LED luminaire is 19 W. The product contains the driver and the LED luminaire including all its parts, such as the heatsink, reflector, high-power blue LED components on a circuit board, and a "remote phosphor" plastic diffuser, whose external surface is coated with a thin layer of luminophores. The LCA is conducted according to the international standards ISO 14040 (2006) andISO 14044 (2006), and a French standard NF P01-010 (2004). The SimaPro version 7.3.2 was used as the LCA software (PRé Consultants).

Goal and scope definition

The goal of this LCA is to study the environmental impacts of an LED downlight luminaire and to determine the life cycle stages and material and energy inputs that are the most significant. In addition, it is determined how sensitive the environmental impacts are to the life of the luminaire.

The LCA includes the manufacturing, transport, installation, use and EoL stages (Figure 2). The system boundaries exclude the lighting, heating and cleaning of the factories; administrative work; transport of the employees; tool manufacturing; infrastructure; and transport infrastructure, such as building and maintenance of roads. The study excluded also the inputs that represented less than 2 % of the total weight of the product and for which there were not life cycle inventory available, according to the rules of the French standard NF P01-010.

The functional unit is the use of the 19 W LED downlight luminaire including the driver during 50 000 h and producing 1140 lm of light with colour rendering index R a of about 80. All the electrical and photometric characteristics were measured by the CSTB lighting laboratory (Martinsons 2009a(Martinsons , 2010)), and the life is based on the information from the manufacturer, since life tests are still in progress at CSTB. The measured luminous efficacy is about 60 lm/W. Since the life of the luminaire is a crucial factor in the LCA, its impact is studied more profoundly. The life of the LED products is allegedly long but there are concerns regarding its realization. The life of the LED downlight luminaire is 50 000 h according to the manufacturer. A life of 50 000 h is a somewhat general approximation for LED luminaires but it is not possible to carry out burning tests of such a long life. The life of the ballast is considered in the LCA to be the same as the life of the luminaire, since it is sold as one unit and the ballast will be replaced together with the rest of the luminaire. At the end of the operating life, the luminaire is expected to provide less than 70 % of initial luminous flux taking into account both the lumen depreciation and product failures.

The life of 50 000 h is used as a base case scenario in the LCA. The base case is compared to the lives of 36 000 h. In case of a shorter life, the share of the manufacturing efforts increases in the total life cycle perspective. 36 000 h life was chosen on the basis of the LM-80 (IESNA 2008) method. The method states that the life of the LED packages and modules shall be estimated based on a burning test of a minimum of 6 000 h. It is recommended to extrapolate the life of the LED package or module based on the burning test only up to six times the test period, i.e., in the case of 6 000 h test, until 36 000 h. It is noted that LM-80 does not apply to LED luminaires as a whole but it was used, nevertheless, due to the lack of a specific method for LED luminaires.

In addition to the life of the luminaire, the annual environmental impacts are dependent on the use scenario. The annual operating hours are not included in the LCA but only the total life is considered. However, the annual environmental impacts depend on the annual operating hours. The annual operating hours vary according to the use of the space, but are approximately, e.g., 2 500 h per year in offices and 5 000 h per year in hotels and retail facilities (EN 15193).

Life cycle inventory analysis

Manufacturing

The manufacturing stage comprises the raw material acquisition of all parts of the luminaire installed in the beginning of the operating life. The manufacturing of the replacement luminaires, in case of the shorter operating life than 50 000 h, is modelled in the use stage according to the NF P01-010 (2004).

Table 1 lists the material and process inputs of the LED downlight luminaire manufacturing. The driver data was retrieved by dismantling the driver and identifying the components. All driver parts include an additional 2 % mass to take into account the material losses of the manufacturing. The printed circuit board (PCB) is modelled as 2-sided board that includes the mounting processes (50 % through-hole technology (THT) and 50 % as surface-mount technology (SMT)). THT components were weighed, while the weight data of the SMT components was retrieved from component datasheets and ZVEI documents (ZVEI 2012). The remote phosphor has recently been the subject of a French Luminosurf research project. In this project, CSTB conducted LCAs on several types of organic and inorganic luminophores used in LED lighting [START_REF] Pradal | Study of Ce-doped Y3Al5O12 nanopowders dispersion[END_REF], Martinsons 2009b). The luminaire assembly process is modelled on the basis of the energy consumption provided by the luminaire manufacturer: approximately 0.03 kWh of electricity per luminaire. The amount of electricity covers only the assembly process and not the other energy consumption of the luminaire factory.

The LED components were modelled as 5 mm LED components for information and communication technology found in the ecoinvent database in which the particular data was from year 2007 (Ecoinvent 2010). However, it is estimated that the diode in the database, a THT component, results to low environmental impacts compared to the actual SMT LED component. Thus, on the basis of the estimations by industry experts, the weight of the SMT LEDs is multiplied by a factor of five in order to take into account the underestimation in the component data. Yet, it is acknowledged that a more accurate and up-to-date life cycle data is needed regarding the LED components used currently in general lighting.

