C*************************************************************
C Test program for subroutine AEBEMA by Stephen Kirkup
C*************************************************************
C
C Copyright 1998- Stephen Kirkup
C School of Computing Engineering and Physical Sciences
C University of Central Lancashire - www.uclan.ac.uk
C smkirkup@uclan.ac.uk
C http://www.researchgate.net/profile/Stephen_Kirkup
C
C This open source code can be found at
C www.boundary-element-method.com/fortran/AEBEMA_T.FOR
C
C Issued under the GNU General Public License 2007,
C see gpl.txt
C
C Part of the the author's open source BEM packages.
C All codes and manuals can be downloaded from
C www.boundary-element-method.com
C
C************************************************************
C This program is a test for the subroutine AEBEMA. The program computes
C the solution to an acoustic/Helmholtz problem exterior to a sphere
C by the boundary element method.
C
C Background
C ----------
C
C The Helmholtz problem arises when harmonic solutions of the wave
C equation
C 2
C __ 2 1 d {\Psi}(p,t)
C \/ {\Psi}(p,t) - ---- --------------- = 0
C 2 2
C c d t
C
C are sought, where {\Psi}(p,t) is the scalar time-dependent velocity
C potential. In the cases where {\Psi} is periodic, it may be
C approximated as a set of frequency components that may be analysed
C independently. For each frequency a component of the form
C
C {\phi}(p) exp(i {\omega} t)
C
C (where {\omega} = 2 * {\pi} * frequency) the wave equation can be
C reduced to the Helmholtz equation
C
C __ 2 2
C \/ {\phi} + k {\phi} = 0
C
C where k (the wavenumber) = {\omega}/c (c=speed of sound in the
C medium). {\phi} is known as the velocity potential.
C
C For the exterior problem, the domain lies exterior to a closed
C boundary S. The boundary condition may be Dirichlet, Robin or
C Neumann. It is assumed to have the following general form
C
C {\alpha}(q) {\phi}(q) + {\beta}(q) v(q) = f(q)
C
C where {\phi}(q) is the velocity potential at the point q on S, v(q)
C is the derivative of {\phi} with respect to the outward normal to S
C at q and {\alpha}, {\beta} and f are complex-valued functions defined
C on S.
C
C Subroutine AEBEMA accepts the wavenumber, a description of the
C boundary of the domain and the position of the exterior points
C where the solution ({\phi}) is sought, the boundary condition and
C returns the solution ({\phi} and v) on S and the value of {\phi}
C at the exterior points.
C
C The test problems
C -----------------
C
C In this test the domain is a sphere of diameter 1 (metre). The acoustic
C medium is air (at 20 celcius and 1 atmosphere, c=344.0 (metres per
C second), density {\rho}=1.205 (kilograms per cubic metre) and the
C solution to the problem with a Dirichlet boundary condition
C ({\alpha}=1, {\beta}=0) and with a Neumann boundary condition
C ({\alpha}=0, beta=1) are sought. For both problems the frequency is
C 400Hz (hence specifying k).
C
C In the R-z plane, the boundary conditions are specified through
C taking the solution to be determined by
C
C {\phi} = sin(k z) ,
C
C which is clearly a solution of the Helmholtz equation.
C
C
C The boundary is described by a set of NS=32 elements of equal size,
C so that each side comprises eight elements. The boundary solution
C points are the centres of the elements.
C The *s show the exterior points at which the solution is sought;
C the points (0.025,0.025), (0.075,0.025), (0.025,0.075),
C (0.075,0.075), and (0.05,0.05).
C----------------------------------------------------------------------
C The PARAMETER statement
C -----------------------
C There are four components in the PARAMETER statement.
C integer MAXNS : The limit on the number of boundary elements.
C integer MAXNV : The limit on the number of vertices.
C integer MAXNPE : The limit on the number of exterior points.
