Accurate Treatment of a General Sloping Interface in a Finite-Element 3d Narrow-Angle Pe Model

Abstract

Parabolic equation models in 3D usually apply the "staircase" approximation to general rangevarying interfaces between adjacent layers. This is the simplest technique available: it consists in neglecting range and azimuthal derivatives in the associated interface conditions. Our aim in this paper is to analyze the influence of the stair-step approximation technique, common to most 3D PE models, on a one-way sound wave propagation problem. We present a new finite-element 3D narrowangle PE model which accurately treats the variable interface conditions. This is accomplished by using (i) an appropriate parabolized condition of the same aperture as the parabolic equation used, and (ii) a new change-of-variable technique which does not require any homotheticity condition of the layers as in previous works. Numerical simulations for the 3D wedge problem are presented. The convergence of the numerical solutions with respect to the azimuth is investigated. Unlike other 3D PE models working in cylindrical coordinates, the convergence tests have been carried out using a range-dependent number of points in azimuth. Numerical solutions obtained with the newly developed model are compared with a reference solution based on the image source and with a solution obtained with a 3D PE model that uses a stair-step technique.

An iterative three-dimensional parabolic equation solver for propagation above irregular boundaries

“…For the 3D PE, a marching scheme can be derived for V m ¼ vecðW m Þ, which is a vector of size ðN Â N g Þ obtained by stacking the columns of W m (Sturm and Kampanis, 2007), i.e.,…”

“…Numerical solutions of 3D problems are significantly more challenging to derive and implement than for 2D problems. When the three-dimensional parabolic equation (3D PE) is solved through direct methods, either with finitedifferences (Cheng et al, 2009) or finite-elements (Sturm and Kampanis, 2007), the resulting linear system involves very large sparse matrices. Such linear systems are computationally intensive to solve and typically require the use of iterative methods, such as the Generalized Minimal Residual (GMRES) or the Bi-Conjugate Gradient (BiCG) (Saad, 2003).…”

Developments of parabolic equation method in the period of 2000–2016

“…This mapping replaced a sloping interface with a sloping surface, because it was relatively easy to handle the case of a pressurerelease boundary condition. In 2007, Sturm and Kampanis [20] analyzed the influence of "staircase" approximation on the one-way sound wave propagation problem. They presented a new finite-element 3D narrow-angle PE model which accurately treated the variable interface conditions by using an appropriate parabolized condition and a new change-of-variable technique which does not require any homotheticity condition of the layers.…”