Nathalie Verdière
email: verdiern@univ-lehavre.fr

Valentina Lanza

Rodolphe Charrier

Damienne Provitolo

Edwige Dubos-Paillard

Cyrille Bertelle

Aziz Alaoui

Mathematical modeling of human behaviors during catastrophic events

Keywords: Modeling, catastrophic event, human behavior

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Introduction

Nowadays, management of disasters has become a major issue, due to their huge financial and human costs. In fact, our societies, independently from their development level, are still not sufficiently prepared to a natural or anthropic catastrophe and to possible domino effects. However, there is an increasing trend concerning the number of disasters, ranking from a hundred in 1960 to over 800 in 2000, and the trend is not expected to be inverted in the future, due to the population growth and densification in the risk zones [START_REF] Provitolo | Risques et catastrophes. Observer, spatialiser, comprendre, gérer[END_REF].

A fundamental lever for reducing human vulnerability in the face of such events is definitely the population training, in order to adapt their behav-iors to extreme situations. In fact, during several catastrophic events, controlled and uncontrolled behaviors in either individuals, small groups or crowds have been observed. These reactions depend not only on the event and its temporality (nature, unexpectedness, presence of alert), but also on the population characteristics (density, composition, preparation level) [START_REF] Crocq | La psychologie des catastrophes et les blessés psychiques[END_REF].

In this paper, our aim is to model the collective behaviors that take place in a crowd during a catastrophe, in order to better apprehend and handle the collective reactions. In the literature, several models at different scales have been proposed for modeling crowd dynamics, also in extreme scenarios as in a panic situation [START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF]. At microscopic level we encounter cellular automata or agent based models [START_REF] Pan | A multiagent based framework for the simulation of human and social behaviors during emergency evacuations[END_REF], where each individual of the population is modeled as single entity. Moreover, especially for the study of pedestrian flows, some microscopic models consider the pedestrians as particles subject to a mixture of socio-psychologically and physical forces [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF]. This approach permits to take into account the heterogeneity of the population but this means also high computational requirements and sometimes a difficulty in transferring the microscopic properties at a macroscopic level [START_REF] Rahmandad | Heterogeneity and network structure in the dynamics of diffusion: Comparing agent-based and differential equation models[END_REF]. At macroscopic level, the models of crowd dynamics consist in partial differential equations that describe the evolution in time and space of the density and mean velocity of the crowd flow. In particular, interactions of crowds and structures in panic situations have been considered [START_REF] Venel | A mathematical framework for a crowd motion model[END_REF][START_REF] Goatin | A macroscopic model for pedestrian flows in panic situations[END_REF]. Finally, at mesoscopic level, between the microscopic and the macroscopic ones, we have the models that exploit the approach of the kinetic theory, through the Boltzmann or Vlasov equations, depending on the different range of interactions [START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF].

However, these mathematical models consider only the panic reaction, and do not take into account what is well-known by now in human sciences [START_REF] Dubos-Paillard | Emergent human behaviour during a disaster: thematic versus complex systems approaches[END_REF][START_REF] Provitolo | Structural and dynamical complexities of risk and catastrophe systems: an approach by System Dynamics Modelling[END_REF]: in a catastrophe, the population can exhibit different concurrent reactions, not only the panic one, and each individual does not keep the same behavior during all the event. It is what we propose to do in this paper in exploiting the potential of SIR models [START_REF] James | Mathematical Biology I: An Introduction[END_REF] that are widely used in epidemics. Indeed, in these models, one can decompose the population in several subpopulations categorized in compartments. Furthermore, different types of transition between these compartments can be easily considered. In this paper, we omit the spatial dynamics in order to focus first of all on the different psychological behaviors in a crowd, and the processes of transition from one behavior to an other during a disaster.

The collective behaviors that have been observed in the impact and in the destruction zone [START_REF] Crocq | La psychologie des catastrophes et les blessés psychiques[END_REF] can be classified in two main categories:

• the instinctive behaviors, managed by the reptilian zone of the brain, that handle with the impulsive and urged behaviors [START_REF] Laborit | La légende des comportements[END_REF] • the controlled behaviors, where the pre-frontal cortex adapts in a more reflexive way the reactions to an external perturbation [START_REF] George | Premières impressions. L'essentiel Cerveaux et Psycho: Les émotions au pouvoir Science[END_REF].

In particular, in the first group we have all the behaviors of instinctive escape and fight, the panic, but also the behaviors as a sort of automaton [START_REF] Vermeiren | Psychologie des foules : éléments de compréhension et de prévention en situation de catastrophe collective[END_REF], while in the second one we have all the persons that keep calm and self-control.

In our case, we have subdivided the population in situation of catastrophe in three groups corresponding to the three following collective reactions: reflex behaviors except the panic one, panic and controlled behaviors. Indeed, according to the specific status of the panic, this reaction has been differentiated from the others reflex behaviors. Moreover, according to [START_REF] Hatfield | Emotional Contagion[END_REF], we suppose that the interactions among these classes of population can be realized through imitation and emotional contagion processes. In fact, it is well-known that in a crowd the perception of an emotional state causes in an observer an automatic imitation of this expression.

The paper is structured as follows. In part 2, our choice of the thee groups of compartmental reactions is discussed and from this discussion, the mathematical model is deduced. In part 3, the available data present in the literature are presented and our strategy for calibrating the model is given. In part 4, numerical simulations attest the relevance of our suggested approach.

2 The mathematical model

Choice of three groups of compartmental reactions

In this paper, we consider the human behaviors in the impact zone of a catastrophe, with a fast dynamic and no alert to the population. We suppose that the effect of surprise is total and there are no precursor signs or warnings that allow the population to adopt preventive behaviors. To give an example, there may be an earthquake or a local tsunami. We have distinguished three different types of behaviors in such situation.

The first type consists in the reflex behaviors and concerns the reptilian brain. In our case, it corresponds to the set of instinctive behaviors except panic. This mechanism permits to react quickly, either by running away as fast as possible or by being flabbergasted and being physically unable to move in space. It can take the form of sideration and automate behaviors for example [START_REF] Crocq | La psychologie des catastrophes et les blessés psychiques[END_REF]. In our model, we have decided to globalize all these reflex behaviors, despite their diversity.

The second one corresponds the panic behavior. Panic has a particular status since, even if it is not always adopted (as, for example, during an earthquake in prepared regions as Japan), this behavior is the most feared. Indeed, this mechanism is difficult to stop once started [START_REF] Crocq | Les paniques collectives[END_REF] and can provoke dangerous situations in a crowd, due to trampling and crushing. Moreover, the extinction of collective panic is more linked to internal dynamics than to the remoteness of the danger [START_REF] Crocq | La psychologie des catastrophes et les blessés psychiques[END_REF]. Thus, even if it belongs to reflex behaviors, we consider it apart due to its particular nature. Furthermore, in our model, the collective panic can propagate via imitation and contagion mechanisms [START_REF] Hatfield | Emotional Contagion[END_REF].

