Fabio Mancinelli
email: fabio.mancinelli@inria.fr

Leveraging the Web Platform for Ambient Computing: An Experience

Keywords: Web Platform, Ambient Computing, Interoperability

This paper explores a very simple idea: what can be achieved by using the principles and the technologies of the Web Platform 1 when they are applied to Ambient Computing? To answer this question we will present an experience that addresses some of the problems of Ambient Computing by making use of the technologies and the common practices of today's Web Platform. The aim is to provide an architecture that tries to lower the deployment costs while guaranteeing accessibility, interoperability and extendibility.

Introduction

Ambient computing proposes a paradigm where devices present in a physical environment collaborate in order to support people in carrying out their daily tasks. Though many technologies have been introduced in order to implement this paradigm, there are still a lot of open issues that must be addressed and solved. In particular many of these issues concern the "ubiquitousness" of these solution and their interoperability with the multitude of heterogeneous devices that might interact with such a kind of systems. In this paper we explore a very simple idea: what can be achieved by using the principles and the technologies of the Web Platform when they are applied to Ambient Computing? This idea is motivated by the fact the by leveraging the Web Platform we can address interoperability issues by relying on a uniform platform that has a well defined set of protocols and and semantics. This is very important because different solutions must interoperate in order to provide a better experience to the endusers. By using the Web Platform as a common ground, and by building on top of it, many of the interoperability issues are solved forefront. In this paper, basically, we took the Web Platform as the reference platform, and built on top of it a simple system for Ambient Computing paradigm. Our aim was to describe the experience we did in building an actual system and to present what are the advantages that, in our opinion, derive from following such a kind of approach.

In this section we will briefly introduce the Web Platform and what it consists of. We used the term platform because nowadays developers are using the Web, its architecture and the technologies it is based on, as an actual platform for engineering, developing and deploying their applications2 . The idea of the Web as a platform is also corroborated by all the companies that are producing advanced tools and solutions for easily building complex application without taking care of all the low-level details [START_REF]Google Web Toolkit[END_REF][START_REF] Ruby | Agile Web Development with Rails, Third Edition[END_REF] The Web Platform has an architecture that is based on a well defined set of principles and constraints [START_REF] Fielding | Principled design of the modern web architecture[END_REF] that are implemented by a set of standard and widely deployed communication protocols [START_REF] Fielding | Hypertext transfer protocol -HTTP/1.1[END_REF], and makes use of commonly used data formats for exchanging information [START_REF]IANA, Iternet Assigned Numbers Authority: MIME media types[END_REF].

The key principles and constraints described in [START_REF] Fielding | Principled design of the modern web architecture[END_REF], that are relevant for the purpose of this paper are:

-Addressability: Every resource must addressable by a well defined resource identifier. -Uniform interface: A single interface, with a well defined semantics, must be used for accessing and manipulating resources. -Hypermedia as the engine of application state: By leveraging addressability, the execution of the application is obtained by following hypermedia links to resources that represent the "next state" of the application.

These principles are implemented in the Web Platform in the following way: addressability is given by using Uniform Resource Identifiers (URIs) that provide a "simple and extensible way for identifying an abstract or physical resource" [START_REF] Berners-Lee | Uniform resource identifiers (URI): Generic syntax[END_REF]. URIs are used extensively in order to address any resource 3 , and to operate on their representations. URIs also enable hypermedia features that are used as the primary mechanism to make applications change their state.

The uniform interface is implemented by the HTTP protocol that provides five standard methods with a well defined semantics 4 for interacting with resources.

The HTTP protocol is a very simple request-response protocol that exchanges messages consisting of a set of headers and a body. Headers contain metainformation for further describing the HTTP message; the body contains data to be associated to HTTP messages depending on their nature. HTTP headers are essential for making messages self descriptive and amenable to advanced pre/post processing.

Another important aspect of the HTTP protocol are their response code. Each HTTP request comes with a response containing a code used to communicate the result of the request. There are several response codes, grouped in classes: meta, success, redirection, client-side error and server-side error classes. For the purpose of this paper, the redirection class will have a particular relevance, as it will described below.

To conclude this section we would like to point out that the most important part of the Web Platform consists of all the standard format, languages and clients that are used to create leverage for the web applications. In particular the XML [START_REF] Bray | Extensible markup language (XML) 1.0[END_REF] and XHTML [START_REF]W3C HTML Working Group: XHTML 1.0 the extensible hypertext markup language[END_REF] formats, the JavaScript [START_REF]ECMAScript language specification[END_REF] language and the standard-compliant Web browsers. These elements, of course, will be central in the experience presented in this paper.

Architecture

In Ambient Computing, there are several problems that should be addressed in order to build an effective system. First of all there should be a way for defining, discovering and identifying what is available in the surroundings and, of course, there should be a way for interacting with the found entities.

These requirements entail several others that can be summarized as the following:

-Networking: How can the entities that are present in a given environment exchange information among them? -Layout: What is exactly the surrounding environment and how could it be defined and identified? -Addressing: How can we address entities that are in a given environment? -Management: How can an entity providing a service declare itself, and how other entities can be aware of it? -Interaction: How can entities co-operate, and what kind of interaction entities can have in order to fulfill a given task?

The proposed architecture is presented in Figure 1 and, by having in mind the previous list, we will describe in the following sections its structure and how these requirements are met.

Networking

For our architecture we need a flexible networking infrastructure that should be widely supported and easy to manage. The choice is to use the TCP/IP networking on top of WiFi connectivity. Basically the target infrastructure will consist of TCP/IP (private) networks that are pervasively available on a physical environment through WiFi hotspots. An hotspot defines an environment called an environment that logically regroups all the devices connected to the hotspot.

This choice might seem restrictive because it presupposes, in order to be compatible with the infrastructure, the availability of built-in WiFi support on the devices. However many current-generation smart-phones, PDAs, and appliances have already this support built-in and in the future this support will become more and more widespread.

Since we want to deploy the architecture on top of the Web Platform, another assumption we are making is that the devices should be equipped at least with a Web browser, i.e., a component capable of understanding the HTTP protocol. This again should not be too restrictive: a WiFi-equipped device surely has a browser built-in (or installable as a third party software), so we can suppose that if a device is able to connect to the WiFi infrastructure, it will also able to "browse" what is available on that infrastructure as well.

Finally we don't make any assumption on the capabilities of the browser, unless for the fact that it should support basic XHTML capabilities like receiving and sending forms, and support redirection.

