A novel integral formulation for the solution of maxwell equations (original) (raw)
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International Journal for Research in Applied Science and Engineering Technology -IJRASET, 2020
This work is dedicated to the numerical results and the implementation of the method coupling a discontinuous Galerkin with an integral representation (CDGIR). The originality of this work lies in the choice of discretization by discontinuous Galerkin element and a mixed form for Maxwell's equations. The numerical tests justify the effectiveness of the proposed approach. Keywords: Finite element method, Maxwell equations, Discontinuous Galerkin method, fictitious domain, integral representation, time-harmonic. I. INTRODUCTION Mathematically, the phenomenon of the electromagnetic waves propagation is generally modeled by the system of equations known as the Maxwell equations. There are two modes of the Maxwell equations to be treated, a first mode that is known by the time domain Maxwell equations in which the evolution of electromagnetic fields is studied as a function of time and the second mode that is known by the frequency domain Maxwell equations where one studies the behavior of electromagnetic fields when the source term follows a harmonic dependence in time. Numerical modeling has become the most important and widely used tool in various fields such as scientific research. The finite-difference methods (FDM), the finite element methods (FEM) and the finite volume methods (FVM) are the three classes of methods known for the numerical resolution of the problems of electromagnetic waves propagation. In 1966, Yee cited the first efficient method in [42] which is the finite-difference methods in the time domain (FDMTD). When diffraction problems are posed in unbounded domain, the use of these methods induces a problem. In order to solve it, two techniques are used. The first consists in reducing to a bounded domain by truncating the computational domain, then it is necessary to impose an artificial condition on the boundary on the truncation boundary. The second technique consists in writing an equivalent problem posed on the boundary of the obstacle, it is therefore what is called the theory of integral equations. The numerical resolution can then be done by discretizing the problem by collocation (method of moments, method of singularity) or by a finite element discretization of the boundary. In 1980, Nedelec introduces the edge finite element method developed in [31] which is also available in [29, 30]. With the conservation of energy, this method also possesses several advantages; it allows to treat unstructured meshes (complex geometries) as it can be used with high orders (see [41, 24, 29]). In recent years, research has revealed a new technique known as Discontinuous Galerkin Methods (GDM); this strategy is based on combining the advantages of FEM and FVM methods since it approaches the field in each cell by a local basis of functions by treating the discontinuity between neighboring cells by approximation FVM on the flows. Initially, these methods have been proposed to treat the scalar equation of neutron transport (see [35]). In the field of wave propagation, precisely for the resolution of the Maxwell equations in the time domain, many schemes are based on two forms of formulations: a concentrated flux formulation (see [16, 34]) and an upwind flux formulation (see [22, 12]). Discontinuous Galerkin methods have shown their effectiveness in studying the problem with discrete eigenvalues (see [23]). In frequency domain, for the resolution of Maxwell equations, the majority consider the second order formulation (see [25, 32, 33]), as others study the formulation of the first order as in [6, 20]. This strategy of the CDGIR method allows us to write a problem in an unbounded domain into an equivalent problem in a domain bounded by a fictitious boundary where a transparent condition is imposed. This transparent condition is based on the use of the integral form of the electric and magnetic fields using the Stratton-chu formulas (see [7]). This process has been studied, in the
An integral equation is developed for determining the time-dependent current distribution on a wire structure excited by an arbitrary time-varying electric field. The subsectional collocation form of the method of moments is used to reduce this integral equation to a form that can be evaluated on a digital computer as an initial value problem. A Lagrangian interpolation scheme is introduced so that the dependent variables can be accurately evaluated at any point in the spacetime cone; thus, only mild restrictions on the space and time sample density are required. The integral equation relating present values of the current to previously computed values is presented in a form that can be directly converted into a computer code. Expressions are developed for the computer time and the relative advantages of time-domain and frequency-domain calculations are discussed, providing impetus for analyses in the time domain in certain cases. Part II of this paper will present well-validated numerical results obtained using the technique described.
