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2.1 **The** **Helmholtz** boundary-value problem
**The** DG variational formulations **of** **the** **Helmholtz** **equation** (
f., for example, [22, 20, 24℄) are generally obtained by writing **the** wave **equation** **in** **the** form **of** a rst-order PDEs system. Most **of** **the** studies dedi
ated **to** **the** **solution** **of** this problem by this kind **of** te
hniques (**in** addition **to** **the** previous referen
es, see, for example, [10, 7, 42, 33℄) deal with **the** **Helmholtz** **equation** with
onstant
oe
ients. If a
ousti
s is taken as **the**
on
rete shape **to** **the** problem being dealt with, this amounts **to** assuming that **the** equations governing **the** a
ousti
u
tuations **of** pressure and velo
ity
orrespond **to** **the** propagation **of** an a
ousti
wave **in** an ideal stagnant and uniform uid (
f., for example. [39, Chap. 2℄). We follow here a more general path and
onsider as **in** [28℄ that **the** propagation is related **to** an ideal stagnant uid but not ne
essarily uniform. **The** a
ousti
system for su
h a
onguration
an be written as follows (
f., for example, [31, Eqs. (64.5) and (64.3) ℄)

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tic fields with n microphones for **the** purpose **of** reconstructing this field over a region **of** interest Ω contained **in** a larger **domain** D **in** which **the** acoustic field propagates. **In** many applied settings, **the** shape **of** D and **the** bound- ary conditions on its border are unknown. Our reconstruction **method** is based on **the** approximation **of** a general **solution** u by linear combinations **of** Fourier-Bessel functions or plane waves. We analyze **the** convergence **of** **the** least-squares estimates **to** u using these families **of** functions based on **the** samples (u(x i )) i=1,...,n . Our analysis describes **the** amount **of** regular-

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Here we are going **to** use **the** point **of** view **of** J.-F. Bony (see [Bon]). He considers **the** case **of** a source which concentrates on one or two points (with V1 6= 0) using a time-dependant **method** based on a BKW approximation **of** **the** propagator **to** prove that, microlocally, **the** **solution** **of** **the** **Helmholtz** **equation** is a finite sum **of** lagrangian distributions. **In** particular, abstract estimates **of** **the** **solution** are only used for **the** large times control, and this part **of** **the** **solution** has no contribution for **the** semiclassical measure, so **the** measure is actually constructed explicitely. Moreover, this **method** requires a geometrical assumption weaker than **the** Virial hypothesis used **in** **the** previous works.

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II. R ECONSTRUCTION **METHOD**
Our goal here, given a **solution** **to** **the** **Helmholtz** **equation** (2) **in** a **domain** D ⊂ R d , d = 2 or 3, is **to** reconstruct it **in** a **domain** Ω ⊂ D from a limited number **of** punctual measurements, without knowing **the** shape **of** D or **the** boundary conditions on ∂D. **The** reconstruction scheme we use is based on **the** Vekua theory and least- squares approximations, and has already been shown **to** compare favorably with existing methods such as OMP using sparsity **in** **the** Fourier **domain** [1], and **to** give good results **in** experimental settings [2].

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At **the** same time, **the** idea **to** post a material layer around **the** **domain** with absorbing properties was also developed. Recently, B´erenger [5] proposed **to** use layers **of** a totally artificial, perfectly matched material such that no artificial reflection was present for any plane wave with any incidence at **the** continuous level. **In** this **method**, each component **of** **the** perturbed electromagnetic field is split into two artificial subcomponents, making this formulation a split-field formulation, very different from Maxwell equations. **The** **method** had a deep impact on **the** community and was adapted **to** different engineering problems involving waves [12, 23, 37]. It was also enhanced **in** many ways, by getting rid **of** possible **unbounded** linear growth [1, 4] **in** particular via so-called unsplit formulations [28, 34, 37], where **the** equations for perturbed fields can be seen as perturbations **of** Maxwell equations. This also allows **the** introduction **of** metallic conductors through **the** UPML regions, like an infinite feeding line for instance.

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C. Spectral log-derivative propagation
**In** spite **of** **the** sparse character **of** **the** discretized Hamil- tonian, memory can become a limiting factor for systems described by a large number **of** collision channels. **In** this case, it may be necessary **to** split **the** full propagation interval **in** smaller intervals, each comprising, for instance, only one ele- ment. **The** scattering **equation** is solved **in** any given element **to** determine at each point a matrix **of** linearly independent solutions F a with elements F αa (I) labeled by column index I and

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As **the** well-known Rellich’s uniqueness theorem [8] and succeeding results (see, e.g., [9]), this theorem points out a forbidden behavior **of** solutions **to** **the** **Helmholtz** **equation** **in** an **unbounded** **domain**. **In** particular, it is not related **to** boundary conditions (no assumption is made on **the** behavior **of** u near **the** boundary **of** Ω). Most Rellich type results involve a particular Besov space related **to** **the** boundedness **of** **the** energy flux and lead **to** **the** uniqueness **of** **the** **solution** **to** scattering problems. Our theorem involves a more restrictive functional framework: **the** assumption u ∈ L 2 (Ω) rather expresses **the** boundedness **of** **the**