Layout and discovery

With the term layout, we refer to the definition of how a physical environment, i.e, the set of all the devices that are linked to a WiFi hotspot are organized. In order to do so we introduce the notion of space: a logical environment that is used to group devices that belong to the same "domain". The semantics of a domain is defined by the administrator and can, for example, reflect the organization of the physical environment (i.e., each space might correspond to a physical room), or to some administrative subdivision (i.e., a space for the Computer Science Department).

The entry-point of the system, from the point of view of a device, is the WiFi hotspot where it connects to. Once it is connected, an IP address is assigned to it and it can start to communicate with other entities present on the same network.

The problem now is how can this device discover the layout that is defined for a given environment? Discovery protocols such as SLP [START_REF] Guttman | Service location protocol[END_REF] or SSDP used in the Universal Plug and Play framework [START_REF] Goland | Simple service discovery protocol/1.0: Operating without an arbiter[END_REF], use the underlying network topology for delivering multicast packets containing announcement information about the device present in the network. This announcement also contains information the actual logical location of a given resource (e.g., printer-location: 'Room 3').

This approach has two consequences: first, a program listening for packets on a given multicast port should run on the mobile device; and second, the device should embed some logic in order to understand the messages and, for example, filtering only the messages that refers to devices in "Room 3". Since these messages are completely arbitrary, an ad-hoc application is needed for this.

In our architecture we would like to impose the least requirements on the device using the infrastructure (i.e., only they should be equipped only with a Web browser), so the previous approach is not suitable.

When a device wants to interact with the infrastructure it has to obtain an IP address from one of the available WiFi hotspots. This is usually done using the DHCP [START_REF] Droms | Dynamic host configuration protocol[END_REF] protocol that configures the device's gateway, DNS and so on. So the idea is to configure the WiFi network for hosting a transparent proxy that will redirect all the traffic generated by the browser to a given server that can provide, despite of the actual address the user typed in her browser, an XHTML page that represent the entry point for all the environments registered at that hotspot.

This is basically the role of the Environment Manager pictured in Figure 1. What this page actually contains and how it is built will be detailed in Sections 3.4 and 3.5. Of course the content of this page will reflect the logical layout of the devices that are present.

This setup implements a transparent discovery mechanism that can be used without using any additional software. We would like to point out that what is discovered here is an entry point that will give directions on how to reach actual devices, while in SLP the target devices are directly discovered. Therefore, our approach implies that there is a central authority, i.e., the Environment Manager that is behind the WiFi hotspot, that should keep track, in a centralized way, of all the device that are present in a given environment. The procedure for doing so will be described in Section 3.4.

Addressing

When we reason about addressing the first thing to decide is what are the resources in our domain that needs to be addressed, and how these addresses are structured. In our architecture we will address two kinds of resources: spaces and service devices. The first ones are used for the logical organization of devices, in order to reflect their actual disposition in physical space or in administrative domains; the second ones are devices that provide the actual services. As we are leveraging the Web Platform, we will use URIs for addressing resources, and the address template has the following form: http://environment/{space}/{device} 5 .

From the address template it is clear that it is not possible to specify space nesting. In fact, in our architecture, an address like http://campus/building/hall/room/printer would not be recognized.

However we allow the definition of a logical hierarchy of spaces by associating to each space a parent addressable using the special address http://place/{space}/parent.

The fact that service devices can be addressed using an URI, implies that every service device must have a web proxy that can be used in order to interact with it. This web proxy, that can be located on the devices itself or somewhere else, provides the actual interface, adapted to the Web Platform, and that will be used for controlling the device itself. Usually, as we will show in Section 3.5, this interface will consist of an XHTML page presenting a form with all the relevant attributes exposed by the device.

Management

In order to organize the layout of the environment, service devices need to be able to perform two sets of operations: logical space creation/removal/re-parenting, and device registration/unregistration/re-parenting in a given environment.

All these operation are carried out by leveraging the HTTP protocol.

Space operations are quite straightforward: a space is created or re-parented by PUTting the space name and the (optional) parent name to the environment entrypoint, whose location is given by http://environment. Re-parenting is done in the case where the space already exists.

A space is removed by DELETEing the space address. The current mean by which they are performed are XHTML forms that are presented in documents that are served by the environment manager. These forms provide the interface for performing the previous operations, and are available as resources at well defined URI 6 . The attributes for identifying the space name and parent are arbitrary since they are provided through the form by the same entity (i.e., the environment entry point) that will re-interpret them when they will be received as POSTed data. This is fine is the POSTer is a human that fills out the form, however if we imagine to automatize this process, then these attributes should be well defined and immutable.

The attentive reader might notice that XHTML forms only supports the GET and POST methods 7 . However it is possible to overload the POST method by specifying the actual HTTP method in an hidden field of the form. In this way the POST request will be treated as if it were a request that used the HTTP method specified in the POSTed data.

Device registration, unregistration and re-parenting are done in a similar way. What changes is the set of attributes that are contained in the POSTed data. The device relevant attributes are: its name, the optional logical space where it is located, and the URI for its representation. (f) Service device registration.

Interaction

Fig. 2. Interaction with the system.

In this section we will present a classic interaction of a device with the system. This description is based on a real case study that we have implemented as a proof of concept, and that is available at [START_REF] Mancinelli | Leveraging the web platform for ambient computing: An experience -case study[END_REF] A user enters a space and with its mobile phones looks for WiFi hotspots that are available. Once she has found a suitable one, she connects to it and opens her browser.

At this point she doesn't know what address to type in the address bar, however no matter what she will type the Environment Manager will intercept all her requests and reply as if they were addressed to itself. Since many browser have a predefined home page that is displayed when the browser is opened, this operation will be totally transparent for the user and what she will see after connecting to the hotspot is just the environment entry point as presented in Figure 2(a). Of course this means that all the user will not be able to reach the "outer" internet, however this interception mechanism could be put in place only for the first user request. In this way the user will be able to bookmark the environment entry point page, and will be free to use the internet as usual afterwards.

From the environment entry point, the user can navigate the structure of the environment by following hyperlinks; for example she might go to the room she is currently. Once the space page is displayed, the user has a view of all the devices that are present in the given space (Figure 2(b)). A service device called "MP3-player" is available for playing music in the current room. Clicking on the service device will present the user an interface for operating the player. This interface is served by the web proxy associated to the actual MP3-player.

We point out that the actual web proxy could be located everywhere in the network. For example, it could be running in the data centers of the campus (i.e., in the "outer world" pictured in Figure 1). What happens when a device requests a http://environment/{space}/{device} address is that the Environment Manager redirects the device to a new address that is the one of the actual web proxy for the service device. This is possible because of the HTTP protocol that supports, as described in Section 2, redirection response codes.