Finally, the third type includes all the controlled behaviors. They are governed by the prefrontal cortex, which takes over the reptilian brain. Thus, reflex reactions are substituted by controlled, intelligent and reasoned reactions. They can take different forms in a catastrophe, as, for example, evacuation, leak, containment, sheltering, research for help, pillage, theft... As for the first type, we have decided to globalize all these controlled behaviors, despite their variety.

It is worth noting that the three previous behaviors do not all occur at the same time and respect a certain order. Indeed, the first behavior of an individual in the face of danger is a reflex one followed, in a second step, by controlled or panic behavior [START_REF] George | Premières impressions. L'essentiel Cerveaux et Psycho: Les émotions au pouvoir Science[END_REF].

Formalization of the human behavior

In this paper, we propose a SIR-based mathematical model composed of four classes, one constitutes daily behaviors, and the three others correspond to the three previous behaviors described at Section 2.1. Thus, first of all, we suppose to have a class named Q composed of individuals in a daily behavior and that, during the event, no death nor birth takes place. Hence, globally the population is constant and composed by N individuals. Moreover, during the catastrophe, Q is the sum of two subpopulations:

• Q 1 (t): it designs the number of individuals with routine behaviors. Clearly, just before the catastrophic event occurs, all the population is in this state, therefore

Q 1 (0) = N ,
• Q 2 (t): it designs the number of individuals who come back to normal lifestyle after the outbreak of the disaster. We expect that at the end of the event, all the individuals will be in this state, thus Q 2 (t end ) = N.

According to Section 2.1, the population during the catastrophe is decomposed into 3 subpopulations who are represented by the following variables:

• x(t) = number of persons with reflex behaviors,

• y(t) = number of persons with controlled behaviors,

• z(t) = number of persons with panic behaviors.

Since we suppose to be in presence of a sudden and unpredictable event, all the involved population will have firstly a reflex reaction, corresponding to instinctive comportments. Thus, the routine behaviors, represented here with the variable Q 1 (t), can only be transformed in reflex behaviors, that is in x(t). Hereafter, reflex behaviors can become controlled or panic behaviors. Since Q 2 (t) represents the number of individuals who come back to normal lifestyle, it can be alimented only by the controlled behaviors y(t). In fact, an individual needs to recover self-control in order to regain the everyday routine. Moreover, we suppose that, once they have come back to normality, they maintain their habitual behaviors. Thus, the individuals in Q 2 cannot pass in Q 1 and re-enter in the loop.

Furthermore, according to our psychological and geographical references [START_REF] Dubos-Paillard | Emergent human behaviour during a disaster: thematic versus complex systems approaches[END_REF][START_REF] Hatfield | Emotional Contagion[END_REF][START_REF] Crocq | Les paniques collectives[END_REF][START_REF] Boyd | Réactions psychologiques des victimes de catastrophes[END_REF], during catastrophic events we have interactions and transitions between the different behaviors, as represented in Figure 1. The exterior event, that is the catastro-

Q(t) = daily behaviors Q1(t) Q2(t) y(t) = controlled behaviors x(t) = reflex behaviors z(t) = panic behaviors γ(t) ϕ(t) s1 s2 B1 α B2 δ C1 C2 µ
instinctive behavior, managed by the reptilian zone of the brain controlled behaviors, managed by the pre-frontal cortex Figure 1: Graphic modeling of three types of human behaviors in context of disasters phe onset, is represented by a forcing function γ which can be discrete or continuous, depending on the type of the event under study. For example, an event such as a local tsunami can be modeled by a discrete function, whereas an inundation can be modeled by a continuous function since it can be announced fewer hours before its start. In our case, we suppose to be in the first situation and the brutality and the speed of the catastrophic event is modeled through a logistic function.

For taking into account the possible continuation or repetition of the catastrophe perception stress, the arrows labeled s 1 and s 2 are added, where s 1 and s 2 are supposed to be constant parameters. Once the population is in a reflex behavior, they can evolve in a controlled or panic one according to the parameters B 1 and B 2 , respectively. In the same way, a part of the controlled population can evolve to a panic behavior and reciprocally according to the coefficients C 1 and C 2 , respectively. All the previous transitions are causality links. However, some processes of imitation and contagion exist and are modeled by the arrows labeled α, µ, δ. In the graphic, α transposes the process of imitation between x and y which is realized in both directions. This process is modeled as an epidemiological propagation and has the following form: α•f 1 (x(t))•y(t). This modeling permits to favor the imitation in one direction, in particular from x to y. Indeed, in our numerical tests, we have assumed that there must be at least 55% of reflex behaviors for that controlled individual behaviors imitate reflex ones. In the same manner, the constant δ traduces the imitation processes between x and z and is mod-eled by the function δ • f 2 (x(t)) • z(t). Finally, the constant µ traduces the imitation processes between controlled and panic individuals behavior, knowing that the imitation is essentially in the sens panic towards controlled individuals behavior. It is modeled by the term µ • g(y(t)) • z(t). From the graphical modeling in Figure 1, the mathematical model is deduced:

                                                             dx dt = γ(t)Q 1 (t) 1 - x(t) x m -(B 1 + B 2 )x(t) +αf 1 (x(t))y(t) + δf 2 (x(t))z(t) + s 1 y(t) + s 2 z(t), dy dt = B 1 x(t) -αf 1 (x(t))y(t) + C 1 z(t) -s 1 y(t) -C 2 y(t) -ϕ(t)y(t) 1 - Q 2 (t) Q 2 m + µg(y(t))z, dz dt = B 2 x(t) -s 2 z(t) -δf 2 (x(t))z(t) -C 1 z(t) +C 2 y(t) -µg(y)z, dQ 1 dt = -γ(t)Q 1 (t) 1 - x(t) x m , dQ 2 dt = ϕ(t)y(t) 1 - Q 2 (t) Q 2m (1)
Since the concerned population is supposed to be constant, that is the equality

Q 1 (t) + Q 2 (t) + x(t) + y(t) + z(t) = N for all t ∈ [0, T ] is verified, system (
1) can be reduced to four equations and rewritten as:

                                                         dx dt = γ(t)Q 1 (t) 1 - x(t) x m -(B 1 + B 2 )x(t) +αf 1 (x(t))y(t) + δf 2 (x(t))z(t) + s 1 y(t) + s 2 z(t), dy dt = B 1 x(t) -αf 1 (x(t))y(t) + C 1 z(t) -s 1 y(t) -C 2 y(t) -ϕ(t)y(t) 1 - N -Q 1 (t) -x(t) Q 2 m -y(t) -z(t) + µg(y(t))z(t), dz dt = B 2 x(t) -s 2 z(t) -δf 2 (x(t))z(t) -C 1 z(t) +C 2 y(t) -µg(y)z(t), dQ 1 dt = -γ(t)Q 1 (t) 1 - x(t) x m .
(

) 2
3 Calibration of the model

Unfortunately, in the literature, the available data to calibrate the model are scarce. However, one can distinguish two groups of quantitative data. The first one concerns the percentages of the population adopting a certain type of behavior and the second one relates on the duration of such behaviors.

