Ioannis Kanellos
email: ioannis.kanellos@enst-bretagne.fr

Thomas Le Bras
email: thomas.lebras@enst-bretagne.fr

Ioana Suciu
email: ioana.suciu]@enst-bretagne.fr

Interpretative e-learning personalization: Methodology, formal aspects and generic scenarios of individual/group dynamics. A case of a course in art history

Introduction

Most of today's e-learning development focus on knowledge formats and production, supposing perhaps that reception can be seen as the symmetric case of production in a courseware. In other words, that the point of view of the teacher may be transposed towards the learner. Nevertheless, understanding is an interpretation-dependent process, lying mainly on reading strategies adopted by the learner. We may thus wonder whether it is yet possible to conceive systems whose architecture is driven by hermeneutical principles (i.e. where the interpretational activity is considered as a priority, and the role of the receptor of the course in the very constitution of the course is at least as important as this of the teacher) inherited from [START_REF] Rastier | Sémantique Interprétative[END_REF]. A reading strategy may be represented as a knowledge path built up from information put at disposal in the framework of a course. The aim of our work is to furnish some modeling issues in such a direction, using as case study academic course in art history. In collaboration with the Art Diagnosis Centre of Ormylia (Greece), some of us contributed, in the framework of various projects, to the setting up of a fine art ontology (over 30 000 concepts) able to cover the knowledge of large range iconographic corpora (see, for instance, [START_REF]ARTeFaCT: Generic Platform for the Creation of Interactive Art Experience in Mixed Reality[END_REF]).

Designing a personalized course in Art History

Methodology

The design of the system follows a progressive three-steps methodology:

• Level 1 (data): a reservoir of images (reproductions of paintings), texts (liturgical, the Bible, patristic texts, critical), audio and video files etc. given by different resources.

• Level 2 (information): the ontologies and points of view ( [START_REF] Kanellos | Le concept de genre comme point de départ pour une modélisation sémantique du document électronique[END_REF]) associated to the data of the first level. Information is managed following a point-of-view split, and available under alternative versions: local ontologies form non-extensional spaces [START_REF] Zaldivar-Carrillo | Contributions à la formalisation de la notion de contexte[END_REF].

• Level 3 (knowledge): the path constitution itself. A path is the result of a choice operated over parts of ontologies of the second level. Ready-to-use paths can be defined by teachers, corresponding to typical (normative) learning itineraries, they may be suggestive or prescriptive, depending on the type of the course. Paths represent (not yet a course but) the acquisition result of a reading progression. On the basis of this structure, we define intensional identity between paths (same components, same order), extensional identity (same components, not specially same order), intensional and extensional variation (not the same components, but the same references to ontologies), partial and total completion, saturation, extension and reduction, union and intersection, proximity and similarity. We suppose that to each path P i is attached an assessment method, A Pi .

Personalization at individual level

Some scenarios are of recurrent interest in individual personalization: Upgrading and improvement: The objective is to improve the responses to the assessment. The system makes explicit the list of failed parts of a given A Pc and suggests paradigmatic solutions. Deepening and refinement: With or without failure in assessment, it is always possible to analyze the answers of a learner and suggest refinements. They are given by alternative courses and completions/extensions of the paths they involve. Erudition and expertise: They are the upper stages one can envisages following the previous scenario. By successive completions and extensions, the erudition learning objective tends to cover the whole knowledge at disposal in the local ontologies. Pleasure reading and discovery: The learner is driven by a personal desire to discover. Starting from initial data (not even from a P c , nor under the obligation of an A Pc ) she/he navigates through resources and associated knowledge, visiting parts of interest for her/him and configuring her/his path dynamically.

Issues in individual/group learning dynamics

Here, we study in particular three different adaptive group learning scenarios, set up by three types of group learning contextual cooperation forms (we draw our inspiration from [START_REF] Schmidt | Cooperative Work: A Conceptual Framework[END_REF]): Augmentative learning is based on the idea that group members are limited in their cognitive resources. By combining their abilities and aggregating their learning efforts, the group members can, under specific external intervention, access to a new knowledge (parts of some ontologies). Integrative learning is based on the ongoing process of differentiation of individual skills. The group members define their personal, idiosyncratic points of view. Each member has the opportunity to reach the knowledge of a complete path. Debative learning deals with the case where the learning limitation is due to individual limited rationality. The same (or versioned) information is available for all the group members, but different strategies are applied by each of them in order to reach their learning objectives (pleasure, erudition…).


C Laurent
email: laurent@cerfacs.fr

M Bauerheim

T Poinsot

F Nicoud

A novel modal expansion method for low-order modeling of thermoacoustic instabilities in complex geometries

Keywords:

This work proposes an improvement to existing methods based on modal expansions used for the prediction of thermoacoustic instabilities in zero Mach number flow conditions. Whereas the orthogonal basis made of the acoustic eigenmodes of the domain bounded by rigid walls is classically used, an alternative method based on a modal expansion onto an over-complete set of acoustic eigenmodes is proposed. This allows avoiding the misrepresentation of the acoustic velocity field often observed near non rigid-wall boundaries. A Low Order Model network utilizing a state-space framework is then built upon this novel type of modal expansion. Several test cases, going from non reacting ducts to a complex geometry with combustion, are studied to assess the potential of the approach. The methodology not only successfully mitigates the misrepresentation in the acoustic field in the presence of non-rigid-wall boundaries, but it also drastically improves the convergence speed. The modularity of the method and its ability to handle complex geometries are illustrated by considering a configuration featuring an annular chamber, an annular plenum, as well as multiple burners. This novel technique is expected to bring worthy improvements to existing Low Order Models using modal expansions for the prediction of combustion instabilities.

Introduction

Since their early study by Rayleigh [START_REF] Rayleigh | The explanation of certain acoustical phenomena[END_REF] , thermoacoustic instabilities have been a subject of primary scientific interest as well as a major concern for a number of industrial projects. Experiments with increasingly complex setups and advanced diagnostics were carried out over the years to study this intricate interplay between flame dynamics and acoustic waves. Progresses in experimental works were accompanied by considerable efforts in both numerical and theoretical study of thermoacoustic instabilities. Regarding the former, Large Eddy Simulation (LES) was proved as the most accurate tool for the analysis of instabilities in combustors featuring complex geometries [START_REF] Poinsot | Prediction and control of combustion instabilities in real engines[END_REF] . Yet, the high cost of these full scale simulations led to a growing popularity of alternative and cheaper numerical methods relying on a separation of the acoustic flow and the complex flame dynamics. The acoustic field is solution of the Helmholtz equation in the frequency domain, while the flame response to acoustic perturbations is often embedded into a Flame Transfer Function (FTF) [START_REF] Crocco | Aspects of combustion stability in liquid propellant rocket motors part I: fundamentals. low frequency instability with monopropellants[END_REF] , or a Flame Describing Function (FDF) representing nonlinear effects [START_REF] Noiray | A unified framework for nonlinear combustion instability analysis based on the flame describing function[END_REF] .

One of the key aspects for the resolution of thermoacoustic eigenmodes is the ability of the method to accurately account for complex geometries that are encountered in industrial combustors. One of the most straightforward approaches is the direct discretization of the Helmholtz equation that is then solved thanks to a Finite Element Method (FEM) solver. State-of-the-art FEM Helmholtz solvers are able to solve for thermaoustic eigenmodes in complex geometries comprising active flames and dissipative effects [START_REF] Nicoud | Acoustic modes in combustors with complex impedances and multidimensional active flames[END_REF][START_REF] Ni | Including flow-acoustic interactions in the Helmholtz computations of industrial combustors[END_REF] , and can also incorporate the FDF formalism to capture nonlinear limit-cycle behaviors [START_REF] Silva | Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor[END_REF][START_REF] Laera | Flame describing function analysis of spinning and standing modes in an annular combustor and comparison with experiments[END_REF] . However, direct discretization FEM Helmholtz solvers often result in a large number of Degrees of Freedom (DoF), synonym of a considerable computational cost, and only permit little modularity, as any change in the geometrical parameters requires a new geometry and mesh generation. In order to circumvent these shortcomings, numerous research groups have opted for the development of Low Order Models (LOMs) enabling even cheaper resolution of thermoacoustic instabilities. Low O rder M odeling resides in two basic ideas: [START_REF] Rayleigh | The explanation of certain acoustical phenomena[END_REF] the number of DoF should be reduced as much as possible in order to permit fast computations, and (2) the model should be flexible and highly modular, in the sense that it should allow for Fig. 1. Schematic representation of an acoustic network. Hashed lines show rigid walls. Gray dotted lines are opening to the atmosphere. The geometry is split into distinct subsystems, that are connected together. The subsystem ( i ) is defined by its volume i , its rigid wall boundary S wi , its boundary opened to the atmosphere S ai , and its boundaries S ci that are to be connected to other subsystems in the acoustic network. The interface S ci is split into several surface elements S 0 j located at the connection points x 0 j . It also contains a volume heat source H i .

the straightforward modification of most geometrical or physical parameters. The latter point is often achieved thanks to a divide and conquer strategy, where complex geometries are decomposed into an acoustic network of simpler subdomains (see Fig. 1 ). Fast and modular LOMs have been promisingly applied to intensive tasks demanding a large number of repeated resolutions, such as Monte Carlo Uncertainty Quantification [START_REF] Bauerheim | Symmetry breaking of azimuthal thermoacoustic modes: the UQ perspective[END_REF][START_REF] Avdonin | Uncertainty quantification and sensitivity analysis of thermoacoustic stability with non-intrusive polynomial chaos expansion[END_REF] , or passive control through adjoint geometrical optimization [START_REF] Aguilar | Adjoint methods for elimination of thermoacoustic oscillations in a model annular combustor via small geometry modifications[END_REF] .

Existing thermoacoustic LOMs can be classified into four main categories, according to the method employed to describe the acoustic field.

