EMSolve Selected Results

Coaxial Waveguide and Numerical Dispersion

Computation of Eigenfrequencies and Eigenmodes of Three-Dimensional, Inhomogeneous Resonant Cavities

Eigenmodes of a Stanford Linear Accelerator Center (SLAC) Induction Cell

Full-Wave Simulation of Substrate Coupling Effects in Integrated Circuits

High-Order Spatial Discretization

Low-Frequency Signal Propagation in a Mountain Environment

Optical Trapping of Dielectric Microspheres

Photonic Band-Gap Waveguides

Radar Propagation in Buildings: A 10-Billion-Element Simulation

RF Signal Propagation in a Cave

Simulation of Accelerator Wakefields

Single-Mode Optical Fiber

TITAN Petawatt Facility EMP Modeling

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Coaxial Waveguide and Numerical Dispersion

In this example, we use EMSolve to simulate the propagation of an electromagnetic wave along a coaxial waveguide. The problem is excited with a time-dependent voltage source boundary condition applied to the input cap of the mesh representing the nondispersive TEM01 mode. The voltage source has a temporal profile equal to a ramped sine wave function and a spatial profile proportional to the inverse of the radial coordinate. A PEC boundary condition is applied to the inner and outer cylindrical walls while an absorbing boundary condition (ABC) is applied to the end cap of the mesh. An analytic (or exact) solution to this problem exists and is simply the value of the time- and space-dependent voltage source at the input boundary evaluated at the retarded time t' = t - c/z, where c is the speed of light in the guide and z is the propagation direction. This allows for a normed error analysis of the method, thus providing quantitative insight into the dispersion properties of the method. Due to numerical dispersion, the computed solution will gradually get out of phase with the exact solution.

The images above show two meshes of the coaxial waveguide, a fine mesh (left) and a coarse mesh (right) with an 8x reduction in the number of elements. The image below shows a magnitude plot of the computed electric field along with a sliced vector plot of the computed magnetic field.   Click on the image to see a movie of the simulation.

In the image below we plot the maximum computed L2 error as a function of the discrete time step for two different simulations: one using first-order (p = 1) basis functions on the fine mesh and the other using high-order (p = 2) basis functions on the coarse mesh with curvilinear surface elements (s = 2) on the inner and outer cylindrical walls. The error in the approximate electric field is computed for each element in the mesh using the L2 volume norm. Note that in both cases, the maximum global phase error due to numerical dispersion increases as a function of time, but the p-refined simulation yields a much slower rate of growth.

In the next image, we plot the base 10 log of the computed error as a function of propagation distance along the coarse mesh for a fixed time step value. We do this for the three cases: p = 1, 2, and 3. Again, note that as the degree of the basis functions p is increased, the maximum value and the growth rate of the phase error due to numerical dispersion is drastically decreased. Also note that for the p = 1 case, the phase error begins to decrease at around z = 60; this is because the computed wave is now a full 180 degrees out of phase with the exact wave. It should be noted that the improved performance of p-refinement comes at a corresponding increase in computational cost: the total number of problem unknowns for the p = 1 case is 14,910; for p = 2 there are 111,692 unknowns, and for p = 3 there are 368,466 unknowns.

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January 20, 2009