In this contribution a novel fast-converging integral equation method is introduced that can be used to solve the mixed transmission/scattering problems by composite structures including multiple domains and metallic coatings that can contain junctions. This is achieved by combining the global multi-trace method with the quotient space discretisation of the multi-screen boundary integral equation. The method is illustrated by means of a sufficiently general geometry, the discretisation is discussed, and an effective Calderon multiplicative preconditioner is introduced. Numerical results corroborate the correctness and efficiency of the method.@en

We demonstrate how the coupling of a full-wave time-domain boundary element method (BEM) solver with a circuit solver can be used to model 1) the generation of high frequency oscillations in resonant tunneling diode (RTD) oscillators, and 2) the mutual coupling and synchronization of non-identical RTDs with significant differences in frequencies to achieve coherent power combination. Numerical simulations show a combined output power of up to 3.7 times a single oscillator in synchronized devices. The non-differential conductance of the RTD is modeled as a lumped component with a non-linear current-voltage relationship. The lumped element is coupled to the radiating structure using a finite-gap model in a consistent and discretisation independent manner. The resulting circuit equations are solved simultaneously and consistently with time-domain electric field integral equations that model the transient scattering of electromagnetic (EM) fields from conducting surfaces that make up the device. This paper introduces three novel elements: (i) the application of a mesh independent feed line to the modelling of feed lines of RTD devices, (ii) the coupling of the radiating system to a strongly non-linear component with negative differential resistance, and (iii) the verification of this model with circuit models where applicable and against the experimental observation of synchronisation when two RTDs are placed in close proximity. These three elements provide a methodology that create the capacity to model RTD sources and related technology.@en

In this contribution, a well-conditioned method for the modelling of scattering by so-called multi-screens or PEC sheets including junctions is introduced. The method starts from the inflated screen approach by Claeys and Hiptmair. We introduce a Calderón preconditioner and a suitable discretisation scheme. The resulting scheme contains many more DoFs than strictly required. We will show how almost all redundancy can be removed without significant loss of effectiveness of the method.@en

In this paper, a global multi-trace method for the scattering of time-harmonic waves by a structure that can contain, in addition to dielectric and perfectly conducting regions also perfectly conducting thin sheets, is presented. The method is direct in the sense that the unknowns are traces and jumps of the fields in the structure. The flexibility of the method and the correctness of the solution will be demonstrated by realistic examples.@en

We have identified the sources of the different problems plaguing the EFIE at low frequencies in both the frequency and the TD, as well as their traditional cures. Despite their apparent effectiveness, these techniques have been shown to have a limited applicability because they introduce their own set of problems which include the high computational burden of the LS decomposition and its effect on the high-refinement conditioning of the FD-EFIE and the numerical instabilities introduced by the treatment of the TD-EFIE. Techniques leveraging qH projectors, immune from the aforementioned side-effects, have been introduced to address the different aspects of the low-frequency breakdown of the FD formulation and of the large time step breakdown of its TD counterpart. In case of the FD, using projectors allows the same re-scaling of the solenoidal and non-solenoidal parts of the RWG space as traditional LS, but it has the added benefits of not requiring identification of the global loops of the structure as well as not introducing any further high-refinement ill-conditioning. In the TD case, the projectors are still used to separate the loop and star parts of the discretized space, but this separation is used to apply the correct derivative and integrative terms to the different parts of the operators. Coupled with an adequate mixed time-discretization scheme, this technique fully addresses the low-frequency limitations of the TD-EFIE. Along with presenting these purely theoretical concepts, we have provided implemen-tation related hints, allowing the techniques presented in this chapter to be reliably and readily implemented into existing solvers. Finally, while we have addressed their low-frequency breakdown, both EFIE formulations still suffer from a high-refinement breakdown. While in standard low-frequency scenarios, a curing of low-frequency issues may suffice, for more pathological cases techniques addressing both break-downs may be required. Strategies based on qH projectors and Calderon identities have recently been introduced for the frequency and TD formulations [23, 40] and should be used in this case.

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This contribution introduces an integral equation based domain decomposition method for the modelling of scattering of time harmonic electromagnetic waves by a penetrable obstacle. It is a single source method constructed from the gap idea that lies at the heart of the global multi-trace formulation. The advantages compared to the existing global multi-trace formulations are that it only requires half of the unknowns, and that the system matrix is well-conditioned without requiring further regularisation or preconditioning.@en

We present a method for the numerical evaluation of 6D singular integrals appearing in Volume Integral Equations. It is an extension of the Sauter-Schwab/Taylor-Duffy strategy for singular triangle-triangle interaction integrals to singular tetrahedron-tetrahedron interaction integrals. This general approach allows to use different kinds of kernel and basis functions. It also works on curvilinear domains. Our approach is based on relative coordinates and splitting the integration domain into subdomains for which quadrature rules can be constructed. Further, we show how to build these tensor-product quadrature rules economically using quadrature rules defined over 2D, 3D and 4D simplices. Compared to the existing approach where the integral is computed as a sequence of 1D integrations significant speedup can be achieved. The accuracy and convergence properties of the method are demonstrated by numerical experiments.@en

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following shortcomings: 1) ill-conditioning when the frequency is low; 2) ill-conditioning when the discretization density is high; 3) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); 4) incorrect solution at low frequencies due to a loss of significant digits; and 5) the presence of spurious resonances. In this article, quasi-Helmholtz projectors are leveraged to obtain magnetic field integral equation (MFIE) that is immune to drawbacks 1)-4). Moreover, when this new MFIE is combined with a regularized electric field integral equation (EFIE), a new quasi-Helmholtz projector-combined field integral equation (CFIE) is obtained that also is immune to 5). The numerical results corroborate the theory and show the practical impact of the newly proposed formulations.

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The time domain-electric field integral equation (TD-EFIE) and its differentiated version are widely used to simulate the transient scattering of a time dependent electromagnetic field by a perfect electric conductor (PEC). The time discretization of the TD-EFIE can be achieved by a space-time Galerkin approach or, as it is considered in this contribution, by a convolution quadrature using implicit Runge-Kutta methods. The solution is then computed using the marching-on-in-time (MOT) algorithm. The differentiated TD-EFIE has two problems: 1) the system matrix suffers from ill-conditioning when the time step increases (low frequency breakdown) and 2) it suffers from the DC instability, i.e., the formulation allows for the existence of spurious solenoidal currents that grow slowly in the solution. In this article, we show that 1) and 2) can be alleviated by leveraging quasi-Helmholtz projectors to separate the Helmholtz components of the induced current and rescale them independently. The efficacy of the approach is demonstrated by numerical examples including benchmarks and real-life applications.

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