1. The first class of LOM is a wave-based 1D network approach, where the acoustic pressure and velocity are written in function of the Riemann invariants A + and A -. This method was first successfully used in the LOTAN tool [START_REF] Dowling | Acoustic analysis of gas turbine combustors[END_REF] , designed to resolve in the frequency domain linearly unstable thermoacoustic modes in simple configurations. More recently and in a similar fashion, the open-source LOM solver Oscilos [START_REF] Li | OSCILOS: the open source combustion instability low order simulator[END_REF] developed at Imperial College, London, was used to perform for instance time-domain simulations of thermoacoustic limit cycles in longitudinal combustors [START_REF] Li | Time domain simulations of nonlinear thermoacoustic behaviour in a simple combustor using a wave-based approach[END_REF][START_REF] Han | Prediction of combustion instability limit cycle oscillations by combining flame describing function simulations with a thermoacoustic network model[END_REF] comparable to the Rijke tube [START_REF] Rijke | Notice of a new method of causing a vibration of the air contained in a tube open at both ends[END_REF] . Wave-based low order modeling was also generalized to more complex cases, including azimuthal modes in configurations comprising an annular combustion chamber linked to an annular plenum through multiple burners. This procedure allowed Bauerheim et al. to conduct a series of studies based on a family of analytical solutions for azimuthal modes in annular combustors [START_REF] Bauerheim | Symmetry breaking of azimuthal thermoacoustic modes: the UQ perspective[END_REF][START_REF] Bauerheim | An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum[END_REF][START_REF] Bauerheim | A theoretical study of mean azimuthal flow and asymmetry effects on thermo-acoustic modes in annular combustors[END_REF] . Even though wave-based LOMs are the most adequate to deal with networks of longitudinal elements where acoustic waves can be assumed as planar, they also suffer strict limitations: they are indeed unable to capture nonplanar modes in complex geometries. 2. The second class of LOM relies on modal (or Galerkin) expansions, to express the acoustic pressure field as a combination of known acoustic modes. Modal expansion was first introduced and formalized in an acoustic context by Morse and Ingard in their influential book Theoretical Acoustics [START_REF] Morse | Theoretical acoustics[END_REF] dated from 1968. In the field of thermoacoustics, Zinn et al. [START_REF] Zin | Application of the Galerkin method in the solution of nonlinear axial combustion instability problems in liquid rockets[END_REF] and Culick [START_REF] Culick | Nonlinear behavior of acoustic waves in combustion chambers -II[END_REF][START_REF] Culick | Combustion instabilities in liquid-fueled propulsion systemsdan overview[END_REF][START_REF] Culick | Overview of combustion instabilities in liquid-propellant rocket engines[END_REF][START_REF] Culick | Unsteady motions in combustion chambers for propulsion systems[END_REF] were among the first to use it to study combustion instabilities in liquid fuel rocket engines. Similarly to the wave-based approach, multiple studies utilizing modal expansions are dealing with the Rijke tube: for example by Ju-niper [START_REF] Juniper | Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition[END_REF] , Waugh and Juniper [START_REF] Waugh | Triggering in a thermoacoustic system with stochastic noise[END_REF] , and Balasubramanian and Sujith [START_REF] Balasubramanian | Non-normality and nonlinearity in combustion-acoustic interaction in diffusion flames[END_REF] . Simplified annular configurations were also examined thanks to pressure modal expansion: Noiray and coworkers [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF][START_REF] Noiray | Deterministic quantities characterizing noise driven Hopf bifurcations in gas turbine combustors[END_REF][START_REF] Noiray | On the dynamic nature of azimuthal thermoacoustic modes in annular gas turbine combustion chambers[END_REF] and Ghirardo et al. [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] conducted a series of theoretical studies in such geometries. More complex modal expansion-based networks were developed for multiburners chamber-plenum geometries, by Stow and Dowling [START_REF] Stow | A time-domain network model for nonlinear thermoacoustic oscillations[END_REF] , Schuermans and co-workers [START_REF] Schuermans | Thermoacoustic modeling and control of multi burner combustion systems[END_REF][START_REF] Schuermans | Modeling and control of thermoacoustic instabilities[END_REF] , and Belluci et al. [START_REF] Bellucci | Thermoacoustic modeling of a gas turbine combustor equipped with acoustic dampers[END_REF] .

Their strategy is to perform modal expansions for the pressure in the chamber/plenum and to assume acoustically compact burners that can be lumped and represented by simple transfer matrices. Unlike wave-based low-order modeling, this method is not limited to planar acoustic waves, and can resolve both azimuthal and longitudinal chamber modes. Even though their approach does not rely on an acoustic network decomposition, Bethke et al. [START_REF] Bethke | Stability analysis of gas-turbine combustion chamber[END_REF] showed that arbitrarily complex geometries can be incorporated in a thermoacoustic LOM by expanding the pressure onto a set of basis functions computed in a preliminary step thanks to a FEM Helmholtz solver. Although they appear more general than wave-based LOMs, modal expansion-based LOMs are also subjected to strict limitations, which mainly resides in the choice of the modal basis employed to expand the acoustic pressure. This point is the main object of this paper, and is discussed in more details later. 3. The third class of thermoacoustic LOM is a mixed method, that combines both Riemann invariants and modal expansions. The former are used to account for longitudinal propagation, while the latter are used to account for multi-dimensional geometries. This mixed method was employed by Evesque and Polifke [START_REF] Evesque | Low-order acoustic modelling for annular combustors: validation and inclusion of modal coupling[END_REF] to model azimuthal modes in multi-burners annular chamber/plenum configurations. In more recent works [START_REF] Orchini | Effects of nonlinear modal interactions on the thermoacoustic stability of annular combustors[END_REF][START_REF] Dong | Low-order network modeling for annular combustors exhibiting longitudinal and circumferential modes[END_REF] the mixed strategy was further developed to investigate nonlinear spinning/standing limit cycles in the MICCA annular combustor. More precisely, in [START_REF] Orchini | Effects of nonlinear modal interactions on the thermoacoustic stability of annular combustors[END_REF] the planar acoustic field in the ducted burners is represented by Riemann invariants, while it is represented through modal expansions in the chamber and in the plenum. 4. Finally, the last class of LOMs consists in those based on a direct spatial discretization. For example, Sayadi et al. [START_REF] Sayadi | Thermoacoustic instability -a dynamical system and time domain analysis[END_REF] made use of a finite difference scheme to build a dynamical system representation of a one-dimensional thermoacoustic system comprising a volumetric heat source localized within the domain. A more generic direct discretization LOM is the taX Low-Order Model developed at Technical University of Munich [START_REF] Emmert | Linear state space interconnect modeling of acoustic systems[END_REF] . Its modularity lies in the ability to combine in a same thermoacoustic network one-dimensional elements discretized by finite difference, geometrically complex elements discretized through FEM, and other types of elements such as FTFs and scattering matrices. Since this LOM is built upon direct discretization of linearized partial differential equations, it can also potentially incorporate richer physics, including mean flow effects or acoustic-vortex interactions. However, the price to pay for this high modularity is a large number of DoF: for instance, in the taX LOM, about O (10 5 ) DoF were needed to obtain acoustic eigenmodes of an annular combustor comprising 12 ducted injectors (but no active flame).

Interestingly, most modal expansion LOMs used the same type of eigenmodes basis, namely the basis composed of the rigid-wall cavity modes , or in other words acoustic eigenmodes satisfying homogeneous Neumann boundary conditions ( i.e. zero normal velocity) over the entire boundaries of the domain (and without internal volume sources). This paper focuses on the nature of the acoustic eigenmodes basis used to decompose the pressure, and the convergence properties resulting from this expansion. Although the rigidwall modal basis presents the huge advantage of being orthogonal, many actual systems obviously comprise frontiers with far more complex boundary conditions than just a homogeneous Neumann condition (for example an inlet or an outlet where the impedance has a finite value). The use of such basis then appears paradoxical: how is it possible that a solution expressed as a rigid-wall modes series converges towards a solution satisfying a non rigid-wall boundary condition? This singularity in the pressure modal expansion was already noticed by Morse and Ingard [START_REF] Morse | Theoretical acoustics[END_REF] , but they did not study its impact on the convergence of the whole method. Later, Culick [START_REF] Culick | Unsteady motions in combustion chambers for propulsion systems[END_REF] provided more explanations about this singularity: the modal expansion does not converge uniformly over the domain, but only in the less restrictive sense of the Hilbert norm ( L 2 norm); as a result, even though each individual term of the expansion does not satisfy the appropriate boundary condition, the infinite sum of these terms may satisfy it. In other words, "the limit of the sum is not equal to the sum of the limit" in the neighborhood of the boundary. An interesting examination regarding this singularity is provided in a recent work by Ghirardo et al. [START_REF] Ghirardo | The effect of the flame phase on thermoacoustic instabilities[END_REF] , where a projection onto a mode satisfying a specific non rigid-wall inlet boundary condition was discussed. The convergence issue arising from rigid-wall modal expansion is even more problematic in the case of acoustic LOMs where the geometry is decomposed into a network of subsystems that need to be coupled together at their boundaries (see Fig. 1 ). For each individual subdomain, these coupling boundaries are not rigid-wall but each term of the basis corresponds to a rigid-wall: the convergence singularity may arise at each one of the coupling interfaces. Although Culick [START_REF] Culick | Unsteady motions in combustion chambers for propulsion systems[END_REF] proposed an explanation to this singularity based on a local justification, only very few studies deal with global effects, such as for example convergence speed of the eigenfrequencies.

