The Origin of Spurious Solutions in Computational Electromagnetics

A discontinuous Galerkin least-squares method for div–curl systems

This study investigates and evaluates applications of the adjoint problem and its solution in frequency-domain computational electromagnetics (CEM). The study establishes and validates adjoint-based applications including higher order parameter sampling, a posteriori error estimate evaluation, and p- and h-refinements. These applications can improve the efficiency, automation, and robustness of CEM methods. We employ a 1-D finite-element-method scattering solver, simplifying the implementation, replicability, clarity, and intuitiveness of analysis results and conclusions, which then extend naturally to higher dimensional solvers and more complicated CEM problems. While demonstrated with a higher order solver, the derived techniques apply to low-order methodology as well. This is the first demonstration of the applicability of adjoint-based a posteriori error estimation techniques to adaptive discretization refinement in frequency-domain CEM with arbitrary-order basis functions. This work introduces the application of dual-weighted residual error estimation and selective adaptivity based on error cancellation. The proposed targeted, adaptive mesh/model p- and/or h-refinement heuristics informed by adjoint element-wise error contribution estimates show near-monotonic reduction of the quantity-of-interest error with increased numbers of refined elements. In general, adjoint techniques are underutilized in CEM, and another goal of this work is to promote their future use for refinement, optimization, and uncertainty quantification.

A fast non-similarity wavelet transform method, proposed earlier by the authors, is further studied. Combined with conventional techniques such as the method of moments, the boundary element method and the finite element method, the wavelet transform method can effectively handle a wide range of complicated problems which involve various complex media and conductors. The computational complexity and the CPU time for the proposed technique are presented. Studies further show that the use of non-similarity nonorthonormal cardinal spline wavelet (NS-NCSW) transform, although does not seem to reduce the computational complexity, can save a great amount of CPU time and computer memory. Numerical experiments are presented to demonstrate the merits of the proposed technique.

A new solution to Maxwell's differential equations is proposed. A new approach for writing solutions to these equations under consideration uses quaternions. The equations are written as a kind of generalization of the Cauchy-Riemann equations and have a form of partial differential equation of first order. The Green's function was found for direct (without potentials) solutions of Maxwell's equations. To calculate Green's function, we use factorization of the d'Alembert operator and the fact that Green's function for the d'Alembert operator is known. Three examples of determining the electromagnetic field strength were considered. This is an example of finding the strength of the electromagnetic field created by the charge q moving with constant speed v along the axis x1. The example of finding the electric field strength created by a uniformly charged thin rod at a point perpendicular to the rod at a distance R from the rod. The example of finding the electric field strength created by a dipole at a point located perpendicular to the middle of the dipole at a distance R from the middle of the dipole.

The solution of Helmholtz and Maxwell equations by integral formulations (kernel in exp(i kr)/r) leads to large dense linear systems. Using direct solvers requires large computational costs in O(N(3)). Using iterative solvers, the computational cost is reduced to large matrix-vector products. The fast multipole method provides a fast numerical way to compute convolution integrals. Its application to Maxwell and Helmholtz equations was initiated by Rokhlin, based on a multipole expansion of the interaction kernel. A second version, proposed by Chew, is based on a plane-wave expansion of the kernel. We propose a third approach, the stable-plane-wave expansion, which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane-wave expansion. The computational complexity is Nlog N as with the other methods.

9 - Computational Electromagnetics: The Finite-Difference Time-Domain Method

The finite-difference time domain (FDTD) method is a numerical technique based on the finite difference concept. It is employed to solve Maxwell's equations for the electric and magnetic field distributions in both the time and spatial domains. The FDTD method utilizes the central difference approximation to discretize two of Maxwell's curl equations, namely, Faraday's and Ampere's laws, in both the time and spatial domains, and then solve the resulting equations numerically to derive the electric and magnetic field distributions at each time step and spatial point using the explicit leap-frog scheme. The FDTD solution, thus derived, is second-order accurate, although the difference formulation is first order, and is stable if the time step size is chosen to satisfy the special criterion.

