Stretched coordinate PML in TLM
As with all differential equation based numerical methods, open boundary problems in TLM require special boundary treatments to be applied at the edges of the computational domain in order to accurately simulate the conditions of an infinite propagating medium. Particular consideration must be given...
| Main Author: | |
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://eprints.nottingham.ac.uk/68933/ |
| _version_ | 1848800516210950144 |
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| author | Odeyemi, Jomiloju |
| author_facet | Odeyemi, Jomiloju |
| author_sort | Odeyemi, Jomiloju |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | As with all differential equation based numerical methods, open boundary problems in TLM require special boundary treatments to be applied at the edges of the computational domain in order to accurately simulate the conditions of an infinite propagating medium. Particular consideration must be given to the choice of the domain truncation technique employed since this can result in the computation of inaccurate field solutions. Various techniques have been employed over the years to address this problem, where each method has shown varying degree of success depending on the nature of the problem under study. To date, the most popular methods employed are the matched boundary, analytical absorbing boundary conditions (ABCs) and the Perfectly Matched Layer (PML). Due to the low absorption capability of the matched boundary and analytical ABCs a significant distance must exist between the boundary and the features of the problem in order to ensure that an accurate solution is obtained. This substantially increases the overall computational burden. On the other hand, as extensively demonstrated in the Finite Difference Time Domain (FDTD) method, minimal reflections can be achieved with the PML over a wider frequency range and for wider angles of incidence. However, to date, only a handful of PML formulations have been demonstrated within the framework of the TLM method and, due to the instabilities observed, their application is not widely reported. The advancement of the PML theory has enabled the study of more complex geometries and media, especially within the FDTD and Finite Element (FE) methods. It can be argued that the advent of the PML within these numerical methods has contributed significantly to their overall usability since a higher accuracy can be achieved without compromising on the computational costs. It is imperative that such benefits are also realized in the TLM method.
This thesis therefore aims to develop a PML formulation in TLM which demonstrates high effectiveness in a broad class of electromagnetic applications. Motivated by its suitability to general media the stretched coordinate PML theory will be basis of the PML formulation developed. The PML method developed in this thesis is referred to as the mapped TLM-PML due to the implementation approach taken which avoids the direct discretization of the PML equations but follows more closely to the classical TLM mapping of wave equations to equivalent transmission line quantities. In this manner the highly desired unconditionally stability of the TLM algorithm is maintained. Based on the mapping approach a direct stretching from real to complex space is thus applied to the transmission line parameters. This is shown to result in a complex propagation delay and complex frequency dependent line admittances/impedances. Consequently, this modifies the connect and scatter equations.
A comprehensive derivation of the mapped TLM-PML theory is provided for the 2D and 3D TLM method. The 2D mapped TLM-PML formulation is demonstrated through a mapping of the shunt node. For the 3D case a process of mapping the Symmetrical Condensed Node (SCN) is formulated. The reflection performance of both the 2D and 3D formulations is characterised using the canonical rectangular waveguide application. Further investigation of the capability of the developed method in 3D TLM simulations is demonstrated by applying the mapped TLM-PML in: (i) the simulation of planar-periodic structures, (ii) radiation and scattering applications, and (iii) in terminating materially inhomogeneous domains. A performance comparison with previously proposed TLM-PML schemes demonstrates the superior temporal stability of the mapped TLM-PML. |
| first_indexed | 2025-11-14T20:52:48Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-68933 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:52:48Z |
