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1
Meagher, Timothy P. "A New Finite Difference Time Domain Method to Solve Maxwell's Equations." PDXScholar, 2018. https://pdxscholar.library.pdx.edu/open_access_etds/4389.
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We have constructed a new finite-difference time-domain (FDTD) method in this project. Our new algorithm focuses on the most important and more challenging transverse electric (TE) case. In this case, the electric field is discontinuous across the interface between different dielectric media. We use an electric permittivity that stays as a constant in each medium, and magnetic permittivity that is constant in the whole domain. To handle the interface between different media, we introduce new effective permittivities that incorporates electromagnetic fields boundary conditions. That is, across the interface between two different media, the tangential component, Er(x,y), of the electric field and the normal component, Dn(x,y), of the electric displacement are continuous. Meanwhile, the magnetic field, H(x,y), stays as continuous in the whole domain. Our new algorithm is built based upon the integral version of the Maxwell's equations as well as the above continuity conditions. The theoretical analysis shows that the new algorithm can reach second-order convergence O(∆x2)with mesh size ∆x. The subsequent numerical results demonstrate this algorithm is very stable and its convergence order can reach very close to second order, considering accumulation of some unexpected numerical approximation and truncation errors. In fact, our algorithm has clearly demonstrated significant improvement over all related FDTD methods using effective permittivities reported in the literature. Therefore, our new algorithm turns out to be the most effective and stable FDTD method to solve Maxwell's equations involving multiple media.
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Brookes, P. J. "Time domain methods for the solution of Maxwell's equations on unstructured grids." Thesis, Swansea University, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.636158.
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Designers of aerospace vehicle have recently highlighted computational simulations of electromagnetic systems as a key phase of the design process. Problems of interest involve the simulation of electromagnetic waves, over a wide frequency range, interacting with complex geometries of varying electrical length. This thesis represents the investigation and development of efficient numerical techniques for the simulation of time dependent electromagnetic phenomena. Unstructured grid based algorithms, which have already been successfully employed in the simulation of steady inviscid fluid flows, are applied to the solution of Maxwell's linear curl equations. Finite element time domain solution procedures employing element and edge based data structures are investigated and developed, with a view to extending the range of wave frequencies involved in scattering problems. A two-step Taylor-Galerkin procedure is modified to incorporate a capability to model the wave scattering effects of thin wires. In addition, a hybridisation of the Yee finite difference time domain algorithm and a finite volume time domain procedure is shown to alleviate the restriction of employing Cartesian grids to approximate complex geometries, whilst maintaining an attractively low operation court. Current high performance computing resources are exploited through an efficient parallel implementation of an existing edge based solution algorithm. The extended solution capabilities are demonstrated by the simulation of the scattering effects of a complete aircraft.
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Kim, Joonshik. "Finite Element Time Domain Techniques for Maxwell's Equations Based on Differential Forms." The Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1293588301.
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Edelvik, Fredrik. "Hybrid Solvers for the Maxwell Equations in Time-Domain." Doctoral thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-2156.
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The most commonly used method for the time-domain Maxwell equations is the Finite-Difference Time-Domain method (FDTD). This is an explicit, second-order accurate method, which is used on a staggered Cartesian grid. The main drawback with the FDTD method is its inability to accurately model curved objects and small geometrical features. This is due to the Cartesian grid, which leads to a staircase approximation of the geometry and small details are not resolved at all. This thesis presents different ways to circumvent this drawback, but still take advantage of the benefits of the FDTD method. An approach to avoid staircasing errors but still retain the efficiency of the FDTD method is to use a hybrid grid. A few layers of unstructured cells are used close to curved objects and a Cartesian grid is used for the rest of the domain. For the choice of solver on the unstructured grid two different alternatives are compared: an explicit Finite-Volume Time-Domain (FVTD) solver and an implicit Finite-Element Time-Domain (FETD) solver. The hybrid solvers calculate the scattering from complex objects much more efficiently compared to using FDTD on highly resolved Cartesian grids. For the same accuracy in the solution roughly a factor of 10 in memory requirements and a factor of 20 in execution time are gained. The ability to model features that are small relative to the cell size is often important in electromagnetic simulations. In this thesis a technique to generalize a well-known subcell model for thin wires, in order to take arbitrarily oriented wires in FETD and FDTD into account, is proposed. The method gives considerable modeling flexibility compared to earlier methods and is proven stable. The results show excellent consistency and very good accuracy on different antenna configurations. The recursive convolution method is often used to model frequency dispersive materials in FDTD. This method is used to enable modeling of such materials in the unstructured FVTD and FETD solvers. The stability of both solvers is analyzed and their accuracy is demonstrated by computing the radar cross section for homogeneous as well as layered spheres with frequency dependent permittivity.
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Dosopoulos, Stylianos. "Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell's Equations." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1337787922.
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Andersson, Ulf. "Time-Domain Methods for the Maxwell Equations." Doctoral thesis, Stockholm : Tekniska högsk, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3094.
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Niegemann, Jens [Verfasser], and K. [Akademischer Betreuer] Busch. "Higher-Order Methods for Solving Maxwell's Equations in the Time-Domain / Jens Niegemann. Betreuer: K. Busch." Karlsruhe : KIT-Bibliothek, 2009. http://d-nb.info/1014099129/34.
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Boat, Matthew. "The time-domain numerical solution of Maxwell's electromagnetic equations, via the fourth order Runge-Kutta discontinuous Galerkin method." Thesis, Swansea University, 2008. https://cronfa.swan.ac.uk/Record/cronfa42532.
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This thesis presents a high-order numerical method for the Time-Domain solution of Maxwell's Electromagnetic equations in both one- and two-dimensional space. The thesis discuses the validity of high-order representation and improved boundary representation. The majority of the theory is concerned with the formulation of a high-order scheme which is capable of providing a numerical solution for specific two-dimensional scattering problems. Specifics of the theory involve the selection of a suitable numerical flux, the choice of appropriate boundary conditions, mapping between coordinate systems and basis functions. The effectiveness of the method is then demonstrated through a series of examples.
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Eng, Ju-Ling. "Higher order finite-difference time-domain method." Connect to resource, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1165607826.
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Kung, Christopher W. "Development of a time domain hybrid finite difference/finite element method for solutions to Maxwell's equations in anisotropic media." Columbus, Ohio : Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1238024768.
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Edelvik, Fredrik. "Finite volume solvers for the Maxwell equations in time domain." Licentiate thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2000. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-86389.
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Two unstructured finite volume solvers for the Maxwell equations in 2D and 3D are introduced. The solvers are a generalization of FD–TD to unstructured grids and they use a third-order staggered Adams–Bashforth scheme for time discretization. Analysis and experiments of this time integrator reveal that we achieve a long term stable solution on general triangular grids. A Fourier analysis shows that the 2D solver has excellent dispersion characteristics on uniform triangular grids. In 3D a spatial filter of Laplace type is introduced to enable long simulations without suffering from late time instability. The recursive convolution method proposed by Luebbers et al. to extend FD–TD to permit frequency dispersive materials is here generalized to the 3D solver. A better modelling of materials which have a strong frequency dependence in their constitutive parameters is obtained through the use of a general material model. The finite volume solvers are not intended to be stand-alone solvers but one part in two hybrid solvers with FD–TD. The numerical examples in 2D and 3D demonstrate that the hybrid solvers are superior to stand-alone FD–TD in terms of accuracy and efficiency.
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Jeong, Jaehoon. "Analytical time domain electromagnetic field propagators and closed-form solutions for transmission lines." [College Station, Tex. : Texas A&M University, 2006. http://hdl.handle.net/1969.1/ETD-TAMU-1105.
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Benoit, Jaume. "Identification de sources temporelles pour les simulations numériques des équations de Maxwell." Thesis, Clermont-Ferrand 2, 2012. http://www.theses.fr/2012CLF22314.
