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This review analyzes following numerical methods of a solution of problems of a sound diffraction on ideal and elastic scatterers of a non-analytical form: a method of integral equations, a method of Green’s functions, a method of finite elements, a boundary elements method, a method of Kupradze, a T-matrix method and a method of a geometrical theory of a diffraction.

There are a large number of numerical methods of a sound scattering studies on ideal and elastic bodies of a non-analytical form. The article presents a theory and a nu- merical experiment of seven such methods. These results were obtained as authors of a review and other researchers.

We discern the ideal non-analytical scatterer in the form of the terminal cylinder with the semi-sphres on its ends (see

The pressure

where Q―the point of the surface of the scatterer.

Then (1) for the Dirichlet condition at the surface is accepting the appearance:

For the Neimann’s condition:

The function

The integral to the left of (4) must understand in the sense of the main meaning.

With the help Ψ we are find and the scattered pressure

For the Neimann’s condition, we are bring the function

The scattered pressure in the point

The scattered pressure

The surface S consists of S_{2} and the surfaces S_{1} и S_{3} (see

For the calculation of the integrals (2), (3) and (5), (6) on surface

At

We are going to spread the method of the integral equations, used in [

In the quality of such scatterer, we are going to consider the terminal isotropic elastic cylindrical shell with the semi-spheres on its ends (see _{1}, the Lame’s coefficients―λ and μ. The shell was filled in the internal liquid medium with the density ρ_{2} and the sound velocity C_{3} and it was placed in the external liquid medium with the density ρ_{0} and the sound velocity C_{0}.

At the shell falls the plane harmonic wave with pressure p_{i} under the angle Θ_{0} and with the wave vector

As was shown in [

where

The second integral equation presents the Kirchhoff integral for the diffracted pressure

where_{1}; C(P_{1}) is the numerical coefficient, equal 2π, if _{1} out S_{a}; S_{a} is the external surface of the shell; Q is the point of the external surface of the shell.

For the pressure _{1} is got the third integral equation:

where

S_{b} is the internal surface of the shell.

To the integral Equations (8), (9) and (10) are added the boundary conditions on the external (S_{a}) and internal (S_{b}) surfaces of the shell [

For choosing boundary conditions we will have the integrals of the two types: the integrals with the isolated special point and the integrals which are considered of the sense of the principal meaning. The method of the calculation of the second types was described in [

Applying the normal modes and image sources methods for a harmonic signal in the planar waveguide is equivalent [

The spectrum

where:

The spectrum

The spectrum of the scattered (reflected or transmitted) signal

Similarly to (23) with the use of

Considering the elastic shell into the liquid layer with the thickness H and the constant sound velocity the boundary conditions will be at follows: at the upper boundary of the waveguide Dirichlet condition is fulfilled, at the lower boundary―Neumann condition [

The scatterer centre is fixed at the distance of

The scattered field for the non-analytical form elastic shell has been determined either with the help of the method of the integral equations [

We consider non-analytical body, the surface of which does not apply to coordinate symones with divided variables in the scalar Helmholtz equation. We examine this non-analytical scatterer in the form of a finite circular cylinder bounded on the sides of the hemispheres (

Sound pressure, scattered by this body, can be found one of the numerical methods for the solution of diffraction problems [_{2} and the surface of hemispheres S_{1 }and S_{3} (

where p_{s}(P)―the sound pressure scattered by the body, P―the point of observation, which has a spherical coordinates:_{s}(Q)―the sound pressure in the point Q; G(P, Q)―Green’s function of the free space, satisfying the inhomogeneous Helmholtz equation.

In the (15) Green’s function is selected as a potential point source:

where

Using relative arbitrariness of the choice of Green’s function, you can get the Kirchhoff formula options, consisting of a single member:

By using formulas (17), (18) is considerably simplified computational procedure: you want to define only one of the parameters (p_{s}(Q) or dp_{s}(Q)/dn) on the surface S. However, in this case, the match of the surface S with a coordinate the surface one of coordinate systems in which it is possible separation of variables is necessary. Thus, application of Green’s functions for analytical surfaces (infinite cylinder and sphere) faces of these surfaces, interconnected is the main feature of this method.

