Multisector parabolic-equation approach to compute acoustic scattering by noncanonically shaped impenetrable objects

Parabolic equation (PE) methods have long been used to efficiently and accurately model wave phenomena described by hyperbolic partial differential equations. A lesser-known but powerful application of parabolic equation methods is to the target scattering problem. In this paper, we use noncanonically shaped objects to establish the limits of applicability of the traditional approach and introduce wide-angle and multiple-scattering approaches to allow accurate treatment of concave scatterers. The PE calculations are benchmarked against finite-element results, with good agreement obtained for convex scatterers in the traditional approach, and for concave scatterers with our modified approach. We demonstrate that the PE-based method is significantly more computationally efficient than the finite-element method at higher frequencies where objects are several or more wavelengths long.

Figure 1

Schematic of coordinate systems for the multisector parabolic equation method. Subfigures show the cases where the scattered paraxial direction $x$ is at an angle of (a) $\varphi =0$ and (b) $\varphi =\pi /6$ with respect to the incident direction $x^{\prime }$. The marching occurs in the $xz$ plane (defined by the paraxial direction $x$), with the object and incident wave defined in the $x^{\prime }{z}^{\prime }$ plane. The paraxial cone designates the angular range around the paraxial direction for which the PE is valid.

Figure 2

Two-dimensional target strength a circle with impedance [as defined in Eq. (6) with $\alpha =1$ and $\beta /ik=1$] boundary conditions for an incident plane wave for (a) $ka=4\pi$ and (b) $ka=10\pi$. Dashed blue lines are from the multisector PE method, and solid red are finite-element results.

Figure 3

Two-dimensional target strength a circle with impedance [as defined in Eq. (6) with $\alpha =1$ and $\beta /ik=0.5$] boundary conditions for an incident plane wave for (a) $ka=4\pi$ and (b) $ka=10\pi$. Dashed blue lines are from the multisector PE method, and solid red are finite-element results.

Figure 4

Two-dimensional target strength of an ellipse with $a_x=10\mathrm{m}$, $a_z=2\mathrm{m}$ ($k{a}_x=20\pi$) with (a) soft, (b) hard, and (c) impedance [as defined in Eq. (6) with $\alpha =1$ and $\beta =ik$] boundary conditions for an incident plane wave. Dashed blue lines are from the multisector PE method, and solid red are finite-element results.

Figure 5

Three-dimensional target strength of an ellipsoid with $a_x=5\mathrm{m}$ and $a_y=a_z=2\mathrm{m}$ ($k{a}_x=10\pi$) with (a) soft, (b) hard, and (c) impedance [as defined in Eq. (6) with $\alpha =1$ and $\beta =ik$] boundary conditions for end-on plane-wave incidence. Dashed blue lines are from the multisector PE method, and solid red are finite-element results.

Figure 6

Two-dimensional target strength of an ellipse angled at ${45}^{\circ }$ with $a_x=5,a_z=2$ with (a) soft, (b) hard, and (c) impedance [as defined in Eq. (6) with $\alpha =1$ and $\beta =ik$] boundary conditions for an incident plane wave. Dashed blue lines are from the multisector PE method, and solid red are finite-element results.

Figure 7

Three-dimensional target strength of an ellipsoid with $a_x=5\mathrm{m}$ and $a_y=a_z=2\mathrm{m}$ with (a) soft, (b) hard, and (c) impedance [as defined in Eq. (6) with $\alpha =1$ and $\beta =ik$] boundary conditions for ${45}^{\circ }$ plane-wave incidence. Dashed blue ($xz$ plane) and green ($xy$ plane) lines are from the multisector PE method, and solid red ($xz$ plane) and purple ($xy$ plane) are finite-element results.

Figure 8

Three-dimensional target strength of a finite circular cylinder with $a=2\mathrm{m}$ and $h=5\mathrm{m}$ for broadside plane-wave incidence, with (a) soft and (b) hard boundary conditions. Dashed blue ($xz$ plane) and green ($xy$ plane) lines are from the multisector PE method, and solid red ($xz$ plane) and purple ($xy$ plane) are finite-element results.

