Derivation of high order absorbing boundary conditions for the Helmholtz equation in 2D

We present high order absorbing boundary conditions (ABC)for the Helmholtz equation in 2D, that can adapt to any regular shapedsurfaces. The new ABCs are derived by using the technique ofmicro-diagonalisation to approximate the Dirichlet-to-Neumann map.Numerical results on different shapes illustrate the behavior of thenew ABCs along with high-order finite elements

High Order Absorbing Boundary Conditions for the 2D Helmholtz Equation

International audienceWe present high order absorbing boundary conditions (ABC) for the 2D Helmholtz equation that can adapt to any regular shaped surface. The new ABCs are derived by using the technique of micro-diagonalisation to approximate the Dirichlet-to-Neumann map. Numerical results on different shapes illustrate the behavior of the new ABCs along with high-order finite elements

Conservative Semi-Lagrangian solvers on mapped meshes

International audienceWe are interested in the numerical solution of the collisionless kinetic or gyrokinetic equations of Vlasov type needed for example for many problems in plasma physics. Different numerical methods are classically used, the most used is the Particle In Cell method, but Eulerian and Semi- Lagrangian (SL) methods that use a grid of phase space are also very interesting for some applications. Rather than using a uniform mesh of phase space which is mostly done, the structure of the solution, as a large variation of the gradients on different parts of phase space or a strong anisotropy of the solution, can sometimes be such that it is more interesting to use a more complex mesh. This is the case in particular for gyrokinetic simulations for magnetic fusion applications. We develop here a generalization of the Semi-Lagrangian method on mapped meshes. Classical Backward Semi-Lagrangian methods (BSL), Conservative Semi-Lagrangian methods based on one-dimensional splitting or Forward Semi- Lagrangian methods (FSL) have to be revisited in this case of mapped meshes. A first use of the classical advective BSL method on a mapped mesh has been described in 1. We consider here the problematic of conserving exactly some equilibrium of the distribution function, by using an adapted mapped mesh, which fits on the isolines of the Hamiltonian. This could be useful in particular for Tokamak simulations where instabilities around some equilibrium are investigated. We also consider the problem of mass conservation. In the cartesian framework, the FSL method automatically conserves the mass, as the advective and conservative form are shown to be equivalent. This does not remain true in the general curvilinear case. Numerical results are given on some gyrokinetic simulations performed with the GYSELA code and show the benefit of using a mass conservative scheme like the conservative version of the FSL scheme

Multi-dimensional modeling and simulation of semiconductor nanophotonic devices

Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonic devices is an important tool in the development of future integrated light sources and quantum devices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically explore their optimization potential. The efficient implementation of multi-dimensional numerical simulations for computer-aided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in device-scale modeling of quantum dot based single-photon sources and laser diodes by self-consistently coupling the optical Maxwell equations with semiclassical carrier transport models using semi-classical and fully quantum mechanical descriptions of the optically active region, respectively. For the simulation of realistic devices with complex, multi-dimensional geometries, we have developed a novel hp-adaptive finite element approach for the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties of the photonic structures. For electrically driven devices, we introduced novel discretization and parameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on various applications, including vertical-cavity surface-emitting lasers, grating couplers and single-photon sources

Éléments finis d'ordre élevé pour maillages hybrides - Application à la résolution de systèmes hyperboliques linéaires en régimes harmonique et temporel

In this thesis, we are interested in the construction of high-order finite elements adapted to hybrid meshes for the resolution of time-dependent and time-harmonic linear hyperbolic systems. We paid a special attention to the construction of pyramidal elements. We search optimal finite elements for three different formulations, the optimality being in the sense of the convergence in the norm of the space considered for the formulation. For H^1 and H(curl) formulations, optimal nodal and hp finite elements are constructed. The elementary matrices are evaluated with appropriate quadrature formula, and error estimates are performed to check the convergence of the constructed optimal elements. For the LDG (Local Discontinuous Galerkin) formulation, we present finite elements using orthogonal basis functions that allow us to design fast construction of the mass matrix and matrix-vector product. In the three cases, we present numerical properties of the elements, we check numerically that we get the theoretical convergence, and we compare our elements with other elements found in the literature. Finally, numerical experiments in 3D are conducted with time-dependent and time-harmonic equations (wave or Helmholtz equation, and Maxwell's equations). We show the efficiency of hybrid meshes compared to pure tetrahedral meshes or hexahedral meshes obtained by splitting tetrahedra into four hexahedra.Dans cette thèse, nous nous intéressons à la construction d'éléments finis d'ordre élevé adaptés aux maillages hybrides, pour la résolution de systèmes hyperboliques linéaires en régimes harmonique et temporel. L'accent est plus particulièrement porté sur la construction d'éléments pyramidaux. On étudie trois formulations pour lesquelles on cherche des éléments finis "optimaux" au sens de la convergence dans la norme de l'espace considéré pour la formulation. Pour les formulations H^1 et H(rot), on construit des éléments finis "optimaux" nodaux et hp. Les matrices élémentaires sont évaluées grâce à des formules de quadrature adaptées et des estimations d'erreur sont effectuées pour vérifier la convergence des éléments optimaux construits. Pour la formulation discontinue LDG (Local Discontinuous Galerkin), on présente des éléments utilisant des fonctions de base orthogonales permettant de mettre au point une construction de la matrice de masse et un produit matrice-vecteur rapides. Dans le cas des trois formulations, on étudie les propriétés numériques des éléments construits, on vérifie que l'on retrouve bien numériquement la convergence théorique et on compare nos éléments avec d'autres éléments trouvés dans la littérature. Finalement, on présente des expériences numériques en 3D avec l'équation des ondes ou de Helmholtz, et les équations de Maxwell dans le cas des régimes temporels et harmoniques. On montre ainsi l'efficacité des maillages hybrides par rapport aux maillages purement tétraédriques ou aux maillages hexaédriques obtenus en découpant chaque tétraèdre d'un maillage purement tétraédrique en quatre hexaèdres