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0 2 3
Figure 9. Reference model with a mantle viscosity **of** 1Cl20Pa .s. Geometry **of** plates, second invariant **of** **the** deviatoric stresses **in** **the** plates,
dynamic pressure and velocity fields **in** **the** mantle at d i fferent stages.
li mit **to** **the** upholding **of** **the** slab's coherency. Laboratory experiments on olivine samp l es produce similar values for temperatures around 1000 0 ( and high pressure [Weidner et al., 2001), while others obtained plas tic yielding above 2.5 GPa [Schubnel et al., 2013). Our modeled slabs exceed 2 GPa rather locally **in** **the** sharp bends **of** folds at 400-660 km depth, which gives us confidence that **the** slab's mechanical coherency is pre served (even though brittle failure and nonlinear flow mechanisms obviously occur during slab folding). And if **the** slab does 'break' completely, **the** remnant upper part **of** **the** slab will still pile up over its previ ously deposited slice, leading **to** a similar geometry at **the** large scale. A future step **of** our study is obviously **to** account for more complex and se l f-cons i stent rheology.

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accuracy **of** **the** numerical **solution** at **the** boundary depends on **the** number N and **the** angle φ [3]. Because **of** **the** spatial derivative **in** **equation** (2), additional boundary conditions must be prescribed on **the** auxiliary ﬁelds at **the** boundary **of** **the** edges (i.e. at **the** corner **of** **the** **domain**) **to** close **the** sys- tem. Following [4], we introduce new relations that ensure **the** compatibility **of** **the** system without any supplementary approximation. With these relations, **the** auxiliary ﬁelds deﬁned on adjacent edges are coupled at **the** common corner. For **the** ﬁelds {u f ,i } N i=1 deﬁned on Γ f , having an adjacent edge Γ f 0 , **the**

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2 Oliver Bodart, Val´ erie Cayol, Farshid Dabaghi and Jonas Koko
and/or location **of** **the** crack, **the** variation **of** **the** latter implies a remeshing and assembling **of** all **the** matrices **of** **the** problem.
Using a **domain** decomposition technique then appears as **the** natural solu- tion **to** these problems. **In** [1], a first step was made with **the** development **of** a direct solver implementing a **domain** decomposition **method**. **The** present work represents a step further with **the** use **of** such a solver, which has been improved since **the** publication **of** [1], **to** solve inverse problems **in** **the** field **of** earth sciences. **To** our knowledge, this is **the** first work using these kind **of** techniques **in** this field **of** **application**. **The** next step **of** our project will be **the** shape optimization problem **to** identify **the** shape and location **of** **the** crack. Let Ω be a bounded open set **in** R d , d = 2, 3 with smooth boundary ∂Ω := ΓD ∪ ΓN where ΓD and ΓN are **of** nonzero measure and ΓD ∩ ΓN = ∅. We assume that Ω is occupied by an elastic solid and we denote by u **the** displacement field **of** **the** solid and **the** density **of** external forces by f ∈ L 2 (Ω). **The** Cauchy stress σ(u) and strain ε(u) are given by

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towards **the** **solution** **of** **the** continuous problem. **The** incident wave is completely reflected by **the** obstacle and **the** scattered waves generated by **the** extremities **of** **the** crack are well approximated (see figure 10-(b)).
As **in** **the** scalar case, **the** enrichment **of** **the** M h approximation space introduces spurious modes **in** **the** solu-
tion. Although **the** amplitude **of** these non-physical waves goes **to** zero with **the** size **of** **the** discretization step, it is still significant for a usual choice **of** **the** discretization parameters, typically corresponding **to** 20 points per wavelength. These spurious modes are for example visible **in** **the** results presented **in** figure 11-(a) where we have amplified by a factor eight **the** results **of** figure 10-(b). **In** order **to** study **in** more detail these phe- nomena we represent **in** figure 12 **the** evolution **in** time **of** **the** modulus **of** **the** velocity field at three points:

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i = tan 2 (iπ/M) and M = 2N + 1. **The**
accuracy **of** **the** numerical **solution** at **the** boundary depends on **the** number N and **the** angle φ [3]. Because **of** **the** spatial derivative **in** **equation** (2), additional boundary conditions must be prescribed on **the** auxiliary fields at **the** boundary **of** **the** edges (i.e. at **the** corner **of** **the** **domain**) **to** close **the** system. Following [4, 5], we introduce new relations that ensure **the** compatibility **of** **the** system without any supplementary approximation. With these relations, **the** auxiliary fields defined on adjacent edges are coupled at **the** common corner. For **the** fields {u f ,i } N i=1 defined on Γ f , having an adjacent edge Γ f 0 , **the**