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Les travaux effectués durant cette thèse s’inscrivent dans le cadre d’une collaboration entre l’équipe CEM de l’Institut Pascal et l’équipe EDPAN du Laboratoire de Mathématiques de l’Université Blaise Pascal de Clermont-Ferrand. Nous présentons ici une étude qui, partant de l’analyse du processus de Retournement Temporel en électromagnétisme, a débouché sur le développement d’une méthode originale baptisée Linear Combination of Configuration Fields (LCCF) ou, en français, Combinaison Linéaire de Configurations de Champs. Après avoir introduit l’ensemble des outils et méthodes utilisés dans ces travaux, ce mémoire détaille le processus de Retournement Temporel de base ainsi qu’un ajout apporté à celui-ci. Par la suite, la méthode LCCF s’étant révélée applicable à plusieurs problèmes d’identification de sources en électromagnétisme, nous nous consacrons à la présentation détaillée des différentes variantes de celle-ci et nous illustrons son utilisation sur de nombreux exemples numériquesThis Ph.D thesis is the result of a collaboration between the CEM team of Pascal Institute and the EDPAN team of the Laboratory of Mathematics of the Blaise Pascal University in Clermont-Ferrand. We present here a study based on Time Reversal process in Electromagnetics. This work led to the development of a novel method called Linear Combination of Configuration Field (LCCF). This thesis first introduces the tools and the numerical methods used during this work. Then, we describe the Time Reversal process and a possible improvement to the basic technic. Afterwards, several possible applications of the LCCF method to electromagnetic source identification problems are detailed and we illustrate each of it on various numerical examples
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Rawat, Vineet. "Finite Element Domain Decomposition with Second Order Transmission Conditions for Time-Harmonic Electromagnetic Problems." The Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1243360543.
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Abenius, Erik. "Direct and Inverse Methods for Waveguides and Scattering Problems in the Time Domain." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-6013.
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Vegh, Viktor. "Numerical modelling of industrial microwave heating." Thesis, Queensland University of Technology, 2003. https://eprints.qut.edu.au/37144/7/37144_Digitised%20Thesis.pdf.
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The numerical modelling of electromagnetic waves has been the focus of many research areas in the past. Some specific applications of electromagnetic wave scattering are in the fields of Microwave Heating and Radar Communication Systems. The equations that govern the fundamental behaviour of electromagnetic wave propagation in waveguides and cavities are the Maxwell's equations. In the literature, a number of methods have been employed to solve these equations. Of these methods, the classical Finite-Difference Time-Domain scheme, which uses a staggered time and space discretisation, is the most well known and widely used. However, it is complicated to implement this method on an irregular computational domain using an unstructured mesh.
In this work, a coupled method is introduced for the solution of Maxwell's equations. It is proposed that the free-space component of the solution is computed in the time domain, whilst the load is resolved using the frequency dependent electric field Helmholtz equation. This methodology results in a timefrequency domain hybrid scheme. For the Helmholtz equation, boundary conditions are generated from the time dependent free-space solutions. The boundary information is mapped into the frequency domain using the Discrete Fourier Transform. The solution for the electric field components is obtained by solving a sparse-complex system of linear equations. The hybrid method has been tested for both waveguide and cavity configurations. Numerical tests performed on waveguides and cavities for inhomogeneous lossy materials highlight the accuracy and computational efficiency of the newly proposed hybrid computational electromagnetic strategy.
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Xie, Zhongqiang. "Fourth-order finite difference methods for the time-domain Maxwell equations with applications to scattering by rough surfaces and interfaces." Thesis, Coventry University, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.369842.
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Lee, Richard Todd. "A novel method for incorporating periodic boundaries into the FDTD method and the application to the study of structural color of insects." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29772.
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Thesis (Ph.D)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009.Committee Chair: Smith, Glenn; Committee Member: Buck, John; Committee Member: Goldsztein, Guillermo; Committee Member: Peterson, Andrew; Committee Member: Scott, Waymond. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Woyna, Irene [Verfasser], Thomas [Akademischer Betreuer] Weiland, and Irina [Akademischer Betreuer] Munteanu. "Wideband Impedance Boundary Conditions for FE/DG Methods for Solving Maxwell Equations in Time Domain / Irene Woyna. Betreuer: Thomas Weiland ; Irina Munteanu." Darmstadt : Universitäts- und Landesbibliothek Darmstadt, 2014. http://d-nb.info/1110792905/34.
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Hassan, Emadeldeen. "Topology optimization of antennas and waveguide transitions." Doctoral thesis, Umeå universitet, Institutionen för datavetenskap, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-102505.
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This thesis introduces a topology optimization approach to design, from scratch, efficient microwave devices, such as antennas and waveguide transitions. The design of these devices is formulated as a general optimization problem that aims to build the whole layout of the device in order to extremize a chosen objective function. The objective function quantifies some required performance and is evaluated using numerical solutions to the 3D~Maxwell's equations by the finite-difference time-domain (FDTD) method. The design variables are the local conductivity at each Yee~edge in a given design domain, and a gradient-based optimization method is used to solve the optimization problem. In all design problems, objective function gradients are computed based on solutions to adjoint-field problems, which are also FDTD discretization of Maxwell's equations but solved with different source excitations. For any number of design variables, the computation of the objective function gradient requires one solution to the original field problem and one solution to the associated adjoint-field problem. The optimization problem is solved iteratively using the globally convergent Method of Moving Asymptotes (GCMMA). By the proposed approach, various design problems, including tens of thousands of design variables, are formulated and solved in a few hundred iterations. Examples of solved design problems are the design of wideband antennas, dual-band microstrip antennas, wideband directive antennas, and wideband coaxial-to-waveguide transitions. The fact that the proposed approach allows a fine-grained control over the whole layout of such devices results in novel devices with favourable performance. The optimization results are successfully verified with a commercial software package. Moreover, some devices are fabricated and their performance is successfully validated by experiments.
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Chicaud, Damien. "Analysis of time-harmonic electromagnetic problems in elliptic anisotropic media." Electronic Thesis or Diss., Institut polytechnique de Paris, 2021. http://www.theses.fr/2021IPPAE014.
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La simulation numérique de problèmes électromagnétiques dans des configurations physiques complexes est largement utilisée pour de nombreuses applications scientifiques et industrielles, telles que la conception de métamatériaux optiques ou l'étude des plasmas froids. L'analyse mathématique et numérique des problèmes de Maxwell est bien connue dans des contextes physiques simples, où les paramètres du milieu sont isotropes. Des résultats en milieux anisotropes existent, mais se limitent généralement au cas des tenseurs réels symétriques (ou complexes hermitiens) définis positifs. Cependant, pour certains milieux plus complexes, les problèmes ne sont pas couverts par la théorie standard. De nouveaux outils mathématiques doivent donc être développés pour analyser ces problèmes.Dans cette thèse, nous analysons des problèmes électromagnétiques harmoniques en temps pour une classe générale de tenseurs matériels anisotropes, appelés elliptiques. Nous développons un cadre fonctionnel étendu adapté à ces problèmes avec conditions limites de Dirichlet, Neumann ou Robin. Dans le cas de Robin, un intérêt particulier est porté à la caractérisation des espaces pour les traces de Robin. Nous étudions la régularité de la solution et de son rotationnel, et donnons des éléments d'analyse numérique. Dans la perspective de l'utilisation de méthodes de décomposition de domaine (DDM) pour une résolution accélérée, nous proposons et étudions différentes formulations décomposées, en nous focalisant sur leurs espaces fonctionnels et leur équivalence avec le problème global. Quelques expérimentations numériques sur la DDM complètent ce travailThe numerical simulation of electromagnetic problems in complex physical settings is a trending topic which conveys many scientific and industrial applications, such as the design of optical metamaterials, or the study of cold plasmas. The mathematical and numerical analysis of Maxwell problems is wellknown in simple physical contexts, when the material parameters are isotropic. Some results in anisotropic media exist, but they generally tend to focus on the case where the material tensors are real symmetric (or complex) Hermitian) definite positive. However, problems in more complex media are not covered by the standard theory. Therefore, new mathematical tools need to be developped to analyse thses problems. This thesis aims at analysing time-harmonic electromagnetic problems for a general class of complex anisotropic material tensors. These are called ellopptic materials. We derive an extended functional framework well-suited for these anisotropic problems, generalizing well-known results. We study the well-posedness of Maxwell boundary value problems for Dirichlet, Neumann, and Robin boundary conditions. For the Robin case, the characterization of appropriate function spaces for Robin traces is addressed. The regularity of the solution and its curl is studied, and elements of numerical analysis for edge finite elements are provided. In the perspective of the use of Domain Decomposition Methods (DDM) for accelerated numerical computing, various decomposed formulations are proposed and studied, focusing on their right meaning in terms of function spaces and equivalence with the global problem. These results are complemented with some numerical DDM experimentations in anisotropic media
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Ritzenthaler, Valentin. "Stratégies de couplage des méthodes Compatible Discrete Operators appliquées aux équations de Maxwell dans le domaine temporel." Electronic Thesis or Diss., Toulouse, ISAE, 2024. http://www.theses.fr/2024ESAE0060.