The main contribution of this work is a reformulation of the modal expansion method under the zero Mach number flow assumption, allowing for decompositions of the pressure onto an over-complete family of modes , gathering rigid-wall ( u = 0 ) and pressure release ( p = 0 ) modes, instead of the classical rigid-wall modes basis. This paper is organized as follows: in Section 2 , the original mathematical derivation of pressure expansion onto the rigid-wall modal basis is recalled. The problem is then reformulated using expansions onto an over-complete set of modes, also called a frame . In this frame, non rigid-wall conditions are used for certain elements. Section 3 briefly describes the state-space formalism proposed in [START_REF] Schuermans | Thermoacoustic modeling and control of multi burner combustion systems[END_REF][START_REF] Schuermans | Modeling and control of thermoacoustic instabilities[END_REF] to couple subsystems together and build a modular acoustic LOM network. This state-space formal-ism is then extended to account for the frame of non-orthogonal modes. In Section 4 , the convergence properties are evaluated and compared for both types of modal expansions on a canonical case which consists of a long duct with a sharp cross-section change. Finally in Section 5 , the modularity of the proposed LOM and its ability to deal with cases involving combustion instabilities in complex configurations is demonstrated by considering an annular chamber/plenum configuration featuring multiple burners.

Acoustic pressure modal expansion

In acoustic LOM networks for complex configurations, the system is split into smaller geometric subsystems. Let us consider a subsystem (i) as in Fig. 1 , defined as a bounded domain i delimited by ∂ i = S wi ∪ S ai ∪ S ci . The primary variable of interest is the acoustic pressure p( x , t ) expressed in the physical space, but its frequency-domain counterpart ˆ p ( x , ω) is often introduced to ease the analysis, especially when dealing with linear acoustics. These two quantities are related by the (inverse) Fourier transform as follows:

ˆ p ( x , ω) = + ∞ -∞ e -jωt p( x , t ) dt , p( x , t ) = + ∞ -∞ e jωt ˆ p ( x , ω) dω (1)
For the sake of simplicity, in the following the sound speed field is assumed uniform and the baseline flow to be at rest. Note however that these hypotheses are not necessary and could be omitted. The frequency-domain acoustic pressure ˆ p ( x , ω) in the subsystem i is then solution of the following Helmholtz equation:

⎧ ⎪ ⎨ ⎪ ⎩ c 2 0 ∇ 2 ˆ p ( x , ω) -jαω ˆ p ( x , ω) + ω 2 ˆ p ( x , ω) = ˆ h ( x , ω) for x ∈ i ∇ s ˆ p = 0 for x s ∈ S wi , ˆ p = 0 for x s ∈ S ai ∇ s ˆ p = ˆ f ( x s , ω) for x s ∈ S ci ( 2
)
where the notation

∇ s ˆ p = ∇ ˆ p ( x s
ˆ h ( x , ω) = -jω ( γ -1 ) ˆ ω T ( x , ω) = -jω ( γ -1 ) H i ( x ) ˆ Q (ω) ( 3
)
where γ is the heat capacity ratio and ˆ ω T ( x , ω) is the local fluctuating heat release rate resulting from flame dynamics. In Eq. ( 3) , ˆ ω T ( x , ω) is decomposed into a global heat release rate ˆ Q (ω) and a spatial volume density H i ( x ) representing the flame shape (the integral of H i ( x ) over i is unity). In the following, only one flame in the subdomain i is considered for conciseness, but the reasoning can be extended without difficulty to any number of distinct and independent flames located in a same subdomain i .

In modal expansion based LOMs, on each subsystem i the pressure is decomposed onto a family formed of known acoustic eigenmodes of i . The purpose is then to derive a set of governing equations for the corresponding modal amplitudes. This step usually makes use of the inner product defined for any functions f ( x ) and g( x ) as:

f, g = f ( x ) g( x ) d 3 x (4) The associated L 2 norm is noted || f || 2 = f, f 1 / 2 .

The classical rigid-wall modal expansion

The usual derivation consists in writing the decomposition of the acoustic pressure onto an orthogonal basis of known acoustic modes ( ψ n ( x ) ) n 1 of the subdomain i , as ˆ p ( x , ω)

= n ˆ γ n (ω) ψ n ( x ) . The set ( ψ n ( x )
) n 1 is classically chosen as the rigid-wall eigenmodes of the subsystem i (without volume sources and acoustic damping). These eigenmodes verify rigid-wall conditions ( i.e. zero normal velocity) over S wi , but also over the connection boundary S ci . In the presence of boundaries that are known to be opened to the atmosphere ( S ai in Fig. 1 ), the eigenmodes basis can be chosen to satisfy the appropriate condition on S ai ( i.e. zero pressure), without further difficulty since the expansion basis is still orthogonal. The set ( ψ n ( x ) ) n 1 is solution of the following eigenvalue problem: [START_REF] Nicoud | Acoustic modes in combustors with complex impedances and multidimensional active flames[END_REF] where ω n is the eigen-pulsation of the n th eigenmode. By making use of the second Green's identity (reminded in Supplemental Material A), it can be shown that the set ( ψ n ( x ) ) n 1 defined by Eq. ( 5) is indeed an orthogonal basis, that is ψ n , ψ m = 0 for any

⎧ ⎨ ⎩ c 2 0 ∇ 2 ψ n + ω 2 n ψ n = 0 for x ∈ i ∇ s ψ n = 0 for x ∈ S wi , ψ n = 0 for x ∈ S ai ∇ s ψ n = 0 for x ∈ S ci
n = m .
For conciseness, the successive steps required to obtain the modal amplitudes ˆ γ n (ω) are not detailed here, but the interested reader is referred to Supplemental Material A, as well as [START_REF] Morse | Theoretical acoustics[END_REF][START_REF] Morse | Methods of theoretical physics[END_REF] . To ease the formalism, the connection surface S ci is split into M S elements S 0 j connecting i with the adjacent subdomains j at the boundary points x 0 j . Note that the subdomains j are not necessarily distinct, since there may exist several connection points between i and a same neighbor ( i.e. we can have x 0 j1 = x 0 j2 , but j1 = j2 ). An example of such surface splitting is shown in Fig. 1 . After projecting the pressure field onto the orthogonal basis ( ψ n ( x ) ) n 1 thanks to the inner product defined in Eq. ( 4) , the modal amplitudes ˆ γ n (ω) are found to be solutions of the following equation:

(ω 2 -jωα -ω 2 n ) ˆ γ n (ω) jω = M S j=1 ρ 0 c 2 0 S 0 j ψ n ( x 0 j ) n ˆ u j s ( x 0 j ) - ( γ -1 ) H (n ) i n ˆ Q (ω) (6)
where

n = || ψ n || 2 2 , H (n ) i = H i , ψ n is the projection of the flame shape H i ( x ) onto ψ n ( x ) , and ˆ u j s ( x 0 j , ω) = -ˆ f ( x 0 j , ω ) / (ρ 0 jω ) is
the normal velocity forcing imposed by adjacent subsystems j at the boundary points x 0 j . Since time derivative in the physical space corresponds to multiplying by j ω in the Fourier domain, it is possible to recast Eq. ( 6) into the time-domain. In doing so, it proves useful to introduce

ˆ n = ˆ γ n / ( jω) ( i.e. ˙ n (t) = γ n (t)
) and then use the inverse Fourier transform to obtain:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p( x , t ) = ∞ n =1 ˙ n (t) ψ n ( x ) ¨ n (t) = -α ˙ n (t) -ω 2 n n (t) - M S j=1 ρ 0 c 2 0 S 0 j ψ n ( x 0 j ) n u j s ( x 0 j , t ) + ( γ -1 ) H (n ) i n Q (t) (7)
This dynamical system governs the temporal evolution of the pressure field in the subdomain i , under the normal velocity forcing u j s ( x 0 j , t ) imposed by adjacent subsystems j , and under the volume forcing Q ( t ) imposed by fluctuating flames contained within i . This set of equations was used for example in [START_REF] Stow | A time-domain network model for nonlinear thermoacoustic oscillations[END_REF][START_REF] Schuermans | Thermoacoustic modeling and control of multi burner combustion systems[END_REF] , where the infinite series was truncated up to a finite order N . It is also worth noting that the acoustic velocity can be calculated from the knowledge of the modal amplitudes n ( t ) as

u (x, t ) = -n n (t) ∇ ψ n ( x ) /ρ 0 .
A state-space approach can then be used to couple together the subsystems defining the whole thermoacoustic system of interest. This formalism will be detailed in Section 3 .

Finally, since the acoustic pressure is a linear combination of the modal basis vectors ψ n , it necessarily verifies the same boundary conditions, in particular ∇ s p = 0 , viz. u . n s = 0 on S ci . Since the acoustic velocity should not be zero over the boundaries of the (arbitrarily chosen) sub-domain i , this may result in a singularity in the representation of the acoustic velocity field. The impact of this singularity on the convergence properties of the method is discussed in Section 4 . The following section proposes a mathematical reformulation of the pressure modal expansion to mitigate this undesirable feature.

Modal expansion onto an over-complete frame of acoustic eigenmodes

The purpose is now to introduce a modal expansion of the acoustic pressure that would allow for satisfying any boundary condition on S ci (and not ∇ s p = 0 only). In this matter, it is necessary to retain in the modal expansion an additional degree of freedom, such that both acoustic pressure and normal acoustic velocity at the connection boundary S ci remain a priori undetermined.