Transport of parts is included in each part accordingly, a total of approximately 2 tkm by lorry and 12 tkm by transoceanic freight. Likewise, packaging is taken into account and modelled in each part. Packaging is modelled mainly as cardboard that accounts for approximately 10 % of the weight of the product parts if precise data was unavailable. The cardboard used in the packaging of the luminaire parts is estimated to total 114 g and the package of the final product 210 g. The waste treatment in the manufacturing stage covers the transport and recycling of the cardboard packages.

Transport

Transport stage includes the transport from the manufacturer site to the place it is used. Transport data was gathered from the luminaire manufacturer, and when such data was not available, the default scenarios for the product environmental profile (PEP) (Association P.E.P. 2011) were used as reference. In PEP, the global transport is estimated to be 19 000 km by ship and 1 000 km by lorry, the intercontinental transport 3 500 km by lorry, and local transport 1 000 km by lorry.

Installation

The installation of the luminaire considers the transport and the final storage of the package materials. The transport is modelled as 30 km by lorry. The final waste disposal is modelled as municipal solid waste deposition.

Use

The use of the luminaire is modelled on the basis of the energy consumption. The downlight luminaire consumes 19 W per hour. The French average mix is used for electricity production. The French electricity is modelled to be produced by nuclear power (77 %), hydropower (12 %), coal (4 %), natural gas (3 %), and oil (1 %), and imported (2 %) (Ecoinvent 2010).

End-of-Life

On the basis of the information from the French lamp recycling organization Recylum, two EoL scenarios of the LED luminaire were created: current and prospective treatment scenarios. The current scenario is estimated to be 95 % of landfill and 5 % of recycling treatment, while the prospective scenario includes 40 % of landfill and 60 % of recycling treatment. The landfill disposal takes into account the transport of the waste and the deposition on the landfill. The recycling treatment includes the transport of the product from the point of use to the collection point and further to the storage centre and finally to the treatment centre. The recycling treatment contains also the shredding process and separation of the fractions, the transport of the fractions, and their recycling and disposal.

Life cycle impact assessment

The life cycle impact assessment (LCIA) introduces the environmental impact categories and calculation methods used in the LCA. Ecoinvent and European Reference Life cycle Database were used as databases in the LCIA (Ecoinvent 2010;[START_REF] Ec | Environmental management -Life cycle assessment -Principles and framework ISO 14044 (2006) Environmental management -Life cycle assessment -Requirements[END_REF]. This assessment uses the MatFrance method, version 2.03, that is based on a French standard NF P01-010 ( 2004) and AIMCC (2009). The NF P01-010 is not compulsory but it is widely used in the environmental assessments in the building sector in France. The environmental impact categories are introduced in Table 2. It is mainly based on CML 2001 method (Guinée et al. 2002) but contains some differences: e.g., the photochemical ozone creation potential takes into account only the hydrocarbons (NF P01-010 2004).

Four scenarios of the product system were chosen to analyse the impact of the product life and the EoL scenario. 1) luminaire life 36 000 h (2 products needed during 50 000 h), current recycling scenario in the EoL 2) luminaire life 36 000 h (2 products needed during 50 000 h), prospective recycling scenario in the EoL 3) luminaire life 50 000 h (1 product needed during 50 000 h), current recycling scenario in the EoL 4) luminaire life 50 000 h (1 product needed during 50 000 h), prospective recycling scenario in the EoL

Results

The environmental impacts of the manufacturing of an LED downlight luminaire were divided between the driver, LED package, aluminium parts, remote phosphor, luminaire assembly process, waste treatment of the packaging materials, and other materials (Figure 3). The LED components and the circuit board they are attached to, i.e., the LED array, accounted for 3 % of the total weight of the luminaire, while their environmental impact ranged between 4 % and 50 % depending on the impact category. The aluminium parts, such as the heatsink, accounted for 11-42 % of the environmental impacts. The environmental impacts of the driver ranged between 19 % and 66 %. The PCB of the driver has notable share in most environmental impact categories, but otherwise the environmental impacts of the driver are scattered across a number of parts. The most critical parts of the driver are the electrolyte capacitors in inert waste and ozone depletion, unspecified capacitors in water pollution and photochemical ozone creation, steel in water consumption, and glass diodes in photochemical ozone creation and hazardous waste.