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Dans le domaine de la simulation numérique des équations de Maxwell, l'un des principaux objectifs consiste à rendre compte numériquement de la réalité physique des champs électromagnétiques avec une haute précision et un faible coût calcul. Il existe aujourd'hui de nombreuses méthodes permettant de résoudre le système de Maxwell en domaine temporel, présentant chacune, en fonction des situations, des qualités et des défauts. Dans cette thèse, on s’intéresse à deux stratégies de couplagedes méthodes Compatible Discrete Operators (CDO) appliquées aux équations de Maxwell dans le domaine temporel. La première, consiste à définir localement la métrique du schéma en fonction de la géométrie du maillage. La seconde, consiste à partitionner le domaine de calcul en deux sous-domaines et à coupler les méthodes par la définition d'opérateurs sur l’interface. Pour cela, les équations de Maxwell sont étudiées en deux parties : les relations topologiques, d’une part, et les relations constitutives, d’autre part. Dans le cadre CDO, les relations topologiques sont formulées au moyen d’opérateurs différentiels discrets correspondant à la discrétisation des opérateurs vectoriels classiques. Afin de prendre en compte des conditions au bord non homogènes, ces opérateurs sont étendus au bord. Les relations constitutives sont quant à elles formulées au moyen d’opérateurs de Hodge discrets. Ils définissent la métrique du schéma et dépendent des paramètres matériels. Le schéma discret en espace et en temps est alors analysé en terme de stabilité et de consistance. Il est ensuite testé numériquement sur différentes configurations de maillages hybridesIn numerical simulations of Maxwell's equations, one of the main goals is to accurately represent the physical reality of electromagnetic fields while keeping a low computational cost. Numerous methods exist for solving the system in the time domain, each with its own strengths and weaknesses, depending on the situation. In this thesis, we focus on two coupling strategies of Compatible Discrete Operators (CDO) schemes applied to Maxwell's equations in time domain. The first consists in locally defining the metric of the scheme by considering the mesh geometry. In the second approach, the computational domain is partitioned in two subdomains and the coupling is achieved by defining operators on the interface. To this end, Maxwell's equations are studied in two parts: the topological relations and the constitutive relations. In the CDO framework, the topological relations are formulated using discrete differential operators corresponding to the discretization of the classical vector operators. In order to take into account non-homogeneous boundary conditions, these operators are extended using a dual boundary mesh. The constitutive relations are formulated using discrete Hodge operators. They define the metric of the scheme and depend on the material parameters. The discrete scheme in space and time is then analyzed in terms of stability and consistency. We then test it on different configurations using hybrid meshes
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Nilsson, Martin. "Iterative solution of Maxwell's equations in frequency domain." Licentiate thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-86390.
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We have developed an iterative solver for the Moment Method. It computes a matrix–vector product with the multilevel Fast Multipole Method, which makes the method scale with the number of unknowns. The iterative solver is of Block Quasi-Minimum Residual type and can handle several right-hand sides at once. The linear system is preconditioned with a Sparse Approximate Inverse, which is modified to handle dense matrices. The solver is parallelized on shared memory machines using OpenMP. To verify the method some tests are conducted on varying geometries. We use simple geometries to show that the method works. We show that the method scales on several processors of a shared memory machine. To prove that the method works for real life problems, we do some tests on large scale aircrafts. The largest test is a one million unknown simulation on a full scale model of a fighter aircraft.
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Schwarzbach, Christoph. "Stability of finite element solutions to Maxwell's equations in frequency domain." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2009. http://nbn-resolving.de/urn:nbn:de:bsz:105-24780.
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Eine Standardformulierung der Randwertaufgabe für die Beschreibung zeitharmonischer elektromagnetischer Phänomene hat die Vektor-Helmholtzgleichung für das elektrische Feld zur Grundlage. Bei niedrigen Frequenzen führt der große Nullraum des Rotationsoperators zu einem instabilen Lösungsverhalten. Wird die Randwertaufgabe zum Beispiel mit Hilfe der Methode der Finiten Elemente in ein lineares Gleichungssystem überführt, äußert sich die Instabilität in einer schlechten Konditionszahl ihrer Koeffizientenmatrix. Eine stabilere Formulierung wird durch die explizite Berücksichtigung der Kontinuitätsgleichung erreicht. Zur numerischen Lösung der Randwertaufgaben wurde eine Finite-Elemente-Software erstellt. Sie berücksichtigt unter anderem unstrukturierte Gitter, räumlich variable, anisotrope Materialparameter sowie die Erweiterung der Maxwell-Gleichungen durch Perfectly Matched Layers. Die Software wurde anhand von Anwendungen in der marinen Geophysik erfolgreich getestet. Insbesondere demonstriert die Einbeziehung von Seebodentopographie in Form einer stetigen Oberflächentriangulierung die geometrische Flexibilität der SoftwareThe physics of time-harmonic electromagnetic phenomena can be mathematically described by boundary value problems. A standard approach is based on the vector Helmholtz equation in terms of the electric field. The curl operator involved has a large, non-trivial kernel which leads to an instable solution behaviour at low frequencies. If the boundary value problem is solved approximately using, e. g., the finite element method, the instability expresses itself by a badly conditioned coefficient matrix of the ensuing system of linear equations. A stable formulation is obtained by taking the continuity equation explicitly into account. In order to solve the boundary value problem numerically a finite element software package has been implemented. Its features comprise, amongst others, the treatment of unstructured meshes and piecewise polynomial, anisotropic constitutive parameters as well as the extension of Maxwell’s equations to the Perfectly Matched Layer. Successful application of the software is demonstrated with examples from marine geophysics. In particular, the incorporation of seafloor topography by a continuous surface triangulation illustrates the geometric flexibility of the software
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Marchand, Renier Gustav. "Finite element tearing and interconnecting for the electromagnetic vector wave equation in two dimensions." Thesis, Stellenbosch : University of Stellenbosch, 2007. http://hdl.handle.net/10019.1/2471.
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Thesis (MScEng (Electrical and Electronic Engineering))--University of Stellenbosch, 2007.The finite element tearing and interconnect(FETI) domain decomposition(DD) method is investigated in terms of the 2D transverse electric(TEz) finite element method(FEM). The FETI is for the first time rigorously derived using the weighted residual framework from which important insights are gained. The FETI is used in a novel way to implement a total-/scattered field decomposition and is shown to give excellent results. The FETI is newly formulated for the time domain(FETI-TD), its feasibility is tested and it is further formulated and tested for implementation on a distributed computer architecture.
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Rihani, Mahran. "Maxwell's equations in presence of metamaterials." Electronic Thesis or Diss., Institut polytechnique de Paris, 2022. https://theses.hal.science/tel-03670420.
