Final Report Summary - TRANSEXMAT (Transient Analysis of Exotic Materials for Electromagnetics)
The three main goals of the project are:
1. The design of accurate and stable time domain boundary element methods that can model the interaction of electromagnetic fields with piecewise homogeneous materials that can be conducting, dielectric, paramagnetic, diamagnetic, lossy, chiral, or combinations thereof.
2. The design of a hybrid time domain FEM/BEM solver combining said BEM solver with a mature FEM implementation without jeopardizing stability and accuracy.
3. The parallel implementation of said hybrid solver to be able to be executed in a distributed heterogeneous and unreliable computing environment.
Up to this stage the following progress has been made towards these goals:
1. A novel discretization scheme for time domain boundary integral equations has been designed. The new scheme is based on a truly space-time variational approach to the discretisation of time domain boundary integral equations. Classically, the temporal discretization always has been of collocation type. These schemes are easier to implement and concerns regarding discrete causality are non-existent. Unfortunately, stability of these schemes is not guaranteed, and indeed, many papers report on the sensitivity of the stability properties on implementation details and the exact nature of the scattering problem. Our new formulation demonstrates some clear advantages over this classic approach:
- In numerical experiments, it is shown to give much more accurate results and this over the entire frequency range.
- Moreover, the scheme is much more robust. Even when using approximate integration rules, the method delivers stable results. In fact, in none of the examples of our test suite, instabilities have been observed. This improvement in stability is in accordance with the mathematical analysis of this class of time domain boundary element methods.
- By combining this novel temporal discretisation with advanced techniques for spatial discretization (the so-called mixed discretization), a combined field integral equation based solver has been constructed that in addition to the above advantages is not susceptible to interior resonances.
In all the numerical experiments that we conducted the solver provided stable results. This is not guaranteed to be always the case however. The cause of instabilities in many cases can be traced back to the possibility of constant or linear-in-time spurious current components to creep in the numerical approximation. In this project we managed to build a solver that does not allow this type of solutions. As a result, it is impossible for the new scheme to become unstable when affected by only small numerical deviations such as those resulting from finite machine precision or quadrature errors. Solvers demonstrating this type of robustness have been presented in the past, but in this work the idea has been significantly extended. In particular the solver designed in this project retains its stability properties even if the object that is simulated is complex in the sense that it contains topological loops or handles. In addition a numerical method to model the scattering of transient signals by penetrable objects with the same robustness characteristics has been introduced. This enables for the first time the solution of such electromagnetic interaction problems in the presence of sharp geometric features and otherwise reasonable levels of numerical error.
2. The George Green Institute for Electromagnetics Research within which this project is conducted has been pivotal in the development of the Transmission Line Modelling method (TLM). In particular it has been the GGIEMR that first introduced the unstructured TLM, allowing for the use of only mildly restricted classes of triangulations to model the geometry. This removes up to very large extend the so called stair casing error encountered in the structured TLM but also in other traditional methods such as FDTD. A scheme to couple the Unstructured TLM and the time domain boundary element method described above has been successfully designed. The so called Boundary Element / Unstructured TML (BEUT) is based on finding a one dimensional representation theorem for every boundary cell separating the BEM and TLM governed regions. Collecting these single cell representation theorems in a block diagonal matrix gives a representation theorem for the interior region valid for a single time step. At this point well-known recipes for the construction of integral equations for the transmission problem can be applied. Both the correctness and efficiency of the BEUT have been tested on benchmark examples and in real life scenarios. The main strength, and main reason for its development is the ability to model interaction of electromagnetic waves with complex devices containing interior regions filled by exotic natural or man-made materials and exterior regions where the field configuration is radiation dominated. The designed solvers does just that; these claims have been verified on the simulation of a system of Luneburg lenses.
3. Coupling the time domain boundary element method with the unstructured TLM method provides major improvements in terms of memory and time efficiency. Indeed homogeneous regions of the device and its surroundings do not need to be subdivided in simplices and as a result these regions do not count towards the number of degrees of freedom. Nevertheless the assembly of the interaction matrices and the computation of space-time convolutions during the marching-on-in-time solution remains a costly process. This is especially true for two-dimensional simulations, where the Green’s function (the response of the system to a singular source) has infinite extend in time. In the literature effective methods to mitigate this problem have been suggested such as the fast Hilbert transform based 2D plane wave time domain algorithm introduced by E. Michielssen et al., and the oblivious convolution quadrature method presented by M. Lopes-Fernandez et al.. The incorporation of these methods is was not achieved within this project. However in order to complete large scale simulations using the BEUT (such as the simulation of the Luneburg system), the matrix assembly process has been accelerated by multithreading. The resulting code has been deployed on a cloud based cluster to create our results.
Parts of the code developed in this project have been released to the public under the form of open source packages. The aim is to lower the threshold of novice researchers in the field and to shorten their initial training and induction. Implementing time domain boundary element methods is notoriously sensitive to implementation details. The packages made available are:
• https://github.com/krcools/WiltonInts84.jl : computation of potentials radiated by piecewise polynomial signals supported by a triangle. These routines provide the basis for most of today’s time domain boundary element methods.
• https://github.com/dan-phd/BEUT : Matlab implementation of the hybrid BEUT solver designed in this project. The package is modular to ease extension and contains code to run demos demonstrating the capabilities.
• https://github.com/dan-phd/2DTDBEM : C++ programme for the assemble of 2D boundary element method interactions matrices. This code is openMP enabled and can be compiled for deployment on *Nix/Windows machines.
