Numerical solution of inhomogeneous and non-linear ordinary differential equations using the TLM multicompartment model

The paper presents a simple algorithm for solving a system of inhomogeneous high order differential equations with variable coefficients. The method also provides a numerical solution to non-linear ordinary differential equations. The technique is based on reducing the high order equations into a system of first order rate equations. Through a simple translation process, the variables in the reduced set of equations are solved simultaneously by an iterative scheme using the TLM multicompartment model. The numerical technique is demonstrated by solving well-known second order differential equations. The numerical solutions are compared with the analytical solutions to the differential equations.     

A hybrid TLM‐FDTD method for the modelling of diffusion

A new hybrid TLM‐FDTD algorithm for solving diffusion problems is described. The method utilizes the transmission line model to define the time step and the FDTD's leap‐frog algorithm to determine the voltages and currents of the network analogue of the diffusion equation. Unlike the standard TLM method, the proposed one does not generate spurious oscillations. The method is explicit and can be used to solve highly non‐linear problems without the need to solve non‐linear equations. The implementation of a simple adaptive time‐stepping algorithm is also described. Copyright © 2000 John Wiley & Sons, Ltd.     

Circuit model for a symmetrical condensed TLM node

An equivalent circuit for a symmetrical condensed TLM node is presented. The basic circuit has twelve ports and consists of 24 1:1 ideal transformers. Three additional conductances model the conductivity of the medium. The circuit is extended with additional ideal transformers giving six ports for short-circuit and open-circuit stubs which model the permittivity and permeability of the inhomogeneous medium. The scattering matrix of the 18-port network is determined symbolically by solving a set of linear circuit equations. The matrix agrees with previously published results for a symmetrical condensed TLM node, obtained from Maxwell's equations.     

The efficiency of transmission-line matrix modelling—a rigorous viewpoint

The difference equations of the scalar linear transmission-line matrix (TLM) routine as introduced by Johns for numerically solving the diffusion equation are shown to be isomorphic to Goldstein's correlated random walk model of diffusion. For the infinite homogeneous bar their exact solution is derived algebraically and given in the form of Jacobi polynomials. This puts the TLM algorithm on a sounder mathematical and physical basis. The accuracy in solving the diffusion equation is investigated in general form and thus its astonishing efficiency explained. Several other basic questions of this numerical technique are also discussed.     

A method for numerical modelling of convection–reaction–diffusion using electrical analogues

In this paper, a new method, called the lumped‐component circuit method (LCM), is developed for one‐dimensioal and two‐dimensional convection–reaction–diffusion with low to moderate Peclet numbers, tested for modelling both steady‐state and transient problems, and compared with standard finite volume method (FVM) schemes. The method has been developed principally for solving equations with piecewise‐constant coefficients using nodes that are not positioned to correspond to the coefficient discontinuities. In such situations, the FVM solutions do not converge consistently as the node spacing is decreased, but LCM solutions do. In general, the LCM method is more accurate than the FVM schemes tested, and, while the computational cost of LCM is higher, results suggest that it can be more efficient. Like the transmission line method (TLM), it is an indirect scheme in which the problem to be solved is first represented by an analogous transmission line (TL). Unlike with TLM, however, the TL is then modelled using a lumped‐component circuit, the voltages at nodes within that circuit being calculated. Copyright © 2015 John Wiley & Sons, Ltd.     

Dispersion relation calculation of photonic crystals using the transmission line matrix method

In this paper we present the use of the transmission line matrix (TLM) method as a modelling tool for the computation of the dispersion relation of photonic crystals (PCs). First, the formulation of a complex‐valued TLM algorithm for the implementation of the periodicity condition on the sidewalls of the simulation domain (unit cell), is presented in detail. Then, the stability issues of this formulation are analysed from energy considerations. The analysis shows that the TLM method remains unconditionally stable since the equations enforcing the periodicity condition allow for the energy to be preserved as a constant during the simulations. Finally, the complex‐valued TLM algorithm is validated by simulating two PCs and by comparing the results with those predicted by alternative methods. Copyright © 2004 John Wiley & Sons, Ltd.     