The percentages of population adopting a certain type of behavior

The different types of human behaviors described previously can manifest in variable proportions, in function of the considered catastrophe, the suddenness of the threat, the composition of the group, the individual aptitudes for understanding the danger and the knowledge of the environment. Moreover, [START_REF] Boyd | Réactions psychologiques des victimes de catastrophes[END_REF] considers that in most of the catastrophes, "15% of individuals manifest obvious pathological reactions, 15% keep their cool and 70% manifest an apparently calm behavior but answer in fact to a certain degree of emotional sideration and lost of initiative which reports to a pathological register". These percentage have to be modulated according to the different parameters of our model, which leads us to consider:

• x(t) = 50 to 75% of the population

• y(t) = 12 to 25% of the population

• z(t) = 12 to 25% of the population At our knowledge, no data are available for quantifying transition mechanisms from one state to an other.

The duration of the behavior

The three different reactions have different duration [START_REF] Vermeiren | Psychologie des foules : éléments de compréhension et de prévention en situation de catastrophe collective[END_REF]. The duration of the reflex and panic behaviors varies from few minutes to one hour. Most of the time, these two types of behavior do not exceed 15 minutes. However, for the first one, it may take longer especially if it corresponds to a delay of evacuation in a disaster area. In this case, support and research behaviors for relatives and victims gradually appears [START_REF] Crocq | La psychologie des catastrophes et les blessés psychiques[END_REF]. For the second one, the collective panic behaviors resolves generally spontaneously. However, sometimes, an external intervention permits to the panic population z(t) to come back to an automate behavior x(t), before adopting a controlled behavior y(t).

In general, the duration of the uncontrolled behavior x(t) + z(t) does not last more than 1h30. In this model, we suppose that an individual cannot stay 1 hour in a reflex behavior and another hour in a panic state. The duration of the controlled behavior y(t) varies from few minutes to fewer hours, according to the intervention of the emergency response.

The choice of the parameters will be done in order to find these data.

Numerical examples

For the numerical simulations, we have transformed the model in a dimensionless form, that is, population numbers correspond to fractions of the total population. In the following subsection, the functions intervening in the dimensionless model are specified.

4.1 The functions γ, ϕ, f 1 , f 2 , f 3

In the case of a sudden catastrophe, as a local tsunami, modeled by the function γ, we suppose that the population begins to be rapidly informed that is to say after 1 minute, and that all the concerned population is informed in the 3 following minutes, hence, the shape of the function γ in Figure 2. Clearly, the return to the normality, corresponding to the function ϕ, can not be immediate. We have supposed that it is done after 5 minutes from the outbreak and this return is done very slowly, which leads us to consider the function ϕ as in Figure 2. As we have said before, the form of the curves has to be modulated according to the type of catastrophe event (depending if it is announced or not).

Figure 2: Functions ϕ and γ

The terms αf 1 (x(t))y(t) and δf 2 (x(t))z(t) are the terms of imitation between x(t) and y(t), and x(t) and z(t), respectively. We promote the imitation from x(t) to y(t) in assuming that there must be at least 55% of reflex behaviors for that controlled individuals imitate reflex behavior (see Figure 3). For the imitation term µg(y(t))z, we suppose that the imitation is essentially in the sense from z(t) to y(t) (see Figure 4).

h(s) =              h min for s < s min h max for s > s max - h max -h min 2 cos s -s min s max -s min π + h min + h max 2 else
(3) where [s min , s max ] is the support of the function h and h min (resp. h max ) is its minimum value (resp. maximum value).

Numerical simulations

Different scenarios were made and correspond to different values of parameters.

Simulation 1

The first one corresponds to Figure 5. The chosen values of the parameters permit to find the calibration data presented at Section 3. The areas between each curve and the horizontal axis gives the global percentage of the corresponding population. The global percentage of reflex behaviors (x(t)) equal to 61,41% is included between 50 and 75% otherwise the global percentages of panics and controlled behaviors respectively equal to 18,42% and 20,17% are included between 12% and 25%. Furthermore, the model gives the evolution of these global behaviors distributions.

Simulation 2

In this section, we are interested in the evolution of the model when the parameter B 2 varies, in particular, the possibility to evolve from reflex to panic behaviors. The numerical simulations at Figure 6 shows that for a low value of B 2 , the density of population having a reflex behavior remains important during all the simulation. However, for a high value of this parameter (Figure 7), the density of this population decreases to extinguish rapidly. Furthermore, the densities of panic and controlled populations grow significantly between Figure 6 and7.

Simulation 3

In the following simulations, we force the emergence of panic and controlled behaviors in acting on the parameters B 1 , B 2 , C 1 and C 2 . Figure 8 shows that the return from controlled to daily behavior can be furthered, while Figure 9 induces a high proportion of panic behaviors and a return more difficult to normality as remarked by [START_REF] Crocq | Les paniques collectives[END_REF].

Conclusion

This paper introduces a new step in the modeling of the crowd dynamics in catastrophic events. Indeed, it considers three concurrent behaviors and includes the processes of transition from one behavior to the other. Up to now, the main models consist in modeling the panic which is a fear behavior but it is not always adopted. Furthermore, panic does not necessarily lasts during all the event and, on the contrary, the global behavior of the crowd can change. In this work, two other behaviors have been integrated in the modeling: the reflex one and the controlled one. As seen in human sciences, our simulations show that they can influence the crowd behavior and a return to normality. The next step of this work will consist in doing a mathematical study of this model and integrating it in a diffusion process.

Figure 3 :

 3 Figure 3: Function f 1 and f 2

Figure 4 :

 4 Figure 4: Function g

Figure 5 :

 5 Figure 5: Simulation of the model with the parameters values: x m = 0.75, Q 2m = 1, B 1 = 0.04, B 2 = 0.02, s 1 = 0.01, s 2 = 0.01, C 1 = 0.5, C 2 = 0.5, α = δ = µ = 0.01.

Figure 6 :

 6 Figure 6: Simulation of the model with the parameters values: x m = 0.75, Q 2m = 1, B 1 = 0.04, B 2 = 0.002, s 1 = 0.01, s 2 = 0.01, C 1 = 0.5, C 2 = 0.5, α = δ = µ = 0.01.

Figure 7 :

 7 Figure 7: Simulation of the model with the parameters values: x m = 0.75, Q 2m = 1, B 1 = 0.04, B 2 = 0.2, s 1 = 0.01, s 2 = 0.01, C 1 = 0.5, C 2 = 0.5, α = δ = µ = 0.01.

Figure 8 : 9 :

 89 Figure 8: Simulation of the model with the parameters values:x m = 0.75, Q 2m = 1, B 1 = 0.2, B 2 = 0.04, s 1 = 0.01, s 2 = 0.01, C 1 = 0.5, C 2 = 0.1, α = δ = µ = 0.01.