The comparison in Figure 4 presents two cases of life of the product (36 000 h, and the base case of 50 000 h) and two EoL cases (actual and prospective). Due to the lower manufacturing efforts, the environmental impacts of the base case were 1 to 44 % (17 % on the average) lower than the ones of the case of 36 000 h. The total lifecycle environmental impacts were almost independent from the EoL scenario. In the base case, the total life cycle primary energy consumption was approximately 13 070 MJ and GWP 112 kg CO 2 per the functional unit with either of the EoL scenarios.

Figure 5 shows the environmental impacts of the life cycle divided into manufacturing, transport, installation, and use phase including energy consumption and the manufacturing and transport of the replacement luminaires. Only one luminaire of 50 000 h life is needed for the functional unit (Figure 5a), while two products of 36 000 h life are needed to provide light during 50 000 h (Figure 5b). The share of the manufacturing of the initial product and the replacement products accounted for up to 88 % and 33 % on the average of the life cycle impacts.

Uncertainties

Despite the efforts put in this LCA case, uncertainties and data gaps exist in the assessment. It was not possible to model all inputs of the luminaire life cycle or to obtain representative data of each input. There was no life cycle data available on the LED chip used in general lighting but it was estimated on the basis of an indicator LED. The environmental impacts are somewhat sensitive to the life of the luminaire. In addition, this LCA case study concerns only one example of LED downlight luminaire, and the LED luminaire product range is vast.

Discussion

The environmental impacts of the LED luminaire vary by environmental category. However, the use-stage energy consumption was found out to account for the majority of the environmental impacts in all impact categories except for the non-hazardous and hazardous waste categories. In these two categories, the manufacturing accounted for 79 % and 52 %, of the life cycle impacts, respectively. The high percentages of manufacturing in these waste categories are mainly due to the driver and aluminium parts. The EoL had very little to no effect on the total life cycle impacts due to the dominance of use and manufacturing stages. Yet, the benefits of proper recycling of the LED luminaire are recommended to be subjected to a more profound study.

The environmental impacts of the manufacturing stage are divided mainly between the LED driver, LED array and aluminium parts, which cover together over 80 % of the manufacturing impacts in all impact category. However, there is uncertainty in the LCI data of the LED component: The LCA was conducted using an estimation of an indicator LED to correspond to the LED component by a weight factor of five. Contradictory information has been published on the correlation between the environmental impacts of the indicator LED and a high-power LED component: A US DOE (2012b) report collects thoroughly the material and energy inputs of the manufacturing processes of an LED. It states that the environmental impacts of a high-power LED component represent approximately 94.5 % lower environmental impacts than the 5 mm indicator LED in the ecoinvent database. However, while the US DOE report compared the LED components on the basis of the luminous flux (4 lm to 100 lm), the LCA on the LED downlight luminaire considered the weight of the component, not the luminous flux, as the ecoinvent data was available on weight basis. With these notably different results and methods of comparison, it is clear that more studies and data are needed in order to conclude the environmental impacts of LED products, whether in terms of the weight or luminous flux of the LED component.

The manufacturing efforts of the LED product cannot be neglected in an LCA, since it is implied that the manufacturing may account for a notable share of the environmental impacts. US DOE (2012a) study estimates the manufacturing process energy consumption to range between 0.1 % and 27 % of the total life-cycle energy consumption. US DOE (2012b) report calculated that the raw materials used in manufacturing caused 16.8 % of the total life-cycle impacts on the average. The life of the LED luminaire affects the environmental impacts and their division between the life-cycle stages. The luminaire of the nominal life of 50 000 h causes approximately 17 % lower total environmental impacts compared to the case in which the luminaire had life of 36 000 h. The manufacturing of the LED luminaire causes 23 % of the life cycle impacts in the case of 50 000 h life, while the share of the manufacturing of the initial and replacement products is increased to 33 % in the case of 36 000 h life. Yet, there is significant variation in the sensitivity to the life among the impact categories.

The life cycle of the luminaire resulted to primary energy consumption of 13 070 MJ and GWP of 112 kg CO 2 per functional unit (50 000 h use of the luminaire producing 1140 lm). Several previous LCAs of light sources use lumenhours as the functional unit. The LED downlight consumed approximately 230 MJ primary energy and caused 2,0 kg CO 2 emissions per 1 Mlmh.

Such an LED downlight luminaire has not been subjected to an LCA before. It is not possible to accurately compare this LCA to the ones of LED lamps or LED luminaires of different purposes and LCA boundaries. However, the LCA results show that the use phase energy consumption is dominant but also that the manufacturing stage cannot be neglected. More research is needed in order to study the environmental impacts of the wide range of the LED technology.

Conclusions

The environmental impacts of the LED downlight luminaire in the case study were generally dominated by the energy consumption in the use stage. Thus, it confirms the use of luminous efficacy of a light source as an environmental indicator. However, the results show that the manufacturing efforts cannot be neglected in the LCA of the LED luminaire. Yet, it is acknowledged that there are uncertainties in the LCA calculations, notably in the modelling of the LED components.