Once the user has the device interface displayed on her mobile phone, she can select an MP3 file from her collection and send it to the service device for playing it. Again, this is done by leveraging the HTTP protocol that allows the user to POST data to a given address (i.e., the one of the service device). Figures 2(c) and 2(d) show this interaction whose effect is that of having the (local) MP3 file streamed to the player in the room: the user can then enjoy her favorite music on the HiFi system's speakers in the room.

We would like to point out also the fact that since all the resources are directly addressable, the user has the possibility of bookmarking them for later usage. So, instead of following the link-chain from the environment entry point to the current space and then to the service device, she can directly jump to one or the other.

Finally, the user might also register a service device by using the registration interface provided by the environment manager. This interface, that is presented in Figure 2(f) allows the user to bind, in a given space a service device to a given web-proxy, by giving its address. HTTP already provides authentication mechanisms that can be used to restrict this "administrative" task only to the users that are allowed to perform it (Figure 2(e)).

Conclusion

In this paper we have presented an experience in building a system for Ambient Computing that made use of the Web Platform as its reference architecture. What we presented in this paper is related to previous work presented in [START_REF] Kindberg | A web-based nomadic computing system[END_REF] and [START_REF] Edwards | The case for recombinant computing[END_REF]. In fact there is a big overlap with CoolTown which proposes an infrastructure that is based on the Web Platform, exposes devices through HTML interfaces, and uses URI as the mean for identifying them.

In our system we wanted to stick to basic features available in the Web Platform, without relying on adding any-logic to devices. For example, in CoolTown, extra hardware is used for "sensing" device addresses provided by beacons, and to input them in the web browser (though these addresses might also be directly typed)

The principle we wanted to follow was to stick to the semantics of the HTTP protocol, that provide the uniform interface for accessing every device in the environment, and to build on top of it all the necessary infrastructure for manipulating and accessing the resources provided by these devices.

The experience has proven that this approach is very effective: our system has been built on top of reusable components (i.e., the Web Browser and the transparent proxy8 ), an Environment Manager that consists of less than 500 lines of code and a simple 10 lines Ruby script for implementing the MP3 Player proxy.

Of course our system is very limited at the moment, even when compared to the more advanced CoolTown one, however it is already good enough for a simple ambient computing environment.

The future work will consist in experimenting in the same direction, by integrating other Web technologies and standards in this "base system". For example a direction to explore is that of using syndication formats for handling what is available, and how it can be accessed. There is a lot of work in the Web community around those standard and we think that they can be fruitfully exploited for handling in a seamless and interoperable way device interactions in a pervasive system.

Fig. 1 .

 1 Fig. 1. The system architecture

  A service device accessible from a space. (d) Device communication. (e) Authentication.

With the term "Web Platform" we refer to the ensemble of the protocols and standards the World Wide Web is built upon.1st International Workshop on Ad hoc Ambient Computing (AdhocAmC'08)

This is basically something similar to what happened with the Java Platform[START_REF]Java SE technologies at a glance[END_REF], with the introduction of the Java language, its standard libraries and all their extensions like the "Enterprise Edition" or the "Micro Edition".

The term "resource" is highly generic but, as stated in[START_REF] Richardson | RestFul Web Services[END_REF], we can say that a resource is "anything that is important enough to be referenced as a thing in itself ".

These methods are the well known GET, HEAD, PUT, DELETE, POST, and some other less known ones: OPTIONS, TRACE and CONNECT. Though the semantics for these methods is well defined, POST is an exception because its actual semantics depends on what is specified in the body of the corresponding HTTP request.

This is actually an URI template as defined in[START_REF] Gregorio | Uri template[END_REF].

In our system we publish the forms for registering spaces at http://environment/registration, and the forms for registering service devices at http://environment/{space}/registration

[START_REF] Berners-Lee | Uniform resource identifiers (URI): Generic syntax[END_REF] This limitation will be overcome in XHTML5, but it will take some time before this standard will be widely adopted.

Implemented using by a Squid web caching proxy, and a 10 lines script for URI redirection


E Denimal
email: e.denimal@imperial.ac.uk

F El Haddad
email: f.el-haddad@imperial.ac.uk

C Wong

L Salles
email: l.salles@imperial.ac.uk

TOPOLOGICAL OPTIMIZATION OF UNDER-PLATFORM DAMPERS WITH MOVING MORPHABLE COMPONENTS AND GLOBAL OPTIMIZATION ALGORITHM FOR NONLINEAR FREQUENCY RESPONSE

Keywords:

To limit the risk of High Cycle Fatigue, underplatform dampers are traditionally used in aircraft engines to control the level of vibration. Many studies demonstrate the impact of the geometry of the damper on its efficiency, thus the consideration of topological optimization to find the best layout of the damper seems natural. Because of the nonlinear behaviour of the structure due to the friction contact interface, classical methods of topological optimization are not usable. The present study proposes to optimize the layout of an underplatform damper to reduce the level of nonlinear vibrations computed with the Multi-Harmonic Balance Method. The approach of topological optimization employed is based on the Moving Morphable Components framework together with the Kriging and the Efficient Global Optimization algorithm to solve the optimization problem. The results show that the level of vibration of the structure can be reduced of 30% and allow for the identification of different efficient geometries.

INTRODUCTION

In the aerospace industry, the requirements for the design of engines are becoming more stringent and new engines must be more efficient and lighter. Thus, each component tends to reach its structural limits. Considering the turbine blades, they are submitted to high level of exposure to thermal stresses, centrifugal stresses or vibrational stresses [START_REF] Krack | Vibration prediction of bladed disks coupled by friction joints[END_REF][START_REF] Petrov | State-of-the-art dynamic analysis for non-linear gas turbine structures[END_REF]. The latter can lead to High Cycle Fatigue (HCF) and can potentially lead to a failure of the blades [START_REF] Cowles | High cycle fatigue in aircraft gas tur-(a) M = 0.5M full (b) M = 0.3M full[END_REF]. Because of the large operating speed range and the high modal density of bladed disks, all the critical resonances cannot be avoided. For this reason, it is crucial to limit the level of vibrations at resonance. The most widely used solution are based on friction damping [START_REF] Gaul | The role of friction in mechanical joints[END_REF][START_REF] Feeny | A historical review on dry friction and stick-slip phenomena[END_REF][START_REF] Griffin | A review of friction damping of turbine blade vibration[END_REF].