The possibility of such a method and test calculations of the scattered field were considered in [_{s} (Q) and dp_{s}(Q)/dn on the surface S you can use the following expression:

1) For the homogeneous Dirichlet conditions (ideally soft body), pressure scattered waves on the surface S have the form:

2) For the homogeneous Neumann conditions (ideally rigid body):

where p, (Q)―the sound pressure of the incident wave in point Q. When determining the values p_{t}(Q) you can use the expression for the scalar potential of the plane monochromatic wave single amplitude of the incident on the body from a source located at infinity.

This potential for a perfectly reflective sphere is natural functions in solving the Helmholtz equation in a spherical coordinate system has the following form [

The expression (21) is simplified when considering the axis-symmetric problem (dependence on the coordinate

For scatterer in the form of a perfectly reflecting cylinder scalar potential incident plane harmonic waves unit amplitude of the wave vector,

In the case of the plane problem of the wave vector к perpendicular to the z axis of the cylinder and expression (23) is simplified [

For calculation of integral (17) and (18) on the surface S the quadrature formulas is used. A step of integration over the surface S in the axial and circumferential directions

Using the method of Green’s functions were calculated the equivalent radius

The analysis of equivalent radiuses

1) The angular position of reflecting and diffraction lobes totally correspond to the physical representations;

2) The angular characteristics

At all figures clearly observed diffraction (shadow) petal, and it grows and shrinks with increasing frequency. On Figures 7-9, the mirror petal is shows, which is similar to the shadow petal with increasing frequency, but in contrast, limited asymptotically. You may notice that the angular diagrams of the non-analytical scatterer are very similar to the angular characteristics of the scattering elongated spheroids (ideal and elastic) with the ratio of the semi-axes 1:10 [

The solution of the problem of the sound scattering by an elastic shell of the non- analytical form is based an article [

As non-analytical bodies considered two structures:

1) A finite-length circular cylindrical elastic shell limited an the ends by the two halves of a prolate spheroidal shell (

2) The same end cylindrical shell bounded the ends by the two halves of a spherical shell (

In article, [

We consider a compound elastic shell formed by a finite cylindrical shell whose ends are closed by two hemi spherical shells of the same diameter (

The method of Green’s functions in combination with analytical methods can be used for the solution of tasks of diffraction of plane sound wave on elastic isotropic scatter of non-analytical form, that consists of circular cylindrical shell of terminated length L and radius r_{0}, bounded at the butts by the halves of elongated spheroidal shell [

The internal surface of the spheroidal shell is given by coordinate _{0}) , and external-by coordinate

The shell material is isotropic, with the density

The scatterer is placed in ideal compressed liquid with density

The amplitude-phase distribution of the sound pressure and of normal component of vibrating velocity in the points of this non-analytical surface is found from the strict solution of tridimensional boundary tasks of dynamic theory of elasticity on the endless

elastic cylindrical surface and elastic spheroidal surfaces, respectively.

The schemes of solution of axially symmetric tasks of diffraction on elastic cylinder and spheroid are rather similar: in both cases the vector potential

Although on the spheroid these coefficients are found not in the enclosed form, but using the method of truncation from the infinite system of equations.

What is more, while finding the solution of tridimensional task of diffraction on elastic speroidal scatterer the vector potential

where

The sound pressure in the far-field can be found by one of the numerical methods, among those is the comfortable method based on the usage of mathematical formula of Helmholtz-Huygens ‘Principle (integral of Kirchhoff) [

where r―is the distance between the point Q on the surface of the shell and the point P with coordinates r_{1}, θ_{1}, φ_{1} in the far-field.

