Time-Domain Stability of Artificial Boundary Condition Coupled with Finite Element for Dynamic and Wave Problems in Unbounded Media

The finite element modeling of the dynamic and wave problems in unbounded media requires an artificial boundary condition to simulate the truncated infinite domain. The Dirichlet-to-Neumann boundary condition has been transformed from frequency to time domain by using the rational function approximation and auxiliary variable technique. It is extended to three-dimensional layer problem here. The resulting artificial boundary condition is stable itself in time domain, whereas the time-domain instability of the artificial boundary condition coupled with the finite element method is found for the foundation vibration recently and for the wave propagation here. A simple and effective method that introduces the damping proportional to the stiffness matrix in the finite element method is given to cure such coupling instability completely. The stabilized damping is so small that it does not affect the solution accuracy nearly. The numerical examples show the instability phenomenon and indicate the effectiveness of the damping method. The time-domain stability studies here can be a reference for the other artificial boundary conditions.

Transmission Line and Its Implementation

In unbounded media, wave propagation is supposed to be unguided. The existence of uniform plane wave is considered to be all through the space. Electromagnetic energy related with the wave stretched over a broad area. In TV and radio broadcasting, unbounded medium propagation of the wave is required. Here transmission of information is destined for one and all who may be interested. Another way of transmitting information is by guided media. Guided media acts to direct the transmission of energy from transmitter to receiver. Transmission lines are usually used in low frequency power distribution and in high frequency communications as well as in the ethernet and internet in computer networks. Two or more parallel conductors may be used to construct a transmission line, which connects source to a load. Typical transmission lines consist of coaxial line, waveguide, microstrip line, coplanar waveguide, etc. In this chapter, problems related with transmission lines are solved with the help of EM field theory and electric circuit theory.

A Comparative Study of Absorbing Layer Methods to Model Radiating Boundary Conditions for the Wave Propagation in Infinite Medium

Radiating boundary condition is an important consideration in the finite element modelling of unbounded media. Absorbing layer techniquessuch as Perfectly Matched Layers (PML) and Absorbing Layers by Increasing Damping (ALID) becoming popular as they are efficient in absorbing outward propagating waves energy. In this study, a comparative analysis has been carried out between PML and ALID+VABC (Absorbing Boundary conditions for Viscoelastic materials) methods. The methods are analyzedusing LS-DYNAexplicit solver and the efficiency is compared with standard solutions.The study concluded that PML requires less number of elements to model the boundary conditions when compared with ALID+VABC. But PMLrequires a smaller element length which increases overall computational time. Both the methods are efficient in absorbing the wave energy. However, PML requires additional implementation cost to solve the complex equations.