Dry friction can be introduced in different locations of the structure (shrouds, roots, blade tips), but the most efficient solution relies on the use of Underplatform Dampers (UPD) [START_REF] Griffin | A review of friction damping of turbine blade vibration[END_REF][START_REF] Szwedowicz | Numerical and experimental damping assessment of a thin-walled friction damper in the rotating setup with high pressure turbine blades[END_REF]. They are located in the groove under the platforms of two adjacent blades and consist in a small metal device. They remain in contact with the platforms due to the centrifugal loading. When the blades vibrate, the relative displacement between the platforms and the UPD generates friction and so energy is dissipated [START_REF] Sanliturk | Underplatform dampers for turbine blades: theoretical modeling, analysis, and comparison with experimental data[END_REF][START_REF] Sanliturk | Friction damper optimization: simulation of rainbow tests[END_REF].

The geometry of the UPD has a significant impact on its effectiveness. Indeed, many studies are related to the study of the comparison of different geometries. The most classical geometries are based on a wedge geometry [START_REF] Petrov | Advanced modeling of underplatform friction dampers for analysis of bladed disk vibration[END_REF][START_REF] Panning | Optimization of interblade friction damper design[END_REF][START_REF] Pesaresi | Modelling the nonlinear behaviour of an underplatform damper test rig for turbine applications[END_REF], a cylindrical geometry [START_REF] Panning | Optimization of interblade friction damper design[END_REF][START_REF] Csaba | Modelling of a microslip friction damper subjected to translation and rotation[END_REF] or a mix of them [START_REF] Zucca | Range of variability in the dynamics of semi-cylindrical friction dampers for turbine blades[END_REF]. From those studies, it appears that the shape and the geometry of the damper has a high impact on its efficiency.

Topological Optimization (TO) for continuum structures consists in the identification of a sub-domain occupied by the structure in a total available domain where only the boundary conditions are known [START_REF] Rozvany | Layout and generalized shape optimization by iterative coc methods[END_REF][START_REF] Eschenauer | Topology optimization of continuum structures: a review[END_REF]. The topology of a component is defined by its internal and external boundaries and by the number of inner holes. These characteristics are optimized simultaneously with respect to a predefined objective and constraint functions. TO is mostly used in a pre-design stage to identify the most efficient layout of the structure. The two most classical methods of TO are the density-based approach and the Level Set Method (LSM). The first method consists in the optimization of the element densities of a mesh describing the design space [START_REF] Bendsøe | Optimal shape design as a material distribution problem[END_REF][START_REF] Philip | Generating optimal topologies in structural design using a homogenization method[END_REF]. The second one is based on an implicit description of the material boundaries with a Level Set Function (LSF), which is often propagated by solving a Hamilton-Jacobi equation [START_REF] Van Dijk | Level-set methods for structural topology optimization: a review[END_REF][START_REF] Allaire | Structural optimization using sensitivity analysis and a levelset method[END_REF]. These two approaches require the sensitivities, of the densities or of the shape, with respect to the objective function and the constraints. These sensitivities might be impossible to determine analytically for some problems and are too expensive to be determined numerically.

A recent approach, called the Moving Morphable Components (MMC) framework, was proposed by Guo et al. [START_REF] Guo | Doing topology optimization explicitly and geometrically-a new moving morphable components based framework[END_REF]. The basic idea relies on the fact that the topology can be described by the association of elementary beams. Hence, the material domain is still described by an LSF which is decomposed in local LSFs that describe each beam. The expression of these local LSF is explicit and depends on a few parameters. Hence, the size of the optimization problem is drastically reduced, and it gives the opportunity to consider global optimisation methods. Thus, in 2019, Raponi et al. [START_REF] Raponi | Kriging-assisted topology optimization of crash structures[END_REF], proposed to use the Kriging meta-modelling method together with the Efficient Global Opti-mization (EGO) algorithm [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] to solve a problem of topological optimisation for crash-worthiness where the sensitivities cannot be determined.

Considering vibration issues, TO has been applied to the optimisation of eigenvalues in a linear context in [START_REF] Sun | Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components[END_REF][START_REF] Bendsøe | Optimization of material properties for improved frequency response[END_REF][START_REF] Díaaz | Solutions to shape and topology eigenvalue optimization problems using a homogenization method[END_REF][START_REF] Allaire | A level-set method for vibration and multiple loads structural optimization[END_REF] with the different methods presented previously. The extension to the non-linear case is done in [START_REF] Dou | Optimization of nonlinear structural resonance using the incremental harmonic balance method[END_REF][START_REF] Dou | Structural optimization for nonlinear dynamic response[END_REF] where the width of a clamped-clamped beam is optimised (constant thickness) and geometric non-linearities are considered. Gradient-based optimisation is used for the optimisation. Considering contact problems, the problem of TO has been addressed in a quasi-static context in [START_REF] Myśliński | Topology optimization of quasistatic contact problems[END_REF][START_REF] Strömberg | Topology optimization of structures in unilateral contact[END_REF][START_REF] Maury | Shape optimisation with the level set method for contact problems in linearised elasticity[END_REF]. The objective of this study is to perform TO for the optimisation of nonlinear dynamic systems in the presence of a friction interface, which has never been done before according to the knowledge of the authors. Here, TO is applied on a UPD to identify the damper's layout that reduces to the maximum the levels of the nonlinear vibrations of the blades on a given frequency range. The optimization process must take into consideration the friction contact interface between the damper and the structure, but also the fact that the load applied on the UPD depends on its mass, and so on its geometry. Furthermore, the objective function of the problem is expected to be strongly non-convex with many local minima. For this reason, global optimisation is considered and the MMC framework with Krigingbased optimization is used to solve the current problem.

The objective of this paper is to investigate the capacity of the MMC framework coupled with global optimisation algorithm to optimise the topology of an UPD in order to reduce the amplitude of the nonlinear vibrations of the blades. Hence, the methodology is applied on a 2D system that simulates the behaviour of a real HPT blade as a preliminary work to examine the potential of the approach. The paper is organized as follows. First the optimisation problem is introduced. The mechanical system, the problem geometry and the methodology used to determine the nonlinear vibrations of the structure are presented. Secondly, the MMC framework and the optimization process based on Kriging and on the EGO algorithm are presented. Finally, the numerical results are presented.

PROBLEM PRESENTATION

The objective of this section is to present the physical problem that will be optimized. Hence, the mechanical system as well as the modelling choices and the numerical strategy for the prediction of the nonlinear vibrations are presented.