The Green’s functions in (26) are taken in the form of potential of the point-source. The quadrature formulas are used for finding of this integral, and here the integration step (sampling) of the surface of the scatterer should not exceed 0.5l_{0}, where l_{0}―is the length of the plane monochromatic wave, falling from the liquid onto the surface of the scatterer.

The Green’s functions in (26) are taken in the form of potential of the point-source. The quadrature formulas are used for finding of this integral, and here the integration step (sampling) of the surface of the scatterer should not exceed 0.5l_{0}, where l_{0}―is the length of the plane monochromatic wave, falling from the liquid onto the surface of the scatterer.

The other way of solving the task of sound diffraction on elastic isotropic body with non-analytical surface is based on the method of Kupradze [

where

From the common characteristics of wave and elastic potentials we find that

where

The usage of method of T-matrix in the task of sound diffraction on elastic bodies of non-analytical form is examined in [

The equation for

where the vector

The solution of the received system of the two integral equations is found by the method of T-matrix with the use of boundary conditions.

The particular place in the tasks of diffraction on the bodies with non-analytical surface in the sphere of high frequencies is occupied by the geometrical theory of diffraction (GTD), based on the asymptotic methods [

In comparison to ray acoustics (RA), the GTD regards the process of formation of diffraction rays along with reflection and refraction. When the wave falls onto the body or its edge, the boundary shade-light appears for geometrical rays, i.e. the geometro- acoustic solution faces the break that states the formation of additional diffraction fields, compensation this break.

The four main additional laws are considered in GTD, in comparison to RA: two of them determine the direction of diffraction rays, two others―their amplitudes:

1) The range of diffraction rays is produced not by all of the falling rays, but:

а) by the rays falling onto the non-homogeneous parts of the body S―edges, ridges, lines of discontinuity in the curvature (

2) When the ray of the primary field falls onto the edge of the body (

The law of formation of diffraction rays in the sphere of shade of plain convex body differs from the law of rays’ formation at edges and ribs. In this case, the diffraction rays get away from the surface of the shade part of the body and form the slippage waves.

Each ray of the falling wave which contacts the body, from the “surface ray”

3) The amplitude of diffraction ray is proportionate to the amplitude of the formatting primary ray in the impact point. The diffraction fields can be presented in the formula:

where S―is the eikonal (function, determing the phase structure of the field―the system of the wave front); J―is the Jacobian of transition to the ray coordinates―the parameter, proportionate to the area of cross-section of rays of the tube;

The parameter

4) The coefficient (matrix) of diffraction D is determined by local features of body’s geometry in the area of the falling ray (in the case of edges and ribs) or in the area of the surface ray between the falling point and take-off point of diffraction ray (in the plain body).

In terms of physics

The notion of diffraction coefficient in the general case can be defined more precisely. Thus, if the amplitude of the primary field converts into zero in the ray that produces diffraction rays, then the amplitude of diffraction field will not be equal to zero, i.e. the formula (18) cannot be used. In this case the diffraction field will be proportionate to the value of the first derivative of the amplitude of the primary field along the front.

The algorithm of solution of the task of diffraction consisting of 3 rules is based on the laws of GTD:

1) The solution is found as the result of number of the fields of ray type:

One of the components is the primary field. Each of the fields differs from zero in the area with the boundary of the body surface and the shade-light boundary of the field.

2) All the sum (33) components, except the primary field (it is considered to be defined) are determined from the primary field according to the laws of RA and of GTD.

It should be kept in mind that the reflected, refracted and diffraction fields can be formed from the primary field not only directly but also as the result of the complex sequence of reflections, refractions and diffractions.

1) The question of the choice of the coefficients of diffraction is important in the process of calculations of diffraction fields. According to the fourth law of GTD, the coefficients of diffraction are equal to all bodies that possess equal local geometrical features and the features of geometry of the falling wave. Here logically appears the third rule of algorithm of diffraction task.