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Le sujet principal de cette thèse est l’étude de la propagation des ondes électromagnétiques, en régime harmonique, dans un milieu hétérogène composé d’un diélectrique et d’un matériau négatif (c’est-à-dire avec une permittivité diélectrique négative ε et/ou une perméabilité magnétique négative μ) qui sont séparés par une interface avec une pointe conique. En raison du changement de signe de ε et/ou μ, les équations de Maxwell peuvent être mal posées dans les cadres classiques (basés sur l’espace L2). D’autre part, nous savons que lorsque les deux problèmes scalaires associés, impliquant respectivement ε et μ, sont bien posés dans H1, les équations de Maxwell sont bien posées. En combinant la méthode de la T-coercivité avec l’analyse de Mellin dans les espaces de Sobolev à poids, nous présentons, dans la première partie de ce travail, une étude détaillée de ces problèmes scalaires. Nous prouvons que pour chacun d’entre eux, le caractère bien posé dans H1 est perdu si et seulement si le contraste associé appartient à un ensemble critique appelé intervalle critique. Ces intervalles correspondent aux ensembles de contrastes négatifs pour lesquels des singularités propagatives, aussi appelées ondes de trou noir, apparaissent à l’extrémité de la pointe. Contrairement au cas d’un coin 2D, pour une pointe 3D, plusieurs ondes de trou noir peuvent exister. Des expressions explicites de ces intervalles critiques sont obtenues pour le cas particulier des pointes coniques circulaires. Pour les contrastes critiques, en utilisant le principe de radiation de Mandelstam, nous construisons des cadres fonctionnels dans lesquels le caractère bien posé des problèmes scalaires est restauré. Le cadre physiquement pertinent est sélectionné par un principe d’absorption limite. En outre, nous présentons, dans la deuxième partie de ce travail, une nouvelle méthode numérique pour les problèmes scalaires dans le cas des contrastes non-critiques. Cette approche, contrairement aux techniques existantes, ne nécessite pas d’hypothèses supplémentaires sur le maillage au voisinage de l’interface. La troisième partie de la thèse concerne l’étude des équations de Maxwell avec un ou deux coefficients critiques. En utilisant de nouveaux résultats de potentiels vecteurs dans des espaces de Sobolev à poids, nous expliquons comment construire de nouveaux cadres fonctionnels pour les problèmes électrique et magnétique, qui sont directement liés à ceux obtenus pour les deux problèmes scalaires associés. Si l’on utilise le cadre qui respecte le principe d’absorption limite pour les problèmes scalaires, alors les cadres fournis pour les problèmes électrique et magnétique sont également cohérents avec le principe d’absorption limite. Enfin, la dernière partie porte sur des résultats d’homogénéisation des équations de Maxwell harmoniques et des problèmes scalaires associés dans un domaine 3D qui contient une distribution périodique d’inclusions faites de matériau négatif. En utilisant l’approche de la T-coercivité, nous obtenons des conditions sur les contrastes telles que le processus d’homogénéisation est possible pour les problèmes scalaires et vectoriels. De façon peu intuitive, nous montrons que les matrices homogénéisées associées auxproblèmes limites sont soit définies positives, soit définies négativesThe main subject of this thesis is the study of time-harmonic electromagnetic waves in a heterogeneous medium composed of a dielectric and a negative material (i.e. with a negative dielectric permittivity ε and/or a negative magnetic permeability μ) which are separated by an interface with a conical tip. Because of the sign-change in ε and/or μ, the Maxwell’s equations can be ill-posed in the classical L2 −frameworks. On the other hand, we know that when the two associated scalar problems, involving respectively ε and μ, are well-posed in H1, the Maxwell’s equations are well-posed. By combining the T-coercivity approach with the Mellin analysis in weighted Sobolev spaces, we present, in the first part of this work, a detailed study of these scalar problems. We prove that for each of them, the well-posedeness in H1 is lost iff the associated contrast belong to some critical set called the critical interval. These intervals correspond to the sets of negative contrasts for which propagating singularities, also known as black hole waves, appear at the tip. Contrary to the case of a 2D corner, for a 3D tip, several black hole waves can exist. Explicit expressions of these critical intervals are obtained for the particular case of circular conical tips. For critical contrasts, using the Mandelstam radiation principle, we construct functional frameworks in which well-posedness of the scalar problems is restored. The physically relevant framework is selected by a limiting absorption principle. In the process, we present a new numerical strategy for 2D/3D scalar problems in the non-critical case. This approach, presented in the second part of this work, contrary to existing ones, does not require additional assumptions on the mesh near the interface. The third part of the thesis concerns Maxwell’s equations with one or two critical coefficients. By using new results of vector potentials in weighted Sobolev spaces, we explain how to construct new functional frameworks for the electric and magnetic problems, directly related to the ones obtained for the two associated scalar problems. If one uses the setting that respects the limiting absorption principle for the scalar problems, then the settings provided for the electric and magnetic problems are also coherent with the limiting absorption principle. Finally, the last part is devoted to the homogenization process for time-harmonic Maxwell’s equations and associated scalar problems in a 3D domain that contains a periodic distribution of inclusions made of negative material. Using the T-coercivity approach, we obtain conditions on the contrasts such that the homogenization results is possible for both the scalar and the vector problems. Interestingly, we show that the homogenized matrices associated with the limit problems are either positive definite or negative definite
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Viquerat, Jonathan. "Simulation de la propagation d'ondes électromagnétiques en nano-optique par une méthode Galerkine discontinue d'ordre élevé." Thesis, Nice, 2015. http://www.theses.fr/2015NICE4109/document.
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L’objectif de cette thèse est de développer une méthode Galerkine discontinue d’ordre élevé capable de prendre en considération des simulations réalistes liées à la nanophotonique. Au cours des dernières décennies, l’évolution des techniques de lithographie a permis la création de structure géométriques de tailles nanométriques, révélant ainsi une large gamme de phénomènes nouveaux nés de l’interaction lumière-matière à ces échelles. Ces effets apparaissent généralement pour des objets de taille égale ou (très) inférieure à la longueur d’onde du champ incident. Ce travail repose sur le développement et l’implémentation de modèles de dispersion appropriés (principalement pour les métaux), ainsi que sur un large éventail de méthodes computationnelles classiques. Deux développements méthodologiques majeurs sont présentés et étudiés en détails: (i) les éléments courbes, et (ii) l’ordre d’approximation local. Ces études sont accompagnées de plusieurs cas-tests réalistes tirés de la nanophotoniqueThe goal of this thesis is to develop a discontinuous Galerkin time-domain method to be able to handle realistic nanophotonics computations. During the last decades, the evolution of lithography techniques allowed the creation of geometrical structures at the nanometer scale, thus unveiling a variety of new phenomena arising from light-matter interactions at such levels. These effects usually occur when the device is of comparable size or (much) smaller than the wavelength of the incident field. This work relies on the development and implementation of appropriate models for dispersive materials (mostly metals), as well as on a large panel of classical computational techniques. Two major methodological developments are presented and studied in details: (i) curvilinear elements, and (ii) local order of approximation. This work is complemented with several physical studies of real-life nanophotonics applications
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Schütte, Maria [Verfasser]. "On shape sensitivity analysis for 3D time-dependent Maxwell's equations / Maria Schütte." Paderborn : Universitätsbibliothek, 2017. http://d-nb.info/1127109979/34.
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Badia, Ismaïl. "Couplage par décomposition de domaine optimisée de formulations intégrales et éléments finis d’ordre élevé pour l’électromagnétisme." Electronic Thesis or Diss., Université de Lorraine, 2022. http://www.theses.fr/2022LORR0058.