Model Presentation

The model used is based on the geometry of an underplatform test rig used and presented in [START_REF] Pesaresi | Modelling the nonlinear behaviour of an underplatform damper test rig for turbine applications[END_REF]. It was designed to simulate the dynamic behaviour of the dynamic response of a real HPT blade and to study the impact of UPD on a blade-like struc- It is composed of two pseudo-beams fixed on a common base which simulates a rigid disk. To be consistent with the work done before in [START_REF] Pesaresi | Modelling the nonlinear behaviour of an underplatform damper test rig for turbine applications[END_REF], the boundary conditions remain the same and the right side of the base is clamped for later experimental validations. A 2D geometry may seems simplistic, but the main concern of this study is to investigate the interest and the possibilities of using krigingbased optimisation for TO of nonlinear vibrations with friction. This simple case allows for much lower simulation times and a lower number of optimisation parameters. The mesh of the blades is composed of 3324 8-node biquadratic plane strain elements. The structure is made of steel with a Young modulus of 197 GPa and a density equal to 7800 kg/m 3 . The study focuses on the first bending mode of the blades. Since there are two blades, their motion can be in-phase or out-of-phase. These two modes are represented in Figure 2 and the natural frequencies are equal to 246.73 Hz and 247.51 Hz, respectively. It is worth noticing they are determined without the damper.

As a reference case, a wedge damper is located between the two platforms of the blades as represented in Figure 1. The centrifugal loading keeps it in place during operation. The mesh of the damper is represented in Figure 3 and is composed of 3604 8node biquadratic plane strain elements. The mesh is constructed so that the contact points between the damper and the blades are matching. The number of elements may seem huge for this component, but this mesh will be used later in the process of the topological optimization and corresponds to the discretisation of the problem.

A node-to-node contact formulation is used in the problem and 51 contact points are present on each side of the damper. The dynamic behaviour at each contact node is modelled with a 1D friction contact element. It consists of one Jenkins element and one other spring to allow normal load variations. Each element is characterized by four parameters, namely the friction coefficient µ, the tangential stiffness k t , the normal stiffness k n and the initial pre-load N 0 . The normal pre-load and the stiffness are normalized by unit of area, so each contact element has its own characteristics. During the cycle, the normal loading is kept constant and no separation is possible.

The contact pressure due to the centrifugal loading C F is modelled by a constant value [START_REF] Petrov | Advanced modeling of underplatform friction dampers for analysis of bladed disk vibration[END_REF] and the initial pressure σ 0 on each side of the damper is given by:

σ 0 = 1 2 C F A(cos α + µ sin α) ( 1
)
where A is the contact area on each side, µ is the friction coefficient and α is the damper angle. The load C F , aimed to simulate the centrifugal loading, is given by C F = m damper × R × ω 2 where m damper is the damper mass, R the radius and ω the rotational speed.

Multi-Harmonic Balance Method

The friction interface brings nonlinearity in the system and specific numerical strategies must be employed to determine the vibration levels. Thus, the method of the multi-harmonics balance (MHBM) coupled with a model reduction technique in-FIGURE 3: Mesh of the full damper cluded in the in-house code FORSE (FORce Response SuitE) is used in the present study. The latter has been presented in details in [START_REF] Petrov | Explicit finite element models of friction dampers in forced response analysis of bladed disks[END_REF][START_REF] Petrov | A high-accuracy model reduction for analysis of nonlinear vibrations in structures with contact interfaces[END_REF]. The reduction of the model is based on a FRF matrix representation of the model. This FRF matrix is determined by calculating its exact value at some frequency points and adding a second term that describes its variation in a range of frequencies [START_REF] Petrov | A high-accuracy model reduction for analysis of nonlinear vibrations in structures with contact interfaces[END_REF].

The dynamic equation of the system can be written:

M ü + C u + Ku + f nl (u, u) = p(t) (2)
where M, C and K are the mass, damping and stiffness matrices respectively. f nl corresponds to the nonlinear forces vector due to the friction contact, p is the external forces vector and u is the displacements vector. The dot represents the derivative with respect to the time. As the external forces are periodic, it is natural to search for a periodic response for (2). Hence, the idea of the MHBM, is to write the problem on a Fourier series composed of N h retained harmonics. So, the response can be written:

q(t) = q 0 + N h ∑ i=1 q C i cos(k i ωt) + q S i sin(k i ωt) (3)
where q 0 is the constant term in the Fourier series, q C i are the cosine coefficients, q S i the sine coefficients and k i the i th retained harmonic. The harmonics retained are chosen a priori by the user. By injecting the latter in Equation 2, and using the orthogonality of the Fourier series, the problem can be approximated by:

Z(ω) q + fnl = p ( 4
)
where Z is the dynamic stiffness matrix, q = q 0 , q C 1 , q S 1 , . . . , q C n , q S n is the vector of the Fourier coefficients, fnl and p the nonlinear and the external efforts expressed in the Fourier basis, respectively. An AFT algorithm is used for the determination of the nonlinear forces Fourier coefficients [START_REF] Cameron | The orthogonal development of non-linear functionals in series of fourierhermite functionals[END_REF]. The problem ( 4) is solved by using a Newton-Raphson algorithm. To find a continuous response curve with regard to the pulsation of excitation ω, continuation is used. Here a secant predictor together with an arc-length corrector is used. In the following of the paper, harmonics 1 and 3 are retained for the MHBM simulations. This choice is based on a convergence study, not presented here for the sake of consistency.

Application on the Initial Test Case

The method of the MHBM is then used to compute the nonlinear dynamic response of the structure. The system is excited at the bottom of the left blade (blue point in Figure 1) and the displacements at the blades tips are determined (green points in Figure 1). A friction coefficient equal to 0.5 is chosen and the contact stiffness are taken equal to k t = k n = 0.5 N/mm 3 . Different excitation levels are considered, from 1e -5 to 1e -2 N and the nonlinear receptances of the point at the tip of the left blade are given in Figure 4. For the full damper, the centrifugal loading C F is equal to 9.54e -4 N.

For an excitation force equals to 1e -5 N or 1e -4 N, two resonant frequencies appear at 246.6 Hz and 248.6 Hz (curves superimposed in Figure 4). When the excitation force increases to 1e -3 N, a damping effect appears, and the two peaks are cut. If the excitation force is increased to 5e -3 N, the two peaks are even more damped, but the second peak is shifted to 248 Hz. Finally, if the excitation force is again increased to 1e -2 N, the receptance for the first peak increases and the resonant frequency is slightly shifted of 0.1 Hz. For the second peak, the resonant frequency continues to be shifted and is equal to 247.5 Hz and the receptance also increases.