2) The coefficient of diffraction in the present task is found from the analysis of the precise solution of the simple (model) task, close in geometry. For example, in the case of diffraction of the field

The usage of GTD is limited in the present number of solutions because the coefficients of diffraction D are determined in the model tasks. In the present the model tasks for numerous two-dimensional cases-fir diffraction on the wedge, plain cylinder, etc.― are solved. In the model three-dimensional tasks, we frequently have to use the approximate results.

The method of the finite elements (MFE) and its variations allow getting the solutions of tasks of sound radiation of elastic bodies of nearly all forms. Here we consider, for example, the possibility of common use of MFE with the method of Grin functions for numerical solution of task of the distant field of sound radiation by extended spheroid shell, under the influence of the point sources on its surface (

The choice of this body is caused by the definite solution of the task of sound radiation of spheroid shell under the influence of point sources on its surface that was received using the theorem of reciprocity from the three-dimensional boundary task of

diffraction of plane monochromatic wave on this shell; the solution is presented in [

The geometrical and physical parameters of the shell are similar to those presented on

The numerical solution of the task is made in two stages [

1) The calculation of the values of sound wave potential and its gradient on the closed test area in the nearest field (of the Fresnel zone), that are created by the point sources;

2) Re-calculation of received results in the distant field (in the Frauenhofer zone).

On the first stage it is necessary to conjugate the solutions of FEM on the surface of area S in the sphere V_{1}, adjoining the shell, with the point analytical solution of equation of Helmholtz in the external endless sphere V_{2} (

The functional of the full energy if the system “shell-fluid” will have the form:

where г д е P and T―are potential and kinetic energies of the shell; w―normal displacement of the shell to the surface _{1} and V_{2 }correspondently, that meet the equation of Helmholtz and the condition of radiation on infinity in the volume V_{2}.

The condition of stationary state of E functional leads to fulfillment of equation of the shell movement and of equation of Helmholtz in the sphere V_{1}, as well to equality

of the normal speeds

The axial symmetry of the shell and sources of radiation leads to the fact that the displacements of the shell and potentials of liquid speed will not be subject to the surrounding coordinate φ.

The substitution of the forms that approximate the shell displacements and potentials Φ_{1} and Φ_{2} in conditions of stationary state of E functional leads to the lineal algebraic system of the solving equations.

For the solving of this task is necessary to use the circular finite elements for: the shell, liquid and filling gas, for conjugation of these elements between one another, as well as the finite elements for modeling of the sphere V_{2}. Using these elements is possible to calculate the nearest field for the arbitrary sources in the form of the shells if spinning. In this case, the numerical calculations are made for the finite-elementary net that consists of 131 elements and 410 focal points (

The AFD of the sound pressure and normal component of vibration speed in the focal points of control surface in the nearest field is initial for the second stage of the task solution.

The sound pressure in the distant field is found with Kirchhoff’s integral, as well as in (26).

The two variants of the forms of control surface were used in the calculations:

Non-analytical in the form of cylinder with hemispheres on the edges (as the most useful from the measuring process organization point of view,

On the basis of the proposed algorithms in [

For the solving of tasks of radiation and diffraction for the bodies of non-analytical surface the boundary element method (BEM) is successfully used during the last years; the numerous scientific works are published on this research area, where the theoretical basis of the method as well as the different aspects of the application are stated [

The bibliography analysis shows that BEM is one of the most relevant and widely used among the other numerical methods of solving the boundary tasks.

The following advantages of BEM (in comparison with FEM, for instance) can be considered when solving the tasks:

1) The sampling of the boundary of the sphere with scatterer, not of the whole sphere, when as the result the additional measures for realization of condition of radiation at infinity are not demanded;

2) The reduction of initial differential equation to the boundary integral equation that presents the exact formulation of the stated task. Here the accumulation of error comes in the process of numerical solution of integral equations in the result of sampling, approximation and calculation;

3) The use of analytical method, true for the whole sphere, provides the potentially higher accuracy than FEM does, where the approximation is committed in every area. In the common case for scatterers of general geometric form the boundary surface is presented in the form of collection of elementary areas [