Let us then introduce two distinct families

( ξ m ) m ≥ 1 and ( ζ k ) k ≥ 1
of acoustic eigenmodes of the subsystem i , characterized by the two following eigenproblems: Consider now the eigenmodes family (

⎧ ⎨ ⎩ c 2 0 ∇ 2 ξ m + ω 2 m ξ m = 0 for x ∈ i ∇ s ξ m = 0 for x ∈ S wi , ξ m = 0 for x ∈ S ai ∇ s ξ m = 0 for x ∈ S ci (8) ⎧ ⎨ ⎩ c 2 0 ∇ 2 ζ k + ω 2 k ζ k = 0 for x ∈ i ∇ s ζ k = 0 for x ∈ S wi , ζ k = 0 for x ∈ S ai ζ k = 0 for x ∈ S ci
φ n ) n ≥ 1 formed as the concatenation of ( ξ m ) m ≥ 1 and ( ζ k ) k ≥ 1 : (φ n ) n 1 = (ξ m ) m 1 ∪ (ζ k ) k 1 . The eigenpulsations associated to the eigenmodes φ n are noted (ω n ) n 1 = (ω m ) m 1 ∪ (ω k ) k 1 .
As a concatenation of two orthogonal bases, ( φ n ) n ≥ 1 is not a basis but is instead an overcomplete set of eigenmodes, also called a frame [START_REF] Daubechies | Painless nonorthogonal expansions[END_REF] . The concept of frame was introduced in the context of nonharmonic Fourier analysis [START_REF] Young | An introduction to non-harmonic fourier series[END_REF] , and later used in a number of fields ranging from wavelet analysis, to digital image processing and time-series forecasting. The use of an over-complete frame to perform modal expansions is the most crucial element of the proposed method. Let us precise that the frame ( φ n ) n ≥ 1 could be built from the concatenation of other sets of eigenmodes, as long as those do not a priori impose any constraint between pressure and velocity at the boundary S ci . However, using the concatenation of the rigid-wall basis ( ξ m ) m ≥ 1 and the open atmosphere basis ( ζ k ) k ≥ 1 has two main advantages: (1) they are usually the easiest to obtain analytically or numerically, and (2) the frame ( φ n ) n ≥ 1 formed by their concatenation verifies a generalized Perseval's identity which ensures the well-posedness of modal expansions [START_REF] Daubechies | Painless nonorthogonal expansions[END_REF] .

In the following, a compact vectorial notation is introduced to avoid the use of multiple summation symbols: for any indexed quantity ( f i ) i ≥ 1 we note f the column-vector such that:

t f = ( f 0 f 1 f 2 . . . )
, where t () designates the vector transpose.

For a doubly indexed quantity ( f ij ) i,j ≥ 1 , we note f the matrix whose coefficients are the f ij . Conversely, ( f ) n is the n th component of the vector f . Similarly to Section 2.1 , the pressure modal expansion is sought under the form ˆ

p ( x , ω) = n ˆ γ n (ω) φ n ( x ) , that is, using the vectorial notation ˆ p ( x , ω) = t ˆ γ (ω) φ( x )
. The analytical derivation of the modal amplitudes ˆ γ n (ω) is rather long and its details are not necessary to understand the remaining of the paper; it is thus not included here but made available in Supplemental Material B.

As an outcome of this analytical derivation, the modal amplitudes ˆ γ n (ω) are found to be solutions of the following equation:

(ω 2 -jωα -ω 2 n ) ˆ γ n (ω) jω = M S j=1 ρ 0 c 2 0 φ ⊥ n ( x 0 j ) ˆ u j s ( x 0 j ) - M S j=1 ρ 0 c 2 0 ∇ s φ ⊥ n ( x 0 j ) ˆ ϕ j ( x 0 j ) -( γ -1 ) H ⊥ i,n ˆ Q (ω) ( 10
)
where:

φ ⊥ n ( x 0 j ) = S 0 j -1 φ( x 0 j ) n , ∇ s φ ⊥ n ( x 0 j ) = S 0 j -1 ∇ s φ( x 0 j ) n , H ⊥ i,n = -1 H i n ( 11
)
As previously the surface S ci has been decomposed into M S plane surface elements S 0 j located at x 0 j . In Eq. ( 11) , = t φ, φ is the matrix whose coefficients are mn = φ m , φ n . This matrix is the Gram matrix associated to the over-complete frame ( φ n ) n ≥ 1 : it reduces to a diagonal matrix in the case of an orthogonal basis. The volume source ˆ h ( x , ω) is expressed according to Eq. ( 3) , and H i is the column vector containing all the projections of the flame shape

H i ( x ) onto the elements of the over-complete frame (φ n ( x )) n 1 .
In Eq. ( 10) , ˆ u j s ( x 0 j ) is the velocity forcing imposed onto i by the adjacent subsystems j at the connection points x 0 j , while ˆ ϕ j ( x 0 j ) = -ˆ p j ( x 0 j ) / (ρ 0 jω) is an acoustic potential forcing imposed onto i at these boundary points. One may note that the second term in the right-hand side of Eq. ( 10) has no counterpart in Eq. ( 6) , contrary to the first and third terms. As in Section 2.1 for Eq. ( 6) , Eq. ( 10) can then be recast into the time-domain by introducing n ( t ) such that γ n (t) = n (t) ; this leads to:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p( x , t ) = N n =1 ˙ n (t) φ n ( x ) ¨ n (t) = -α ˙ n (t) -ω 2 n n (t) - M S j=1 ρ 0 c 2 0 φ ⊥ n ( x 0 j ) u j s ( x 0 j , t ) + M S j=1 ρ 0 c 2 0 ∇ s φ ⊥ n ( x 0 j ) ϕ j ( x 0 j , t ) + (γ -1) H ⊥ i,n Q (t) ( 12
)
which is an extension of Eq. ( 7) when the expansion is performed on the over-complete frame ( φ n ) n ≥ 1 instead of the orthogonal basis ( ψ n ) n ≥ 1 . This dynamical system governs the temporal evolution of the pressure field in the domain i , under normal velocity forcing u j s ( x 0 s , t ) and acoustic potential forcing ϕ j ( x 0 s , t ) imposed by adjacent subsystems, as well as the volume forcing Q ( t ) due to the presence of active flames within i . In the case of the orthogonal modal basis ( ψ n ) n ≥ 1 used in the previous section, the terms ∇ s ψ ⊥ n ( x 0 j ) vanish (since the eigenmodes satisfy rigidwall boundary conditions on S ci ), and we have the simple relations

ψ ⊥ n ( x 0 j ) = S 0 j ψ n ( x 0 j ) / n and H ⊥ i,n = H (n ) i / n .
Because of the use of the over-complete frame ( φ n ) n ≥ 1 , the acoustic pressure and velocity are free to evolve independently on the boundary S ci , which is the major improvement of the method compared to classical formalism where the normal acoustic velocity u . n s is necessarily zero on the connection boundary S ci . The governing dynamical system of Eq. ( 12) is a projection of the wave equation ( Eq. ( 2) ) onto the modal frame ( φ n ) n ≥ 0 .

However, as this frame is over-complete such projection is illconditioned, which constitutes one of the major pitfalls of the proposed method. The first consequence is a numerical difficulty to compute the inverse of the frame Gram matrix does not noticeably increase the computational cost in comparison to the classical rigid-wall modal expansion. Secondly, even though is accurately inverted, the frame over-completeness may still result in poorly conditioned spurious components in Eq. [START_REF] Dowling | Acoustic analysis of gas turbine combustors[END_REF] . However, the approach described above produces well-behaved expansions for the pressure, that is expansions where the terms with the highest energy are physically meaningful while low energy terms represent spurious fluctuations. Thus, an energetic criterion for the robust and automatic identification of these spurious components was designed, as discussed in more details in Appendix C .

State-space formalism for subsystems coupling

The dynamical system derived in Section 2.2 allows us to solve for the acoustic pressure in each individual subdomain i . In order to resolve the acoustic flow in the whole geometry, individual subdomains need to be connected together. An elegant formulation to connect subdomains is to use a state-space approach. This method, already used in [START_REF] Stow | A time-domain network model for nonlinear thermoacoustic oscillations[END_REF][START_REF] Schuermans | Thermoacoustic modeling and control of multi burner combustion systems[END_REF][START_REF] Schuermans | Modeling and control of thermoacoustic instabilities[END_REF][START_REF] Emmert | Linear state space interconnect modeling of acoustic systems[END_REF] , is adopted in this work but requires to be adapted. Some implementation details are given below; further developments relative to state-space representations can be found in control theory textbooks [START_REF] Friedland | Control system design: an introduction to state-space methods[END_REF] . For any physical system described by a set of coordinates X ( t ) in a phase-space, we call linear state-space representation of this system a set of equations under the form: ˙ [START_REF] Li | OSCILOS: the open source combustion instability low order simulator[END_REF] where X ( t ) is the coordinates vector in the phase-space, also called state vector, A is the dynamics matrix, B is the input matrix, U ( t ) the input vector, Y ( t ) the output vector, C the output matrix, and D is the action, or feedthrough matrix. The first equation of the statespace representation governs the dynamical evolution of the state vector under the forcing exertedby the input vector. The second equation defines a way to compute any desired outputsfrom the knowledge of the state vector and the forcing term. Note that the output Y ( t ) depends on the state X ( t ), but the reverse is not true: X ( t ) evolves independently of the selected output Y ( t ). The statespace formalism, through the Redheffer star-product [START_REF] Redheffer | On a certain linear fractional transformation[END_REF] (Supplemental Material C), provides a direct way to connect two systems represented by their state-space realizations, by relating their respective inputs and outputs. A simple example of this operation is given in Section 4 .

X (t) = A X (t) + B U (t) Y (t) = C X (t) + D U (t)
By considering the dynamical system of Eq. ( 12) , it is straightforward to build a state-space representation for a subdomain i . A convenient choice is to build the state-vector X ( i ) ( t ) from the modal amplitudes ˙ n (t) and their temporal primitives n ( t ).

The input vector U ( i ) ( t ) contains the normal acoustic velocities u j s ( x 0 j , t ) and the acoustic potentials ϕ j ( x 0 j , t ) imposed by adjacent subdomains j . The heat release volume source Q ( t ) is also included in the input vector. Beside, the computed output vector Y ( i ) ( t ) consists of the pressures p( x 0 j , t ) and the normal acoustic velocities u s ( x 0 j , t ) at every point x 0 j on the connection boundary S ci . In addition, any quantity of interest that can be computed from the modal amplitudes contained in the state-vector X ( i ) ( t ) may also be added in the output vector Y ( i ) ( t ). For conciseness, the detailed expressions of the state-space matrices for a subdomain i are not given here, but can be found in Appendix A .