Hélène Barucq -Rabia

Djellouli -Anne-Gaëlle Saint-Guirons

Hélène Barucq

Rabia Djellouli

Anne-Gaëlle Saint-Guirons

Construction and performance analysis of local DtN absorbing boundary conditions for exterior Helmholtz problems. Part I : Elliptical-shaped boundaries

Keywords: Absorbing boundary conditions, elliptical coordinates, Dirichlet-to-Neumann operator, scattering problems Conditions aux limites absorbantes, coordonées elliptiques, opérateur Dirichletto-Neumann, problèmes de scattering

We propose a new class of approximate local DtN boundary conditions to be applied on prolate spheroid-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are : (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior elliptical-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. Moreover, these conditions coincide with the classical local DtN condition designed for spherical-shaped boundaries. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions when applied for solving radiators and scattering problems. We also compare their performance to the second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroidal coordinates. The analysis reveals that, in the low frequency regime, the new second order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values.

Introduction

Exterior Helmholtz problems are classical mathematical models for studying scattering problems arising in many applications such as sonar, radar, geophysical exploration, nondestructive testing, etc... Despite their simplicity, this class of problems is not completely solved particularly from a numerical point of view. For example, the computation of the solutions of these problems requires first to limit it to a finite domain. This is often achieved by surrounding the given scatterer(s) (or radiator) by an artificial boundary that is located at some distance (measured in multiples of wavelength of interest) from its surface. A so-called "nonreflecting" boundary condition is then prescribed on the artificial boundary to represent the "far-field" behavior of the scattered field. The challenge here is the development of a simple but reliable as well as cost-effective computational procedure for representing the far-field behavior of the scattered. The quest for such conditions is ongoing (see, e.g., the recent review by Turkel in the book [START_REF] Turkel | Iterative methods for the exterior Helmholtz equation including absorbing boundary, conditions[END_REF]).

We propose in this work new two-dimensional approximate local DtN boundary conditions to be employed on elliptical-shaped boundaries that are primary candidates for surrounding elongated scatterers. The idea for constructing such conditions is driven by several considerations chief among them the following two reasons. First, the widely-used second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel for circular-shaped boundaries [START_REF] Bayliss | Boundary conditions for the numerical solution of elliptic equations in exterior regions[END_REF] performs poorly when it is expressed in elliptical coordinates and applied to elliptical-shaped boundaries in the low frequency regime [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]. The accuracy deteriorates significantly for large eccentricity values of the boundaries as observed in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]. The damping effect introduced to this condition [START_REF] Reiner | Improvement of the performance of the BGT2 condition for low frequency acoustic scattering problems[END_REF] improves the performance for small eccentricity values. However, the modified BGT2 still performs poorly for eccentricity values larger than 0.6 in the (relatively) low frequency regime (see Figure 6 in [START_REF] Reiner | Improvement of the performance of the BGT2 condition for low frequency acoustic scattering problems[END_REF]). Hence, there is a need for constructing local absorbing boundary conditions (ABC) that extend the range of satisfactory performance. Second, the two-dimensional approximate local DtN conditions designed for circular-shaped boundaries [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF] outperforms significantly BGT conditions, particularly for low wavenumber values as reported in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]. However, using these conditions on circular-shaped exterior boundaries when solving scattering problems by elongated scatterer often leads to larger than needed computational domains, which hampers computational efficiency. This suggests that approximate local DtN boundary conditions designed for elliptical-shaped boundaries is an attractive alternative for improving the computational performance.

Given that, this work is devoted to the construction of theses conditions and to the assessment of their performance when employed for solving two-dimensional acoustic scattering problems by elliptical-shaped obstacles. The idea of constructing two-dimensional approximate local DtN boundary conditions is not new. Indeed, as stated earlier, such conditions have been already derived for circular-shaped boundaries [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF]. The construction procedure adopted in [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF] is based on the localization of the truncated global DtN boundary condition [START_REF] Keller | Exact nonreflecting boundary conditions[END_REF]. The key ingredient of this procedure is the trigonometric identities that express high order derivatives of sine and cosine functions (see for example Eq. (A4) p. 276 in [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF]). However, this property is not satisfied by the periodic Mathieu functions (see p. 376 in [START_REF] Stratton | Electromagnetic theory[END_REF]).

Consequently, the procedure used in [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF] is no longer applicable to the truncated global DtN boundary operator when expressed in elliptical coordinates [START_REF] Grote | On nonreflecting boundary conditions[END_REF][START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF]. Hence, the construction methodology we propose for deriving the class of approximate local DtN boundary conditions in elliptical coordinates can be viewed as an inverse-type approach. More specifically, we start from a Robin-type boundary condition with unknown coefficients. Unlike the case of polar coordinates, these coefficients depend on the angle θ of the elliptical coordinates. Such dependence is necessary to preserve the symmetry and local nature of the resulting boundary conditions. Then, we require that the considered condition to be an exact representation of the first modes. Consequently, the coefficients are the unique solution of a linear algebraic system.

We assess mathematically and numerically the performance of the constructed approximate local DtN boundary conditions. More specifically, we analyze the effect of low wavenumber and the eccentricity on the performance of these conditions in the case of twodimensional scattering problems. We adopt the on-surface radiation condition formulation (OSRC) [START_REF] Kriegsmann | A new formulation of electromagnetic wave scattering using an on-surface radiation boundarycondition approach[END_REF] in order to perform analytically this investigation. We note that such formulation is not appropriate for high frequency regime as observed previously in [START_REF] Antoine | Fast approximate computation of a time-harmonic scattered field using the On-Surface Radiation method[END_REF]. The main interest in the following analyses is to evaluate the performance of the proposed approximate local DtN conditions at low wavenumber to see if relatively small computational domains can be employed in order to avoid excessive computational cost. The OSRC formulation must be viewed as an extreme case while an exterior elliptical-shaped boundary surrounding an elongated scatterer would be less "demanding" on the boundary condition. The analysis herein shows that the constructed second-order local DtN condition retains a good level of accuracy in the low frequency regime for all eccentricity values of the elliptical-shaped boundaries.

Preliminaries

Throughout this paper, we use elliptical coordinates (ξ, θ) which are related to the rectangular cartesian ones (x, y) by:

x = a cos θ , y = b sin θ (1)
where θ ∈ [0, 2π). The parameters a and b respectively represent the semi-major and semiminor axes of the ellipse, and are given by:

a = f cosh ξ , b = f sinh ξ (2)
where ξ is a strictly positive real number and f is the interfocal distance given by:

f = a 2 -b 2 (3)

INRIA

We also define the eccentricity e on the ellipse ξ = ξ 0 by:

e = 1 cosh ξ 0 = 1 - b 2 a 2 (4)
The eccentricity e characterizes the slenderness of the boundary. It satisfies 0 < e < 1.

