Summary
International Symposium on Antennas and Propagation
2009
Session Number:2F2
Session:
Number:2F2-5
Stabilizing 3D Volterra Time Domain Integral Equation Algorithms
Ahmed Y. Al-Jarro, Phillip D. Sewell, Trevor M. Benson, Ana Vukovic,
pp.711-714
Publication Date:2009/10/21
Online ISSN:2188-5079
Summary:
An accurate and stable 3D Volterra Time Domain Integral Equation TDIE algorithm capable of running on both structured rectangular grids and unstructured tetrahedral meshes is presented here for the first time. This development is in response to recent demands for numerical methods to solve volume integral equations in the time domain. The derivation of a flexible, stable, accurate and effective numerical tool is presented and moreover it is demonstrated that it is capable of accurately approximating the solution of problems accommodating arbitrarily shaped dielectric structures with complex features and time varying material properties. Canonical structures are considered here to demonstrate stability and accuracy of the algorithm. Substantial progress has been reported in order to make TDIE algorithms computationally efficient in comparison to established numerical methods that are based on the differential form of the Maxwell equations such as the FDTD [1] and TLM [2]. As in the case of FDTD [3], it has also been observed that almost all numerical approximation of higher dimensional TDIE solutions suffer from late time instabilities [4]. In the case of MoM [5] and FE [6] approaches, compensatory numerical treatment has focused on a combination of careful design and the choice of spatial and temporal bases functions. Nevertheless, complete removal of these late time oscillations has met with limited success in the general case. Consequently, it is still often required that special attention must be given to the geometry of the structure, its physical parameters and the mesh used. The problems of instabilities are often attributed to an accumulation on numerical errors due to the discretisation on the integrals involved. Consideration of the stability of computer algorithms based upon the Volterra TDIE formulation in 1D has previously been reported and their solution was significantly improved from the earlier implementations by means of both a semiimplicit formulation and a central difference Crank-Nicholson technique [7]. Also their solution has been demonstrated on a modified rectangular space-time meshes [7], and on unstructured triangular space-time meshes [8]. These simple modifications were found to radically increase the flexibility of the computer implementations of the algorithm, allowing numerical solutions that are both stable and accurate for various media without any reservation on the structure, its mesh or permittivity contrast between the discontinuity region and background [7, 8]. However, in the 3D case, a more general treatment of such numerical instabilities is needed, and this can be achieved by employing low pass filtering. A digital filter approach is used to successfully remove the components of the solution which are prone to instability [9]. Here we find that the use of the simple low pass filtering technique, also known as an averaging scheme, is adequate to stabilise the computations. This only requires the use of three field values at any given moment in time, and therefore saves on the overall computational overhead required to obtain the solution. Higher order, finite impulse response FIR filtering, [9] can also be considered; a future step towards complete characterisation of this approach which is promising to yield an unconditionally stable and accurate algorithm.