# The PML Method in Acoustics

Acoustic scattering problems are often solved numerically by means of the finite element method. The Finite Element Method (FEM) is a general computational technique for finding approximate solutions of partial differential equations, but in itself it can only be used in finite computational domains. A significant subset of acoustic scattering and radiation problems needs to be simulated with open boundaries, which involved the development of various extensions that intend to make the method capable of treating infinite domains.

Since finite element calculations can only be performed in finite computational domains, one possible solution is to truncate the infinite region and set up an artificial boundary that encloses the region of interest. Then onto this imaginary surface a narrow layer is attached, in which the governing equations undergo some modifications in order to make the layer absorbent for outgoing waves.

This thesis presents a possible solution using absorbing layers, by implementing the Perfectly Matched Layer (PML) method under MATLAB environment. The wave equation is constructed in an anisotropic form that makes the solution decay continuously inside the layer. A wave outgoing from the physical domain enters the layer without any reflection because of the analytic continuation in the complex plane, and inside the PML it decays continuously. At the outer boundary of the PML it is reflected back, it decays continuously again, and by the time it reaches the boundary of the physical domain it is too weak to cause any error.

The thesis presents the deduction of the analytic and discrete forms of the one-, two- and three-dimensional PML, its implementation and analysis under MATLAB environment. The realized PML has been compared against two alternative methods that are used for similar problems, paying attention to computational speed and numerical error at the same time.

The basic formulation can only be applied to domains with simple boundaries. Therefore a new solution has been developed that uses only one local absorbing operator instead of the global absorbing functions that depend on global spatial coordinates. Using the new transformation it is feasible to handle PML around arbitrary boundaries.

The global and the local realizations of the PML have been incorporated into the MATLAB toolbox Nihu, which is developed by the BUTE Department of Telecommunications. I also have implemented a family of routines, which aid the construction and handling of arbitrary parameterized absorbing layers.

I have measured a real problem to validate the simulation model. Excitation was created on a car model, and both the acceleration on the surface and the sound pressure in the near field of the model was measured in an anechoic room. Then the measured and the computed sound pressure was compared on a hemisphere surface.