Haut PDF Application of fictitious domain method to the solution of the Helmholtz equation in unbounded domain

Application of fictitious domain method to the solution of the Helmholtz equation in unbounded domain

Application of fictitious domain method to the solution of the Helmholtz equation in unbounded domain

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A Symmetric Trefftz-DG Formulation based on a Local Boundary Element Method for the Solution of the Helmholtz Equation

A Symmetric Trefftz-DG Formulation based on a Local Boundary Element Method for the Solution of the Helmholtz Equation

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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Sampling and reconstruction of solutions to the Helmholtz equation

Sampling and reconstruction of solutions to the Helmholtz equation

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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Semiclassical measure for the solution of the dissipative Helmholtz equation

Semiclassical measure for the solution of the dissipative Helmholtz equation

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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Reconstruction of solutions to the Helmholtz equation from punctual measurements

Reconstruction of solutions to the Helmholtz equation from punctual measurements

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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Discontinuous Galerkin Time-Domain solution of Maxwell's equations on locally-refined grids with fictitious domains

Discontinuous Galerkin Time-Domain solution of Maxwell's equations on locally-refined grids with fictitious domains

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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Application of the spectral element method to the solution of the multichannel Schroedinger equation

Application of the spectral element method to the solution of the multichannel Schroedinger equation

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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A Rellich type theorem for the Helmholtz equation in a conical domain

A Rellich type theorem for the Helmholtz equation in a conical domain

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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A fictitious domain method for lithosphere-asthenosphere interaction: Application to periodic slab folding in the upper mantle

A fictitious domain method for lithosphere-asthenosphere interaction: Application to periodic slab folding in the upper mantle

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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An efficient domain decomposition method with cross-point treatment for Helmholtz problems

An efficient domain decomposition method with cross-point treatment for Helmholtz problems

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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Fictitious domain method for an inverse problem in volcanoes

Fictitious domain method for an inverse problem in volcanoes

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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Domain decomposition methods for the diffusion equation with low-regularity solution

Domain decomposition methods for the diffusion equation with low-regularity solution

EQUATION WITH LOW-REGULARITY SOLUTION P. CIARLET, JR, E. JAMELOT AND F. D. KPADONOU Abstract. We analyze matching and non-matching domain decomposition methods for the numerical approximation of the mixed diffusion equations. Special attention is paid to the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. The domain decomposi- tion method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. We prove well-posedness of the discrete problem, that is solvability of the corresponding linear system, provided two algebraic conditions are fulfilled. If moreover the conditions hold independently of the discretization, convergence is ensured.
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On the convergence of the fictitious domain method for wave equation problems

On the convergence of the fictitious domain method for wave equation problems

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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Continuity of the trace operator with respect to the domain and application to shape optimization

Continuity of the trace operator with respect to the domain and application to shape optimization

to be imposed on Γ. The function g is given and u is the solution of a partial differential equation on Ω. In the case where the unknown boundary Γ is the graph of a function, continuity results for this type of functionals with a suitable topology on F have been obtained in [6, 10, 24]. However, in many physical problems this assumption on the unknown boundary is too restrictive. The situation where the unknown boundary can not be the graph of a function occurs in many engineering problems such as, for example, the dam problem in non-homogeneous porous media [10, 19, 22], the Stefan problem [20], optimal insulating and electro-chemistry [3, 17] and the semiconductor problem [27, 16]. Here, we use a general parameterization of the unknown boundary in order to preserve the physical general information on this boundary. The topology we use on Θ ad is just
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An efficient domain decomposition method with cross-point treatment for Helmholtz problems

An efficient domain decomposition method with cross-point treatment for Helmholtz problems

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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Generalized Combined Field Integral Equations for the iterative solution of the three-dimensional Helmholtz equation

Generalized Combined Field Integral Equations for the iterative solution of the three-dimensional Helmholtz equation

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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Involving the Application Domain Expert in the Construction of Systems of Systems

Involving the Application Domain Expert in the Construction of Systems of Systems

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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Internal controllability of the Korteweg-de Vries equation on a bounded domain

Internal controllability of the Korteweg-de Vries equation on a bounded domain

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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A general fictitious domain method with immersed jumps and multilevel nested structured meshes

A general fictitious domain method with immersed jumps and multilevel nested structured meshes

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

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Involving the application domain expert in the construction of systems of system

Involving the application domain expert in the construction of systems of system

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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