Thomas Rylander

Stable FEM-FDTD Hybrid Method for Maxwell's Equations

In this thesis edge elements are applied to solve several problems in computational electromagnetics. In particular, a hybrid scheme joining the Finite Element Method (FEM) and the Finite-Difference Time-Domain (FDTD) algorithm is developed, tested and exploited. The hybrid scheme combines the efficiency of FDTD with the ability of the FEM to model complex geometry. The hybrid scheme is rigorously proven to be stable up to the maximal FDTD time step without added dissipation and it is free from spurious solutions. The reflection from the FEM-FDTD interface is low. The hybrid scheme has been tested by computing the Radar Cross Section (RCS) for a Perfect Electric Conducting (PEC) sphere and the NASA almond. It has also been used for extensive parameter studies of patch antennas and transitions from a waveguide to a microstrip.

Application of stable FEM-FDTD hybrid to scattering problems

A recently developed, stable, finite-element method (FEM), finite-difference time-domain (FDTD) hybrid that eliminates the staircase approximation of complex geometries is tested by convergence studies for radar cross-sections. For a conducting sphere, 1 dB accuracy in all directions is obtained with nine cells per wavelength, whereas the NASA almond requires a higher resolution of about 15 cells per wavelength. For scatterers with a smooth boundary, the results converge quadratically with the mesh size, but for a horizontally polarized wave incident on the NASA almond, the order of convergence is lower because of singular fields at the tip.

Perfectly matched layer in three dimensions for the time-domain finite element method applied to radiation problems

A new perfectly matched layer (PML) formulation for the time-domain finite element method is described and tested for problems in three-dimensional space. An integral part of this PML algorithm is a novel time integration scheme based on Galerkins method with a piecewise linear temporal expansion of the electric field. Our time integration procedure is constructed by forming a linear combination of exact and trapezoidal integration applied to the temporal weak form, which reduces to the well-known Newmark scheme in the case where the PML is not present. This technique allows for a consistent time discretization of the electric field vector wave equation augmented with the PML, which includes terms that involve the electric field itself, time derivatives thereof and exponential functions temporally convolved with the electric field. The auxiliary variables that are associated with such convolutions are efficiently updated in an explicit manner by matrix-vector products. The new PML formulation is successfully tested on canonical problems. The PML can very efficiently absorb propagating electromagnetic waves, which is demonstrated by reflections coefficients as low as -100 dB. The PML formulation is capable of giving highly accurate results for very broad frequency bands which is confirmed by tests that show an error smaller than 2% over more than six octaves in frequency. Finally, we apply the PML formulation to a bow-tie antenna and comparisons with measurements show good agreement.

A brick-tetrahedron finite-element interface with stable hybrid explicit-implicit time-stepping for Maxwell's equations

A new brick-tetrahedron finite-element interface with stable hybrid explicit-implicit time-stepping for Maxwells equations is described and tested. The tetrahedrons are connected directly to the bricks, as opposed to previous curl-conforming formulations that use an intermediate layer of pyramids. The electric field is expanded in linear edge elements, which yields a discontinuous tangential electric field at the brick-tetrahedron interface and this discontinuity is treated by Nitsches method. In addition, tangential continuity for an arbitrary constant electric field is imposed in the strong sense at the interface, which makes it possible to avoid penalization that perturbs the frequency spectrum. This hybridization preserves the null-space of the curl-curl operator and is free from non-physical spurious modes, which is confirmed by numerical tests. The implicit Newmark time-stepping scheme is employed for the tetrahedrons, which allows for local mesh refinement without reduced time-step. For the brick elements, spatial lumping and explicit time-stepping is employed, which yields the standard finite-difference time-domain scheme. Furthermore, we prove that the explicit-implicit time-stepping employed at the hybrid interface is stable for time-steps up to the Courant limit of the explicitly time-stepped bricks. Second order of convergence is achieved for field solutions without singular behavior. The reflection from the brick-tetrahedron interface is small and scattering from a thin layer of tetrahedrons indicates levels at approximately -49dB for a resolution of 14 cells per wavelength.

