Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Finite element and boundary element analysis of electromagnetic NDE phenomena

The endeavor to produce quality products coupled with a drive to minimize failure in major industries such as aerospace, power and transportation is the driving force behind studies of electromagnetic nondestructive evaluation (NDE) methods. Popular domain and integral methods used for the purpose of modeling electromagnetic NDE phenomena include the finite element and boundary element methods. However no single numerical modeling technique has emerged as the optimal choice for all electromagnetic NDE processes. In a computer aided design environment, where the choice of an optimum modeling technique is critical, an evaluation of the various aspects of different numerical approaches is extremely helpful;In this dissertation, a comparison is made of the relative advantages and disadvantages of the finite element (FE) and boundary element (BE) methods as applied to the DC and AC Potential drop (DCPD and ACPD) methods for characterizing fatigue cracks. The comparison covers aspects of robustness, computer resource requirements and ease of numerically implementing the methods. Two dimensional FE and BE models are used to model an infinitely thin fatigue crack using the ACPD method, and a two and three dimensional FE and BE model is used to study the compact tension and single edge notch specimen using the DCPD method. Calibration curves and field plots in the specimen are compared to experimental and analytical data. The FE and BE methods are complementary numerical techniques and are combined to exploit their individual merits in the latter part of this dissertation. A three dimensional hybrid formulation to model eddy current NDE is then developed which discretizes the interior with finite elements and the exterior with boundary elements. The three dimensional model is applied to an absolute eddy current coil over a finite block. A feasibility study to confirm the validity of the formulation is undertaken by comparing the numerical results for probe lift-off and coil impedance measurements with published data;This comparative study outlined above indicates that when the solution is required at discrete points, as in the potential drop methods, or the model needs to handle infinite boundaries, as in eddy current NDE, the boundary element model is more suitable. Since it is based 011 the Green\u27s function, the BE method is limited to linear problems. Finite element analysis gives full field solutions, making it ideal for studying energy/defect interactions. The hybrid FE/BE formulation handles non-linearity and infinite boundaries naturally, thus utilizing the best of both worlds

Discretisations and Preconditioners for Magnetohydrodynamics Models

The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In the first part of this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretisation of the $\mathbf{B}$-$\mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. Our approach relies on specialised parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. In the second part, we focus on incompressible, resistive Hall MHD models and derive structure-preserving finite element methods for these equations. We present a variational formulation of Hall MHD that enforces the magnetic Gauss's law precisely (up to solver tolerances) and prove the well-posedness of a Picard linearisation. For the transient problem, we present time discretisations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. In the third part, we investigate anisothermal MHD models. We start by performing a bifurcation analysis for a magnetic Rayleigh--B\'enard problem at a high coupling number $S=1{,}000$ by choosing the Rayleigh number in the range between 0 and $100{,}000$ as the bifurcation parameter. We study the effect of the coupling number on the bifurcation diagram and outline how we create initial guesses to obtain complex solution patterns and disconnected branches for high coupling numbers.Comment: Doctoral thesis, Mathematical Institute, University of Oxford. 174 page

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

Incorporation of feed-network and circuit modeling into the time-domain finite element analysis of antenna arrays and microwave circuits

In this dissertation, accurate and efficient numerical algorithms are developed to incorporate the feed-network and circuit modeling into the time-domain finite element analysis of antenna arrays and microwave circuits. First, simulation of an antenna system requires accurate modeling of interactions between the radiating elements and the associated feeding network. In this work, a feed network is represented in terms of its scattering matrix in a rational function form in the frequency domain that enables its interfacing with the time-domain finite element modeling of the antenna elements through a fast recursive time-convolution algorithm. The exchange of information between the antenna elements and the feed network occurs through the incident and reflected modal voltages/currents at properly defined port interfaces. The proposed numerical scheme allows a full utilization of the advanced antenna simulation techniques, and significantly extends the current antenna modeling capability to the system level. Second, a hybrid field-circuit solver that combines the capabilities of the time-domain finite element method and a lumped circuit analysis is developed for accurate and efficient characterization of complicated microwave circuits that include both distributive and lumped-circuit components. The distributive portion of the device is modeled by the time-domain finite element method to generate a finite element subsystem, while the lumped circuits are analyzed by a SPICE-like circuit solver to generate a circuit subsystem. A global system for both the finite-element and circuit unknowns is established by combining the two subsystems through coupling matrices to model their interactions. For simulations of even more complicated mixed-scale circuit systems that contain pre-characterized blocks of discrete circuit elements, the hybrid field-circuit analysis implemented a systematic and efficient algorithm to incorporate multiport lumped networks in terms of frequency-dependent admittance matrices. Other advanced features in the hybrid field-circuit solver include application of the tree-cotree splitting algorithm and introduction of a flexible time-stepping scheme. Various numerical examples are presented to validate the implementation and demonstrate the accuracy, efficiency, and applications of the proposed numerical algorithms

Coupled structural, thermal, phase-change and electromagnetic analysis for superconductors, volume 2

Two families of parametrized mixed variational principles for linear electromagnetodynamics are constructed. The first family is applicable when the current density distribution is known a priori. Its six independent fields are magnetic intensity and flux density, magnetic potential, electric intensity and flux density and electric potential. Through appropriate specialization of parameters the first principle reduces to more conventional principles proposed in the literature. The second family is appropriate when the current density distribution and a conjugate Lagrange multiplier field are adjoined, giving a total of eight independently varied fields. In this case it is shown that a conventional variational principle exists only in the time-independent (static) case. Several static functionals with reduced number of varied fields are presented. The application of one of these principles to construct finite elements with current prediction capabilities is illustrated with a numerical example