The importance of the manufacturing stage will be further increased when the energy production is shifted to the less-polluting energy production. In that case, the significance of the EoL stage may also increase. The results showed that the EoL treatment had nearly no impact from the life cycle point of view. Yet, the recycling of the LED luminaire may increase along with the increase of the collection and treatment targets of the waste electrical and electronic equipment.

Further studies are suggested to include a wide range of LED products. In case of comparing light sources of different shapes and sizes, the functional unit needs to be appropriate. It is also advised to analyse the sensitivity of the LCA to the used energy source. In addition, the future sustainability studies may consider the fast increase of the luminous flux and decrease of the costs of the LED products. EP XP P01-020 kg PO 4 eq. Table 2 Environmental impact categories chosen for the LCA of LED downlight luminaire. CFC-11 refers to trichlorofluoromethane.

Tables 1 and2 Figures

Fig. 1 Fig. 2

 12 Fig. 1 Simplified diagram of the LED downlight luminaire

Fig.

  Fig. Division of the environmental impacts of the manufacturing stage of the LED downlight luminaire

Table 1

 1 Raw material and process inputs of the LED downlight luminaire manufacturing.

	Tables
	Part Driver	Raw material, product or process input PCB, 2-sided, 50 % SMT, 50 % THT, lead-free, including mounting processes	Amount 0.009 m	2
		Capacitors (electrolyte, film or unspecified)	18 g
		Diodes			0.6 g
		Resistors			2 g
		Transformers			48 g
		Integrated circuits		0.1 g
		Transistors			0.3 g
		Other components (active, passive or unspecified)	0.7 g
		Steel			4 g
		Plastics			130 g
		Connectors (65 % nylon 66, 27 % copper, 7 % iron, 1 % tin)	5 g
	LED array	Light-emitting diodes (16 pcs)		28 g
		Silicone product; thermoforming		3.74 g
		Aluminium			23 g
	Aluminium parts	Aluminium (heatsink and reflector), processing Coating	700 g 0.17 m	2
	Other parts	Steel			17 g
		Plastics; injection moulding		26 g
		Cable			7 g
		Paper			3 g
	Remote phosphor cover	YAG (yttrium aluminium garnet) coating	0.2 g
		Electricity, French mix		0.002 kWh
		Aluminium oxide		0.1 g
		Organic chemicals		0.1 g
		Plastics, injection moulding		7 g
	Assembly	Electricity, French mix		0,029 kWh
	Waste treatment (packaging) Recycling of the intermediary cardboard packages	175 g
	Environmental impact category	Abbreviation	Method	Unit (eq.=equivalent)
	Primary energy		PE	NF P01-010	MJ
	Renewable energy		RE	NF P01-010	MJ
	Non-renewable energy		NRE	NF P01-010	MJ
	Abiotic depletion potential		ADP	NF P01-010	kg Sb eq.
	Water consumption		WaC	NF P01-010	l
	Hazardous waste		HW	NF P01-010	kg
	Non-hazardous waste		NHW	NF P01-010	kg
	Inert waste		IW	NF P01-010	kg
	Radioactive waste		RW	NF P01-010	kg
	Global warming potential		GWP	NF P01-010	kg CO 2 eq.
	Acidification potential		AP	NF P01-010	kg SO 2 eq.
	Air pollution		AiP	NF P01-010	m 3
	Water pollution		WaP	NF P01-010	m 3
	Ozone depletion potential		ODP	NF P01-010	kg CFC-11 eq.
	Photochemical ozone creation potential	POCP	NF P01-010	kg C 2 H 4 eq.
	Eutrophication potential

Acknowledgement

The work is part of the CITADEL research project carried out by CSTB (Centre Scientifique et Technique du Bâtiment) in Grenoble in collaboration with LAPLACE (Laboratory on Plasma and Conversion of Energy) in University Paul Sabatier, Toulouse, and other French laboratories. This project is funded by ADEME, the French Agency for the Environment and Energy Management. In addition, the work is funded by the doctoral program in electrical energy engineering (DPEEE) in Finland.