After iteratively applying the Redheffer star-product to connect together state-space representations of every subsystems, the full state-space of the whole geometry is obtained as: [START_REF] Li | Time domain simulations of nonlinear thermoacoustic behaviour in a simple combustor using a wave-based approach[END_REF] where U f ( t ) is an external forcing that can be either a surface or a volume source term. In the former case, coefficients in the external input matrix B f contains terms similar to the first 2 M S columns B ( i ) in Eq. (A.1) , while in the latter case it contains terms similar to the last column of B ( i ) in Eq. (A.1) . Two approaches are then possible: (1) Eq. ( 14) can be integrated over time to obtain the temporal evolution of the acoustic flow under the external forcing U f ( t ), or (2) the complex eigenvalues and eigenvectors of the dynamics matrix A f can be solved for, yielding the global acoustic eigenfrequencies and eigenmodes of the whole domain. If λ n = 2 πσ n + j2 π f n is the n th complex eigenvalue of the matrix A f , then f n is the eigenfrequency of the n th acoustic mode of the whole geometry. In the absence of acoustic losses, volume sources or complex boundary impedances, σ n is zero. Conversely, if the system comprises acoustic sources, then 2 πσ n , is the growth-rate of the n th acoustic mode of the whole geometry: σ n > 0 (resp. σ n < 0) implies that the mode is unstable (resp. stable). The mode shape can also be reconstructed from the modal components contained in the eigenvector v n associated to the eigenvalue λ n .

˙ X f (t) = A f X f (t) + B f U f (t)

Convergence properties

In this section both modal expansions presented in Section 2 are implemented within the LOM state-space framework introduced in Section 3 , and used to study a canonical case, namely a long quasi-one-dimensional tube comprising a sharp cross-section change. The goal is to show the limits of the rigid-wall modes decomposition and to prove the performances of the over-complete frame approach in a case where both types of modal expansions can be evaluated and compared to an analytical solution. In the following, superscripts OB (resp. FR ) refer to results obtained with the use of the orthogonal basis ( ψ n ) n ≥ 1 introduced in Section 2.1 (resp. the over-complete frame ( φ n ) n ≥ 1 introduced in Section 2.2 ). Superscripts A designate analytical solutions used for comparison.

In this example, the long duct with a sudden cross-section change represented in Fig. 2 is considered. Both ends of the duct are closed by rigid walls. It is decomposed into 3 subsystems, including 2 long ducts ( 1 and 2 ) with constant cross-sections S 1 and S 2 , and a third subsystem sc of length L sc enclosing the region in the neighborhood of the cross-section variation.

Since both tubes 1 and 2 are long ( D 1 L 1 , D 2 L 2 ), only plane longitudinal acoustic waves are considered here. The rigidwall orthogonal bases of both ducts are then:

⎧ ⎪ ⎨ ⎪ ⎩
ψ (1) n (x 1 ) n N 1

= cos nπ x 1 L 1 n N 1 ψ (2) n (x 2 ) n N 2 = cos nπ x 2 L 2 n N 2 (15)
where superscript (1) (resp. (2) ) refers to the modal basis in 1 (resp. 2 ), x 1 and x 2 are the longitudinal coordinates in the two ducts ( Fig. 2 ), and N 1 (resp. N 2 ) is the number of modes used for the pressure modal expansion in 1 (resp. 2 ). Similarly the overcomplete frames introduced in Section 2.2 for both ducts are given by:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ φ (1) n (x 1 ) n N 1 = cos nπ x 1 L 1 n N 1 / 2 cos (2 n + 1) π x 1 2 L 1 n N 1 / 2 φ (2) n (x 2 ) n N 2 = cos nπ x 2 L 2 n N 2 / 2 sin (2 n + 1) π x 2 2 L 2 n N 2 / 2 (16)
These orthogonal bases ( Eq. ( 15) ) and over-complete frames ( Eq. ( 16) ) are consistent with rigid-wall conditions at both end of the long duct ( x 1 = 0 and x 2 = L 2 ). However, the orthogonal bases also impose zero velocity near the cross section change (at x 1 = L 1 and x 2 = 0 ), while the over-complete frames retain an additional degree of freedom such that both velocity and pressure are a priori undetermined and free to evolve independently near the crosssection change. In the following, the same number N of eigenmodes are used for modal expansions in both ducts (

N 1 = N 2 = N).
Additionally, all comparisons between orthogonal basis and overcomplete frame expansions are carried out with the same total numbers of modes N : in other words results from any orthogonal basis containing an even number of vectors N are compared to results from a frame composed of two subfamilies of size N /2. Note also that, since only plane longitudinal waves are considered here, the connection boundaries S c 1 and S c 2 do not need to be discretized into several surface elements S 0 j : therefore M S = 1 for each subdomain 1 and 2 , and M S = 2 for sc .

Modal expansion is not performed for the subdomain sc enclosing the cross-section change as its exact geometry has only very little effect on the global eigenmodes of the long duct: instead, volume-averaged conservation equations are used to derive a state-space representation of this subsystem. More details are given in Appendix B . The Redheffer star-product (Supplemental Material C) is then applied recursively to connect state-space representations of subsystems 1 , sc , and 2 . The state-space real- results in low relative errors, ranging between 10 -7 and 10 -8 for all modes 1, 9 and 11, even with small values of N . For instance, only about 10 modes in each frames are necessary to accurately capture the eigenfrequency f 11 . The condition number of the frame Gram matrix, which is an indicator of the expansion over-completeness, is C( ) = 10 8 for N = 6 . It then progressively deteriorates and reaches 10 19 at N = 50 , and saturates to this value for large N . This deterioration of the frame conditioning does not result into a degradation of the numerical results. However, it can be related to a saturation of the error, since increasing the size of the frame beyond N = 20 does not result in smaller errors.

Finally, it is worth emphasizing that Gibbs fringes have also been reported in earlier studies employing orthogonal basis modal expansions, for instance by Sayadi et al. [START_REF] Sayadi | Thermoacoustic instability -a dynamical system and time domain analysis[END_REF] . However, the origin of these Gibbs oscillations was fundamentally different from those observed in the present example: in previous works, this phenomenon was caused by the presence of an infinitely thin region of fluctuating heat release located in the interior of the domain of interest. This assumption results in an exact solution for the velocity field that is discontinuous at the flame location, and that cannot be accurately represented by a classical Galerkin decomposition. The over-complete frame expansion presented in this work aims at suppressing Gibbs oscillations due to a misrepresentation of the acoustic fields at the boundaries of the subdomains, and does not have the ability to handle Gibbs fringes produced by a discontinuity in the interior of the subdomains. Nonetheless, a simple workaround to handle this latter case is to replace the in- 15) ), while gray lines are results obtained with frames modal expansions ( Eq. ( 16) ).

finitely thin flame by a slightly thicker region of heat release: this yields a solution for the velocity field that has a sharp yet continuous spatial variation in the vicinity of the flame. Note also that infinitely thin flame are interesting conceptually and from an analytical perspective; however, physical flames are always finite and sometimes thick enough to produce non-compact effects on thermoacoustic instabilities [START_REF] Subramanian | Distributed time lag response functions for the modelling of combustion dynamics[END_REF] .

Application to thermoacoustic instabilities in an annular combustor

In order to demonstrate its ability to predict thermoacoustic instabilities in complex geometries, the state-space LOM based on generalized frame modal expansions (FR) is now applied to a more advanced academic configuration comprising active flames in an annular chamber. Results obtained with the classical orthogonal rigid-wall basis (OB) are also provided and compared to those computed with a 3D Finite Element (FE) solver called AVSP [START_REF] Nicoud | Acoustic modes in combustors with complex impedances and multidimensional active flames[END_REF] . The geometry studied is displayed in Fig. 6 (a), and the correspond-ing low-order acoustic network is represented in Fig. 6 (b). It comprises an annular plenum (denoted with the subscript P ), an annular chamber (subscript C ), and four identical ducted burners (subscript B ) of length L B where the active flames are located. Rigid-wall boundary conditions are assumed at the plenum backplane and the chamber outlet plane.

The AVSP unstructured mesh consists of 3 × 10 6 tetrahedral cells, while the acoustic network contains 14 subdomains (1 plenum P , 1 chamber C , 4 burners Bi , and 8 cross-section changes sc i ), with the addition of 4 active flames. The flames H i are located at the coordinate αL B within each burner, and are considered as planar volume source of thickness δ. Note that the cross-section area S B of the ducted burners is much smaller than the area of the plenum exit plane and of the chamber backplane. This allows for the simplification of the plenum-burner and burner-chamber junctions, by only considering discrete point-like connections. In other words, the chamber (or the plenum) is connected to the burner B i through the subdomain sc i at a single point, implying that M S = 4 for the chamber and the plenum (a single discrete connection surface S 0 j = S B is used for each one of the 4 burners). The subdomain sc i is similar to the cross-section change described in the previous section, as it essentially enforces continuity of pressure and acoustic flux between the burner end and the backplane of the chamber (or the exit plane of the plenum). This simplification also allows us to consider rigid-wall boundary conditions at the chamber backplane and at the plenum exit plane when defining the eigenmodes for these subdomains (since the velocity should actually be non-zero only at 4 point-like locations of infinitely small spatial extent). The Gibbs phenomenon evidenced in the previous section is then not expected to appear in the chamber and in the plenum, and it is therefore valid to employ orthogonal rigid-wall bases in these two subdomains. On the contrary, boundary conditions at both ends of the burners are expected to differ from rigid-wall or open atmosphere, and it is therefore necessary to employ over-complete frame expansions in these subdomains in order to mitigate the Gibbs phenomenon that may appear. Thus, the plenum is modeled as a 2D annular subdomain of coordinates ( x P , θ P ), whose orthogonal basis is: Pressure in the chamber could be expanded onto an analogous analytical modal basis. However, this one is deliberately assumed analytically unknown, and a modal basis computed thanks to a preliminary AVSP simulation of the isolated chamber (without the burners and the plenum) is used instead. Note that it is not necessary to perform an AVSP simulation for each LOM simulation: it is indeed preferable to generate the chamber modal basis and all related quantities (including the matrix -1 ) in a single preliminary AVSP simulation, and to assemble the state-space realization of this subdomain, which can then be employed in as many LOM simulations as desired. In the present example, the use of a numerically computed modal basis demonstrates the ability of the present framework to combine in an acoustic network subdomains of different types and thus to handle efficiently arbitrarily complex systems.