Optimisation Problem

The objective is to optimise the topology of the damper in order to identify the geometry that gives the minimum level of vibrations at blade tips. Thus, the maximum of vibration amplitudes observed at blades tips over the frequency range [100, 400] Hz is chosen as the objective function of the problem. The frequency range is chosen in order to cover all the possible variation of the resonance frequencies that may happen during the optimisation problem and to ensure a good initialisation of the MHBM. The limits of the design space are the limits of the full damper presented in Figure 3. The excitation amplitude is fixed to 1e -3 N and is constant during the whole study. The loading range is directly related to the damper mass can take value between 0 N (damper of null mass) and 9.54e -4 N (full damper).

MOVING MORPHABLE COMPONENTS FRAMEWORK AND OPTIMIZATION PROCESS

The objective of the following section is to present the methodology employed to perform topological optimization. The methodology is based on the KG-LSM [START_REF] Raponi | Kriging-assisted topology optimization of crash structures[END_REF], in other words the structure is represented implicitly by a Level-Set Function (LSF). The Kriging is used to surrogate this LSF and to update it to identify the best layout.

Parametrization of the Problem

Let's consider a bounded domain D ∈ R 2 which is the domain of all the admissible shapes. Ω denotes the region of the domain D occupied by the material. The LSF Φ that describes Ω is defined as:

   Φ(v) > 0, v ∈ Ω, Φ(v) = 0, v ∈ δ Ω, Φ(v) < 0, v ∈ D \ Ω, (5)
where v = (x, y) T ∈ D is a point of the domain, δ Ω is the interface between the material and void. The 0-iso-line defines precisely the limit between material and void.

The idea of the Moving Morphable Components (MMC) framework, proposed by Guo et al. [START_REF] Guo | Doing topology optimization explicitly and geometrically-a new moving morphable components based framework[END_REF], is to decompose this global LSF into several local LSFs that describe an elementary beam component which can be moved and deformed in the design space. The idea relies on the fact that the optimal topologies can often be decomposed in an ensemble of interconnected beams. Hence, each local LSF Φ i describes a domain Ω i occupied by the component i. And the total domain Ω occupied is defined by Ω = ∪ N i=1 Ω i . And for D = R 2 , each component i is described by an explicit formulation of the LSF given by [START_REF] Guo | Doing topology optimization explicitly and geometrically-a new moving morphable components based framework[END_REF]:

Φ i (u) = - cos θ i (x -x 0i ) + sin θ i (y -y 0i ) l i /2 m + -sin θ i (x -x 0i ) + cos θ i (y -y 0i ) t i /2 m -1 (6)
where (x 0i , y 0i ) is the centre position of the component, θ i is the inclination angle, L the length and t the thickness. m is a relatively large even number taken equal to 6 here, as defined in [START_REF] Guo | Doing topology optimization explicitly and geometrically-a new moving morphable components based framework[END_REF].

Hence, each component is described by a set of five parameters, namely (x 0i , y 0i , θ i , L,t), as illustrated in Figure 5 where the parametrization of a component is displayed. The corresponding local LSF is given in Figure 5b, where the negative values have been set to zero for display considerations. By changing these five parameters, each component can move and be distorted, and the assembly of the different components describes the general structure.

Once the structure is described by the LSF, this description must be mapped on the mechanical model and more precisely on the mesh. A classical method consists in having a fixed mesh on which the LSF is mapped with the use of an ersatz material: for the regions occupied by the material, the density of the material are used, but for the region composed of void, an ersatz material is used (i.e. with a density and a Young modulus largely lower than the material used). This allows to work on a fixed mesh and to always have a well-defined mechanical problem. This approach is largely used in the community [START_REF] Van Dijk | Level-set methods for structural topology optimization: a review[END_REF][START_REF] Allaire | Structural optimization using sensitivity analysis and a levelset method[END_REF][START_REF] Guo | Doing topology optimization explicitly and geometrically-a new moving morphable components based framework[END_REF][START_REF] Raponi | Kriging-assisted topology optimization of crash structures[END_REF][START_REF] Challis | A discrete level-set topology optimization code written in matlab[END_REF]. However, in the computation of the eigenmodes of a structure, some can be localised in the ersatz material. To avoid this problem, the void elements are removed [START_REF] Li | Eliminate localized eigenmodes in level set based topology optimization for the maximization of the first eigenfrequency of vibration[END_REF]. Hence, in the present study, the elements of the mesh that are not occupied by the material are removed. Moreover, the LSF is discretized in the centre of the elements [START_REF] Challis | A discrete level-set topology optimization code written in matlab[END_REF], i.e. if the LSF is positive at the centre of an element of the mesh then this element is composed of material, but if the LSF is negative then the element is removed. This discretisation might seem a bit coarse, however the mesh can be refined (without additional numerical cost) for a more accurate description of the geometry of the damper. A second argument is that the topological optimization process is used to identify types of geometries that might be more efficient, but other design considerations need to be considered after and will define the final layout. To ensure a realisable component, the connectivity of the component is first checked with a 2D graph as proposed by Raponi et al. [START_REF] Raponi | Kriging-assisted topology optimization of crash structures[END_REF], and if check-board effect takes place then the thickness of the components is increased. An example of the global process is illustrated in Figure 6, where in Figure 6a given in black. On Figure 6b, the 0-iso-contour of each LSF is represented and finally the corresponding mesh is represented in Figure 6c.

Optimization with Kriging and EGO Algorithm

The optimization problem considered in this paper can be written under the form:

min x f obj (x) (7)
where f obj is the objective function of the problem, namely the maximum of displacements at the blades tips. x ∈ R n is the vector of the design parameters that describes the geometry of the problem, which in this case corresponds to the number of LSF used multiplied by 5, which are the describing parameters. The optimization problem is written without a constraint on the volume of the damper. This choice is driven by the fact that the most important goal in this study is to get a more efficient geometry, than a lighter one. Hence, no volume constraint is considered since it could lead to the missing of some efficient geometries. The problem considered is strongly nonlinear, non-convex with many local minima, and the determination of the shape sensitivities of the objective function is not conceivable. Hence, an optimization strategy based on global optimization is considered. More precisely, the methodology is based on the use of a Kriging meta-model together with the Efficient Global Optimization (EGO) algorithm [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF].

