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The need to analyze aircraft noise over ground with general properties occurs in various applications, most notably in environmental engineering, where the analysis of the sound pressure level (SPL) distribution near the ground due to aircraft noise is desired. Since the human hearing range is very wide, the determination of the SPL distribution for a given source spectrum is not trivial and may be regarded as a multiscale problem. One has to solve repeatedly, for many different wave numbers, the Helmholtz equation in the upper half space while imposing the given impedance boundary condition on the possibly non-flat ground. An efficient Helmholtz solver must therefore be incorporated in the scheme that finds the SPL distribution. Three totally different computational methods that may be used to solve this problem are considered here: A fictitious sources method, a parabolic approximation method and a wave-enriched finite element method (with two versions: PUM and GFEM). The three methods are compared in terms of their computational properties, and numerical examples are presented to demonstrate their performance.

A semi-analytic solution is presented for multiple inhomogeneous inclusions and cracks in a half-space under elastohydrodynamic lubrication contact. In formulating the governing equations, each inhomogeneous inclusion embedded under the contacting surfaces is modeled as a homogeneous inclusion with initial eigenstrains plus unknown equivalent eigenstrains by employing Eshelby's equivalent inclusion method, while each crack of mixed modes I and II is treated as a distribution of climb and glide dislocations with unknown densities according to the dislocation distribution technique. Such a treatment converts the problem into a homogeneous lubricated contact with disturbed deformation due to the inclusions and cracks. The unknowns in the governing equations are integrated by a numerical algorithm and determined iteratively by utilizing a modified conjugate gradient method. The iterative process is performed until the convergence of the half-space surface displacements, which involve the displacements due to the inhomogeneous inclusions and cracks as well as the fluid pressure. Samples are presented to demonstrate the generality of the solution.