ψ (P)
In the following, the size of the modal bases in the chamber and in the plenum is fixed to N = 12 modes. For cases where the rigid-wall basis is used in the ducted burners, this one is the same as in the previous section ( Eq. ( 15) ). If an over-complete frame is used, it is given by:

φ (B ) n (x B ) n N = cos nπ x B L B n N/ 2 sin nπ x B L B n N/ 2 (23)
Note the difference with the over-complete frame of Eq. ( 16) , as the present one allows pressure and velocity to evolve independently from one another at both ends of the duct. Active flames are located within each burner, and the flame shape H i (x B ) is the rectangular function of thickness δ centered around x B = αL B . The flame response is modeled thanks to a classical Flame Transfer Function (FTF), relating the fluctuations of heat release to the fluctuations of acoustic velocity at a reference point located at x

(re f ) B = βL B .
The flame reference point is located in the burners, near the plenum exit-plane ( β = 0 . 05 ). A simple constantdelay FTF is assumed, such that in the frequency domain the fluctuating heat release rate reads:

ˆ Q (ω) = Q e -jωτ ˆ u (x B = βL B , ω) u ( 24
)
where Q is the flame power, τ is the flame delay, and u is the mean flow speed through the injector. A state-space realization of the time-delay e -jωτ is generated thanks to a Multi-Pole expansion:

e -jωτ ≈ M PBF k =1 -2 a k jω ω 2 + 2 c k jω -ω 2 0 k ( 25
)
where each term in the sum is called a Pole Base Function (PBF).

The coefficients a k , c k , w 0 k are determined thanks to a recursive fitting algorithm recently proposed by Douasbin et al. [START_REF] Douasbin | Delayed-time domain impedance boundary conditions (D-TDIBC)[END_REF] . By making use of the inverse Fourier transform, it is then straightforward to convert this frequency domain transfer function into a timedomain state-space realization of size 2 M PBF × 2 M PBF , whose expression is provided in Appendix D . This procedure to generate a state-space realization of a FTF was already used by Ghirardo et al. [START_REF] Ghirardo | State-space realization of a describing function[END_REF] . For the flame-delay considered here, 12 PBFs were observed to be sufficient to accurately fit the term e -jωτ , yielding 24 DoF for each flame in the state-space representation of the whole system. Finally, the 18 state-space representations of the 14 acoustic subdomains and 4 active flames are connected together.

As mentioned earlier, the Gibbs phenomenon is expected to occur at both ends of the ducted burners. A particular attention is therefore paid to the type of modal expansion carried out in these subdomains. A total of 4 LOM simulations are performed, and results are compared to the ones computed with the FE solver AVSP, which are used as reference. Results for 10 of the first modes of the combustor are summarized in Table 1 . First of all, FR and OB expansions with N = 10 modes are both compared to AVSP. Then these computations are repeated with a number of modes increased to N = 30 . Mode 1 is the combustor Helmholtz mode, and its frequency and growth rate were observed to be very sensitive to the addition of a correction length to the ducted burners. As the determination of the optimal correction length is out of the scope of this paper, differences regarding the Helmholtz mode are not further discussed. The FR expansion with N = 10 appears to successfully resolve the frequencies and growth rates of the modes considered, with relative errors compared to AVSP below 10%. On the contrary, the OB expansion onto N = 10 modes largely fails at resolving the growth rates of all but one of the considered modes, with relative errors up to 348% (Mode 5). Mode 3 is the first unstable azimuthal mode of the combustor, and is therefore of particular interest: FR expansion yields an error of only 6% for the growth rate of this mode, whereas OB expansion produces an error of 45%. When the size of the expansion basis/frame is increased, results are globally improved for both FR and OB cases. Yet, even with N = 30 the OB approach still fails at achieving an acceptable accuracy for Modes 3, 5, and 7. The error on the growth rate of the first unstable azimuthal mode is of 0.3% in the FR case, while it is still overestimated by 18% in the OB case. Modes 9 and 10 are mixed modes, similar to those described by Evesque and Polifke [START_REF] Evesque | Low-order acoustic modelling for annular combustors: validation and inclusion of modal coupling[END_REF] . In mode 9 the plenum first longitudinal mode prevails, while mode 10 is the mixed 1 st -azimuthal-1 st -longitudinal plenum mode, coupled with the 5 th azimuthal chamber mode. Previous comments also applies to these mixed modes: the over-complete frame ex-pansion of size N = 10 yields an excellent agreement with the FE solver, and outperforms even larger orthogonal basis expansions. Note that in the FR case, the condition number of the frame Gram matrix used in the 4 ducted burners is of order 10 8 for N = 10 , and increases to 10 18 for N = 30 . It was verified that increasing the frame size does not further deteriorates its conditioning: it instead saturates at 10 18 even for large N .

The relatively poor accuracy of the OB expansion LOM is explained by a closer examination of the modes shapes in the burners. Figure 7 shows the shape of the first unstable azimuthal mode of the combustor (Mode 3), plotted over a line starting from the bottom of the plenum, passing through a burner, and ending at the chamber outlet plane. With N = 30 , both FR and OB pressure mode shapes ( Fig. 7 (a) and (c)) are relatively close to the AVSP computation, except for 3D effects in the neighborhood of the subdomains connections that cannot be captured. On the contrary, Fig. 7 (d) shows that the OB expansion produces significant Gibbs oscillations of the velocity at both ends of the burner. Note that the Gibbs phenomenon is only present at the ends of the burners, and does not affect the velocity field within the chamber and the plenum. Conversely, Fig. 7 (b) shows that the frame expansion successfully mitigates this Gibbs phenomenon.

The conjugation of these spurious oscillations with the presence of active flames responding to velocity fluctuations explains the large discrepancies observed in the growth rates. Indeed, the closeup view displayed in Fig. 7 (d) reveals that the flame reference point (represented by a star) lies in a region where the velocity is strongly affected by Gibbs oscillations. In contrast, the FR expansion ( Fig. 7 (b)) yields a reference velocity close to the AVSP reference velocity. As heat release fluctuations are directly proportional to the reference velocity ( Eq. ( 24) ), any misprediction of the velocity in the burner results in a potentially erroneous growth rate. Thus, should the point of reference lies in a region where numerical oscillations are present, the computed thermoacoustic modes may strongly depend on unphysical and uncontrolled details such as the relative position of the point of reference and the Gibbs oscillations. Consequently, the orthogonal rigid-wall basis expansion results in a LOM that is highly sensitive to the location of the flame reference point, which is a highly undesirable feature of a numerical model.

This example demonstrates the modularity of the proposed LOM, which can combine in a same thermoacoustic network active flames, one-dimensional subdomains (the burners), 2D subdomains (the plenum), and complex 3D subdomains of arbitrary shape (the chamber) for which the modal basis is numerically computed. Obviously, the approach is not limited to azimuthal eigenmodes, but is also able to capture any other form of thermoacoustic eigenmodes. It is also worth comparing the cost associated to the overcomplete frame expansion LOM to existing Low-Order Models. As shown above, N = 10 modes were sufficient to achieve a satisfactory resolution (with error below 10%) of the first 20 modes of the combustors (not all shown in Table 1 ). The state-space of the whole system comprises 248 DoF: 2 × 24 for the plenum and the chamber, 4 × 20 for the straight ducts, 8 × 3 for the cross-section changes, and 4 × 24 for the active flames. After the preliminary computation of the chamber modal basis with AVSP (160 CPU seconds for 12 modes), the LOM computation of all the eigenmodes was performed in a few CPU seconds only. This is comparable to the 300 DoF necessary to treat a similar annular configuration in the work of Schuermans and co-workers [START_REF] Schuermans | Thermoacoustic modeling and control of multi burner combustion systems[END_REF][START_REF] Bellucci | Thermoacoustic modeling of a gas turbine combustor equipped with acoustic dampers[END_REF] . However, unlike this latter method, the present example did not assume acoustically compact injectors represented as lumped elements, and the acoustic field is fully resolved within the burners. LOMs relying on direct discretization of the flow domain, although more straightforward to put into application, appears to result in more DoF and higher cost: Emmert et al. [START_REF] Emmert | Linear state space interconnect modeling of acoustic systems[END_REF] required 10 5 DoF and 38 CPUs to compute 5 eigenmodes of a 12-injectors annular geometry without any active flames. Finally, in contrast to mixed-method LOMs based on modal expansions alongside Riemann invariants A + /A - [START_REF] Li | Time domain simulations of nonlinear thermoacoustic behaviour in a simple combustor using a wave-based approach[END_REF] , the proposed Low-Order Model ( Eq. ( 14) ) can be directly integrated in time for temporal simulations of thermoacoustic acoustic instabilities, whereas the time-domain translation of Riemann invariants appears to be somehow constraining [START_REF] Meindl | Efficient calculation of thermoacoustic modes utilizing state-space models[END_REF] .

Conclusions

This work addressed a known issue in LOMs for thermoacoustics, namely the misrepresentation of the acoustic field arising from modal expansions onto rigid-wall eigenmodes bases, already reported by numerous earlier studies [START_REF] Culick | Unsteady motions in combustion chambers for propulsion systems[END_REF] . Under the assumption of zero Mach number flow, a reformulation of the classical modal expansion making use of over-complete frames of eigenmodes was proposed for the first time. Two major observations were drawn from the analysis of a simple one-dimensional acoustic problem: (1) the modal expansion singularity was clearly identified as a Gibbs-like phenomenon affecting the acoustic velocity, resulting in slow convergence speeds and erroneous velocity values near junctions between subdomains; (2) the over-complete frame expansion was shown to successfully suppress the Gibbslike phenomenon and to yield significantly improved convergence speeds. The only pitfall stemming from the frame expansion, lies in its over-completeness that may entail ill-conditioned features, which may produce spurious, non-physical dynamics of the pressure evolution. Thus, a specific inversion procedure is used to compute -1 , the inverse of the frame Gram matrix. This approach ensures that these spurious components stay negligible in comparison to physically meaningful components. More rigorously, in Appendix C it is demonstrated that spurious eigenmodes arising from the frame over-completeness have low energy, and a criterion is derived to systematically identify them.

