Introduction of Discrete Elements and Thin Wires in the TLM Method
3.1. Introduction of discrete elements [BIS 99]
3.1.1. History of 2D TLM
The idea of introducing localized nonlinear elements into the TLM was initially demonstrated by Johns and O’Brien [JOH 80], who used line sections to connect the element to the nodes of the mesh. Kosmopoulos [KOS 89] and Voelker and Lomax [VOL 90] modeled nonlinearities by using variable (with voltage) admittance line sections, whose value was updated at each time step.
Russer et al. [RUS 91] linked nonlinear elements to the TLM network using transmission lines, which were matched at the connection point of the admittance to the node, which led to an implied integration of the nonlinear equations defining the device. In all of these approaches, the time step used to update the coefficient values of the nonlinear equations were linked to the TLM time step. In order to ensure a high level of precision and stability in the solving of differential equations, the time step of the TLM had to be quite fine. However, as the “rigidity” of differential equations varies over time, this may lead to periods where the update time for the values of the coefficients is excessive and therefore the algorithm is not efficient. On the other hand, relaxing the sampling interval may be the cause of erroneous solutions and computation instabilities. The approach presented in this chapter avoids these limitations since it ensures the uncoupling of the discrete ...