An adjoint field approach to Fisher information based sensitivity analysis in electrical impedance tomography

An adjoint field approach is used to formulate a general numerical framework for Fisher information-based sensitivity analysis in electrical impedance tomography. General expressions are given for the gradients used in standard least-squares optimization, i.e. the Jacobian related to the forward problem, and it is shown that these gradient expressions are compatible with commonly used electrode models such as the shunt model and the complete electrode model. By using the adjoint field formulations together with a variational analysis, it is also shown how the computation of the Fisher information can be integrated with the gradient calculations used for optimization. It is furthermore described how the Fisher information analysis and the related sensitivity map can be used in a preconditioning strategy to obtain a well-balanced parameter sensitivity and improved performance for gradient-based quasi-Newton optimization algorithms in electrical impedance tomography. Numerical simulations as well as reconstructions based on experimental data are used to illustrate the sensitivity analysis and the performance of the improved inversion algorithm in a four-electrode measurement set-up.

Iterative solution of global electromagnetic wavefields with finite elements

The time-independent Maxwell equations are solved iteratively in 2D geometry for 3D global waves in plasma physics. Krylov space methods, such as the generalized- or the quasi-minimal residuals (GMRES or QMR), are applied together with an incomplete factorization (ILU) preconditioning to a formulation using nodal elements for the electromagnetic scalar and vector potentials. The plasma response is represented as a complex, frequency dependent, dielectric tensor operator and can be used for a variety of applications involving low frequency waves in a tokamak. The iterative approach does not only result in considerable memory savings, but it is also more efficient than a direct solution and paves the way for the parallelization of global wave and stability codes.

A linear nonconforming finite element method for Maxwell's equations in two dimensions. Part I: Frequency domain

We suggest a linear nonconforming triangular element for Maxwells equations and test it in the context of the vector Helmholtz equation. The element uses discontinuous normal fields and tangential fields with continuity at the midpoint of the element sides, an approximation related to the Crouzeix-Raviart element for Stokes. The element is stabilized using the jump of the tangential fields, giving us a free parameter to decide. We give dispersion relations for different stability parameters and give some numerical examples, where the results converge quadratically with the mesh size for problems with smooth boundaries. The proposed element is free from spurious solutions and, for cavity eigenvalue problems, the eigenfrequencies that correspond to well-resolved eigenmodes are reproduced with the correct multiplicity.

Shape optimization of the total scattering cross section for cylindrical scatterers

We propose and test a gradient-based shape optimization algorithm for the total scattering cross section of infinitely long cylinders, by means of changing the shape of the cylinders cross section. On the basis of the optical theorem, we derive sensitivity expressions for both dielectric and metal cylinders given an incident plane wave, where the wave vector is perpendicular to the cylinder axis. Both the transverse electric (TE) case and the transverse magnetic case are considered. The sensitivity expressions are based on the continuum form of Maxwells equations, and they provide the sensitivity with respect to an arbitrary number of shape parameters in terms of the field solution of the original scattering problem and an adjoint scattering problem. These results are used to construct a gradient-based optimization algorithm that we exploit for the reduction of the total scattering cross section in the TE case for metal cylinders, e.g., struts used in reflector antennas. We present optimized cross sections that are oblong in the direction of the incident wave vector, and some of these designs feature corrugations that are parallel to the cylinder axis. We show designs with asymmetric cross sections that yield a low monostatic scattering cross section for certain directions in combination with a low total scattering cross section, which can be used to reduce the noise temperature contributions from the upper strut in an inverted Y tripod reflector antenna.

Error analysis of singularity cancellation quadrature on curvilinear triangles

In computational electromagnetics, when using the method of moments (MoM) to solve surface integral equations, numerical integration of near-singularities is required. Here, a brief overview of a theoretical error analysis for the recently proposed Arcsinh transformation-based quadrature scheme, generalized to curvilinear triangle domains, is given. Gaussian product rule quadrature is also considered in this context. Accurate error prediction is demonstrated. Insights gained into the error mechanisms of the Arcsinh scheme enable one to use it with confidence where applicable. Such situations are mild near-singularities and especially, extreme near-singularities. These occur within the MoM.

Microwave resonator sensor for detection of dielectric objects in metal pipes

We present a microwave sensor for detection of dielectric objects in material mixtures that are transported in metal pipes. The sensor exploits several resonant modes that operate below the cut-off frequency of the dominant waveguide mode in the pipe. This non-invasive measurement technique is particularly attractive for detection of undesirable objects in certain industries. Initial measurements on a prototype sensor show good agreement with electromagnetic simulations and indicate favorable detection properties. A detection algorithm based on the likelihood-ratio test is evaluated based on synthetically generated data from a finite-element model that simulates inhomogeneous background materials. We conclude that the multi-mode feature of the sensor is advantageous for the ability to distinguish between undesirable objects and ordinary fluctuations in the background material.