Note that when e → 0, the ellipse degenerates into a circle and when e → 1, the ellipse degenerates into a line with length 2f . We recall that the n-th elliptic cylindrical mode mode u n is given by [START_REF] Senior | Scalar diffraction by a prolate spheroid at low frequencies[END_REF] : 3) n (kf, cosh ξ)Se n (kf, cos θ), n ≥ 0 (for even mode)

u n =    Re (
Ro (3) n (kf, cosh ξ)So n (kf, cos θ), n ≥ 1 (for odd mode) [START_REF] Bowman | Electromagnetic and acoustic scattering by simple shapes[END_REF] where Se n (resp. So n ) are the even (resp. odd) periodic Mathieu functions, and Re (3) n (resp. Ro (3) n ) are the even (resp. odd) radial Mathieu functions of the third kind (see p. 376 in [START_REF] Stratton | Electromagnetic theory[END_REF]). In addition, the periodic functions Se n and So n satisfy the differential equation (see Eq. ( 15)-( 16), p. 377 in Ref. [START_REF] Flammer | Spheroidal Functions[END_REF]):

∂ 2 v n ∂θ 2 + c n - (kf ) 2 2 cos 2 θ v n = 0 (6)
where c n is called the characteristic value (see p. 376 in Ref. [START_REF] Flammer | Spheroidal Functions[END_REF] or p. 721 in Ref. [START_REF] Abramovitz | Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables[END_REF]). For even modes v n = Se n , we have c n = a n := a n (kf ), and for odd modes v n = So n , we have

c n = b n := b n (kf ).
3 The new approximate local boundary conditions and their derivation

Next, we construct new local DtN absorbing boundary conditions in elliptical coordinates, and describe their properties. Note that, because of the nature of the modes (even and odd), it is possible to construct conditions that are exact representation of the first two even modes or the first two odd modes [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF]. We can also construct conditions that are exact, in the least-squares sense, for the first three modes [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF]. In the following, we derive and analyze the conditions that are exact representation of the even modes. The construction of the other conditions can be found in [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF]. We must point out that the performance of all these conditions is comparable as demonstrated in [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF].

The approximate local DtN boundary conditions in elliptical coordinates

The two-dimensional first-(DtN1) and second-order (DtN2) local Dirichlet-to-Neumann boundary conditions, defined on the elliptical-shaped surface ξ = ξ 0 , are given by:

DtN1 : ∂ u ∂ξ = √ 1 -e 2 e r 0 u (7)
DtN2 :

∂u ∂ξ = √ 1 -e 2 (a 0 -a 1 )e a 0 r 1 -a 1 r 0 -(r 0 -r 1 ) (eka) 2 2 cos 2θ u + (r 1 -r 0 ) ∂ 2 u ∂θ 2 (8)
where the coefficients r n depend on the even radial Mathieu functions of the third kind Re (3) n , the wavenumber ka, and the eccentricity e as follows:

r n = ∂Re (3) n ∂ξ (eka, e -1 )
Re (3) n (eka, e -1 ) ; n ≥ 0

The following four remarks are noteworthy:

First, the Robin-type boundary conditions ( 7)-( 8) are, by construction, an exact representation of the first elliptical mode u 0 given by Eq. ( 5). The boundary condition ( 8) is an exact of the second even mode u 1 (see Eq. ( 5)). On can find in Ref. [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF] the construction of a second-order DtN boundary condition that is exact for the first mode u 0 and the second odd mode u 1 , as well as a second-order DtN boundary conditions that is exact, in the least-squares sense, for the first mode u 0 , the second even mode u 1 , and the second odd mode u 1 . The performance of all these second-order DtN conditions are comparable if not identical as demonstrated in [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF].

Second, the boundary conditions ( 7), ( 8) are called local DtN conditions because they result from a localization process of the truncated global DtN boundary operator defined in [START_REF] Grote | On nonreflecting boundary conditions[END_REF][START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF]. The local feature of these conditions is of a great interest from a numerical view point. Indeed, the incorporation of these conditions in any finite element code introduces only mass-and stiffness-type matrices defined on the exterior boundary. The coefficients a n and r n can be computed once for all at the preprocessing level.

Third, it must be point out that when e = 0 (the ellipse becomes a circle), conditions [START_REF] Geers | Doubly asymptotic approximations for transient motions of submerged structures[END_REF] and ( 8) are identical to the two-dimensional DtN conditions designed for circular shaped boundaries [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF][START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]. This property can be easily established using the asymptotic behavior of the even radial Mathieu functions of the third kind Re (3) n and the even characteristic values a n .

Last, approximate local DtN boundary conditions of order higher than two are inappropriate for conventional finite element implementations since they require regularity higher than C 0 . Consequently, we consider only the first (DtN1) and second order (DtN2) conditions in the following.

INRIA

The procedure for constructing the approximate local DtN conditions

As stated earlier in the introduction, the idea of constructing approximate local DtN boundary conditions is not new. Indeed, such conditions have been already derived for sphericalshaped boundaries in [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF]. The construction procedure adopted in [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] is based on the localization of the truncated global DtN boundary condition [START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF]. The key ingredient of this procedure is the trigonometric identities that express high order derivatives of sine and cosine functions (see for example Eq. (A4) p. 276 in [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF]). However, this property is not satisfied by the angular Mathieu functions (see Eq. ( 6)), and therefore, the procedure used in [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] cannot be applied to the truncated global DtN boundary operator when expressed in elliptical coordinates [START_REF] Grote | On nonreflecting boundary conditions[END_REF][START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF]. The approach we propose for constructing these approximate local DtN boundary condition in the case of elliptical-shaped boundaries can be viewed as an inverse-type methodology. More specifically, we start from a Robin-type boundary condition with unknown coefficients. These coefficients depend on the elliptical characteristic values. In addition, we require that the considered condition is an exact representation of the first modes. Consequently, the coefficients are the unique solution of a linear algebraic system. Next, we describe this procedure for construction the approximate local DtN1 and DtN2 conditions given by Eqs. ( 7)-( 8).

i. Construction of the first order DtN1 boundary condition

We consider a Robin-type boundary condition that is an exact representation of the first elliptical mode u 0 given by Eq. ( 5). Hence, we set:

∂u ∂ξ = Au (10)
where the constant A satisfies sinh ξ ∂Re

(3) 0 ∂ξ (kf, cosh ξ) Se 0 (kf, cos θ) = A Re

(3) 0 (kf, cosh ξ) Se 0 (kf, cos θ) (11) Then, we substitute, at ξ = ξ 0 , Eqs. ( 2), (3), (4) into Eq. ( 11), and deduce that:

A = √ 1 -e 2 e r 0 (12)
where r 0 is given by Eq. ( 9) for n = 0.

ii. Construction of the even second order DtN2 boundary condition

The goal here is to construct a Robin-type boundary condition that is an exact representation of the first two modes u 0 and u 1,even given by Eq. ( 5). Hence, we set:

∂u ∂ξ = C u + D ∂ 2 ∂θ 2 - (eka) 2 2 cos 2θ u (13)
where C and D are constant (independent of θ) to be determined. Note that, unlike DtN2 boundary condition for the circular-shaped boundaries, the coefficients of this condition depend on the angular variable θ. Such dependence is necessary for constructing a symmetric boundary condition since the angular Mathieu functions satisfied differential equation given by Eq. ( 6). Indeed, it is easy to verify that, for all even modes u n , we have:

∂ 2 u n ∂θ 2 = -a n + (eka) 2 2 cos 2θ u n (14)
In order to determine the constants C and D, we assume that, at ξ = ξ 0 , we have:

∂u n ∂ξ = C u n + D ∂ 2 ∂θ 2 -( (eka) 2
2 cos 2θ u n ; n = 0 and n = 1(even mode) (15) Then, using Eq. ( 14), it follows that (C, D) is the unique solution of the folllowing 2 × 2 linear system:

         C -D a 0 = √ 1 -e 2 e r 0 C -D a 1 = √ 1 -e 2 e r 1 (16)
where the coefficients r n are given by Eq. ( 9). The DtN2 even boundary condition given by Eq. ( 8) is a direct consequence of solving the system [START_REF] Kriegsmann | A new formulation of electromagnetic wave scattering using an on-surface radiation boundarycondition approach[END_REF] and then substituting the expressions of (C, D) into Eq. ( 13).

The performance of the new approximate local DtN conditions

In the following, we assess analytically and numerically the performance of the approximate local DtN1 and DtN2e boundary conditions given by ( 7) and [START_REF] Geers | Third-order doubly asymptotic approximations for computational acoustics[END_REF]. More specifically, we analyze the effect of low wavenumber ka and the eccentricity e on the performance of DtN1 and DtN2e in the case of both radiator and scattering problems. We adopt the onsurface radiation condition formulation (OSRC) [START_REF] Kriegsmann | A new formulation of electromagnetic wave scattering using an on-surface radiation boundarycondition approach[END_REF] in order to perform this investigation analytically. We note that such formulation is not appropriate for high frequency regime as observed previously in [START_REF] Antoine | Fast approximate computation of a time-harmonic scattered field using the On-Surface Radiation method[END_REF]. The main interest in the following analyses is to evaluate the performance of the proposed approximate local DtN conditions at low ka to see if relatively small computational domains can be employed in order to avoid excessive computational cost. The OSRC formulation is an extreme case while an exterior elliptical-shaped artificial boundary surrounding an elongated scatterer would be less "demanding" on the boundary

INRIA condition.
As in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF][START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF][START_REF] Reiner | Improvement of the performance of the BGT2 condition for low frequency acoustic scattering problems[END_REF], we assess the performance of the ABCs DtN1 and DtN2 using the specific impedance introduced in [START_REF] Geers | Doubly asymptotic approximations for transient motions of submerged structures[END_REF][START_REF] Geers | Third-order doubly asymptotic approximations for computational acoustics[END_REF] as a practical tool for measuring the efficiency of ABCs in the context of the OSRC formulation. This non-dimensional quantity measures the effect of the truncated medium in physical terms. It provides a convenient indicator of the performance of a given approximate representation. In the elliptical coordinates system, the specific impedance can be expressed as follows:

Z = i √ 1 -e 2 ka u ∂u ∂ξ | ξ=ξ0 (17)

The performance of the local DtN conditions for single elliptical mode

Next, we assess analytically and numerically the performance of the approximate local DtN1 and DtN2 boundary conditions given by ( 7) and ( 8) when applied to two-dimensional radiator problems. Recall that the exact specific impedance Z ex2 n , for the nth even mode, on the boundary of an ellipse at ξ = ξ 0 is given by (see Eq. ( 13), p. 3626 in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]):

Z ex2 n = i √ 1 -e 2 ka Re (3) n (eka, e -1 ) ∂Re (3) n ∂ξ (eka, e -1 ) = i eka r n ( 18
)
where the coefficient r n is given by Eq. ( 9). In addition, its asymptotic behavior as ka → 0 (see Eq. ( 17), p. 3627 in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]) is given by:

Z ex2 n ∼            π ka 2 + i ka ln (ka) if n = 0 4π (n!) 2 ka 2 (2n+1) -i ka n if n ≥ 1 (19)

Analytical study

The following lemma states the expression of the approximate specific impedances for the n-th even elliptical mode, on the surface of an ellipse at ξ = ξ 0 . This lemma results from substituting u = u n in Eq. ( 17), and using the boundary conditions ( 7) and ( 8) to evaluate ∂u n ∂ξ (see [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF] for the detailed proof). ) corresponding to the first-order Dirichlet-to-Neumann boundary condition (DtN1) is given by:

Z DtN1 n = Z ex2 0 ( 20
)

RR n 6394

The approximate specific impedance (Z DtN2e n ) corresponding to the second-order Dirichletto-Neumann boundary condition (DtN2e) is given by:

Z DtN2 n = 1 1 Z ex2 0 + a 0 -a n a 0 -a 1 1 Z ex2 1 - 1 Z ex2 0 ( 21
)
where a n := a n (kf ) is the characteristic value corresponding to the nth even periodic Mathieu function Se n (see p. 376 in Ref. [START_REF] Flammer | Spheroidal Functions[END_REF] or p. 721 in Ref. [START_REF] Abramovitz | Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables[END_REF]).

Remark 4.2 Note that when e → 0, that is the ellipse degenerates to a circle, the approximate DtN specific impedances given by Eqs. ( 20) and ( 21) are identical to the ones obtained in the case of circular-shaped radiators [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]. In addition, when n = 0 (resp. n = 1 for even mode), Z DtN2 n coincides with Z ex2 0 (resp. with Z ex2 1 ), as expected.

The next proposition states the asymptotic behavior of the approximate specific impedance Z DtN2 n as ka → 0.

Proposition 4.3

The asymptotic behavior of the DtN2e specific impedance of the nth elliptical even mode as ka → 0 is given by:

Z DtN2 n ∼ π 2n 2 (ka) 3 -i ka n 2 ; n ≥ 2 (22)
Proof of Proposition 4.3. First, observe that (see Eq. (36) p. 120 in Ref. [START_REF] Meixner | Matieusche funktionen and spharoidfunktionen[END_REF]):

a 0 ∼ - (eka) 4 32 when ka → 0 (23)
a n (0) = n 2 for n ≥ 1 (24)

Therefore, when ka → 0, it follows from that the approximate DtN2 specific impedance given by Eq. ( 21) is equivalent to the following function:

φ (ka) = 1 1 π ka 2 + ika ln (ka) + - (eka) 4 32 -n 2 - (eka) 4 32 -1    1 π 2 (ka) 3 -ika - 1 π ka 2 + ika ln (ka)   
; n ≥ 2 (25) and the result of Proposition 4.3 follows from a suitable truncation of a Taylor expansion when ka tends to 0. ; n ≤ 1), which follows from the design of the DtN boundary conditions. Second, for n ≥ 2, the asymptotic behavior for DtN2 is independent of the eccentricity e, and is identical to the case of a circle INRIA (see Eq.(62), p. 27 in [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF]) as well as to the approximate specific impedance corresponding to the second-order BGT condition when applied on elliptical radiator (see Eq. ( 19), p. 3627 in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]). Last, the asymptotic behavior given by Eq. ( 19) and Eq. ( 22) indicates that for higher modes (n ≥ 2), the real part of the exact impedance tends to zero, as ka → 0, faster than the DtN2 specific impedance. This result suggests that DtN2 boundary condition may not be appropriate for single higher modes.