Kriging Surrogate Model

Kriging is an approach to surrogate a function, that is often expensive to compute. In the present case, it surrogates the objective function f obj . It is based on a set of N inputs X = x (1) , . . . , x (N) and their evaluations y = y (1) , . . . , y (N) . In the present case, each y (k) is obtained by computing the displacements FRF at blades tips with the MHBM, considering the damper geometry described by X (k) , over the frequency range [100,400] Hz. The maximum over the frequency range and over the two blades is taken as the evaluation of the objective function. This set of inputs and outputs is called Design of Experiments (DoE). The set of input points is generated with a Latin Hypercube Sampling (LHS) and only the configurations that are connected are retained. Then, the Kriging methodology consists in approximating the function f obj by the function f defined as [START_REF] Roustant | Dicekriging, diceoptim: Two r packages for the analysis of computer experiments by kriging-based metamodeling and optimization[END_REF]:

f (x) = q ∑ j=1 g j (x)β j + Z (x) (8)
where (g j ) j∈ [1,q] is a set of q known real-values functions, (β j ) j∈ [1,q] are the regression coefficients to be determined, and Z is a stationary zero-mean Gaussian process. Its covariance function is given by C(x, x ) = σ 2 R(x, x ) where σ 2 is the unknown variance of the process and R(x, x ) is the correlation function of the process. The functions (g j ) j∈ [1,q] are chosen according to an a priori knowledge of the phenomenon, they are often low order polynomial functions. The function R is usually unknown but is often constructed from a family of kernel functions parametrized by a parameter θ to be optimized [START_REF] Roustant | Dicekriging, diceoptim: Two r packages for the analysis of computer experiments by kriging-based metamodeling and optimization[END_REF].

By assuming that Z is a Gaussian process, then the best unbiased predictor is linear. One can then define the regression matrix G of coefficients G i j = g j (x (i) ), and the correlation matrix R of coefficients R i j = C(x (i) , x ( j) ). The prediction y 0 of the Kriging meta-model at the point x 0 is then given by:

y 0 = f (X 0 ) = g T 0 β + r T 0 R -1 (y -G β ) (9)
where g 0 is the vector of g j (x 0 ) and r 0 is the vector of C(x 0 , x (i) )

and β = (G T R -1 G) -1 G T R -1 y.
The mean square error is given by:

s 2 (x 0 ) = σ 2 1 -r T 0 R -1 r 0 + g 0 -G T R -1 r 0 T G T R -1 G -1 g 0 -G T R -1 r 0 ( 10
)

EGO Algorithm

The basic idea of the EGO algorithm consists in the creation of a Kriging surrogate model from an initial DoE which is updated at each iteration of the algorithm by adding a new point in the DoE [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]. The algorithm ends when a stop criterion is reached. The global work-flow of the method is given in Figure 7. Thus, the point added is based on the Expected Improvement (EI) criterion and is chosen in order to increase the knowledge in the area of the design space where the minimum of the objective function can be located, but also to explore areas where no knowledge is available. This is performed by using the meta-model created at the current step. The stop criterion is often based on a maximum number of iterations.

A modified EI proposed by Raponi et al. [START_REF] Raponi | Kriging-assisted topology optimization of crash structures[END_REF] is considered for this study. It is similar to the classical EI when the design corresponding to the points x is feasible (the connectivity between the components is ensured), and is penalised when the design is not feasible (i.e. when the components are not connected). Hence, at the point x, the EI is defined as [START_REF] Raponi | Kriging-assisted topology optimization of crash structures[END_REF][START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF][START_REF] Roustant | Dicekriging, diceoptim: Two r packages for the analysis of computer experiments by kriging-based metamodeling and optimization[END_REF]:

EI[x] =          (y min -f (x))Φ y min -f (x) s(x) + s(x)φ y min -f (x) s(x)
, if feasible design and s(x) = 0, 0 , if feasible design and s(x) = 0, -P * γ, if infeasible design, (11) where y min is the current minimum, Φ is the Gaussian cumulative distribution function and φ is the Gaussian probability density function, P is the minimum gap distance between the components and γ ∈ R + * is the penalization factor (equal to 1000 here). The new point x new is the solution that maximises the EI. This optimization problem is solved by using a genetic algorithm [START_REF] Goldberg | Genetic algorithms and machine learning[END_REF].

Updating Limit Conditions

For each point initially in the DoE and at each new point added during the process, a new damper geometry is described through the vector x. Hence, the loading and the contact are also impacted and must be updated for each damper geometry. From the mass evolution of the damper, the centrifugal loading C F also evolves, and so the contact pressure and the contact stiffnesses are also impacted. The updating of the contact pressure is done by considering the new value of the centrifugal loading C F , that depends on the damper mass, and also by considering the new contact area A, that is related to the damper geometry, in the Equation 1.

Moreover, the contact stiffnesses depends on the contact loading and so, they are also updated at each iteration by keeping the constant ratio [START_REF] Sextro | Dynamical contact problems with friction[END_REF]:

k (1) σ (1) = k (i) σ (i) (12)
where k (i) is the contact stiffness and σ (i) the contact pressure at the iteration i. As a reminder, k = k t = k n .

Summary Of The Global Process

Finally, the global layout of the methodology can be described as follows. First the initialisation part can be decomposed as follows:

-Initialisation/step 1: a LHS is generated for the initial input points. Only the geometries that are connected are retained [START_REF] Raponi | Kriging-assisted topology optimization of crash structures[END_REF]. -Initialisation/step 2: the LSF corresponding to the design point is constructed and mapped on the initial damper mesh.

The final damper mesh is created by eliminating the void elements. -Initialisation/step 3: for each damper geometry, the nonlinear dynamic response of both the platform and the damper is determined with the MHBM where the contact pressure and contact stiffness have been updated. The excitation amplitude is constant and equal to 1e -3 N. The maximum of displacements at the blades tips over the frequency range [200,400] Hz is retained as the evaluation of the objective function.

and the general optimisation loop can be decomposed as follows:

-Loop/step 1: from the DoE, a surrogate model is generated.

-Loop/step 2: from the surrogate model, the EI is maximized and the new point added to the design. -Loop/step 3: for this point (i.e. new geometry), the objective function is determined (steps 2 and 3 of the initialisation).

The loop is repeated until the stop criterion is reached. Concerning some practical aspects, in the present study, the global work-flow is piloted by Matlab. The package DiceKriging [START_REF] Roustant | Dicekriging, diceoptim: Two r packages for the analysis of computer experiments by kriging-based metamodeling and optimization[END_REF] on R is used for the creation of the different Kriging meta-models. A kernel Matérn 5/2 is used and the regression function is polynomial of order 0.

