Chapter 7

Example Six: Pulsating Sphere

The first exterior problem is the classical fundamental radiation problem ofa pulsating sphere (see Figure 1.1.f ). Key concepts covered are modellingsymmetry and how this affects computational efficiency, appropriate direc-tion of normals for an external problem and the use of CHIEF points in theinterior to improve the condition number of the matrix.

To model a complete sphere select Geometry > Create > Object> Sphere. You will be asked to enter a centre for the sphere. In thecommand line type <0,0,0> and press “enter”. You will then be asked toenter a radius for the sphere. Type <1> and press “enter”. A sphere shouldappear on the screen. To enlarge the view select View > Zoom > Frame(see Figure 7.1).

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7. Example Six: Pulsating Sphere

Figure 7.1: Generated sphere.

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Since the sphere is symmetrical, it can be modelled using half a spherewith a symmetry boundary condition. Half the sphere must therefore bedeleted. Select Geometry > Delete > Volume, click on the light bluelines defining the spherical volume and press “esc”. Select Geometry >Delete > Surface, click on the two surfaces in the negative x-coordinateregion (see Figure 7.2) and press “esc”.

Figure 7.2: Deleting sphere surfaces in the negative x-coordinate region.

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7. Example Six: Pulsating Sphere

Select Geometry > Delete > Line, click on the line in the negativex-coordinate region and press “esc”. You should be left with a semi-sphereas depicted in Figure 7.3

Figure 7.3: Model of half a sphere.

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Save the project in your working folder as “sphere.gid”.The next step is to check the surface normals. For an external bound-

ary element problem all surfaces must face in. Select Utilities > SwapNormals > Surfaces > Select. To see the directions of both surfacenormals clearly you may need to change the rotation of the sphere. To dothis select View > Rotate > Trackball and rotate the view until youare happy that the normals can be clearly seen. Click on all of the surfacesthat have an outward pointing normal until all surfaces are oriented in thecorrect direction (see Figure 7.4). Hit “esc” to leave the “Swap Normals”mode.

Figure 7.4: Orienting the surface normals in the correct direction.

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7. Example Six: Pulsating Sphere

The next step is to define the problem as a Helm3D BEM problem.To do this select Data > Problem type > helm3d. A dialog windowwarning the user that “all data information will be lost” will appear. Select“OK”.

The only boundary conditions required for this problem are velocityboundary conditions. A velocity of 1+0i is needed over the entire surface ofthe sphere. To do this select (Data > Conditions). Click on “Velocity”.Set the real velocity component to 1 and the imaginary velocity componentsto zero. Click on “assign”. Type <:> in the command line to select allsurfaces and select “finish” and then “Close”. The boundary conditionshave now been set.

Select (Data > Problem data). Give the project a title of <sphere>.Ensure the boundary type is selected to be “external”, symmetry is set to“Y-Z”, density is set to 1, speed of sound is set to 6.2832, freq start to 0.1,freq stop to 7 and freq int to 0.1 (see Figure 7.5). Click “Accept data” and“Close” to leave the problem data environment.

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Figure 7.5: Selecting the problem data.

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7. Example Six: Pulsating Sphere

The next step is to mesh the semi-sphere surface. Select (Meshing> Generate). A dialog box will appear asking you to “Enter the sizeof elements to be generated”. Type in <0.2> and press “OK”. A dialogbox will appear which states that 568 triangle elements have been created.Press “OK” and the mesh will appear (Figure 7.6).

Figure 7.6: Meshed semi-sphere.

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A solution to the problem can now be generated by selecting (Calculate> Calculate). A dialog box will appear telling you once the solution isdone. Click “OK”. This calculation will take a lot longer that previous ones.You can view the progress of the calculation by clicking on (Calculate >View process info...).

The file “sphere.fdat” will have been created in your working directory.You can open this using a text editor to review your results. Numerouscolumns of data appear, each labeled with a heading in the first row. Eachrow of data corresponds to one calculation frequency. Of particular interestare the first, second and ninth columns, the frequency, condition numberand pressure amplitude respectively. The condition number is importantas it gives an indication of how well conditioned the matrix to be invertedis. The closer the condition number is to unity, the better the conditioning.The higher the condition number, the more poorly conditioned the matrixis and hence the results obtained at these frequencies are more likely to beincorrect. Note that at certain frequencies the condition number is high.Save the file in a separate folder for comparative purposes which will beexplained later.

One disadvantage to the direct BEM approach is that if the Kirchoff-Helmholtz integral equation is used to represent the sound field on theexterior of a finite volume, at the natural frequencies of the interior of thefinite volume, the exterior problem breaks down and the matrix becomesill-conditioned. This is well documented [9] and many solutions have beenattempted [10, 11]. The CHIEF method [10] is commonly used to over-come the interior natural frequency problem because of its simplicity. Thistechnique solves an overdetermined system of equations formed by placingextra points (CHIEF points) inside the volume of interest. Provided theCHIEF points are not placed at a nodal line of the interior solution, thiswill improve the matrix condition number and allow the matrix to be solvedusing least-squares methods.

Re-enter the preprocessor and alter the problem data to include a CHIEFpoint. Select the “Chief Points” tab, set the “Number of Chief points” tobe 1 and the position of the chief point x1 to be 0. Remesh and resolvethe problem. Save the newly generated *.fdat file for future comparison.Repeat the process with chief point x1 at 0.5.

The analytical solution for the pressure produced by a pulsating sphere,which can be derived from the spherical wave equation, is:

p(r) = (a2

r) iρω

1 + ikae−ik(r−a) (7.1)

where a is the sphere radius, r is the radius at which the pressure is being

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7. Example Six: Pulsating Sphere

calculated and ω is the angular frequency. The characteristic eigenfrequen-cies of the sphere, which are the eigenfrequencies of the interior Dirichletproblem, are given by the equation:

sin(ka) = 0. (7.2)Compare the theoretical and BEM results (saved in the *.fdat files gen-

erated with no CHIEF point, a CHIEF point at the sphere centre and aCHIEF point at half the sphere radius) graphically using your preferredgraphing package by graphing the pressure as a function of frequency or kawhere k is the wavelength and a is the sphere radius. You should obtain aplot similar to Figure 7.7.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

ka

Abs

olut

e su

rfac

e pr

essu

re (

Pa)

no CHIEFCHIEF r=0CHIEF r=0.5Analytical

Figure 7.7: Surface pressure of a pulsating sphere.

The BEM solution with no CHIEF point shows poor agreement withthe analytical solution at ka = π and ka = 2π , where k is the wavenumberand a is the radius of the source. This is due to poor conditioning of thematrix. The placement of a CHIEF point at r/a = 0.5, where r is theradial location from the centre of the sphere, ameliorates the problem atka = π ; however, poor agreement at ka = 2π still occurs due to the CHIEFpoint being on the interior nodal surface corresponding to the characteristiceigenfrequency ka = 2π , meaning that this resonance cannot be cancelled.Placing the CHIEF point at the sphere centre ensures that it does not lieon a nodal surface. The resulting solution is therefore in good agreement

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with the analytical solution. When using BEM to analyse more complexgeometries, the user generally has no prior knowledge of the optimal CHIEFpoint location, and therefore multiple CHIEF points randomly distributedwithin the volume are used.

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