Numerical investigation

We have performed several experiments to investigate numerically the effect of the wavenumber and the slenderness of the boundary on the performance of the second-order DtN boundary condition DtN2 given by Eq. ( 8) when computing single radiating modes. We have compared the results to the ones obtained with BGT2 conditions when applied on ellipticalshaped boundaries (see Figs. [START_REF] Antoine | Fast approximate computation of a time-harmonic scattered field using the onsurface radiation condition method[END_REF] to [START_REF] Geers | Third-order doubly asymptotic approximations for computational acoustics[END_REF] in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]). We report in here the results obtained for three different even modes n = 1, n = 2, n = 3 for illustration (see Figs. ( 1)-( 5)). More numerical results can be found in [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF]. These results have been obtained for three eccentricity values e = 0.1 corresponding to an elliptical-shaped boundary very close to a circle, e = 0.6 corresponding to a "regular" elliptical boundary, and e = 0.9 corresponding to a very elongated elliptical boundary. Because of the dependence of the BGT2 specific impedance on θ ∈ [0, 2π) (see Eq. ( 15) in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]), we have reported the obtained results on the magnitude of the specific impedance in polar coordinates, and the relative errors as a function of θ (see Figs.

(2)-( 5)).

Fig. 1 illustrates the performance of the first-order condition DtN1 versus the BGT1 condition when computing the second even elliptical mode u 1 . One can observe that both conditions perform overall very poorly. When ka = 1 as indicated by the magnitude of the relative error in the right of Fig. 1. Figs. ( 2)-( 5) compare the performance of the approximate second-order DtN2 and BGT2 boundary conditions when computing higher order even modes (u 2 , u 3 ) for two low wavenumber values ka = 0.1 and ka = 1. The results indicate that both conditions perform poorly and therefore are not appropriate for computing higher modes. Note that the second-order DtN2 boundary outperform the BGT2 boundary condition when computing the first two modes u 0 and u 1 (even) since the DtN2 condition is an exact representation of these two modes.

The performance of the local DtN conditions for ellipticalshaped scatterer

Next, we assess analytically and numerically the performance of the approximate local DtN1 and DtN2 boundary conditions given by ( 7) and ( 8) when applied to two-dimensional scattering problems by sound-soft elliptical-shaped obstacle. Recall (see Eq. (33), p. 3635 in Ref. [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]), that the exact specific impedance Z ex2 on the surface of an ellipse at ξ = ξ 0 is given by:

Z ex2 = - i √ 1 -e 2 ka u inc ∂ ∂ξ (u scat )| ξ=ξ0 (26)
where u inc is the incident plane wave with incident angle θ 0 defined by:

u inc = e ikf cosh ξ(cos θ cos θ0+tanh ξ sin θ sin θ0) (27)
and u scat is the acoustic scattered field by a sound-soft elliptical-shaped obstacle. u scat can be represented by the following series [START_REF] Bowman | Electromagnetic and acoustic scattering by simple shapes[END_REF]:

u scat = - √ 8π ∞ m=0 i m Ke m Re (3) m (kf, cosh ξ)Se m (kf, cos θ) + ∞ m=1 i m Ko m Ro (3) m (kf, cosh ξ)So m (kf, cos θ) (28) where Ke m = 1 N (e) m
Re (1) m (eka, e -1 )Se m (eka, cos θ 0 ) Re

Ko m = 1 N (o) m (3) m (eka, e -1 )
Ro (1) m (eka, e -1 )Se m (eka, cos θ 0 ) Ro (3) m (eka, e -

m ) represents the normalization factor of the even (resp. odd) angular Mathieu functions (see Eq. ( 18), p. 377 in Ref. [START_REF] Stratton | Electromagnetic theory[END_REF]). Moreover, we also recall (see Eq.(38), p. 3637 in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]), that the asymptotic behavior of the exact specific impedance Z ex2 of the scattered field on the surface of an ellipse as ka approaches 0 is given by:

Z ex2 ∼ Z ex2 0 ∼ π ka 2 + ika ln(ka) ; ka → 0 (30)
where Z ex2 0 is given by Eq. ( 19).

Analytical study

The next lemma states the expression of the approximate specific impedances on the boundary ξ = ξ 0 of an elliptical-shaped sound-soft scatterer. This lemma results from substituting u = -u inc in Eq. ( 17), and using the boundary conditions ( 7) and ( 8), and the expression of the incident plane wave given by Eq. ( 27) to evaluate ∂u inc ∂ξ at ξ = ξ 0 (see [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF] for the detailed proof).

Lemma 4.5 The approximate specific impedance (Z DtN1,2D ) corresponding to the first-order Dirichlet-to-Neumann boundary condition (DtN1) is given by:

Z DtN1 = Z ex2 0 (31)
The approximate specific impedance (Z DtN2 ) corresponding to the even second-order Dirichletto-Neumann boundary condition (DtN2) is given by:

Z DtN2 = 1 1 Z ex2 0 + 1 a 0 -a 1 (δ + a 0 ) 1 Z ex2 1 - 1 Z ex2 0 (32) where λ = cos θ cos θ 0 + √ 1 -e 2 sin θ sin θ 0 δ = -ikaλ -(ka) 2 ∂λ ∂θ 2 - (eka) 2 2 cos 2θ (33)
Remark 4.6 Note that when e →= 0, that is the ellipse degenerates to a circle, the approximate DtN specific impedances given by Eqs. ( 31) and (32) are identical to the ones obtained in the case of circular-shaped scatterers [START_REF] Harari | Analytical study of the effect of wave number on the performance of local absorbing boundary conditions for acoustic scattering[END_REF].

The next proposition states the asymptotic behavior of the second-order DtN approximate specific impedance Z DtN2 as ka → 0.

Proposition 4.7 The asymptotic behavior of the approximate specific impedance Z DtN2e,2D as ka → 0 is given by: Z DtN2e,2D ∼ Z ex2 0 (34)

Proof of Proposition 4.7. According to (34), we have:

Z DtN2e,2D = Z ex2 0 1 1 + a 0 + δ a 0 -a 1 Z ex2 0 Z ex2 1,even -1 (35)
Next, from the definition of δ, we can write that:

δ = -ikaΨ 1 (ka) with Ψ 1 (ka) → λ when ka → 0 (36)
which implies, using [START_REF] Turkel | Iterative methods for the exterior Helmholtz equation including absorbing boundary, conditions[END_REF], that: 24) and ( 23) imply that:

a 0 + δ = -ikaΨ 2 (ka) with Ψ 2 (ka) → λ when ka → 0 (37) INRIA Next, (
a 1 -a 0 → 1 when ka → 0 (38)
We then use (32) to obtain that when ka → 0,

Z ex2 0 Z ex2 1,even
-1 behaves like Ψ 3 (ka) with:

Ψ 3 (ka) = ln (ka) + i π 2 (39)
Hence, from (37), ( 38) and (39) we deduce that:

a 0 + δ a 0 -a 1 Z ex2 0 Z ex2 1,even -1 → 0 when ka → 0 ( 40
)
and we conclude that:

Z DtN2e,2D ∼ Z ex2 0 ( 41
)
Remark 4.8 First, we note that, unlike the approximate specific impedance coresponding to the secon-order BGT2 condition, the asymptotic behavior, as ka → 0, of Z DtN2 is identical to the behavior of the exact specific impedance Z ex2 (see Eq. ( 30). In addition, when e = 0, the asymptotic of both Z DtN1 and Z DtN2 are identical to the case of circular-shaped soundsoft scatterers.