Figure 9

Time comparison between finite-element (COMSOL) and PE methods for a full angular spectrum solution of scattering from a hard sphere of radius 5 m. The FEM solution has maximum element size $\lambda /6$, while the PE solution has grid spacing $\lambda /20$ in $x$ and $\lambda /10$ in the transverse directions.

Figure 10

Time comparison between finite-element (COMSOL) and PE methods for a full angular spectrum solution of scattering from an ellipsoid of transverse radius 2 m and total length 14 m. The FEM solution times are for the full angular spectrum with maximum element size $\lambda /6$, while the PE calculation times are for the full angular spectrum with grid spacing $\lambda /20$ in $x$ and $\lambda /10$ in the transverse directions.

Figure 11

Bean geometry for three sets of parameters (see text).

Figure 12

Target strength of a soft bean of $ka=8\pi$ for three different sets of parameters (see text) for an incident plane wave. Red solid lines are FEM calculations, blue dotted are narrow-angle PE, and green dashed are wide-angle PE.

Figure 13

Target strength of a hard bean of $ka=8\pi$ for three different sets of parameters (see text) for an incident plane wave. Red solid lines are FEM calculations, blue dotted are narrow-angle PE, and green dashed are wide-angle PE.

Figure 14

$\mathsf{L}$-shaped geometry for three sets of parameters (see text).

Figure 15

Target strength of a soft $\mathsf{L}$ shape for a plane wave of frequency 1500 Hz ($ka\approx 19$) for three different sets of parameters (see text). Red solid lines are FEM calculations, blue dotted are wide-angle PE, and green dashed are wide-angle PE with multiple-scattering contributions.

Figure 16

Target strength of a soft $\mathsf{L}$ shape for a plane wave of frequency 2500 Hz ($ka\approx 31$) for three different sets of parameters (see text). Red solid lines are FEM calculations, blue dotted are wide-angle PE, and green dashed are wide-angle PE with multiple-scattering contributions.

Figure 17

Calculations for a plane wave of frequency 1500 Hz incident on a soft bean rotated by ${20}^{\circ }$. (a) Full-field pressure (absolute value) computed using FEM. (b) Target strength computed using FE and PE methods. Red solid lines are FEM calculations, blue dotted are narrow-angle PE, and green dashed are narrow-angle PE with multiple-scattering contributions.

Figure 18

Calculations for a plane wave of frequency 1500 Hz incident on a forward-pointing chevron with soft boundary conditions. (a) Full-field pressure (absolute value) computed using FEM. (b) Backscattered field (absolute value) computed using the PE method without (left) and with (right) multiple-scattering modification of the incident field. (c) Target strength computed using FE and PE methods. Red solid lines are FEM calculations, blue dotted are wide-angle PE, and green dashed are wide-angle PE with multiple-scattering contributions.

Figure 19

Two-dimensional target strength calculations for a plane wave scattered from acoustically soft circles for (a) $ka=4\pi$ and (b) $ka=10\pi$. Dashed blue lines are from the multisector PE method, and solid red lines are finite-element results.

Figure 20

Two-dimensional target strength calculations for a plane wave scattered from acoustically hard circles for (a) $ka=4\pi$ and (b) $ka=10\pi$. Dashed blue lines are from the multisector PE method, and solid red lines are finite-element results.

Figure 21

Three-dimensional target strength of spheres with soft boundary conditions for plane-wave incidence for (a) $ka=4\pi$ and (b) $ka=10\pi$. Dashed blue lines are from the multisector PE method, and solid red are finite-element results.

Figure 22

Three-dimensional target strength of spheres with hard boundary conditions for plane-wave incidence for (a) $ka=4\pi$ and (b) $ka=10\pi$. Dashed blue lines are from the multisector PE method, and solid red are finite-element results.