Pierre Ropars

Christophe Desceliers

Guy Bonnet

Philippe Jean

Modélisation des incertitudes sur les vibrations d'origine ferroviaire transmises aux bâtiments

Keywords: FEM-BEM, Vibrations ferroviaires, Incertitudes paramétriques/nonparamétriques

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Quantification des incertitudes dans un modèle numérique d'interaction sol-structure soumis à une source ferroviaire

Ce papier porte sur la réponse d'un système dynamique d'interaction sol-structure avec plusieurs sources d'incertitudes. Les forces extérieures sont dues aux vibrations ferroviaires qui se propagent dans le sol jusqu'aux fondations puis dans le bâtiment. Dans des travaux précédents [START_REF] Soize | A nonparametric model of random uncertainties in linear structural dynamics[END_REF], un modèle numérique d'un tel système dynamique, sans incertitude, a été présenté [START_REF] Mondot | Characterization of structure-borne sound sources : The source descriptor and the coupling function[END_REF]. Ce modèle consiste à coupler une représentation déterministe d'un bâtiment avec ces fondations dans un sol proche couplé à un sol lointain. La méthode des éléments finis (FEM) est utilisée pour construire la représentation numérique du bâtiment, des fondations et du sol proche. Le sol éloigné est représenté par des éléments de frontière (BEM). La prédictivité de ce modèle est ici améliorée par la prise en compte des incertitudes reliées à chaque sous systèmes (le sol lointain, le sol proche et les fondations et le bâtiment). Nous proposons de quantifier les incertitudes relatives aux erreurs paramétriques pour chaque méthodes numériques et les incertitudes relatives aux erreurs de modélisation (approche non-paramétrique). De ce fait, un modèle probabiliste mixte paramétrique et non-paramétrique est construit. L'approche probabiliste paramétrique consiste en la modélisation des incertitudes paramétriques par une variable ou un champs aléatoire. L'approche non-paramétrique consiste en la modélisation des matrices globales de masse, raideur et amortissement de chaque modèles numériques par des matrices aléatoires. Les lois de probabilité de ces variables, champs et matrices aléatoires sont construites en utilisant le principe du maximum d'entropie (voir [START_REF] Villot | Modelling the influence of structural modifications on the response of a building to railway vibration[END_REF]). Ce papier traite un exemple numérique dans lequel les fluctuations statistiques de la réponse aléatoire sont déterminées par la méthode de Monte Carlo. La construction du modèle moyen par la méthode des mobilités est présentée dans le paragraphe 2. Dans le paragraphe 3, le modèle moyen est étendu au modèle probabiliste mixte paramétrique et non-paramétrique. Dans la suite, le paragraphe 4 donne les résultats attachés à cet exemple.

Modèle numérique moyen 2.1 Description du système mécanique

Dans ce paragraphe, nous présentons la méthodologie utilisée pour résoudre le problème d'interaction sol-structure dont l'excitation est une source ferroviaire. Cette méthodologie est basée sur la méthode des mobilités qui serra présentée dans le prochain sous-paragraphe. Pour la clarté de la présentation nous présentons un exemple en deux dimensions (voir la Figure 1). Le système étudié est constitué de deux sous-systèmes, le bâtiment et le sol entier incluant les fondations. Le bâtiment et les fondations sont constitués du même matériau. L'excitation ferroviaire est modélisée par une force ponctuelle f T (voir Figure 1), appliquée à la surface du sol, et dans la bande de fréquence [10 -250]

[A s (z, ω)] = -ω 2 [M s (z)] + ω[D s (z)] + [K s (z)] est telle que : [A s (z, ω)] u s (ω) = f s (ω) , (1)
où u s est le vecteur déplacement du modèle éléments finis et f s est le vecteur des forces extérieures. Considérons que les degrés de liberté sont ordonnés comme suit :

[B s ] =   [B 11 s ] [B 12 s ] [B 13 s ] [B 12 s ] T [B 22 s ] [B 23 s ] [B 13 s ] T [B 23 s ] T [B 33 s ]   , u s =   u 1 s u 2 s u 3 s   , f s =   f 1 s f 2 s f 3 s   , (2) où
[B s ] signifie [M s (z)], [K s (z)], [D s (z)] et [A s (z, ω)], et
(z)], [K s (z)], [D s (z)]. Nous obtenons alors [A s (z, ω)] q s (ω) = g s (ω) , (3) où
[A s (z)] = -ω 2 [M s (z)] + ω[D s (z)] + [K s (z)
] est la matrice de raideur dynamique réduite du sol proche avec les fondations, et où

q s =   u 1 s u 2 s q 3 s   , g s =   f 1 s f 2 s [φ s ] T f 3 s   , (4)
dans lequel q 3 s = [M 33 s (z)] -1 [φ s ] T [M 33 s (z)] u 3 s et u 3 s = [φ s ] q 3 s . Soit [S 1 (z, ω)] et [S 2 (z, ω)
] les matrices construites par l'utilisation de la BEM avec une représentation intégrale sur Γ c de façon à modéliser la propagation des ondes dans le sol lointain telle que :

[S 1 (z, ω)]u 2 s (ω) = [S 2 (z, ω)] f 2 s (ω) . (5)
Cette représentation est commune à de nombreux codes BEM-FEM (voir pour notre exemple [START_REF] Jean | Boundary and finite elements for 2D soil-structure interaction problems[END_REF][START_REF] Jean | A 2.5D BEM model for ground-structure interaction[END_REF]).