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La résolution numérique d’un problème de diffraction électromagnétique tridimensionnel en régime harmonique est connue pour être difficile, notamment en haute fréquence et pour des objets diffractants diélectriques et inhomogènes. En effet, elle nécessite de discrétiser un système d’équations aux dérivées partielles posé sur un domaine infini. De plus, le fait de considérer une petite longueur d’onde λ dans ce cas, nécessite naturellement un maillage très fin, ce qui conduit par conséquent à un très grand nombre de degrés de liberté. Une approche standard consiste à combiner une méthode d’équations intégrales pour le domaine extérieur et une formulation variationnelle volumique pour le domaine intérieur (objet diffractant), conduisant à une formulation couplant la méthode des éléments de frontière (BEM) et la méthode des éléments finis (FEM). Bien que naturelle, cette approche présente quelques inconvénients majeurs. Tout d’abord, cette méthode de couplage mène à un système linéaire de très grande taille caractérisé par une matrice composée à la fois de parties creuses et denses. Un tel système est généralement difficile à résoudre et n’est pas directement adapté aux méthodes de compression. Ajouté à cela, il n’est pas possible de combiner facilement deux solveurs pré-existants, à savoir un solveur FEM pour le domaine intérieur et un solveur BEM pour le domaine extérieur, afin de construire un solveur global du problème original. Dans cette thèse, nous présentons un couplage faible bien conditionné entre la méthode des éléments de frontière et celle des éléments finis d’ordre élevé, permettant une simple construction d’un tel solveur. L’approche est basée sur l’utilisation d’une méthode de décomposition de domaine sans recouvrement impliquant des opérateurs de transmission optimaux. Ces derniers sont construits par le biais d’un processus de localisation basé sur des approximations rationnelles complexes de Padé des opérateurs Magnetic-to-Electric non locaux. Le nombre d’itérations nécessaires à la résolution du couplage faible ne dépend que faiblement de la configuration géométrique, de la fréquence, du contraste entre les sous-domaines et du raffinement de maillageIn terms of computational methods, solving three-dimensional time-harmonic electromagnetic scattering problems is known to be a challenging task, most particularly in the high frequency regime and for dielectric and inhomogeneous scatterers. Indeed, it requires to discretize a system of partial differential equations set in an unbounded domain. In addition, considering a small wavelength λ in this case, naturally requires very fine meshes, and therefore leads to very large number of degrees of freedom. A standard approach consists in combining integral equations for the exterior domain and a weak formulation for the interior domain (the scatterer) resulting in a formulation coupling the Boundary Element Method (BEM) and the Finite Element Method (FEM). Although natural, this approach has some major drawbacks. First, this standard coupling method yields a very large system having a matrix with sparse and dense blocks, which is therefore generally hard to solve and not directly adapted to compression methods. Moreover, it is not possible to easily combine two pre-existing solvers, one FEM solver for the interior domain and one BEM solver for the exterior domain, to construct a global solver for the original problem. In this thesis, we present a well-conditioned weak coupling formulation between the boundary element method and the high-order finite element method, allowing the construction of such a solver. The approach is based on the use of a non-overlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Padé approximants of the nonlocal Magnetic-to-Electric operators. The number of iterations required to solve this weak coupling is only slightly dependent on the geometry configuration, the frequency, the contrast between the subdomains and the mesh refinement
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Sturm, Andreas [Verfasser], and M. [Akademischer Betreuer] Hochbruck. "Locally Implicit Time Integration for Linear Maxwell's Equations / Andreas Sturm ; Betreuer: M. Hochbruck." Karlsruhe : KIT-Bibliothek, 2017. http://d-nb.info/1132997453/34.
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Lijoka, Oluwaseun Francis. "Enriched discrete spaces for time domain wave equations." Thesis, Heriot-Watt University, 2017. http://hdl.handle.net/10399/3264.
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The second order linear wave equation is simple in representation but its numerical approximation is challenging, especially when the system contains waves of high frequencies. While 10 grid points per wavelength is regarded as the rule of thumb to achieve tolerable approximation with the standard numerical approach, high resolution or high grid density is often required at high frequency which is often computationally demanding. As a contribution to tackling this problem, we consider in this thesis the discretization of the problem in the framework of the space-time discontinuous Galerkin (DG) method while investigating the solution in a finite dimensional space whose building blocks are waves themselves. The motivation for this approach is to reduce the number of degrees of freedom per wavelength as well as to introduce some analytical features of the problem into its numerical approximation. The developed space-time DG method is able to accommodate any polynomial bases. However, the Trefftz based space-time method proves to be efficient even for a system operating at high frequency. Comparison with polynomial spaces of total degree shows that equivalent orders of convergence are obtainable with fewer degrees of freedom. Moreover, the implementation of the Trefftz based method is cheaper as integration is restricted to the space-time mesh skeleton. We also extend our technique to a more complicated wave problem called the telegraph equation or the damped wave equation. The construction of the Trefftz space for this problem is not trivial. However, the exibility of the DG method enables us to use a special technique of propagating polynomial initial data using a wave-like solution (analytical) formula which gives us the required wave-like local solutions for the construction of the space. This thesis contains important a priori analysis as well as the convergence analysis for the developed space-time method, and extensive numerical experiments.
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Findeisen, Stefan Matthias [Verfasser], and C. [Akademischer Betreuer] Wieners. "A Parallel and Adaptive Space-Time Method for Maxwell's Equations / Stefan Matthias Findeisen. Betreuer: C. Wieners." Karlsruhe : KIT-Bibliothek, 2016. http://d-nb.info/1108452647/34.
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Freese, Jan Philip [Verfasser], and C. [Akademischer Betreuer] Wieners. "Numerical homogenization of time-dependent Maxwell's equations with dispersion effects / Jan Philip Freese ; Betreuer: C. Wieners." Karlsruhe : KIT-Bibliothek, 2021. http://d-nb.info/1227451113/34.
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Kachanovska, Maryna. "Fast, Parallel Techniques for Time-Domain Boundary Integral Equations." Doctoral thesis, Universitätsbibliothek Leipzig, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-132183.
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This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion.
As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems:
the need to construct many dense matrices and to evaluate many singular and near-singular integrals.
The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested.
The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal.
In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.
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Pino, Gabriel. "Fault location in transmission lines using time-domain equations." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/3/3143/tde-28082018-133153/.
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This thesis is a combination of the development of numerical models regarding transient simulation of transmission lines and their advantages associated with fault location methods. The transmission line models presented in this work are in time-domain, which is a new approach considering traditional methods as being the phasor and traveling wave techniques. The use of phasors for this purpose has some technical difficulties: the presence of a damped DC component, greater influence of the fault impedance and metrological equipment layout. This work deals with single-phase AC and mono polar DC transmission lines. The proposed transmission line model has three main differentials compared to the traditional Bergeron model: full distribution of linear resistance; full distribution of leakage conductance; and just one recurrence of historical values. The first point is critical for evaluation of the exponential component of transient short circuit currents. The second point refers to the inclusion of the corona effect in the transmission line modeling. The single recurrence indicated in the third topic is given by the complete resolution of the telegrapher\'s equations, so there is no need of serial composition to improve waveform accuracy. The principle of fault location method is calculating the absolute difference between fault voltages seen by the transmission line ends. This technique guarantees a lower influence of the fault impedance and its electrical parameters.Esta dissertação é uma combinação do desenvolvimento de modelos numéricos para simulação de transitórios eletromagnéticos em linhas de transmissão e suas benesses associadas à localização de faltas. Os modelos de linha de transmissão aqui apresentados estão no domínio do tempo, o que descaracteriza a abordagem tradicional de localização de faltas como técnicas fasoriais e ondas viajantes. A utilização de fasores para esse propósito admite algumas dificuldades técnicas: presença da componente DC amortecida, maior influência da impedância de falta e disposição dos equipamentos metrológicos. Abordam-se linhas de transmissão de sistemas alternado monofásico e contínuo monopolar. A modelagem proposta possui três principais diferenciais frente ao modelo de Bergeron: plena distribuição da resistência linear; plena distribuição da condutância transversal; e apenas uma recorrência a valores históricos. O primeiro ponto é fundamental para avaliação da componente exponencialmente amortecida das correntes transitórias de curto circuito. O segundo ponto se refere à inclusão do efeito corona no modelamento. A recorrência unitária apontada no terceiro tópico apresenta a vantagem de não ser necessária a composição em série do modelo para aprimorar a qualidade das formas de onda. O princípio de localização de faltas se dá pelo cálculo da diferença absoluta entre as tensões instantâneas de falta vistas pelos terminais da linha. Essa técnica garante uma menor influência de impedância de falta e dos parâmetros elétricos.
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Kachanovska, Maryna. "Fast, Parallel Techniques for Time-Domain Boundary Integral Equations." Doctoral thesis, Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2013. https://ul.qucosa.de/id/qucosa%3A12278.