APPLICATION

The methodology presented previously is then applied to optimize the topology of the damper. Seven components are used to describe the geometry of the damper, as shown in Figure 6, which would represent 35 optimization parameters. In order to reduce this number of parameters, some assumptions are done. First, a component is assumed to be horizontal at the top of the design space and aims to seal the platforms. This constraint comes from the fact that the primary function of a UPD is to seal the platforms to ensure the aerodynamic efficiency. Second, the damper is supposed to be symmetric. And finally, one component has its centre position imposed to always have a contact between the damper and the platforms. This restriction ensures just a feasible design. Thanks to these restrictions, the final optimization problem is composed of 14 parameters. As a reminder, the excitation amplitude remains constant and equal to 1e -3 N.

Results

To ensure the validity and the robustness of the method, ten initial DoE are created, and ten simulations are launched. In each case, an initial LHS of 10 5 points is generated, and only the feasible geometries are kept (i.e. no connectivity issue). Depending on the case, it represents [65, 51, 65, 62, 62, 56, 82, 61, 62, 81] initial learning points for the ten cases. In order to get reasonable computation time, a maximum of 250 iterations is chosen. Hence, the evolution of the current minimum at each iteration for the ten cases is given in Figure 8a where each colour corresponds to one case. At the first iteration, the minimum is the one observed in the initial DoE, depending on the case it is included between 6 and 6.8e -4 mm. Then, the evolution of the minima is quite similar for all cases. First the decrease of the minimum is fast. Indeed, in about 40 iterations, all the minima are below 5.5e -4 mm. After 40 iterations, the decrease becomes slower and stabilizes around 150 iterations. For all cases, the algorithm converges to a solution from 5.3 to 5.4e -4 mm. It is worth reminding that for the full damper the maximum level of amplitudes is equal to 8e -4 mm. Hence, the algorithm identifies here ten solutions that are 30% more efficient than the original one. As explained previously, no constraint on the volume is considered in this application. It may sound peculiar since the first objective of topological optimization is to reduce the weight of the structure. In this case, the most important objective is to achieve better design. Considering a constraint on the weight could lead to miss some interesting layouts. However, it is still an interesting argument to consider a solution instead of another one. Hence, in Figure 8b represents the maximum of displacements versus the normalized mass for all the configurations tested with the ten cases. It appears that almost all the configurations tested are located on a front. It shows also that by minimizing the maximum of displacements, the mass increases, i.e. the two functions are antagonist. The last configuration identified in each case, i.e. the current minimum at the iteration 250, is represented on the graph with a triangle. They are all located in the same part of the graph and so have similar masses.

The ten final configurations are given in Figure 9. The green part corresponds to the mesh associated to each geometry, and the borders of the design space are reminded. It can be seen that the geometries are all different. Indeed, some have holes inside the structure (see in Figures 9b,9d,9j), some cut the contact surface into two parts (see in Figures 9a,9e, 9f, 9i). However, some geometries look similar and some groups can be done. For exam- Whereas after optimization, for the second dof, the second peak is the one with the largest amplitude.

In order to observe the efficiency of lighter dampers, the FRF of other geometries tested during the optimization process are considered. Geometries with a mass equal to 50%, 30% and 10% of the total mass M full are considered. One of each case is represented in Figure 11. As previously, for one mass, all the FRF are extremely similar, hence only one FRF is retained in each case for the comparison. The different FRF are represented in Figure 10. The black dashed line corresponds to the initial case (full damper). For the first dof, a decrease of the mass tends to increase the amplitude of the first peak: 1.1e -3 mm of amplitude if M = 0.1M full and 0.54e -3 mm if M = 0.9M full . For the second peak, a lighter damper tends to decrease slightly the amplitudes. Moreover, the all FRF tends to be shifted on the left when the damper is lighter. Considering the second dof (see in Figure 10b), when the damper has a mass M equal to 0.1M full , the two peaks have similar amplitudes. The increase of the mass tends to reduce the amplitude of the first peak, whereas the amplitude of the second peak remains sensibly constant. As for the other dof, the FRF is shifted on the right when the mass increases.

CONCLUSION

In the present study, TO is applied to optimise nonlinear vibrations in the presence of a friction interface, which is completely new for the community from authors knowledge. A simple 2D model composed of two blades and a UPD, where the topology of the damper is optimised, is used as a test case. The TO method is based on the MMC framework to parametrize the LSF that describes the geometry of the system. A Kriging method together with the EGO algorithm is used as optimisation process. The study was aimed to investigate the capacity of the MMC framework coupled with a global optimisation algorithm to perform TO for nonlinear vibrations in a presence of a friction interface. Ten simulations based on different DoE are launched to verify the validity and the robustness of the approach. The results show a large variety of UPD geometries, but they all have very similar effectiveness and almost identical nonlinear dynamic response. The optimised geometries lead to a reduction of about 30% of the nonlinear vibration levels of the structure compared to the case with the full damper. The results obtained in the study tends to show the promising aspect of using the MMC framework coupled with EGO algorithm for TO of UPD. Nevertheless, a deeper investigation must be conducted to take into consideration different mechanical constraints, as the stress, to obtain more realistic layouts that can tolerate the operating conditions.

FIGURE 1 :FIGURE 2 :

 12 FIGURE 1: 2D geometry of the model -( ) clamped -(•) excitation point -(•) output points

2 FIGURE 4 :

 24 FIGURE 4: Nonlinear receptance at the point on the left blade for different excitation levels for a full damper

FIGURE 5 :

 5 FIGURE 5: Parametrization of a component (a) and corresponding LSF (b)

FIGURE 6 :

 6 FIGURE 6: General geometry description: (a) 2D graph for the connectivity check -(b) 0-iso-contour of the different LSF -(c) corresponding mesh

FIGURE 7 :

 7 FIGURE 7: Algorithm EGO scheme

  One color corresponds to one DoE -( ) full damper (reference) -Initial and final geometries for one DoE One color corresponds to one DoE -( ) final solution

FIGURE 8 :

 8 FIGURE 8: (a) Evolution of the objective function (i.e. max of displacements at blades tips) -(b) Objective function versus the normalized mass of the damper for all the different configurations tested during the algorithm

  (a) Damper geometry from DoE1 (b) Damper geometry from DoE2 (c) Damper geometry from DoE3 (d) Damper geometry from DoE4 (e) Damper geometry from DoE5 (f) Damper geometry from DoE6 (g) Damper geometry from DoE7 (h) Damper geometry from DoE8 (i) Damper geometry from DoE9 (j) Damper geometry from DoE10

FIGURE 9 :

 9 FIGURE 9: Damper geometries obtained from the different DoE

ACKNOWLEDGMENT

The authors thank Rolls-Royce plc and the EPSRC for the support under the Prosperity Partnership Grant "Cornerstone: Mechanical Engineering Science to Enable Aero Propulsion Futures", Grant Ref: EP/R004951/1.