A time-domain vector potential formulation for the solution of electromagnetic problems
IEEE Microwave and Guided Wave Letters, 1998
We present an alternative vector potential formulation of Maxwell's equations derived upon introduction of a quantity related to the Hertz potential. Once space and time are discretized, within this formulation the electric field and vector potential components are condensed in the same point in the elementary cell. In three dimensions the formulation offers an alternative to finite-difference time-domain (FDTD) method; when reduced to a two-dimensional (2-D) problem, only two variables, instead of three, are necessary, implying a net memory saving of 1/3 with respect to FDTD.
Journal of Mathematical Analysis and Applications, 2014
We develop and analyze a surface integral equation (SIE) whose solution pertains to numerical simulations of propagating time-harmonic electromagnetic waves in three dimensional dielectric media. The formulae to evaluate the far-field pattern and propagation of the electric and magnetic fields in the interior and exterior of a dielectric body, through surface integrals, require the solution of a 2 × 2 system of weakly-singular SIEs for the two unknown electric and magnetic fields at the interface surface of the dielectric body. The SIE is governed by an operator that is
A New Vector Potential BEM for Magnetic Fields Bounded by Perfect Conductors
IEEE Transactions on Magnetics, 2000
A novel formulation of the boundary integral equation for the magnetic vector potential is presented, where its normal component is imposed to be zero while, instead of enforcing its tangential component, only the circulations of the potential are imposed along any closed paths on the boundary. When the boundary is modelled to be a perfect conductor, these circulations are equal to 0. It is proposed to represent the tangential component of the vector potential as a linear combination of specialized vector functions obtained from the gradients of the nodal element functions. The tangential component of the magnetic induction can be represented in the same way if the boundary is not crossed by electric currents. The integral equation is projected on these vector functions and on an orthogonal set of vector functions simply obtained from the same nodal element functions. This yields an improved conditioning of the system matrix. The number of unknowns is only twice the number of nodes, thus making this method more efficient than existing methods employing the perfect conductor model. The proposed procedure can also straightforwardly be applied to the case when the region outside the perfect conductors is multiply connected, by introducing scalar unknowns associated with the "cuts" employed. It is of practical importance for efficient engineering computations of three-dimensional magnetic fields and inductances in complex conductor systems. Computation examples are given to illustrate the efficiency of the method presented for simply and for multiply connected regions.
Spectral Elements for the Integral Equations of Time-Harmonic Maxwell Problems
IEEE Transactions on Antennas and Propagation, 2000
We present a novel high-order method of moments (MoM) with interpolatory vector functions, on quadrilateral patches. The main advantage of this method is that the Hdiv conforming property is enforced, and at the same time it can be interpreted as a point-based scheme. We apply this method to field integral equations (FIEs) to solve time-harmonic electromagnetic scattering problems. Our approach is applied to the first and second Nédélec families of Hdiv conforming elements. It consists in a specific choice of the degrees of freedom (DOF), made in order to allow a fast integral evaluation. In this paper we describe these two sets of DOF and their corresponding quadrature rules. Sample numerical results on FIE confirm the good properties of our schemes: faster convergence rate and cheap matrix calculation. We also present observations on the choice of the discretization method, depending on the FIE selected.
Solution of Maxwell's equations
Computer Physics Communications, 1992
A numerical approach for the solution of Maxwell's equations is presented. Based on a finite difference Yee lattice the method transforms each of the four Maxwell equations into an equivalent matrix expression that can be subsequently treated by matrix mathematics and suitable numerical methods for solving matrix problems. The algorithm, although derived from integral equations, can be consideredto be a special case of finite difference formalisms. A large variety of two-and three-dimensional field problems can be solved by computer programs based on this approach: electrostatics and magnetostatics, low-frequency eddy currents in solid and laminated iron cores, high-frequency modes in resonators, waves on dielectric or metallic waveguides, transient fields of antennas and waveguide transitions, transient fields of free-moving bunches of charged particles etc.