Les matrices [S 1 (z, ω)] et [S 2 (z, ω)] ne sont pas obligatoirement définies positives. Néanmoins, pour les fréquences régulières, l'inverse de [S 1 (z, ω)] existe, la fonction de réponse en fréquence [H 2 (z, ω)] = [S 1 (z, ω)] -1 [S 2 (z, ω)] est une matrice symétrique et nous avons alors :

u 2 s (ω) = [H 2 (z, ω)] f 2 s (ω) . (6)
Le modèle réduit du sous-système source est définie par les équations (3), ( 4) et (6).

Modèle numérique réduit du sous-système récepteur

Dans ce paragraphe, nous présentons le modèle réduit du sous-système récepteur (le bâtiment). En premier lieu, le modèle d'éléments finis est construit, puis il le modèle numérique réduit est déduit en résolvant un problème aux valeurs propres.

Soit la matrice de masse [M r (z)], la matrice de raideur [K r (z)] et la matrice d'amortissement [D r (z)], les matrices éléments finis modélisant le bâtiment. La matrice de raideur dynamique

[A r (z, ω)] = -ω 2 [M r (z)] + ω[D r (z)] + [K r (z)] est telle que : [A r (z, ω)] u r (ω) = f r (ω) , (7)
où u r est le vecteur des déplacements sur les degrés de liberté du modèle éléments finis et f r et le vecteur éléments finis des forces extérieures appliquées au bâtiment. Nous imposons que les degrés de liberté du sous-système sont ordonnés de la façon suivante : [A r (z, ω)] q r (ω) = g r (ω) ,

[B r ] = [B 11 r ] [B 12 r ] [B 12 r ] T [B 22 r ] , u r = u 1 r u 2 r , f r = f 1 r 0 , (8)
[A r (z)] = -ω 2 [M r (z)] + ω[D r (z)] + [K r (z) (9) où
] est la matrice de raideur dynamique du bâtiment où,

q r = u 1 r q 2 r , g r = f 1 r 0 , ( 10
)
dans lequel q 2 r = [M 22 r (z)] -1 [φ r ] T [M 22 r (z)] u 2 r , de même u 2 r = [Φ r ] q 3 r .

Couplage des modèles réduits des sous-systèmes source et récepteur

Dans ce paragraphe, nous présentons la méthode des mobilités utilisée pour coupler les deux soussystèmes source et récepteur. La méthode des mobilités consiste à calculer f 1 r = -f 1 s , la force appliquée sur le récepteur par la source, en quatre étapes.

Premièrement, la vitesse libre v (ω) = ω u 1 s (ω) est déterminée en résolvant les équations (3), ( 4) et ( 6) pour f 3 s (ω) modélisant la force extérieure f T (ω) (voir Figure 1) et f 1 s (ω) = 0 (interface Γ s libre). En parallèle, les matrices des mobilités de la source [Y Γs (z, ω)], et du récepteur [Y Γr (z, ω)], déterminées sur les degrés de liberté portés par les surfaces Γ s et Γ r (voir Figure 1) doivent être construites. Elles peuvent être facilement obtenues grâce à la décomposition par bloque introduite précédemment. Néanmoins, cela n'est pas toujours possible avec un code commercial. Une approche différente consiste à calculer directement les matrices de mobilité en résolvant un ensemble de problèmes. La mobilité est écrite comme le rapport entre la vitesse et la force appliquée. Dans notre cas, résoudre les équations (3), (4) et (6) pour f 3 s (ω) = 0 (aucune force extérieure à la source) et {f 1 s (ω)} k = δ jk (une force unitaire appliquée sur les degrés de liberté de Γ s ) nous donne : [Y Γs (z, ω)] ij = ω {u 1 s (ω)} i , et résoudre ( 9) et [START_REF] Soize | A nonparametric model of random uncertainties on reduced matrix model in structural dynamics[END_REF] pour {f 1 r (ω)} k = δ jk (force unitaire appliquée sur les degrés de liberté de Γ r ), nous donne : [Y Γr (z, ω)] ij = ω {u 1 r (ω)} i . Puis la force de contact, f p (ω), qui est appliquée par la source sur le système récepteur (lorsque ceux-ci sont couplés) est alors donnée par (pour plus de détails voir [START_REF] Cremer | Structure-Borne Sound structural vibrations and sound radiation at audio frequencies[END_REF][START_REF] Ropars | Modélisation des vibrations d'origine ferroviaire transmises aux bâtiments par le sol[END_REF]) :

f p (ω) = ([Y Γs (z, ω)] + [Y Γr (z, ω)]) -1 v (ω) . (11)
Finalement, la réponse du récepteur est déterminée en résolvant les équations ( 9) et (10) pour f 1 r (ω) = f p (ω). La vitesse moyenne des dalles v d est un indicateur de la gène ressentie par les résidents d'un bâtiment soumit à de telles vibrations [START_REF] Ohlrich | Structure-borne sound sources and their power tranfer[END_REF] et est utilisée ici.

