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From wave equations to complex coordinates: How do perfectly matched layers revolutionize numerical simulation?
In the field of numerical simulation, the application of perfect matching layer (PML) is undoubtedly an important technological breakthrough. This artificial absorber layer is designed specifically for the wave equation and is intended to simulate problems with open boundaries, and in particular plays an important role in the finite difference time domain method (FDTD) and the finite element method (FE).
PML is distinguished from ordinary absorbing materials because it is designed so that waves incident on PML from non-PML media are not reflected at the interface.
In 1994, Berenger pioneered the concept of perfect matching layers, originally targeting Maxwell's equations. Over time, the technique has been reconstructed to accommodate a variety of wave equations, including elastodynamics, linear Euler's equation, Helmholtz's equation, and poroelasticity. Berenger's original formulation was called split field PML, which meant splitting the electromagnetic field into two non-physical fields within the PML region. The later single-axis PML (UPML) became widely adopted due to its simplicity and efficiency.
UPML describes PML as a man-made anisotropic absorbent material.
These different expressions can ultimately be boiled down to a more elegant and general method: the scaling transformation of complex coordinates. This view allows derivation of PML to cover inhomogeneous media such as waveguides, as well as other coordinate systems and wave equations.
Technical description
To design a PML for a rightward propagating wave, one of the variables in the wave equation is removed and transformed into ∂/∂x → 1/(1 + iσ(x)/ω) * ∂/∂ x, where ω is the angular frequency and σ is a function of some x. The purpose of this is to reduce the propagation amplitude of the wave when σ is positive, so that it no longer reflects when it contacts the PML, but disappears in an exponential attenuation manner.
This transformation causes attenuation of the wave whenever x dependence is of the form
e^ikx.
This coordinate transformation can be retained in the transformed wave equation, or combined with the material description to form a UPML description. The absorption coefficient σ of PML is usually frequency dependent, which means that waves of different frequencies will be absorbed to different extents.
Limitations of perfect match layers
Although PML is widely used in computational electromagnetics and has become the absorbing boundary technique of choice, it still has some limitations. In some important cases, PML may not work effectively and lead to unavoidable reflections or exponential growth. After the wave equation is discretized on the computer, small numerical reflections will appear. This is because the design of PML can only work for accurate continuous wave equations.
PML output usually needs to be turned on gradually, and its absorption coefficient σ is usually slowly increased from zero to reduce unnecessary numerical reflections.
Another limitation worth noting is that the performance of PML also suffers when the material exhibits a "backward wave" solution. In some negative index materials, instability occurs, leading to exponential growth. While this problem can be solved by flipping the sign of σ, things get more complicated if the material itself is frequency dependent.
Future exploration
As a technology, perfectly matched layers reshape the boundary conditions of numerical simulations in a unique way. However, as technology advances, we still need to explore and improve its application in special situations, especially when dealing with complex media and heterogeneous structures. Does this mean that the development potential of PML has not yet been fully explored?
