716 research outputs found
Accurate and efficient algorithms for boundary element methods in electromagnetic scattering: a tribute to the work of F. Olyslager
Boundary element methods (BEMs) are an increasingly popular approach to model electromagnetic scattering both by perfect conductors and dielectric objects. Several mathematical, numerical, and computational techniques pullulated from the research into BEMs, enhancing its efficiency and applicability. In designing a viable implementation of the BEM, both theoretical and practical aspects need to be taken into account. Theoretical aspects include the choice of an integral equation for the sought after current densities on the geometry's boundaries and the choice of a discretization strategy (i.e. a finite element space) for this equation. Practical aspects include efficient algorithms to execute the multiplication of the system matrix by a test vector (such as a fast multipole method) and the parallelization of this multiplication algorithm that allows the distribution of the computation and communication requirements between multiple computational nodes. In honor of our former colleague and mentor, F. Olyslager, an overview of the BEMs for large and complex EM problems developed within the Electromagnetics Group at Ghent University is presented. Recent results that ramified from F. Olyslager's scientific endeavors are included in the survey
Compatible finite element methods for numerical weather prediction
This article takes the form of a tutorial on the use of a particular class of
mixed finite element methods, which can be thought of as the finite element
extension of the C-grid staggered finite difference method. The class is often
referred to as compatible finite elements, mimetic finite elements, discrete
differential forms or finite element exterior calculus. We provide an
elementary introduction in the case of the one-dimensional wave equation,
before summarising recent results in applications to the rotating shallow water
equations on the sphere, before taking an outlook towards applications in
three-dimensional compressible dynamical cores.Comment: To appear in ECMWF Seminar proceedings 201
Transformation Optics, Generalized Cloaking and Superlenses
In this paper, transformation optics is presented together with a
generalization of invisibility cloaking: instead of an empty region of space,
an inhomogeneous structure is transformed via Pendry's map in order to give, to
any object hidden in the central hole of the cloak, a completely arbitrary
appearance. Other illusion devices based on superlenses considered from the
point of view of transformation optics are also discussed.Comment: 7 pages (two columns), 9 figures, to appear in IEEE Trans. Mag.,
invited paper in Compumag 2009 (Florianopolis, Brasil), corresponding slides
available on http://www.fresnel.fr/perso/nicolet
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Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods
Discontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and are crafted to be numerically stable by implicitly computing a near-optimal discrete test space as a function of a discrete trial space. Moreover, they are completely general in the sense that they can be applied to a variety of variational formulations, including non-conventional ones that involve non-symmetric functional settings, such as ultraweak variational formulations. In most cases, these properties have been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like in convection-diffusion problems and other singularly perturbed problems.
In this work, other features of DPG methods are systematically exploited and applied to different problems. More specifically, the versatility of DPG methods is elucidated by utilizing the underlying methodology to discretize four distinct variational formulations of the equations of linear elasticity. By taking advantage of interface variables inherent to DPG discretizations, an approach to coupling different variational formulations within the same domain is described and used to solve interesting problems. Moreover, the convenient algebraic structure in DPG methods is harnessed to develop a new family of numerical methods called discrete least-squares (DLS) finite element methods. These involve solving, with improved conditioning properties, a discrete least-squares problem associated with an overdetermined rectangular system of equations, instead of directly solving the usual square systems. Their utility is demonstrated with illustrative examples. Additionally, high-order polygonal DPG (PolyDPG) methods are devised by using the intrinsic discontinuities present in ultraweak formulations. The resulting methods can handle heavily distorted non-convex polygonal elements and discontinuous material properties. A polygonal adaptive strategy was also proposed and compared with standard techniques. Lastly, the natural high-order residual-based a posteriori error estimator ingrained within DPG methods was further applied to problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery.Computational Science, Engineering, and Mathematic
A nonlinear Bloch model for Coulomb interaction in quantum dots
In this paper we first derive a Coulomb Hamiltonian for electron--electron
interaction in quantum dots in the Heisenberg picture. Then we use this
Hamiltonian to enhance a Bloch model, which happens to be nonlinear in the
density matrix. The coupling with Maxwell equations when interaction with an
electromagnetic field is also considered from the Cauchy problem point of view.
The study is completed by numerical results and a discussion about the
advisability of neglecting intra-band coherences, as is done in part of the
literature.Comment: 17 pages. Journal of Mathematical Physics (2014) \`a para\^itr
An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations
We introduce a new numerical method for the time-dependent Maxwell equations
on unstructured meshes in two space dimensions. This relies on the introduction
of a new mesh, which is the barycentric-dual cellular complex of the starting
simplicial mesh, and on approximating two unknown fields with integral
quantities on geometric entities of the two dual complexes. A careful choice of
basis-functions yields cheaply invertible block-diagonal system matrices for
the discrete time-stepping scheme. The main novelty of the present contribution
lies in incorporating arbitrary polynomial degree in the approximating
functional spaces, defined through a new reference cell. The presented method,
albeit a kind of Discontinuous Galerkin approach, requires neither the
introduction of user-tuned penalty parameters for the tangential jump of the
fields, nor numerical dissipation to achieve stability. In fact an exact
electromagnetic energy conservation law for the semi-discrete scheme is proved
and it is shown on several numerical tests that the resulting algorithm
provides spurious-free solutions with the expected order of convergence.Comment: 34 pages, 14 figures, submitte
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