5.6  Test Programs

In this Section the subroutines AEBEM2, AEBEM3 and IBEMA are demonstrated through invoking them and comparing the results with analytic solutions. The corresponding main programs are AEBEM2_T, AEBEM3_T and AEBEMA_T.

5.6.1  Program AEBEM2_T.FOR

The main program AEBEM2_T tests module AEBEM2, the subroutine for computing the solution of the Helmholtz equation exterior to a closed boundary in a two-dimensional domain. In AEBEM2_T the domain lies exterior to a square of side 0.1m with the boundary represented by 32 elements. The representation of the boundary is described in Section 2.2 by the data structures VERTEX and SELV, see Tables 2.A and 2.B and Figure 2.2. Full results are given in file AEBEM2.OUT.

The acoustic medium is air at 20 celcius and 1 atmosphere so the speed of sound is 344m/s. The chosen frequency of the test is 400Hz, hence k = 7.31. For the first two tests, the velocity potential defined by f(p) = i/4H0(kr), with r being the distance from the point p to the centre of the square, is clearly a solution of the Helmholtz equation in the exterior domain. The boundary velocity is given by differentiation this expression for f(p) with respect to the normal to the boundary at p for each collocation point p on S.

In the first two tests the Dirichlet and Neumann boundary conditions arising from this potential is processed by AEBEM2. The solution is given at the points (0,0.15), (0.05,0.15), (0.1,0.15) and (0.05,-0.1). Comparisons between the exact and numerical solutions are given in Table 5.A.

Table 5.A: Results from AEBEM2_T
point exact solution numerical solution numerical solution
to Dirichlet condition to Neumann condition
(0.000,0.150) 0.0176 + i 0.2100 0.0198 + i 0.2079 0.0181 + i 0.2104
(0.050,0.150) 0.0394 + i 0.2177 0.0415 + i 0.2154 0.0397 + i 0.2181
(0.100,0.150) 0.0176 + i 0.2100 0.0198 + i 0.2079 0.0181 + i 0.2104
(0.050,-0.100) -0.0398+ i 0.1804 -0.0375 + i 0.1792 -0.0396 + i 0.1808

5.6.2  Program AEBEM3_T.FOR

The main program AEBEM3_T tests module AEBEM3, the subroutine for computing the solution of the Helmholtz equation exterior to a general closed surface in a three-dimensional domain. In AEBEM3_T the domain is the exterior of a sphere of side 1m with the boundary represented by 36 elements. The representation of the boundary is described fully in Section 2.3 by the data structures VERTEX and SELV, see Tables 2.C and 2.D. Full results are given in file AEBEM3.OUT.

The acoustic medium is air at 20 celcius and 1 atmosphere so that the speed of sound is c = 344m/s. In the first two test problems the acoustic field in the exterior is defined to be

f = ejkr
r
which is a multiple of the Green's function and hence is clearly a solution of the Helmholtz equation. The acoustic frequency is 100Hz, hence k = 1.8265. The numerical and exact solutions at the points (0,0,2), (0,0,4), (0,0,8) and (0,0,-2) are given in Table 5.B.

Table 5.B: Results from AEBEM3_T
point exact solution numerical solution numerical solution
to Dirichlet condition to Neumann condition
(0,0,2) -0.4360 - i 0.2447 -0.4628 - i 0.1897 -0.5011 - i 0.2389
(0,0,4) 0.1302 + i 0.2133 0.1557 + i 0.1960 0.1614 + i 0.2274
(0,0,8) -0.0572 + i 0.1112 -0.0431 + i 0.1175 -0.0549+ i 0.1284
(0,0,-2) -0.4360 - i 0.2447 -0.4628 - i 0.1897 -0.5011 - i 0.2389

In this example a comparison of the computed and exact sound pressures are given. The magnitudes (in decibels) and the phases (in degrees) are given in Table 5.C. The values are computed from the sound pressures in line with the methods described in Section 1.3.

Table 5.C: Results from IBEM3_T
point exact solution numerical solution numerical solution
to Dirichlet condition to Neumann condition
(0,0,2) 72.8dB, -60.7° 72.8dB, -69.8° 73.2dB, -64.7°
(0,0,4) 69.8dB, 148.6° 69.8dB, 139.5° 70.2dB, 144.4°
(0,0,8) 66.8dB, -152.8° 66.8dB, -161.9° 67.2dB, -157.1°
(0,0,-2) 72.8dB, -60.7° 72.8dB, -69.8° 73.2dB, -64.7°

The program AEBEMA_T also shows results from a sphere scattering a point source where in the first case the potential on the sphere is assigned the incident potential in a Dirichlet condition, so that the sphere is effectively invisible. In the second case the sphere represents a rigid (non-vibrating) acoustically hard scatterer. An application similar to these examples is given in the next Section.

5.6.3  Program AEBEMA_T.FOR

The main program AEBEMA_T tests module AEBEMA. In AEBEMA_T the acoustic domain is exterior to a sphere of unit radius. The surface is represented by sixteen truncated conical elements, as described in Section 2.4. The acoustic medium is air at 20 celcius (c=344m/s). Full results are given in file AEBEMA.OUT.


Fig 5.2. Radiation ratio curve for pulsating sphere (- exact, à computed).

In each test problem the frequency ranges from 10 Hz to 1000Hz in 10 Hz steps. The purpose of the first test problem is to plot the radiation ratio of a pulsating sphere. In the test problem the surface velocity is prescribed the value of unity on each element. Note that the radiation ratio refers to the shape of the boundary condition, the amplitude is arbitrary. The results from the test are directed to the file AEBEMA.OUT. The exact radiation ratio for a sphere of radius unity, pulsating at wavenumber k is [(k2)/( k2+1)]. A comparison of the computed and exact radiation ratios are given in Figure 5.2.

Results for an oscillating sphere, that is the sphere vibrating up and down without changing volume, are given in Figure 5.3. These results can be obtained from a test problem similar to AEBEMA by setting the same Neumann surface condition but with

SFVAL(ITEST,ISP)=SELCNT(ISP,2). The exact radiation ratio of an oscillating sphere is [(k4)/( 4+k4)].


Fig 5.3. Radiation ratio curve for oscillating sphere (- exact, à computed).

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