Generation of Higher-Order Polynomial Bases of Nédélec H(curl) Finite Elements for Maxwell's Equations

International audienceThe goal of this study is the automatic construction of a vectorial polynomial basis for Nédélec mixed finite elements, particular, the generation of finite elements without the expression of the polynomial basis functions, using symbolic calculus: the exhibition of basis functions has no practical interest

Generation of Higher-Order Polynomial Bases of Nédélec H(curl) Finite Elements for Maxwell's Equations

International audienceThe goal of this study is the automatic construction of a vectorial polynomial basis for Nédélec mixed finite elements, particular, the generation of finite elements without the expression of the polynomial basis functions, using symbolic calculus: the exhibition of basis functions has no practical interest

Higher-Order Discontinuous Galerkin Method for Pyramidal Elements using Orthogonal Bases

We study arbitrarily high-order finite elements defined on pyramids on discontinuous Galerkin methods. We propose a new family of high-order pyramidal finite element using orthogonal basis functions which can be used in hybrid meshes including hexahedra, tetrahedra, wedges and pyramids. We perform a comparison between these orthogonal functions and nodal functions for affine and non-affine elements. Different strategies for the inversion of mass matrix are also considered and discussed. Numerical experiments are conducted for 3-D Maxwell's equations

Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra

Classical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div)-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover the H(div) proposed spaces are completing the De Rham diagram with optimal elements previously constructed for H1 and H(curl) approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed

Éléments finis d'ordre élevé pour maillages hybrides (Application à la résolution de systèmes hyperboliques linéaires en régimes harmonique et temporel)

Dans cette thèse, nous nous intéressons à la construction d'éléments finis d'ordre élevé adaptés aux maillages hybrides, pour la résolution de systèmes hyperboliques linéaires en régimes harmonique et temporel. L'accent est plus particulièrement porté sur la construction d'éléments pyramidaux. On étudie trois formulations pour lesquelles on cherche des éléments finis optimaux au sens de la convergence dans la norme de l'espace considéré pour la formulation. Pour les formulations H1 et H(rot), on construit des éléments finis optimaux nodaux et hp. Les matrices élémentaires sont évaluées grâce à des formules de quadrature adaptées et des estimations d'erreur sont effectuées pour vérifier la convergence des éléments optimaux construits. Pour la formulation discontinue LDG (Local Discontinuous Galerkin), on présente des éléments utilisant des fonctions de base orthogonales permettant de mettre au point une construction de la matrice de masse et un produit matrice-vecteur rapides. Dans le cas des trois formulations, on étudie les propriétés numériques des éléments construits, on vérifie que l'on retrouve bien numériquement la convergence théorique et on compare nos éléments avec d'autres éléments trouvés dans la littérature. Finalement, on présente des expériences numériques en 3D avec l'équation des ondes ou de Helmholtz, et les équations de Maxwell dans le cas des régimes temporels et harmoniques. On montre ainsi l'efficacité des maillages hybrides par rapport aux maillages purement tétraédriques ou aux maillages hexaédriques obtenus en découpant chaque tétraèdre d'un maillage purement tétraédrique en quatre hexaèdresIn this thesis, we are interested in the construction of high-order finite elements adapted to hybrid meshes for the resolution of time-dependent and time-harmonic linear hyperbolic systems. We paid a special attention to the construction of pyramidal elements.We search optimal finite elements for three different formulations, the optimality being in the sense of the convergence in the norm of the space considered for the formulation. For H1 and H(curl) formulations, optimal nodal and hp finite elements are constructed. The elementary matrices are evaluated with appropriate quadrature formula, and error estimates are performed to check the convergence of the constructed optimal elements. For the LDG (Local Discontinuous Galerkin) formulation, we present finite elements using orthogonal basis functions that allow us to design fast construction of the mass matrix and matrix-vector product. In the three cases, we present numerical properties of the elements, we check numerically that we get the theoretical convergence, and we compare our elements with other elements found in the literature. Finally, numerical experiments in 3D are conducted with time-dependent and time-harmonic equations (wave or Helmholtz equation, and Maxwell's equations). We show the efficiency of hybrid meshes compared to pure tetrahedral meshes or hexahedral meshes obtained by splitting tetrahedra into four hexahedraPARIS-DAUPHINE-BU (751162101) / SudocSudocFranceF