Full textAbstract:
This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion.
As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems:
the need to construct many dense matrices and to evaluate many singular and near-singular integrals.
The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested.
The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal.
In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.
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Volpert, Thibault. "Étude d'un schéma différences finies haute précision et d'un modèle de fil mince oblique pour simuler les perturbations électromagnétiques sur véhicule aérospatial." Thesis, Toulouse, ISAE, 2014. http://www.theses.fr/2014ESAE0042/document.
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Les travaux de cette thèse concerne l’étude d’une méthode élément finis d’ordre spatial élevé que l’on peut assimilé à une extension du schéma de Yee. On parle alors de méthode différences finies d’ordre élevé. Après avoir donné, dans un premier chapitre, un historique non exhaustif des principales méthodes utilisées pour résoudre les équations de Maxwell dans le cadre de problèmes de CEM et montré l’ intérêt de disposer d’un solveur de type "différences finies d’ ordre élevé", nous présentons dans un deuxième chapitre le principe de la méthode. Nous donnons pour cela les caractéristiques du schéma spatial et temporel en précisant les conditions de stabilité de la méthode. En outre, dans une étude purement numérique, nous étudions la convergence du schéma. On se focalise ensuite sur la possibilité d’utiliser des ordres spatiaux variable par cellules dans chaque direction de l’espace. Des comparaisons avec le schéma de Yee et un schéma de Galerkin Discontinu particulier sont ensuite effectuées pour montrer les gains en coûts calcul et mémoire et donc l’intérêt de notre approche par rapport aux deux autres. Dans un troisième chapitre, nous nous intéressons à l’étude de modèles physiques indispensable au traitement d’un problème de CEM. Pour cela, nous nous focalisons particulièrement sur un modèle de fil mince oblique, des modèles de matériaux volumiques et minces et enfin sur la prise en compte de sol parfaitement métallique dans une agression de type onde plane. Chaque modèle est détaillé et validé par comparaison avec des solutions analytiques ou résultant de la littérature, sur des exemples canoniques. Le quatrième chapitre est dédié à une technique d’hybridation entre notre méthode et une approche Galerkin Discontinu en vue de traiter des géométries possédant des courbures. Nous donnons pour cela une stratégie d’hybridation basée sur l’échange de flux qui garantie au niveau continue la conservation d’une énergie. Nous présentons ensuite quelques exemples montrant la validité de notre approche dans une stratégie multi-domaines/multi-méthodes que nous précisons. Enfin le dernier chapitre de cette thèse concerne l’exploitation de notre méthode sur des cas industriels en comparaisons avec d’autres méthodes ou des résultats expérimentauxThis thesis is about the study of a high spatial finite element method whichcan be assimilated at an extension of the Yee schema. In the next, this method is also called high order finite difference method. In the first chapter, we give a non exhaustive recall of the major methods used to treat EMC problems and we show the necessity to have this kind of schema to simulate efficiently some EMC configurations. In the second chapter, the principle of the numerical method is presented and a stability condition is given. A numerical study analysis of the schema convergence is also done. Next, we show the interest to have the possibility to use local spatial order by cell in each direction of the computational domain. Some canonic examples are given to show the advantages interms of CPU time and memory storage of the method by comparison with Yee’s scheme and DG approach. In the third chapter, we define and validate on several examples,some physical models as thin wire, materials and perfectly metallic ground in presence of a plane wave, to have the possibility to treat EMC problems. The fourth chapter is about a hybridization strategy between our high order FDTD method and a DG schema.We focalize our study on a hybrid method which provides an energy conservation of the continuous problem. A numerical example is given to validate the method. Finally, in the last chapter, we present some simulations on industrial problems to show the possibility of the method to treat realistic EMC problems
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Aghabarati, Ali. "Multilevel and algebraic multigrid methods for the higher order finite element analysis of time harmonic Maxwell's equations." Thesis, McGill University, 2014. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=121485.
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The Finite Element Method (FEM) applied to wave scattering and quasi-static vector field problems in the frequency domain leads to sparse, complex-symmetric, linear systems of equations. For large problems with complicated geometries, most of the computer time and memory used by FEM goes to solving the matrix equation. Krylov subspace methods are widely used iterative methods for solving large sparse systems. They depend heavily on preconditioning to accelerate convergence. However, application of conventional preconditioners to the "curl-curl" operator which arises in vector electromagnetics does not result in a satisfactory performance and specialized preconditioning techniques are required. This thesis presents effective Multilevel and Algebraic Multigrid (AMG) preconditioning techniques for p-adaptive FEM analysis. In p-adaption, finite elements of different polynomial orders are present in the mesh and the system matrix can be structured into blocks corresponding to the orders of the basis functions. The new preconditioners are based on a p-type multilevel Schwarz (pMUS) approximate inversion of the block structured system. A V-cycle multilevel correction starts by applying Gauss-Seidel to the highest block level, then the next level down, and so on. On the other side of the V, Gauss-Seidel iterations are applied in the reverse order. At the bottom of the cycle is the lowest order system, which is usually solved exactly with a direct solver. The proposed alternative is to use Auxiliary Space Preconditioning (ASP) at the lowest level and continue the V-cycle downwards, first into a set of auxiliary, node-based spaces, then through a series of progressively smaller matrices generated by an Algebraic Multigrid (AMG). The algebraic coarsening approach is especially useful for problems with fine geometric details, requiring a very large mesh in which the bulk of the elements remain at low order. In addition, for wave problems, a "shifted Laplace" technique is applied, in which part of the ASP/AMG algorithm uses a perturbed, complex frequency. A significant convergence acceleration is achieved. The performance of Krylov algorithms is further enhanced during p-adaption by incorporation of a deflation technique. This projects out from the preconditioned system the eigenvectors corresponding to the smallest eigenvalues. The construction of the deflation subspace is based on efficient estimation of the eigenvectors from information obtained when solving the first problem in a p-adaptive sequence. Extensive numerical experiments have been performed and results are presented for both wave and quasi-static problems. The test cases considered are complicated to solve and the numerical results show the robustness and efficiency of the new preconditioners. Deflated Krylov methods preconditioned with the current Multilevel/ASP/AMG approach are always considerably faster than the reference methods and speedups of up to 10 are achieved for some test problems.La méthode des éléments finis (FEM) appliquée à la dispersion des ondes et aux problèmes de champ de vecteurs quasi-statique dans le domaine fréquentiel mène à des systèmes d'équations linéaires rares, symétriques-complexes. Pour de grands problèmes ayant des géométries complexes, la plupart du temps et de la mémoire d'ordinateur utilisé par FEM va à la résolution de l'équation de la matrice. Les méthodes itératives de Krylov sont celles largement utilisées dans la résolution de grands systèmes creux. Elles dépendent fortement des préconditionnement qui accélèrent la convergence. Toutefois, l'application de préconditionnements conventionnels à l'opérateur "rot-rot" qui surgit en électromagnétisme vectoriel n'aboutit pas à des résultats satisfaisants et des techniques de préconditionnement spécialisés sont exigées.Cette thèse présente des techniques de préconditionnement efficaces multiniveau et multigrilles algébrique (AMG) pour l'analyse p-adaptative FEM. Dans la p-adaptation, des éléments finis de différents ordres polynomiaux sont présents dans le maillage et la matrice du système peut être structurée en blocs correspondant aux ordres des fonctions de base. Les nouveaux préconditionneurs sont basés sur un type d'inversion approximative à multiniveau p Schwarz (pMUS) du système structuré de bloc. Une correction à niveaux multiples en cycle V débute par l'application de Gauss-Seidel au niveau du bloc le plus élevé, suivi par le niveau inférieur, et ainsi de suite. De l'autre côté du V, des itérations de Gauss-Seidel sont appliquées en ordre inverse. Au bas du cycle se trouve le système d'ordre le plus bas, qui est habituellement résolu exactement avec un solveur direct. L'alternative proposée est d'utiliser l'espace auxiliaire de préconditionnement (ASP) au niveau le plus bas et de poursuivre le cycle en V vers le bas, d'abord en un ensemble d'auxiliaires, basé sur les espacements de nœuds, à travers une série de plus en plus petites de matrices générées par un multigrille algébrique (AMG). L'approche de grossissement algébrique est particulièrement utile aux problèmes ayant de fins détails géométriques, nécessitant une très grande maille dans laquelle la majeure partie des éléments restent à un niveau plus bas.En outre, pour des problèmes d'onde, la technique "décalé Laplace" est appliquée, dans laquelle une partie de l'algorithme ASP/AMG utilise une fréquence complexe perturbée. Une accélération de la convergence significative est atteinte. La performance des algorithmes de Krylov est davantage renforcée au cours du p-adaptation par l'incorporation d'une technique de déflation. Cette saillie fait dépasser hors du système préconditionné, les vecteurs propres correspondants aux plus petites valeurs propres. La construction du sous-espace de déflation est basée sur une estimation efficace des vecteurs propres à partir d'informations obtenues lors de la résolution du premier problème dans une séquence p-adaptatif. Des expériences numériques approfondies ont été effectuées et les résultats sont présentés à la fois aux problèmes d'onde et quasi-statiques. Les cas de test sont considérés comme compliqués à résoudre et les résultats numériques montrent la robustesse et l'efficacité des nouveaux préconditionnements. Les méthodes de Krylov de déflation préconditionnés par l'approche multiniveaux/ASP/AMG actuelle sont toujours considérablement plus rapides que les méthodes de référence et des accélérations allant jusqu'à 10 sont atteintes pour certains problèmes de test.