TLM modelling using an SIMD computer

A major limitation of the transmission-line matrix (TLM) method used to solve Maxwell's equations is the long computation time required. The TLM scattering calculations involved can, however, be viewed as parallel in nature. This paper describes an effort to reduce computational time by using an SIMD, DAP multiprocessor computer employed to solve a two-dimensional TLM electromagnetic field formulation. A parallel algorithm based on the TLM scattering algorithm is designed and implemented using FORTRAN- PLUS Enhanced on an AMT DAP 510 machine. Here the connectivity of the DAP is exploited to simulate the intrinsic scattering behaviour on which the TLM algorithm relies. The results show that parallel processing on an SIMD machine such as the DAP is advantageous, especially for higher-order mesh sizes.     

TLM‐based solutions of the Klein–Gordon equation (Part I)

The transmission line matrix (TLM) method has become well established as a numerical solution scheme for wave problems in electromagnetics and, to a lesser extent, in acoustics and mechanics. It has also been applied to diffusion/heat‐conduction problems. Here the technique is extended to solving the Klein–Gordon equation that arises in Quantum Mechanics and in the dynamics of an elastically anchored vibrating string. In Part I, two novel, TLM‐based algorithms are presented and verified. By considering them as solving a special case of the more general ‘forced’ wave equation, they illustrate how, with care, the TLM algorithm can be adapted to model a wide range of effects. Copyright © 2001 John Wiley & Sons, Ltd.     

Transmission‐line matrix models for solving transient problems of diffusion with recombination

Starting from the general telegrapher's equation, we investigate two nodal network constructions for modelling diffusion with recombination by means of the transmission‐line matrix (TLM) method. The diffusion effect is modelled by the series and shunt capacitance in one approach, and by the series inductance and shunt resistance in the other. Both approaches use the series and shunt resistances to model the recombination effect. The constraint of using both TLM networks for solving transient problems of diffusion with recombination is found to be identical in terms of the physics behind the numerical routines. A practical way of determining the spatial resolution and iteration time step for accurate TLM numerical computations is suggested based on a simple frequency analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Dimensional soft tissue thermal injury analysis using transmission line matrix (TLM) method

The skin is sensitive to temperature change and the effect may not be significant while the temperature at the surface is below 44°C. However, higher surface temperatures (above 44°C) will further incur time burning and carbonization so that irreversible damage may happen. An investigation of the heating intensity and the duration of the exposure to the heating source suggested that when the surface temperature is greater than 51°C, the exposure time required to destroy the epidermis is so short that trans‐epidermal necrosis may occur. In this paper, we present one‐ and two‐dimensional numerical models based on transmission line matrix (TLM) method for a quantitative prediction of skin burn injury resulting from the exposure of the skin surface to a high temperature heat source. Transient temperatures are numerically estimated by solving the Pennes' bioheat equation, and the damage function denoting the extent of burn injury is calculated using the Arrhenius assumptions for protein damage rate. The TLM model is used to analyse the effects of exposure time and geometrical dimensions of mutlilayered skin, on the transient temperature distribution and damage extension. TLM results showed good agreement with other numerical sources, suggesting that TLM modelling can be used as a tool for an effective thermal diagnostic of burn injuries. Copyright © 2008 John Wiley & Sons, Ltd.     

Consistent finite difference modelling of Maxwell's equations with lossy symmetrical condensed TLM node

A propagator integral interpretation for general TLM processes is presented and applied to discretized Maxwell's equations. The approach provides exact algebraic solutions of finite difference equations along a quite universal scheme. Specifically, Johns's process based on the symmetrical condensed node is reconstructed and generalized as a clear-cut FD algorithm. The fields computed are exact time-domain solutions of the FD equations, provided that total voltages and currents are evaluated and stub quantities are not externally excited. Losses are very naturally incorporated into the TLM algorithm. The structural similarity of the TLM process (properly operated) with the propagator integral appears just as fundamental as general and should have further applications.     

Two-dimensional FDTD and TLM

Applying the method of moments to Maxwell's equations, Yee's two-dimensional FDTD scheme with central difference approximations and the two-dimensional TLM method are dervied from first principles of field theory. By comparing the eigenvalues of the two methods, the differences between two-dimensional FDTD and TLM are illustrated.     