| publishDate | 2022 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-689332022-07-31T04:41:53Z https://eprints.nottingham.ac.uk/68933/ Stretched coordinate PML in TLM Odeyemi, Jomiloju As with all differential equation based numerical methods, open boundary problems in TLM require special boundary treatments to be applied at the edges of the computational domain in order to accurately simulate the conditions of an infinite propagating medium. Particular consideration must be given to the choice of the domain truncation technique employed since this can result in the computation of inaccurate field solutions. Various techniques have been employed over the years to address this problem, where each method has shown varying degree of success depending on the nature of the problem under study. To date, the most popular methods employed are the matched boundary, analytical absorbing boundary conditions (ABCs) and the Perfectly Matched Layer (PML). Due to the low absorption capability of the matched boundary and analytical ABCs a significant distance must exist between the boundary and the features of the problem in order to ensure that an accurate solution is obtained. This substantially increases the overall computational burden. On the other hand, as extensively demonstrated in the Finite Difference Time Domain (FDTD) method, minimal reflections can be achieved with the PML over a wider frequency range and for wider angles of incidence. However, to date, only a handful of PML formulations have been demonstrated within the framework of the TLM method and, due to the instabilities observed, their application is not widely reported. The advancement of the PML theory has enabled the study of more complex geometries and media, especially within the FDTD and Finite Element (FE) methods. It can be argued that the advent of the PML within these numerical methods has contributed significantly to their overall usability since a higher accuracy can be achieved without compromising on the computational costs. It is imperative that such benefits are also realized in the TLM method. This thesis therefore aims to develop a PML formulation in TLM which demonstrates high effectiveness in a broad class of electromagnetic applications. Motivated by its suitability to general media the stretched coordinate PML theory will be basis of the PML formulation developed. The PML method developed in this thesis is referred to as the mapped TLM-PML due to the implementation approach taken which avoids the direct discretization of the PML equations but follows more closely to the classical TLM mapping of wave equations to equivalent transmission line quantities. In this manner the highly desired unconditionally stability of the TLM algorithm is maintained. Based on the mapping approach a direct stretching from real to complex space is thus applied to the transmission line parameters. This is shown to result in a complex propagation delay and complex frequency dependent line admittances/impedances. Consequently, this modifies the connect and scatter equations. A comprehensive derivation of the mapped TLM-PML theory is provided for the 2D and 3D TLM method. The 2D mapped TLM-PML formulation is demonstrated through a mapping of the shunt node. For the 3D case a process of mapping the Symmetrical Condensed Node (SCN) is formulated. The reflection performance of both the 2D and 3D formulations is characterised using the canonical rectangular waveguide application. Further investigation of the capability of the developed method in 3D TLM simulations is demonstrated by applying the mapped TLM-PML in: (i) the simulation of planar-periodic structures, (ii) radiation and scattering applications, and (iii) in terminating materially inhomogeneous domains. A performance comparison with previously proposed TLM-PML schemes demonstrates the superior temporal stability of the mapped TLM-PML. 2022-07-31 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en cc_by https://eprints.nottingham.ac.uk/68933/1/thesis_resubmission_JOJU_without_corrections_highlighted.pdf Odeyemi, Jomiloju (2022) Stretched coordinate PML in TLM. PhD thesis, University of Nottingham. ABSORBING BOUNDARY CONDITIONS PERFECTLY MATCHED LAYER TRANSMISSION LINE MODELLING METHOD STRETCHED COORDINATE PML |
| spellingShingle | ABSORBING BOUNDARY CONDITIONS PERFECTLY MATCHED LAYER TRANSMISSION LINE MODELLING METHOD STRETCHED COORDINATE PML Odeyemi, Jomiloju Stretched coordinate PML in TLM |
| title | Stretched coordinate PML in TLM |
| title_full | Stretched coordinate PML in TLM |
| title_fullStr | Stretched coordinate PML in TLM |
| title_full_unstemmed | Stretched coordinate PML in TLM |
| title_short | Stretched coordinate PML in TLM |
| title_sort | stretched coordinate pml in tlm |
| topic | ABSORBING BOUNDARY CONDITIONS PERFECTLY MATCHED LAYER TRANSMISSION LINE MODELLING METHOD STRETCHED COORDINATE PML |
| url | https://eprints.nottingham.ac.uk/68933/ |