In a second example, the generalized modal expansion LOM was used to predict thermoacoustic instabilities in an annular combustor. It was shown to yield results close to a finite element solver, whereas the rigid-wall modal expansions failed at accurately predicting the linear growth rates. This second example also demonstrates the modularity of the proposed framework, through its ability to combine in a same acoustic network highly heterogeneous classes of elements such as active flames, one-dimensional ducts, or complex 3D cavities. From a practical point of view, the implementation of the proposed method only requires minimal changes to existing algorithms based on rigid-wall expansions. Namely, apart from the inversion of the Gram matrix and the identification of spurious eigenmodes, the input/output relations of each subdomain composing the acoustic network should be adapted to include both pressure and velocity at the boundaries. Thus, this novel method is expected to be potentially useful in the field of thermoacoutics. subdomain i yields the following state-space realization:

d dt ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 (t) ˙ 1 (t) . . . N (t) ˙ N (t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ X (i ) (t) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 -ω 2 1 -α . . . 0 1 -ω 2 N -α ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ A (i) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 (t) ˙ 1 (t) . . . N (t) ˙ N (t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ + ρ 0 c 2 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 . . . 0 0 0 -φ ⊥ 1 ( x 1 ) ∇ s φ ⊥ 1 ( x 1 ) . . . -φ ⊥ 1 ( x M S ) ∇ s φ ⊥ 1 ( x M S ) γ -1 ρ0 c 2 0 H ⊥ i, 1 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 0 0 -φ ⊥ N ( x 1 ) ∇ s φ ⊥ N ( x 1 ) . . . -φ ⊥ N ( x M S ) ∇ s φ ⊥ N ( x M S ) γ -1 ρ0 c 2 0 H ⊥ i,N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ B (i) × ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ u 1 s ( x 1 , t ) ϕ 1 ( x 1 , t ) . . . u M S s ( x M S , t ) ϕ M S ( x M S , t ) Q (t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ U (i ) (t) (A.1)
In this equation, the state vector X ( i ) ( t ) is of size 2 N , where N is the number of eigenmodes used in the modal expansion. The dynamics matrix A ( i ) is block-diagonal of size 2 N × 2 N . The first 2 M S columns of the input matrix B ( i ) and 2 M S elements of the input vector U ( i ) ( t ) correspond to surface source terms imposed by adjacent subdomains j . The last column of B ( i ) and the last element of U ( i ) ( t ) correspond to the volumetric heat release forcing. For clarity reason, a single source of heat release is considered here, but the input can easily be extended to any number of independent flames.

In the state-space framework that is presented here, any subsystem i outputs both the normal velocity u s ( x 0 j ) = u ( x 0 j ) . n s and the pressure p( x 0 j ) at each one of the M S connection surface elements S 0 j . In addition, it is also possible to incorporate in the output vector (not detailed here) pressure and velocity at any point within i , such that those can then be passed as reference pressure/velocity to an active flame. Thus, the following equation is used to compute the output vector for the subdomain i :

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ u s ( x 1 , t ) p( x 1 , t ) . . . u s ( x M S , t ) p( x M S , t ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Y (i ) (t) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -1 ρ0 ∇ s φ 1 ( x 1 ) 0 . . . -1 ρ0 ∇ s φ N ( x 1 ) 0 0 φ 1 ( x 1 ) . . . 0 φ N ( x 1 ) . . . . . . . . . . . . . . . -1 ρ0 ∇ s φ 1 ( x M S ) 0 . . . -1 ρ0 ∇ s φ N ( x M S ) 0 0 φ 1 ( x M S ) . . . 0 φ N ( x M S ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ C (i) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 (t) ˙ 1 (t) . . . N (t) ˙ N (t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ X (i ) (t) (A.2)
Note that the feedthrough matrix D is zero. At the end, the state-space representation of the acoustics in the subdomain i is defined by Eqs. (A.1) and (A.2) . It is a 2 M S -input 2 M S -output sys-tem (in the absence of heat release source terms), whose dynamics are described by a 2 N × 2 N matrix.

Appendix B. State-space representation for the domain sc enclosing the cross-section change

In this section, the state-space representation for the acoustically compact subdomain sc enclosing the cross-section change between the two ducts is given. It is obtained by volume-averaging the linearized Euler equations in sc , neglecting any acoustic conversion to the vortical mode. The interested reader may consult Supplemental Material D for further details regarding the derivation. The final state-space realization of the subdomain sc is:

d dt u (t) p (t) ϕ (t) X (sc) (t) = 0 0 0 0 0 0 0 -1 /ρ 0 0 A (sc) u (t) p (t) ϕ (t) + ⎛ ⎝ 0 1 ρ 0 L sc 0 -1 ρ 0 L sc 2 S 1 c 2 0 ρ 0 L sc (S 1 + S 2 ) 0 2 S 2 c 2 0 ρ 0 L sc (S 1 + S 2 ) 0 0 0 0 0 ⎞ ⎠ B (sc) ⎛ ⎜ ⎝ u 1 s (t) p 1 (t) u 2 s (t) p 2 (t) ⎞ ⎟ ⎠ U (sc) (t) (B.1)
The state-vector X ( sc ) ( t ) contains the volume-averaged velocity u (t) , pressure p (t) , and acoustic potential ϕ (t) =p (t ) dt /ρ 0 .

The first two components of the input vector U ( sc ) ( t ) are imposed by the duct 1 , while the last two are imposed by the duct 2 . We then note B (sc) 1 (resp. B (sc) 2 ) the matrix consisting of the first two columns (resp. last two columns) of B ( sc ) . In Eq. (B.1) , the first two lines essentially impose acoustic momentum and acoustic volume flux conservation. For low frequencies, these conservation relations reduce to the classical quasi-static jump relations p 1 = p 2 and S 1 u 1 s = -S 2 u 2 s . The third equation of the dynamical system is a time-integrator that facilitates the computation of the output vector, as the two ducts 1 and 2 require normal acoustic velocity and acoustic potential as inputs. Outputs for the state-space representation of sc are computed thanks to first-order approximations:

⎛ ⎜ ⎝ u s (0 , t ) ϕ(0 , t ) u s (L sc , t ) ϕ(L sc , t ) ⎞ ⎟ ⎠ Y (sc) (t) = ⎛ ⎜ ⎝ 1 2 (1 + S 2 S 1 ) 0 0 0 0 1 -1 2 (1 + S 1 S 2 ) 0 0 0 0 1 ⎞ ⎟ ⎠ C (sc) u (t) p (t) ϕ (t) (B.2)
In Eq. (B.2) , the feedthrough matrix D is zero. The first two lines are outputs that are to be imposed to the first duct 1 at the connection boundary S c 1 , while the last two lines are outputs that are to be imposed to the second duct 2 at the connection boundary S c 2 . We then note C (sc) 1 (resp. C (sc)

2

) the matrix formed with the first two rows (resp. last two rows) of C ( sc ) .

Appendix C. Spurious eigenmodes identification

As mentioned in Section 2.2 , the over-completeness of the frame expansion may result in poorly conditioned spurious components of the governing dynamical system ( Eq. ( 12) ). It is therefore primordial to distinguish these spurious components from the physically meaningful ones. The interested reader is reported to [START_REF] Boyd | Chebyshev and Fourier spectral methods[END_REF] for further discussion regarding spurious components identification. The simplest method to achieve it is the brute force approach: after obtaining the eigenmodes of the full-system with an expansion of size N , the computation is repeated with a different value of N . Eigenmodes that are very sensitive to the number of DoF are then considered as spurious. However, the over-complete frame expansion presented in Section 2.2 was observed to produce only low energy spurious components. Therefore, a more efficient identification procedure based on an energetic criterion was implemented. This one is described below.

Let us consider an eigenvector v = t ( . . . a 1 b 1 . . . a N b N . . . ) of the whole system dynamics matrix A f (see Eq. ( 14) and Appendix A ), where the coefficients a 1 , . . . , a N are the state variables involved in the velocity mode shape in the subdomain i (they correspond to the time-dependent modal amplitudes n ( t )), and the coefficients b 1 , . . . , b N are the state variables associated to the pressure mode shape in the subdomain i (they correspond to the time-dependent modal amplitudes ˙ n (t) ). Then, the eigenmode pressure in the subdomain i is reconstructed as ϒ p ( x ) = n b n φ n ( x ) = t b φ( x ) . The L -2 norm of the full system's eigenmode ϒ p ( x ) , or equivalently its energy, is given by:

|| ϒ p || 2 2 = ϒ p ( x ) , ϒ p ( x ) = t b φ( x ) , t φ( x ) b = t b b (C.1)
where is the Gram matrix of the over-complete frame (φ n ( x )) for the subdomain i . Therefore, an eigenmode ϒ p of the whole system has a small energy, say lower than a threshold ε, if and only if: 2) shows that low energy eigenmodes of the full system are directly related to the poor conditioning of the Gram matrix . Indeed, if an orthogonal basis is used for the modal expansion, is well-conditioned (it is in fact diagonal), and no eigenmodes can possibly satisfy Eq. (C.2) : there is therefore no spurious eigenmodes in this case. On the contrary, if an overcomplete frame is used, becomes ill-conditioned, and vectors b lying in singular regions of the spectrum of the quadratic form associated to can exist. Those satisfy the relation of Eq. (C.2) and are then considered as spurious eigenmodes. The threshold was empirically fixed to ε = 10 -4 . However, the procedure showed sensitivity to this parameter: changing the value of ε to 10 -5 or 10 -3 did not affect the modes identified as spurious. This methodology was validated on a number of cases with available reference solutions. It is however not formally proved that it is able to differentiate spurious modes for any given system; if it happens to misidentify those for a particular case, the brute force method should be preferred.