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Moya, Ludovic. "Méthodes Galerkine discontinues localement implicites en domaine temporel pour la propagation des ondes électromagnétiques dans les tissus biologiques." Phd thesis, Université Nice Sophia Antipolis, 2013. http://tel.archives-ouvertes.fr/tel-00950386.
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Cette thèse traite des équations de Maxwell en domaine temporel. Le principal objectif est de proposer des méthodes de type éléments finis d'ordre élevé pour les équations de Maxwell et des schémas d'intégration en temps efficaces sur des maillages localement raffinés. Nous considérons des méthodes GDDT (Galerkine Discontinues en Domaine Temporel) s'appuyant sur une interpolation polynomiale d'ordre arbitrairement élevé des composantes du champ électromagnétique. Les méthodes GDDT pour les équations de Maxwell s'appuient le plus souvent sur des schémas d'intégration en temps explicites dont la condition de stabilité peut être très restrictive pour des maillages raffinés. Pour surmonter cette limitation, nous considérons des schémas en temps qui consistent à appliquer un schéma implicite localement, dans les régions raffinées, tout en préservant un schéma explicite sur le reste du maillage. Nous présentons une étude théorique complète et une comparaison de deux méthodes GDDT localement implicites. Des expériences numériques en 2D et 3D illustrent l'utilité des schémas proposés. Le traitement numérique de milieux de propagation complexes est également l'un des objectifs. Nous considérons l'interaction des ondes électromagnétiques avec les tissus biologiques qui est au cœur de nombreuses applications dans le domaine biomédical. La modélisation numérique nécessite alors de résoudre le système de Maxwell avec des modèles appropriés de dispersion. Nous formulons une méthode GDDT localement implicite pour le modèle de Debye et proposons une analyse théorique et numérique complète du schéma.
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Pa¸ur, Tomislav [Verfasser], and M. [Akademischer Betreuer] Hochbruck. "Error analysis of implicit and exponential time integration of linear Maxwell's equations / Tomislav Pa¸ur. Betreuer: M. Hochbruck." Karlsruhe : KIT-Bibliothek, 2013. http://d-nb.info/1047839822/34.
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Klimek, Mariusz [Verfasser], Sebastian [Akademischer Betreuer] Schöps, and Stefan [Akademischer Betreuer] Kurz. "Space-Time Discretization of Maxwell's Equations in the Setting of Geometric Algebra / Mariusz Klimek ; Sebastian Schöps, Stefan Kurz." Darmstadt : Universitäts- und Landesbibliothek Darmstadt, 2018. http://d-nb.info/1152384236/34.
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Bonazzoli, Marcella. "Méthodes d'ordre élevé et méthodes de décomposition de domaine efficaces pour les équations de Maxwell en régime harmonique." Thesis, Université Côte d'Azur (ComUE), 2017. http://www.theses.fr/2017AZUR4067/document.
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Les équations de Maxwell en régime harmonique comportent plusieurs difficultés lorsque la fréquence est élevée. On peut notamment citer le fait que leur formulation variationnelle n’est pas définie positive et l’effet de pollution qui oblige à utiliser des maillages très fins, ce qui rend problématique la construction de solveurs itératifs. Nous proposons une stratégie de solution précise et rapide, qui associe une discrétisation par des éléments finis d’ordre élevé à des préconditionneurs de type décomposition de domaine. La conception, l’implémentation et l’analyse des deux méthodes sont assez difficiles pour les équations de Maxwell. Les éléments finis adaptés à l’approximation du champ électrique sont les éléments finis H(rot)-conformes ou d’arête. Ici nous revisitons les degrés de liberté classiques définis par Nédélec, afin d’obtenir une expression plus pratique par rapport aux fonctions de base d’ordre élevé choisies. De plus, nous proposons une technique pour restaurer la dualité entre les fonctions de base et les degrés de liberté. Nous décrivons explicitement une stratégie d’implémentation qui a été appliquée dans le langage open source FreeFem++. Ensuite, nous nous concentrons sur les techniques de préconditionnement du système linéaire résultant de la discrétisation par éléments finis. Nous commençons par la validation numérique d’un préconditionneur à un niveau, de type Schwarz avec recouvrement, avec des conditions de transmission d’impédance entre les sous-domaines. Enfin, nous étudions comment des préconditionneurs à deux niveaux, analysés récemment pour l’équation de Helmholtz, se comportent pour les équations de Maxwell, des points de vue théorique et numérique. Nous appliquons ces méthodes à un problème à grande échelle qui découle de la modélisation d’un système d’imagerie micro-onde, pour la détection et le suivi des accidents vasculaires cérébraux. La précision et la vitesse de calcul sont essentielles dans cette applicationThe time-harmonic Maxwell’s equations present several difficulties when the frequency is large, such as the sign-indefiniteness of the variational formulation, the pollution effect and the problematic construction of iterative solvers. We propose a precise and efficient solution strategy that couples high order finite element (FE) discretizations with domain decomposition (DD) preconditioners. High order FE methods make it possible for a given precision to reduce significantly the number of unknowns of the linear system to be solved. DD methods are then used as preconditioners for the iterative solver: the problem defined on the global domain is decomposed into smaller problems on subdomains, which can be solved concurrently and using robust direct solvers. The design, implementation and analysis of both these methods are particularly challenging for Maxwell’s equations. FEs suited for the approximation of the electric field are the curl-conforming or edge finite elements. Here, we revisit the classical degrees of freedom (dofs) defined by Nédélec to obtain a new more friendly expression in terms of the chosen high order basis functions. Moreover, we propose a general technique to restore duality between dofs and basis functions. We explicitly describe an implementation strategy, which we embedded in the open source language FreeFem++. Then we focus on the preconditioning of the linear system, starting with a numerical validation of a one-level overlapping Schwarz preconditioner, with impedance transmission conditions between subdomains. Finally, we investigate how two-level preconditioners recently analyzed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view. We apply these methods to the large scale problem arising from the modeling of a microwave imaging system, for the detection and monitoring of brain strokes. In this application accuracy and computing speed are indeed of paramount importance
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Gläfke, Matthias. "Adaptive methods for time domain boundary integral equations for acoustic scattering." Thesis, Brunel University, 2012. http://bura.brunel.ac.uk/handle/2438/7378.