Use of the two-dimensional TLM method in the electromagnetic simulation of thin semiconductor samples

A two-dimensional lossy shunt TLM node is incorporated into a TLM system and adapted to model for the first time the Maxwell field equations in thin semiconductor samples. Both the characteristics of the node and the TLM system itself are fully described. By considering a parallel-plate structure containing a thin GaAs sample, driven by a voltage source, it is shown, with an example, that this TLM technique can simulate the response of non-stationary electromagnetic fields in a semiconductor to applied excitations.     

A method for algebraic analysis of the TLM algorithm

An analogy is described between the TLM algorithm and discrete state-space control theory. The analogy is used to derive the characteristic equations corresponding to parametrized model structures. Characteristic equations corresponding to several widely used node structures are derived by this means and found to be consistent with the observed behaviour of corresponding TLM models. An example is given of the use of the analogy in teh predictive mode.     

Finite deffernce time-domain approximation of Maxwell's equations with non-orthogonal condensed TLM mesh

A convex hexahedral TLM mesh of arbitrary shape is presented and the transmission-line matrix method extended to any non-orthogonal configuration. The novel mesh constitutes a natural generalization of Johns' condensed node. The associated TLM process is analysed and reconstructed as a genuine finite difference time-domain algorithm. Nodal S-parameters are derived from discretized Maxwell's equations and canonical stability criteria yield the TLM timestep. Unitarity is discussed and energy conservation confirmed in the non-conductive case. A given block-diagonal representation of the S-matrix restrains processing time per node and iteration within the range of traditional methods. The shortcomings of the rigid classical grid, as the need for inaccurate staircasing approximations, are, however, ruled out. Our analysis takes advantage of the recently developed propagator integral approach.     

Electromagnetic transient scattering analysis in time-domain—comparison of TLM and TDIE methods

Here we consider two popular techniques useful for time domain signature analysis of direct scattering problem i.e., given the incident waveform and the target geometry, find the scattered field. We present in this paper a comparison of two popular time domain numerical techniques, viz., the transmission line matrix (TLM) method and the finite-difference time-domain (FDTD) method for electromagnetic scattering problems. Even though there are many similarities between the two methods, the modeling philosophy is different. Whereas in FDTD Maxwell’s equations are solved using a differencing scheme, in TLM a scattering approach akin to Huygens principle is implemented by replacing the space domain with a system of interconnected transmission-lines. The comparison is made via standard canonical shapes, a dielectric cube, and a dielectric sphere, to address the factors affecting accuracy, efficiency, and the required computer resources.     

Simulation of EM wave propagation in magnetoelectric media using TLM

In this paper, the simulation of general frequency‐dependent magnetoelectric material properties in time‐domain TLM is described. The formulation is developed from Maxwell's equations and the constitutive relations using bilinear 𝒵‐transform methods leading to a general Padé system. The approach is applicable to all frequency‐dependent linear materials including those displaying anisotropic and bianisotropic behaviour. The method is validated by the example of a chiral slab having an analytic solution. Copyright © 2001 John Wiley & Sons, Ltd.     

Accuracy of the frequency‐domain TLM method and its application to microwave circuits

This paper investigates the accuracy and convergence of frequency‐domain (FD) TLM solutions and describes a method to identify non‐physical solutions. The numerical dispersion characteristics of various discretization schemes (‘nodes’) are compared. The occurrence of non‐physical solutions when solving three‐dimensional problems is discussed and a method to identify the non‐physical solutions is described. The accuracy of the FDTLM method is shown to be of second order as long as singularities are absent, whereas it is between first and second order if the computational domain includes field singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

The calculation of radar cross-section (RCS) using the TLM method

The calculation of the radar cross-section (RCS) of complex bodies using the symmetrical condensed TLM method is presented. The technique is based on a near-to-far field transformation of the TLM calculated near fields. Several two-dimensional examples are presented which validate the method. The main advantage of utilizing techniques such as TLM for RCS computation lies in the ability to model arbitrary bodies with complex material compositions.