Modèle probabiliste généralisé des incertitudes

Afin d'augmenter la prédictibilité du modèle numérique, plusieurs sources d'incertitudes sont prises en compte. Nous supposons que ρ soil , ρ struct , b soil , b struct , s soil , s struct sont les seuls paramètres incertains de ce modèle. Le modèle probabiliste paramétrique des incertitudes est construit en remplaçant le vecteur z par un vecteur aléatoire Z = (Z 1 , Z 2 , Z 3 , Z 4 , Z 5 , Z 6 ), dont les composantes sont des variables aléatoires dont le modèle probabiliste est construit en utilisant le principe du maximum d'entropie. De ce fait, Z 1 , Z 2 , Z 3 , Z 4 , Z 5 , Z 6 sont des variables aléatoires statistiquement indépendantes ayant une loi de probabilité gamma,leurs fonctions de densités de probabilités sont complètement gouvernées par leurs valeurs moyennes E{Z i } = {z} i et leurs coefficients de dispersions δ Zi [START_REF] Guilleminot | On the statistical depenence for the components of random elasticity tensors exhibing material symmetry properties[END_REF]. Cette loi de probabilité assure, entre autre, que le moment d'ordre deux des composantes du vecteur aléatoire soit fini. En parallèle, l'approche probabiliste généralisée consiste à substituer les matrices aléatoires [B(Z)] par les matrices aléatoires

[B] = [L B (Z)][G B ][L B (Z)] T où [B(Z)]

Résultats et discussions

Incertitudes généralisées sur les valeurs moyennes des vibrations d'une dalle du bâtiment

L'aléa généralisé (Figure 4) apparaît proche de celui généré par les erreurs de modélisation. Du fait de la présence des erreurs de paramètres, l'écart type est ici plus petit que dans le cas où seule les erreurs de modélisation sont présentes.

  Hz. Cette représentation peut paraître simpliste mais cela permet une calibration afin de reproduire exactement des vibrations mesurées in situ [13]. En parallèle, une source plus sophistiquée peut être obtenue par la superposition de forces ponctuelles. Par la suite et dans le cadre de la méthode des mobilités, nous dénommerons le bâtiment par récepteur et le sol, incluant les fondations, par source. Le récepteur est modélisé par la méthode des éléments finis (FEM), la source est, quand à elle, construite en utilisant la méthode des éléments finis couplée aux éléments de frontière (FEM-BEM). Dans cette représentation les paramètres incertains sont la densité de masse du sol ρ soil et de la structure ρ struct (bâtiment et fondations), le module d'élasticité isostatique du sol b soil et de la structure b struct , et le module de cisaillement du sol s soil et de la structure s struct . Nous introduisons un vecteur contenant ces paramètres z = (ρ soil , ρ struct , b soil , b struct , s soil , s struct ).

Figure 1 -

 1 Figure 1 -Représentation de l'exemple

  où [B r ] signifie [M r (z)], [K r (z)], [D r (z)] et [A r (z, ω)], et où u 1 r est le vecteur contenant les degrés de liberté relatifs à Γ r et u 2 r le vecteur relatif aux autres degrés de liberté de la représentation éléments finis du bâtiment. Soit Φ 1 , Φ 2 , . . . les vecteurs propres obtenus par la résolution du problème aux valeurs propres suivant [K 22 r (z)] Φ = λ [M 22 r (z)] Φ. Puisque [M 22 r (z)] et [K 22 r (z)] sont deux matrices symétriques réelles définies positives, les valeurs propres associées 0 < γ 1 ≤ γ 2 ≤ . . . sont positives. Nous définissons [Φ r ] = (Φ 1 Φ 2 . . . Φ m ) comme les matrices modales réduites. La matrice globale de masse [M 22 r (z)] = [φ r ] T [M 22 r (z)] [φ r ], la matrice global de raideur [K 22 r (z)] = [φ s ] T [K 22 r (z)] [φ r ] et la matrice global d'amortissement [D 22 r (z)] = [φ r ] T [D 22 r (z)] [φ r ] sont des matrices m × m réelles, symétriques et définies positives. Nous construisons ainsi la matrice de masse réduite [M r (z)], la matrice de raideur réduite [K r (z)], et la matrice d'amortissement réduite [D r (z)] en substituant les matrices [M 22 r (z)], [K 22 r (z)], [D 22 r (z)] par [M 22 r (z)], [K 22 r (z)], [D 22 r (z)] dans la décomposition par bloque de [M r (z)], [K r (z)], [D r (z)]. Nous obtenons alors :