In this paper, a volume scattering model for a wider variety of scatterers in vegetation remote sensing is proposed by utilizing computational electromagnetics in developing the scattering model of the scatterers. In this proposed model, both Equivalent Principle Algorithm (EPA) and Relaxed Hierarchical Equivalent Source Algorithm (RHESA) are coupled together as the computational electromagnetics method in calculating the backscattering cross section from the scatterers available in vegetation medium. This paper firstly presents the related work and its research gap, followed by the discussion on the proposed approach and the methodology in developing the model. This paper also proposes the technique to validate the developed model.

In this paper, we propose a novel loop extraction algorithm that significantly outperforms other computational electromagnetics methods. Since the algorithm does not involve loop independence tests and the rejection of candidate loops, it is also more efficient than the polynomial-time loop extraction algorithms. Further tests demonstrate that our algorithm can find a spanning tree and associated subminimal loops in linear time for a donut structure consisting of many holes

There exists a great need to develop numerical techniques for efficient solution of multiscale electromagnetic problems, regardless of the computational electromagnetics (CEM) method currently being used to tackle them, be it FEM, FDTD or MoM. Dealing with multiscale objects, which have features that are both small and large compared to the wavelength, is highly challenging and often forces us to compromise the accuracy (relaxing the numerical discretization process when attempting to capture the small-scale features) in order to cope with the limited available resources in terms of CPU memory and time. In this paper, we introduce a new scheme that combines the Recursive Update Frequency Domain (RUFD) method with the Dipole Moment (DM) method, to solve multiscale problems in a numerically efficient manner.

In this work, a hybrid distributed/shared memory parallelization scheme is presented for the domain green’s function method (DGFM). The DGFM is a domain decomposition based computational electromagnetics (CEM) method used for analyzing large disjoint antenna arrays. The array configurations considered consists of identical elements, with regular or irregular array layouts. The hybrid MPI/OpenMP parallelization strategy introduced in this work offers satisfactory speedup performance. This allows for the simulation of large array geometries in a distributed computing environment.

To analyze the distribution of electric intensity of radio-pulse radiation in the plant environment with Colorado potato beetles it was developed the model in the form of the box filled with an isotropic dielectric medium with permittivity and conductivity. The nonstationary problem describing the distribution of electrical intensity of radio-pulse radiation by Maxwell's equations, using Laplace transform reduced to the Helmholtz equation. Solution of Helmholtz equations were obtained by separation of variables. After a series of transformations were obtained by the equation of the electric field components for the plant layer of the Colorado potato beetles. The study of the radio-pulse electromagnetic radiation distribution in the plant environment with Colorado potato beetles and their larvae will determine the necessary biotropic parameters (pulse-modulated frequency, pulse power, pulse period, pulse duration, time of exposure of the plant environment with Colorado potato beetles) radio-pulse energo-informational electromagnetic radiation to suppress reproduction ability of the beetles and destroying their larvae.

Even though electro-magnetic wave can be calculated from Maxwell's equations, the cause of electro-magnetic waves has not been fully understood. This paper proposes a Newton's laws of motion based aether theory to derive identical results as those from Maxwell's equations for free field. The authors suggest that every aether particle has a mass and occupies a volume in space. Every aether particle has translational movement and particle spin movement. The translational movement is similar to the gas particle moving in the air and it does not produce an electro-magnetic wave. The particle spin movement generates shear and a spin wave that will be shown to have the same results as Maxwell's equations. Detailed derivation of electro-magnetic wave solutions from the proposed aether theory and Maxwell's equations is presented in this paper to show the validation of this model.

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The ultimate goal in computational electromagnetics is to develop numerical methods and algorithms that are effective, stable, and give accurate and error-controllable results over a wide range of frequencies, material parameters, geometries, and applications. This article discusses obstacles and challenges in achieving that goal with frequency-domain surface integral-equation methods.