Performance improvement of meshless methods for practical time domain electromagnetic analyses.

Abstract

Solution of Maxwell's equations for practical antennas, filters, waveguides and other microwave devices requires use of numerical methods. Popular numerical methods include the finite difference time domain method (FDTD), finite element method (FEM), method of moments (MoM), and several commercial software tools have been developed based on these. Time domain electromagnetic simulations using FDTD, that are suitable for wideband signals, require large computational resources especially for analyzing complex or multi-scale geometric structures. Recently, a method based on radial basis functions (RBFs) that work with arbitrary node distribution has been introduced to tackle curved geometries in time domain simulations. This research work addresses special challenges in implementing an in-house software using this meshless method for analysing some practical EM problems. The time-domain formulation of Maxwell's equations using scalar RBFs in 2D is im- plemented at first. These RBFs interpolate fields anywhere in the domain from the known values at a set of points within the radius of a support domain with a high accuracy. The simulation domain is assumed as a space of unknown electric and magnetic field vectors wherein all components of these vectors are collocated but the electric and magnetic field locations are interleaved in the space. Components of the electric field at different loca- tions are calculated as an interpolation of magnetic field components within the support domain, and vice-versa. These interpolations depend primarily on the locations of nodes which may be arbi- trarily located, leading to the advantage compared to conventional FDTD, where fields are located only on Cartesian grids. In this interpolation method, RBFs can be used to expand the spatial derivatives of fields and the marching in time by leapfrog method. This time marching of the fields using RBFs is known as time-domain Radial Point Inter- polation Method (RPIM), since it employs radial basis function for interpolation of fields. Furthermore, this formulation can address nodes distributed along a conformal boundary unlike the traditional FDTD. In order to implement RPIM for practical simulations, a node distribution is required. Manually generated entries may lead to overlapping nodes which may cause erroneous so- lutions, especially for complex geometries. In order to automate the process of generating a mesh, a commercial software COMSOL was used to generate the preliminary mesh and an in-house interpreter program was employed to export this mesh geometry for the pri- mary (electric field) and the secondary (magnetic field) nodes were obtained using Voronoi tessellation. During this step material properties are incorporated at each node. This scalar RBF formulation is implemented for a WR-229 based waveguide bend and a quarter circular ring resonator to demonstrate the effectiveness of RPIM implementa- tion to achieve lower computational cost than FDTD. FDTD implementation of circular/ curved geometries require a large number of degrees of freedom since a rectangular do- main superscribing the geometry is to be discretized. On the other hand, only nodes within the interior region and boundary are included in RPIM. The interface boundary conditions between these regions are appropriately derived for this approach. It has been demonstrated in the thesis that a hybrid interface of FDTD and RPIM has the potential to further reduce the computational cost. In scalar formulation for RPIM, the electric fields and magnetic fields are expanded as scalar components of these vector fields. But to ensure the electric and magnetic fields remain divergence-free, the basis function can be modified. These divergence-free RBFs reduce spurious solutions compared to the scalar RBF based RPIM method. In this thesis, these divergence-free RBFs are extended to model uniaxial perfectly matched layer (UPML) in 3D. An in-house code was developed using this formulation for 2D and 3D time domain EM problems. This implementation requires evaluation of optimum shape parameters to calculate parameters of the RBF accurately. This implementation used in practical waveguide problems such as E-plane metal insert fi lter and H-plane waveguide filters. Early formulations of RPIM in electromagnetics use Gaussian basis functions which result in ill-conditioned system matrices leading to inaccuracies in finding the inverse of matrices. Due to the highly ill-conditioned nature, Gaussian RBF results in spurious real parts of eigenvalues and long time energy instability in some problems. In order to effciently solve these issues, other functions such as inverse multiquadric RBFs are explored in this thesis. The divergence free implementation is also extended to use inverse multiquadric RBFs. The signi ficant real parts (of the order of 106 in Gaussian RBF) are reduced to below 10ð€€€2 with IMQ RBF and this also improved the long-time stability. A scheme is proposed to evaluate the optimum value of shape parameter that affects the accuracy and condition number in inverse multiquadric RBF to minimise the computation error. Being an explicit time domain method, the time step in divergence-free RPIM must satisfy the CFL stability criteria as in FDTD. In multi-scale geometries this leads to shorter time steps and hence require longer execution time and more storage. Laguerre polynomial has been reported to convert time into a polynomial order domain and the implicit linear equations are solved by marching in the order of the polynomial. This implicit method using Laguerre polynomials is extended in this research to overcome the above limitations in using the divergence free RBFs for complex problems. This implicit method developed here has shown improvement of time step about eight times the Courant factor for a quarter ring resonator. The results using our simulations are compared with time domain RPIM methods. In summary, this research work focused on time domain numerical methods in elec- tromagnetics to address various challenges to develop in-house EM simulation programs. Results of numerical simulations using the proposed approaches have been compared with other simulations. IMQ RBFs are proposed for efficient calculation with lower condition number and long time stability. Time domain implementation of divergence-free for- mulations of RPIM incorporating non-uniform grids and UPML boundaries have been developed for the analysis and these have been extended further to overcome limits due to the CFL criteria by using Laguerre polynomials.