  Les réalisations statistiquement indépendantes de V d , la moyenne spatiale de la vitesse vibratoire d'une dalle du bâtiment, sont déduites à partir des réalisations indépendantes du vecteur aléatoire Z et des matrices aléatoires[M 33 s ], [D 33 s ], [K 33 s ], [M 22 r ], [D 22 r ], [K 22 r ], [M 11 s (Z)], [M 12 s (Z)], [M 13 s (Z)], [M 22 s (Z)], [M 23 s (Z)], [D 11 s (Z)], [D 12 s (Z)], [D 13 s (Z)], [D 22 s (Z)], [D 23 s (Z)], [K 11 s (Z)], [K 12 s (Z)], [K 13 s (Z)], [K 22 s (Z)], [K 23 s (Z)], [M 11 r (Z)], [M 12 r (Z)], [D 11 r (Z)], [D 12 r (Z)] et [K 11 r (Z)], [K 12 r (Z)], [H 2 (Z, ω)]. Présentement, les résultats sont obtenus avec δ Z1 = δ Z2 = δ Z3 = δ Z4 = δ Z5 = δ Z6 = 0.1 et δ M 33 s , δ K 33 s , δ D 33 s , δ M 22 r , δ K 22 r , δ D 22 r = 0.014.1 Comparaison de l'impact des erreurs paramétriques et de modélisation sur la vitesse moyenne de dalle

Figure 2 -

 2 Figure 2 -Valeurs moyennes et écarts types du déplacement dans la direction x (haut) et dans la direction y (bas) sur la gamme des fréquences ferroviaires ([10-250]Hz)

Figure 3 -

 3 Figure 3 -Valeurs moyennes et écarts types : déplacement dans la direction x (haut) et dans la direction y (bas) sur la gamme des fréquences ferroviaires ([10-250]Hz)

  où u 1 s et u 2 s sont les vecteurs relatifs à Γ s et Γ c respectivement et u 3 s est relatif aux autres degrés de liberté du modèle éléments finis. Soit φ 1 , φ 2 , . . . les vecteurs propres construits par la résolution du problème aux valeurs propres [K 33 s (z)] φ = λ [M 33 s (z)] φ. Puisque [M 33 s (z)] et [K 33 s (z)] sont deux matrices définies positives à composantes réelles, les valeurs propres associées 0 < ω 1 ≤ ω 2 ≤ . . . sont positives. Soit [φ s ] = (φ 1 φ 2 . . . φ n ) la matrice de la base modale tronquée. La matrice globale de masse [M 33 s (z)] = [φ s ] T [M 33 s (z)] [φ s ], la matrice globale de raideur [K 33 s (z)] = [φ s ] T [K 33 s (z)] [φ ] et la matrice globale d'amortissement [D 33 s (z)] = [φ s ] T [D 33 s (z)] [φ s ] sont des matrices n×n symétriques réelles définies positives. Ainsi nous construisons la matrice de masse réduite [M s (z)], la matrice de raideur réduite [K

	s (z)], la matrice d'amortissement
	réduite [D s (z)] en substituant les matrices [M 33 s (z)], [K 33 s (z)], [D 33 s (z)] par [M 33 s (z)], [K 33 s (z)], [D 33 s (z)]
	dans la décomposition par bloque de [M s

  B (Z)] sont telles que [L B (Z)][L B (Z)] T = [B(Z)] et les matrices aléatoires [G B], qui sont statistiquement indépendantes de Z, sont prises dans l'espace SE + introduit dans[START_REF] Soize | A nonparametric model of random uncertainties on reduced matrix model in structural dynamics[END_REF][START_REF] Soize | Maximum entropy approach for modeling random uncertainties in transient elastodynamics[END_REF][START_REF] Villot | Modelling the influence of structural modifications on the response of a building to railway vibration[END_REF]. Le modèle probabiliste de [G B ] a été construit en utilisant le principe du maximum d'entropie et sa fonction de densité de probabilité est complètement définie par un coefficient de dispersion δ B . De ce fait, le modèle probabiliste présenté dans ce document et complètement définie par six valeurs moyennes E{Z 1 } = {z} 1 , . . . , E{Z 6 } = {z} 6 , et par douze coefficients de dispersions δ Z1 , δ Z2 , δ Z3 , δ Z4 , δ Z5 , δ Z6 et

	signifie les matrices [M 33 s (Z)], r (Z)] et où les matrices aléatoires triangulaires inférieures r (Z)], [D 22 r (Z)], [K 22 s (Z)], [M 22 s (Z)], [D 33 [K 33
	[L δ M 33 s , δ K 33 s , δ D 33 s , δ M 22 r , δ K 22 r , δ D 22 r .