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This thesis is concerned with the study of transient scattering of acoustic waves by an obstacle in an infinite domain, where the scattered wave is represented in terms of time domain boundary layer potentials. The problem of finding the unknown solution of the scattering problem is thus reduced to the problem of finding the unknown density of the time domain boundary layer operators on the obstacle’s boundary, subject to the boundary data of the known incident wave. Using a Galerkin approach, the unknown density is replaced by a piecewise polynomial approximation, the coefficients of which can be found by solving a linear system. The entries of the system matrix of this linear system involve, for the case of a two dimensional scattering problem, integrals over four dimensional space-time manifolds. An accurate computation of these integrals is crucial for the stability of this method. Using piecewise polynomials of low order, the two temporal integrals can be evaluated analytically, leading to kernel functions for the spatial integrals with complicated domains of piecewise support. These spatial kernel functions are generalised into a class of admissible kernel functions. A quadrature scheme for the approximation of the two dimensional spatial integrals with admissible kernel functions is presented and proven to converge exponentially by using the theory of countably normed spaces. A priori error estimates for the Galerkin approximation scheme are recalled, enhanced and discussed. In particular, the scattered wave’s energy is studied as an alternative error measure. The numerical schemes are presented in such a way that allows the use of non-uniform meshes in space and time, in order to be used with adaptive methods that are based on a posteriori error indicators and which modify the computational domain according to the values of these error indicators. The theoretical analysis of these schemes demands the study of generalised mapping properties of time domain boundary layer potentials and integral operators, analogously to the well known results for elliptic problems. These mapping properties are shown for both two and three space dimensions. Using the generalised mapping properties, three types of a posteriori error estimators are adopted from the literature on elliptic problems and studied within the context of the two dimensional transient problem. Some comments on the three dimensional case are also given. Advantages and disadvantages of each of these a posteriori error estimates are discussed and compared to the a priori error estimates. The thesis concludes with the presentation of two adaptive schemes for the two dimensional scattering problem and some corresponding numerical experiments.
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Lindhe, Adam. "Reflected Stochastic Differential Equations on a Time-Dependent Non-Smooth Domain." Thesis, KTH, Matematisk statistik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-229073.
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In this thesis we prove existence and uniqueness for reflected stochastic differential equation on a specific non-smooth, time-dependent domain. The domain is the intersection of a finite number of smooth domains that are allowed to vary in time. The reflection is oblique to the domain and at the corners more than one direction of reflection is allowed. The time restrictions on the domain is firstly the existence of a semiconcave family of sets that are C¹;+ in time. Secondly that the distance function to the domain is in W¹;p. The first part of the proof is to construct of three kinds of test functions with desired properties. Using these test functions, existence is proved to the Skorokhod problem. Finally uniqueness is proved for the reflected stochastic differential equation.I den här mastersuppsatsen så bevisar vi existens och entydighet för reflekterade stokastiska differentialekvation på ett icke slätt, tidsberoende område. Området är snittet mellan ett ändligt antal släta områden som tillåts variera i tiden. Reflektionen är ej nödvändigtvis vinkelrät till området och i hörnen finns det mer än en tillåten riktning. Tidsrestriktionen på området är dels existensen av en familj av semikonkava mängder som är C¹;+ i tiden. Dessutom att avståndet till området är W¹;p i tiden. Första delen av beviset är att konstruera tre hjälp funktioner med eftersökta egenskaper. Med hjälp av de här funktionerna så bevisas sedan existens av lösningar till Skorokhod problemet. Slutligen så bevisas entydighet av den reflekterade stokastiska differentialekvationen.
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Hagdahl, Stefan. "Hybrid Methods for Computational Electromagnetics in Frequency Domain." Doctoral thesis, Stockholm : Numerisk analys och datalogi (NADA) ; Tekniska högsk, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-400.
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Tinniswood, Adam D. "Solution of time domain integral equations on distributed memory parallel processing systems." Thesis, University of York, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362022.
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Schwarzbach, Christoph [Verfasser], Klaus [Akademischer Betreuer] Spitzer, Klaus [Gutachter] Spitzer, Peter [Gutachter] Weidelt, and Eldad [Gutachter] Haber. "Stability of finite element solutions to Maxwell's equations in frequency domain / Christoph Schwarzbach ; Gutachter: Klaus Spitzer, Peter Weidelt, Eldad Haber ; Betreuer: Klaus Spitzer." Freiberg : TU Bergakademie Freiberg, 2009. http://d-nb.info/1220836885/34.
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Atle, Andreas. "Numerical approximations of time domain boundary integral equation for wave propagation." Licentiate thesis, KTH, Numerical Analysis and Computer Science, NADA, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-1682.
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Boundary integral equation techniques are useful in thenumerical simulation of scattering problems for wave equations.Their advantage over methods based on partial di.erentialequations comes from the lack of phase errors in the wavepropagation and from the fact that only the boundary of thescattering object needs to be discretized. Boundary integraltechniques are often applied in frequency domain but recentlyseveral time domain integral equation methods are beingdeveloped.
We study time domain integral equation methods for thescalar wave equation with a Galerkin discretization of twodi.erent integral formulations for a Dirichlet scatterer. The.rst method uses the Kirchho. formula for the solution of thescalar wave equation. The method is prone to get unstable modesand the method is stabilized using an averaging .lter on thesolution. The second method uses the integral formulations forthe Helmholtz equation in frequency domain, and this method isstable. The Galerkin formulation for a Neumann scattererarising from Helmholtz equation is implemented, but isunstable.
In the discretizations, integrals are evaluated overtriangles, sectors, segments and circles. Integrals areevaluated analytically and in some cases numerically. Singularintegrands are made .nite, using the Du.y transform.
The Galerkin discretizations uses constant basis functionsin time and nodal linear elements in space. Numericalcomputations verify that the Dirichlet methods are stable, .rstorder accurate in time and second order accurate in space.Tests are performed with a point source illuminating a plateand a plane wave illuminating a sphere.
We investigate the On Surface Radiation Condition, which canbe used as a medium to high frequency approximation of theKirchho. formula, for both Dirichlet and Neumann scatterers.Numerical computations are done for a Dirichlet scatterer.
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Dolean, Victorita. "Algorithmes par decomposition de domaine et méthodes de discrétisation d'ordre elevé pour la résolution des systèmes d'équations aux dérivées partielles. Application aux problèmes issus de la mécanique des fluides et de l'électromagnétisme." Habilitation à diriger des recherches, Université de Nice Sophia-Antipolis, 2009. http://tel.archives-ouvertes.fr/tel-00413574.
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My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.
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Rarata, Zbigniew. "Application and assessment of time-domain DGM for intake acoustics using 3D linearized Euler equations." Thesis, University of Southampton, 2014. https://eprints.soton.ac.uk/371795/.
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Fan noise is one of the major sources of aircraft noise. This can be modelled by means of frequency and time domain CAA methods. Frequency domain methods based on the convected Helmholtz equation are widely used for noise propagation and radiation from turbofan intakes. However, these methods are unsuited to deal easily with turbofan exhaust noise and presently unable to solve large 3D (three-dimensional) problems at high frequencies. In this thesis the application of time-domain Discontinuous Galerkin Methods (DGM) for solving linearized Euler equations is investigated. The research is focused on large 3D problems with arbitrary mean flows. A commercially available DGM code, Actran DGM, is used. An automatic procedure has been developed to perform the DGM simulations for axisymmetric and 3D intake problems by providing simple control of all the parameters (flow, geometry, liners). Moreover, a new method for integrating source predictions obtained from CFD calculations for the fan stage of a turbofan engine with the DGM code to predict tonal noise radiation in the far field has been proposed, implemented and validated. The DGM is validated and benchmarked for intake and exhaust problems against analytical solutions and other numerical methods. The principal properties of the DGM are assessed, best practice is defined, and important issues which relate to the accuracy and stability of the liner model are identified. The accuracy and efficiency of the CFD/CAA coupling are investigated and results obtained are compared to rig test data. The influence of the 3D intake shapes and the mean flow distortion on the sound field is investigated for static rig and flight conditions by using the DGM approach. Moreover, it is shown that the mean flow distortion can have a significant effect on the sound attenuation by a liner.