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Keywords: Acoustic scattering, **Helmholtz** **equation**, second-kind Fredholm integral equa- tion, Krylov iterative **solution**
Résumé
Ce papier propose la construction de nouvelles équations intégrales de type Fredholm de seconde espèce pour la résolution itérative de problèmes de diffraction d’ondes acoustiques tridimensionnelles par un obstacle fermé régulier. L’incorporation au sein de ces équations d’un opérateur racine carré tangentiel vient les régulariser et leur confère de bonnes pro- priétés spectrales. Elles peuvent être vues comme des généralisations de la formulation intégrale de Brakhage-Werner [A. Brakhage and P. Werner, Über das Dirichletsche aus- senraumproblem für die Helmholtzsche schwingungsgleichung, Arch. Math. 16 (1965), pp. 325-329] et de l’équation intégrale en champs combinés (Combined Field Integral **Equation** ou CFIE) [R.F. Harrington and J.R. Mautz, H-field, E-field and combined field **solution** for conducting bodies **of** revolution, Arch. Elektron. Übertragungstech (AEÜ), 32 (4) (1978), pp. 157-164]. Des résultats numériques viennent illustrer leur efficacité.

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I. I NTRODUCTION
During system development process, choices and decisions are made at each stage. For instance, **the** **application** **domain** expert makes decisions on business aspects, while **the** system architect is responsible **of** implementation choices. These two stockholders cover **the** stages concerning **the** requirement definition and **the** design. These two stages are crucial **to** **the** system development as they form its base. Thus, **the** choices made during **the** design must be consistent with **the** decisions made during **the** definition **of** **the** requirements. This is necessary **to** facilitate **the** evolution **of** **the** system [1]. If **the** definition **of** **the** requirements is not formal, **the** risk **of** deviation **of** **the** design from **the** initial objectives is real with each evolution **of** **the** system. **The** system **of** systems (SoS) is a system whose definition is based on pre-existing independent systems **in** **the** runtime environment [2]. **The** latter is **in** perpetual evolution thus forcing a recurrent adaptation **of** **the** SoS. Thus, during their life cycle **the** SoS are very exposed **to** **the** problem related **to** **the** evolution mentioned above. **To** solve this problem, we propose **to** create a strong link between **the** requirement definition stage and **the** SoS design stage. **The** idea is **to** allow **the** **application** **domain** expert **to** define her/his objectives and requirements **in** a formal way that will serve as a guide and a controller **of** **the** choices proposed by **the** system architect during **the** design and evolution stages. We suggest **to** use **the** mission paradigm [3]

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and ku T k L 2 1 L−x dx
are small enough, we obtain that **the** operator F maps B(0, R) into itself. Therefore **the** map F has a fixed point **in** B(0, R) by **the** Banach fixed-point Theorem. **The** proof **of** Theorem 4.9 is complete. Acknowledgments: RC was supported by CNPq and Capes (Brazil) via a Fellowship, Project “ Ciˆencia sem Fronteiras” and Agence Nationale de la Recherche (ANR), Project CISIFS, grant ANR-09-BLAN-0213- 02. LR was partially supported by **the** Agence Nationale de la Recherche (ANR), Project CISIFS, grant ANR-09-BLAN-0213-02. AFP was partially supported by CNPq (Brazil) and **the** Cooperation Agreement Brazil-France.

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Fig. 7. Assessing Risk Abstract Architecture Modeling
Fig. 8. Assessing Risks Concrete Architecture Modeling
maturity (Current, Legacy, Development). **The** authors pro- pose a graphical notation **to** illustrate dependencies between subsystems throw capability dependencies. This work gives some interesting features about capabilities (type and ma- turity). However, this information is directly related **to** **the** constituent systems that will be used at run-time, whereas **the** SoS designer does not have a clear idea about them. This information is more useful during **the** creation **of** **the** concrete architecture **of** **the** SoS. **In** our approach, we consider **the** design stage and use **the** concept **of** Role, which is an abstract concept, **to** specify capabilities independently **of** **the** constituent systems that may exist during **the** execution stage. **The** Department **of** Defense Architecture Framework (DoDAF) [12] is an architecture framework for **the** United States Department **of** Defense (DoD). **In** **the** DoDAF frame- work, there is several views, each **of** which is broken down into products and data: operational view, capability view, systems and services view, etc. **The** operational view aims **to** describe **the** tasks and activities, operational elements, and resource flow exchanges required **to** conduct operations while **the** capability view aims **to** describe **the** mapping between **the** required capabilities and **the** activities that enable those capabilities. **The** Ministry **of** Defense Architecture Frame- work(MoDAF) [13] is an architecture framework for **the** UK Ministry **of** Defense(MOD). Similarly **to** DoDAF, MoDAF provides a set **of** views that provide a standard notation **to**

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