Appendix D. State-space representation for an active flame approximated by a Multi-Pole expansion

For a given active flame, fluctuations of heat release rate Q ( t ) are governed by the FTF of Eq. ( 24) , and the Multi-Pole expansion approximation of the time-delay e -jωτ ( Eq. ( 25) ). Recasting each Pole Base Function into the time-domain leads to the state-space realization of the flame:

d dt ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Z 1 (t) ˙ Z 1 (t) . . . Z M PBF (t) ˙ Z M PBF (t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ X (F TF ) (t) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 -ω 2 01 2 c 1 . . . 0 1 -ω 2 0 M PBF 2 c M PBF ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ A (F TF ) × ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Z 1 (t) ˙ Z 1 (t) . . . Z M PBF (t) ˙ Z M PBF (t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 -1 / u . . . 0 -1 / u ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ B (F TF ) u (re f ) (t) U (F TF ) (t) (D.1)
In Eq. (D.1) , the flame input vector U ( FTF ) ( t ) has a single entry, the reference fluctuating speed u (re f ) (t) = u (x B = βL B , t ) . Note that other formulations including a pressure value as reference are also possible. The state-vector X ( FTF ) ( t ) contains abstract variables that serve as intermediates in the calculation of heat release rate. The state variable ˙ Z k (t) can be interpreted as the proportion of normalized heat release fluctuating at frequencies contained in the band of width 2 c k centered around ω 0 k . This state-space realization is completed by the following output equation:

Q (t) Y (F TF ) (t) = 0 -2 Q a 1 . . . 0 -2 Q a M PBF C (F TF ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ Z 1 (t) ˙ Z 1 (t) . . . Z M PBF (t) ˙ Z M PBF (t) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (D.2)
The flame output vector Y ( FTF ) ( t ) only comprises the heat release rate Q ( t ), which is reconstructed from a linear combination of the individual components ˙ Z k (t) contained in the state-vector.

( 9 )

 9 Both eigenmodes families ( ξ m ) m ≥ 1 and ( ζ k ) k ≥ 1 verify the same rigid-wall (resp. open) boundary conditions on S wi (resp. S ai ). The eigenmodes family ( ξ m ) m ≥ 1 is similar to the orthogonal basis ( ψ n ) n ≥ 1 used in Section 2.1 . Conversely, the eigenmodes family ( ζ k ) k ≥ 1 differs since it verifies open boundary conditions on S ci .

Fig. 3 .

 3 Fig. 3. Pressure mode shape and velocity mode shape of the first mode ( n = 1 ), for N = 4 , 10 , 100 vectors in the OB expansion. Computed solutions (gray dashed lines) are compared to the analytical solutions of Eqs. (19) and (20) (thick dark lines). Closeup views (corresponding to the rectangles) of the pressure and the velocity mode shapes are given on the second row ((d)-(f)) and the fourth row ((j)-(l)) respectively.

Fig. 4 .

 4 Fig. 4. Pressure mode shape (a) and velocity mode shape (b) of the first mode ( n = 1 ) for N = 4 vectors in the frame. Numerical solutions (gray lines with × ) are compared to the analytical solutions of Eqs. (19) and (20) (thick dark lines). The local absolute errors for the mode shapes with N = 4 are also plotted (dashed line), with values indicated on the right axis.

Fig. 5 .

 5 Fig. 5. Comparison between the OB and the FR results in terms of frequency relative error ( E OB/FR f n ), for global modes 1, 9 and 11, in function of the number N of vectors used in the modal expansions. Dark lines are results obtained with rigidwall eigenmodes expansions ( Eq. (15) ), while gray lines are results obtained with frames modal expansions ( Eq. (16) ).

Fig. 6 .

 6 Fig. 6. (a) The unstructured mesh used in the AVSP FE solver. (b) The low-order thermoacoustic network representing this system, which consists of an annular plenum P , 4 burners B i , an annular chamber C , and a set of 8 subdomains sc i containing the cross-section changes. Thick dark lines represent rigid-wall boundary conditions. Gray area ( H i ) are the active flames, and the crosses represent the reference points used in the definition of the flame response. The width of the plenum (resp. chamber) in the radial direction is W P (resp. W C ). All required numerical values for acoustic and flame parameters are indicated in the table.

1 |

 1 left-hand side has been non-dimensionalized by the euclidean norm of the Gram matrix | |, and the euclidean norm of the eigenvector | b | 2 . Eq. (C.

Acknowledgments

The first author would like to thank the French Ministry of Higher Education, Research and Innovation and the Ecole Normale Superieure de Paris-Saclay for funding this work through the CDSN scheme.

Appendix A. State-space realization of an acoustic subdomain

The state-space representation for a subdomain i belonging to an acoustic network is given in this section. The dynamical system of Eq. ( 12) governing the evolution of the acoustic pressure in the

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.combustflame.2019.05. 010 .


Denis Efimov

Andrey Polyakov

Alexander Aleksandrov

Proofs for "Discretization of Homogeneous Systems Using Euler Method with a State-Dependent Step"

Keywords: Homogeneous systems, Discretization, Euler method

This note contains some proofs for the paper "Discretization of Homogeneous Systems Using Euler Method with a State-Dependent Step" of the same authors.

Since V is twice continuously differentiable, then by the Taylor expansion theorem with Lagrange remainder [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] there is θ ∈ (0, 1) such that

V (y i + hf (y i )) = V (y i ) + ∂V (ξ) ∂ξ ξ=yi hf (y i ) + h 2 2 f (y i ) ∂ 2 V (ξ) ∂ξ 2 ξ=yi+θhf (yi) f (y i ),
This work was partially supported by ANR 18 CE40-0008 (Project DIGITSLID), the Government of Russian Federation (Grant 08-08) and Saint Petersburg State University (Project Id 37569826).

Email addresses: denis.efimov@inria.fr (Denis Efimov), andrey.polyakov@inria.fr (Andrey Polyakov), a.u.aleksandrov@spbu.ru (Alexander Aleksandrov).

then

V (x i+1 ) -V (x i ) = x i µ r ∂V (ξ) ∂ξ ξ=yi hf (y i ) + x i µ r h 2 2 f (y i ) ∂ 2 V (ξ) ∂ξ 2 ξ=yi+θhf (yi)
f (y i ).

Note that from (7),

∂V (ξ) ∂ξ ξ=yi hf (y i ) ≤ -ha,
and there is v ∈ (0, +∞) such that sup y∈Sr

sup θ∈(0,1)

f (y) ∂ 2 V (ξ) ∂ξ 2 ξ=y+θf (y) f (y) ≤ v.
Therefore,

V (x i+1 ) -V (x i ) ≤ h x i µ r h 2 v -a
and for all h ∈ (0, h 0 ] with h 0 = min{1, a v } we obtain:

V (x i+1 ) -V (x i ) ≤ - a 2 h x i µ r ≤ -αhV (x i ),
where α = a 2c2 and the property given in (7) was used on the last step. Assume that actually h 0 = min{1, a v , 2 c2 a },

Preprint submitted to Automatica

then V (x i+1 ) ≤ (1 -αh)V (x i ) for all i = 0, 1, . . . and the values of V (x i ) are monotonously decreasing, hence the sequence {x i } ∞ i=0 is converging (the property (b) is substantiated). Global boundedness of {x i } ∞ i=0 (i.e. the property (a)) follows from the properties of the function V for γ = (c -1

Let us evaluate the rate of convergence for different signs of the degree of homogeneity ν of (1) stated in the property (c). Let ν = 0, then trivially t i+1 = t i + h and an exponential rate of convergence is recovered. From (2) we have:

As it has been proven above

for all i = 0, 1, . . . , then for ν < 0 and for any k ≥ 0 an estimate can be derived:

Since |1 -αh| < 1 for the selected h 0 , we get

for all x 0 = 0, which implies a finite-time convergence to the origin of any sequence {x i } ∞ i=0 in (2) but for i → +∞. For ν > 0 we are interesting in the time of convergence from an infinite initial condition to B r (1), then it is necessary to repeat all above arguments in the inverse time, which leads to exactly the same estimate of such a time.

Proof of Proposition 11

Let x i ∈ S r (1) be an arbitrary fixed vector, then the equation (3) can be rewritten as

for any λ > 0 and x ∈ R n , i.e. z is r-homogeneous vector field of degree 0 that is also continuous on S r (1). Denote B(1

z(x i + ∆) -1 then the function hz(x i + •) : B(1) → B( 1) is continuous on the convex compact set B(1). Hence, using the Brouwer fixed-point theorem [START_REF] Leborgne | Calcul différentiel et géométrie[END_REF] we conclude that the last equation has a solution with respect to ∆ ∈ B(1) for any x i ∈ S r (1). The conclusion for any x i ∈ R n follows from Corollary 6 or Proposition 5.

Proof of Theorem 12

Defining x i+1 = Λ r ( x i+1 r )y i+1 with y i+1 ∈ S r (1), we obtain:

with application of the Taylor expansion theorem with Lagrange remainder [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] on the last step. Next, similarly ∂V (ξ) ∂ξ ξ=yi+1 hf (y i+1 ) ≤ -ha from (7), and if the matrix

for any h > 0. If this is not the case, then there is w ∈ (0, +∞) such that sup y∈Sr [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] sup θ∈(0,1)

which results in

and for all h ∈ (0, h 0 ] with h 0 = min{1, a w } we obtain:

where again α = a 2c2 and the property given in (7) was used on the last step. Finally,

for all i = 0, 1, . . . and the values of V (x i ) are monotonously decreasing, hence the sequence {x i } ∞ i=0 is converging (the property (b) is substantiated). Global boundedness of {x i } ∞ i=0 (i.e. the property (a)) follows from the properties of the function V for γ = (c -1 1 c 2 ) 1/µ .

The property (c) can be proven applying the same arguments as in the proof of Theorem 9.