The conception of formation of isoparameter elements allows converting the key coordinates of every point of initial elements

The curvilinear coordinates of every point of the element

where

а) For quadrangular elements:

b) For triangular elements

These correlations present the implicit conversion of surface element to the plane square or the plane equilateral triangle (

The correlations for interpolation of displacements and stress are expressed similarly

on each element [

where

While submitting (38), (39) in the equation and using the rule of numerical integration, we get

where H and

Here, we form the integral equation for diffracted pressure basing on (1.15)

where

While submitting (38), (39) and (40) in (41) and completing the numerical integration, we receive

where T and D―are matrix of coefficients.

On the next stage we solve the system of Equations (40) and (43) using the boundary conditions.

According to the boundary condition the number of indeterminate stress in Equation (40) can be expressed through pressure:

where G and F―are matrix of coefficients, received from matrix

The distributions

In the particular cases, it is possible to use the principle of formation of the net of nodal points for the scatterers possessing the form of axially symmetrical bodies; the principle was used in [

In the process of numerical integration the element of the surface of cylinder with radius

In the process of formation of the net of boundary elements for this task, the discretization step of the boundary surface in the direction of every coordinate should not also exceed 0.5l_{0}.

According to the theorem of Helmholtz, the displacement vector

where

where

For cylinder surface the vector

Due to the axis symmetry of the task for spherical surfaces the vector potential

The potential of diffused wave, as well as the potentials

While all the main physical variables are functions of only two coordinates, the displacement vector would also possess two components.

Using the correlations of the generalized law of Hooke for isotropic sphere, not depending of the choice of coordinate systems, it is possible to present the elastic stress on the finite surface through the deformation components, and then through potentials

а) For cylindrical surface:

where

b) For spherical surface:

where

The following boundary conditions are to be executed at the points of the boundary surface:

a) The normal (radial) component of the displacement vector

where

b) The pressure tangents are missing:

While submitting the component of the displacement vector and elastic stress to the boundary conditions (54)-(56) we receive the algebraic systems of equations for every point on the surface for finding out the indeterminate coefficients in equations of potentials

The indeterminate coefficients are received using the ratio of determinants by the rule of Cramer, that allows to receive subsequently the distributions

The calculation of the diffracted sound pressure

The task solution of diffraction for the elastic isotropic surface does not differ in principle from the examined solution for constant elastic body: the internal boundary (with filler or vacuum inside the shell) is added, and thus the number of indeterminate coefficients and the number of boundary conditions in (49)-(56) increase.

The additional boundary conditions are formed the following way:

4) The normal stress on internal surface of the shell is either missing (the hollow shell) or is equal to the sound gas pressure (gas filled shell);

5) The absence of the tangent stress on internal surface of the shell.

As the illustration of the testing diffraction task for estimation of accuracy of numerical solution by BEM in [

Also the results of calculations of scattered characteristics for elastic scatterer with non- analytical surface form (cylinder with hemispheres at the edges), by FEM, are presented there; the results were approximate to the corresponding results of the precise solution for spheroids, received in [

The article analyzes seven methods of the solution of the problem of the sound scattering on ideal and elastic bodies of the non-analytical forms: method of integral equations, method of Green’s functions, the method of Kupradze, T-matrix method, the method of the geometrical theory diffraction, the method of finite elements and the boundary elements method. This analysis is supplemented with calculations of characteristics of the sound scattering by ideal and elastic scatterers of non-analytical forms.

The author thanks S. L. Il’menkov for writing sections 7, 8 and A. A. Zubova―for the help with calculations.

The work was supported as a part of research under State Contract No. P242 of April 21, 2010, within the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia for the Years 2009-2013”.

Kleshchev, A.A. (2016) Some Methods of Solution of Problems of Sound Diffraction on Bodies of Non- Analytical Form. Open Journal of Acoustics, 6, 45-70. http://dx.doi.org/10.4236/oja.2016.64005