Numerical investigation

We have performed several experiments to investigate numerically the effect of the wavenumber and the slenderness of the boundary on the performance of the second-order DtN boundary condition DtN2 given by Eq. ( 8) when solving sound-soft scattering problems in the OSRC context. We have compared the results to the ones obtained with BGT2 conditions when applied on elliptical-shaped boundaries boundaries (see Figs. [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF] to [START_REF] Harari | Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains[END_REF] in [START_REF] Reiner | The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes[END_REF]). We report in here the results obtained for three different values of the incidence angle θ 0 = 0, π 4 , and π 2 for illustration. More numerical results can be found in [START_REF] Saint-Guirons | Construction et analyse de conditions aux limites absorbantes pour des problèmes de propagation d'ondes[END_REF]. Moreover, since the impedances in this case depend on the observation angle θ ∈ [0, 2π), we have reported in Figs ( 6)-( 8) in polar coordinates the magnitude of the impedances. In addition, we have reported in Figs. 9)- [START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF], clearly indicate that the good performance of DtN2 boundary condition is not sensitive to values of the eccentricity e. These results suggest that the second-order DtN absorbing boundary condition given by Eq. ( 8) is appropriate for elongated boundaries.

ii. The second-order DtN2 absorbing boundary condition clearly outperforms the secondorder BGT2 absorbing boundary condition especially for high eccentricity values. Indeed, there is a significant loss of accuracy for BGt2 boundary condition when e ≥ 0.6 (see Figs. ( 10)-( 11)). This demonstrates that the second-order DtN2 boundary condition extends the range of satisfactory performance to all eccentricity values.

iii. For some particular observation angles, the first-order DtN1 boundary condition performs surprisingly better than the second-order DtN2 boundary condition (see Figs. ( 9)-( 11)). Nevertheless, the total relative error (the average of the relative error over all observation angles) delivered by DtN1 boundary condition is significantly higher than the total relative error corresponding to the second-order DtN2 boundary condition. Relative error of the specific impedance for the DtN1 (dashed/dotted), the DtN2 (crossed), the BGT2 (dashed) for ka = 0.1 (left) and ka = 1 (right) for the incident angles θ 0 = 0 (up), θ 0 = π 4 (middle), θ 0 = π 2 (down).

INRIA 5 Conclusion

We have designed a new class of approximate local ABCs to be applied on elliptical-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are exact for the first radiation modes, they are easy to implement and to parallelize, and they preserve the local structure of the computational finite element scheme. The numerical investigation reveals that in the case of radiator problems, the DtN boundary conditions exhibit a significant loss of accuracy when computing higher mode, as expected from the analytical study. The situation is different when solving acoustic scattering problems. The analysis reveals that the second-order DtN2 boundary condition retains an excellent level of accuracy for low wavenumber values ka and for all eccentricity values e. This clearly demonstrates the superiority of the DtN2 boundary condition over the second-order BGT2 absorbing boundary condition, especially for eccentricity values e ≥ 0.6. Hence, the new second-order absorbing boundary condition extends the range of satisfactory performance to all eccentricity values, which makes this absorbing boundary condition appropriate for very elongated elliptical-shaped boundaries.

Lemma 4 . 1

 41 The approximate specific impedance (Z DtN1 n

Remark 4 . 4

 44 First, we note that the asymptotic behavior of Z DtN1 n (resp. Z DtN2 n ; n ≤ 1) is identical to the behavior of the exact specific impedance Z ex2 0 (resp. Z ex2 n

Figure 1 :

 1 Figure 1: Absolute value (left) and relative error (right) of the specific impedance for the exact (solid), the DtN1 (dash), the BGT1 (circle).

Figure 2 :

 2 Figure 2: Absolute value (left) and relative error (right) of the specific impedance for the exact (solid), the DtN2 (crossed), the BGT2 (dashed).

Figure 3 :

 3 Figure 3: Absolute value (left) and relative error (right) of the specific impedance for the exact (solid), the DtN2 (crossed), the BGT2 (dashed).

Figure 4 :

 4 Figure 4: Absolute value (left) and relative error (right) of the specific impedance for the exact (solid), the DtN2 (crossed), the BGT2 (dashed).

Figure 5 :

 5 Figure 5: Absolute value (left) and relative error (right) of the specific impedance for the exact (solid), the DtN2 (crossed), the BGT2 (dashed).

INRIA

  [START_REF] Givoli | Nonreflecting boundary conditions for elastic waves[END_REF]-[START_REF] Givoli | Exact representations on artificial interfaces and applications in mechanics[END_REF] the relative errors as a function the observation angle θ. These results have been obtained for three eccentricity values e = 0.1 corresponding to a ellipticalshaped boundary very close to a circle (see Fig. (6) and Fig. (9)), e = 0.6 corresponding to a "regular" ellipse (see Fig. (7) and Fig. (10)), and e = 0.9 corresponding to a very elongated ellipse (see Fig. (8) and Fig. (11)). These results indicate that i. Unlike the radiator problem, the second-order DtN2 absorbing boundary condition retains an excellent level of accuracy when solving acoustic problems for low number, as expected (see Proposition 4.7). In addition, the results reported in Figs. (

Figure 9 : 4 (

 94 Figure9: Relative error of the specific impedance for the DtN1 (dashed/dotted), the DtN2 (crossed), the BGT2 (dashed) for ka = 0.1 (left) and ka = 1 (right) for the incident angles θ 0 = 0 (up), θ 0 = π 4 (middle), θ 0 = π 2 (down).

Figure 10 : 4 (

 104 Figure10: Relative error of the specific impedance for the DtN1 (dashed/dotted), the DtN2 (crossed), the BGT2 (dashed) for ka = 0.1 (left) and ka = 1 (right) for the incident angles θ 0 = 0 (up), θ 0 = π 4 (middle), θ 0 = π 2 (down).

Figure 11 :

 11 Figure 11: Relative error of the specific impedance for the DtN1 (dashed/dotted), the DtN2 (crossed), the

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Acknowledgements

The authors acknowledge the support by INRIA/CSUN Associate Team Program. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of INRIA or CSUN.