1. The design of accurate and stable time domain boundary element methods that can model the interaction of electromagnetic fields with piecewise homogeneous materials that can be conducting, dielectric, paramagnetic, diamagnetic, lossy, chiral, or combinations thereof.
2. The design of a hybrid time domain FEM/BEM solver combining said BEM solver with a mature FEM implementation without jeopardizing stability and accuracy.
3. The parallel implementation of said hybrid solver to be able to be executed in a distributed heterogeneous and unreliable computing environment.
Up to this stage the following progress has been made towards these goals:
1. A novel discretization scheme for time domain boundary integral equations has been designed. The new scheme is based on a truly space-time variational approach to the discretisation of time domain boundary integral equations. Classically, the temporal discretization always has been of collocation type. These schemes are easier to implement and concerns regarding discrete causality are non-existent. Unfortunately, stability of these schemes is not guaranteed, and indeed, many papers report on the sensitivity of the stability properties on implementation details and the exact nature of the scattering problem. Our new formulation demonstrates some clear advantages over this classic approach:
- In numerical experiments, it is shown to give much more accurate results and this over the entire frequency range.
- Moreover, the scheme is much more robust. Even when using approximate integration rules, the method delivers stable results. In fact, in none of the examples of our test suite, instabilities have been observed. This improvement in stability is in accordance with the mathematical analysis of this class of time domain boundary element methods.
- By combining this novel temporal discretisation with advanced techniques for spatial discretization (the so-called mixed discretization), a combined field integral equation based solver has been constructed that in addition to the above advantages is not susceptible to interior resonances.
In all the numerical experiments that we conducted the solver provided stable results. This is not guaranteed to be always the case however. The cause of instabilities in many cases can be traced back to the possibility of constant or linear-in-time spurious current components to creep in the numerical approximation. In this project we managed to build a solver that does not allow this type of solutions. As a result, it is impossible for the new scheme to become unstable when affected by only small numerical deviations such as those resulting from finite machine precision or quadrature errors. Solvers demonstrating this type of robustness have been presented in the past, but in this work the idea has been significantly extended. In particular the solver designed in this project retains its stability properties even if the object that is simulated is complex in the sense that it contains topological loops or handles. In addition a numerical method to model the scattering of transient signals by penetrable objects with the same robustness characteristics has been introduced. This enables for the first time the solution of such electromagnetic interaction problems in the presence of sharp geometric features and otherwise reasonable levels of numerical error.
2. The George Green Institute for Electromagnetics Research within which this project is conducted has been pivotal in the development of the Transmission Line Modelling method (TLM). In particular it has been the GGIEMR that first introduced the unstructured TLM, allowing for the use of only mildly restricted classes of triangulations to model the geometry. This removes up to very large extend the so called stair casing error encountered in the structured TLM but also in other traditional methods such as FDTD. A scheme to couple the Unstructured TLM and the time domain boundary element method described above has been successfully designed. The so called Boundary Element / Unstructured TML (BEUT) is based on finding a one dimensional representation theorem for every boundary cell separating the BEM and TLM governed regions. Collecting these single cell representation theorems in a block diagonal matrix gives a representation theorem for the interior region valid for a single time step. At this point well-known recipes for the construction of integral equations for the transmission problem can be applied. Both the correctness and efficiency of the BEUT have been tested on benchmark examples and in real life scenarios. The main strength, and main reason for its development is the ability to model interaction of electromagnetic waves with complex devices containing interior regions filled by exotic natural or man-made materials and exterior regions where the field configuration is radiation dominated. The designed solvers does just that; these claims have been verified on the simulation of a system of Luneburg lenses.
3. Coupling the time domain boundary element method with the unstructured TLM method provides major improvements in terms of memory and time efficiency. Indeed homogeneous regions of the device and its surroundings do not need to be subdivided in simplices and as a result these regions do not count towards the number of degrees of freedom. Nevertheless the assembly of the interaction matrices and the computation of space-time convolutions during the marching-on-in-time solution remains a costly process. This is especially true for two-dimensional simulations, where the Green’s function (the response of the system to a singular source) has infinite extend in time. In the literature effective methods to mitigate this problem have been suggested such as the fast Hilbert transform based 2D plane wave time domain algorithm introduced by E. Michielssen et al., and the oblivious convolution quadrature method presented by M. Lopes-Fernandez et al.. The incorporation of these methods is was not achieved within this project. However in order to complete large scale simulations using the BEUT (such as the simulation of the Luneburg system), the matrix assembly process has been accelerated by multithreading. The resulting code has been deployed on a cloud based cluster to create our results.
Parts of the code developed in this project have been released to the public under the form of open source packages. The aim is to lower the threshold of novice researchers in the field and to shorten their initial training and induction. Implementing time domain boundary element methods is notoriously sensitive to implementation details. The packages made available are:
• https://github.com/krcools/WiltonInts84.jl : computation of potentials radiated by piecewise polynomial signals supported by a triangle. These routines provide the basis for most of today’s time domain boundary element methods.
• https://github.com/dan-phd/BEUT : Matlab implementation of the hybrid BEUT solver designed in this project. The package is modular to ease extension and contains code to run demos demonstrating the capabilities.
• https://github.com/dan-phd/2DTDBEM : C++ programme for the assemble of 2D boundary element method interactions matrices. This code is openMP enabled and can be compiled for deployment on *Nix/Windows machines.