C External modules related to the package
C ---------------------------------------
C subroutine AEBEMA: Subroutine for solving the exterior Helmholtz
C equation. (file AEBEMA.FOR contains AEBEMA and subordinate routines)
C subroutine H2LC: Returns the individual discrete Helmholtz integral
C operators. (file H2LC.FOR contains H2LC and subordinate routines)
C subroutine CGLS: Solves a general linear system of equations.
C (file CGLS.FOR contains CGSL and subordinate routines)
C subroutine FNHANK: This computes Hankel functions of the first kind
C and of order zero and one. (e.g. file FNHANK.FOR)
C file GEOM2D.FOR contains the set of relevant geometric subroutines
C The program
PROGRAM AEBEMAT
IMPLICIT NONE
C VARIABLE DECLARATION
C --------------------
C PARAMETERs for storing the limits on the dimension of arrays
C Limit on the number of elements
INTEGER MAXNS
PARAMETER (MAXNS=32)
C Limit on the number of vertices (equal to the number of elements)
INTEGER MAXNV
PARAMETER (MAXNV=MAXNS+1)
C Limit on the number of test problems
INTEGER MAXTEST
PARAMETER (MAXTEST=200)
C Limit on the number of points exterior to the boundary, where
C acoustic properties are sought
INTEGER MAXNPE
PARAMETER (MAXNPE=1)
C Constants
C Real scalars: 0, 1, 2, pi
REAL*8 ZERO,ONE,TWO,FOUR,PI
C Complex scalars: (0,0), (1,0), (0,1)
COMPLEX*16 CZERO,CONE,CIMAG
C The reference pressure, used to convert units to decibels.
REAL*8 PREREF
C Properties of the acoustic medium
C The speed of sound [standard unit: metres per second]
REAL*8 CVAL(MAXTEST)
C The density [standard unit: kilograms per cubic metre]
REAL*8 RHOVAL(MAXTEST)
C Wavenumber parameter for AEBEMA
REAL*8 K
C Angular frequency
COMPLEX*16 OMEGA
C Geometrical description of the boundary(ies)
C Number of elements and counter
INTEGER NS,IS
C Number of collocation points (on S) and counter
INTEGER NSP,ISP
C Number of vetices and counter
INTEGER NV,IV
C Index of nodal coordinate for defining boundaries (standard unit is
C metres)
REAL*8 VERTEX(MAXNV,2)
C The two nodes that define each element on the boundaries
INTEGER SELV(MAXNS,2)
C The points exterior to the boundary(ies) where the acoustic
C properties are sought and the directional vectors at those points.
C [Only necessary if an exterior solution is sought.]
C Number of exterior points and counter
INTEGER NPE,IPE
C Coordinates of the exterior points
REAL*8 PEXT (MAXNPE,2)
C Number of test problems and counter
INTEGER NTEST,ITEST
C Data structures that contain the parameters that define the test
C problems
C The acoustic frequency for each test. FRVAL(i) is assigned the
C acoustic frequency of the i-th test problem.
REAL*8 FRVAL(MAXTEST)
C The wavenumber for each test. KVAL(i) is assigned the wavenumber
C of the i-th test problem.
REAL*8 KVAL(MAXTEST)
C The nature of the boundary condition is specified by assigning
C values to the data structures SALVAL and SBEVAL.
C SALVAL(i,j) is assigned the value of {\alpha} at the center of the
C j-th element for the i-th test problem.
COMPLEX*16 SALVAL(MAXTEST,MAXNS)
C SBEVAL(i,j) is assigned the value of {\beta} at the center of the
C j-th element for the i-th test problem.
COMPLEX*16 SBEVAL(MAXTEST,MAXNS)
C The actual boundary condition is specified by assigning values to
C the data structure SFVAL.
C SFVAL(i,j) is assigned the value of f at the center of the j-th
C element for the i-th test problem.
COMPLEX*16 SFVAL(MAXTEST,MAXNS)
C The incident potential at the centres of the elements in each test
C case
COMPLEX*16 SPHIIN(MAXTEST,MAXNS)
C The derivative of the incident potential at the centres of the
C elements in each test
COMPLEX*16 SVELIN(MAXTEST,MAXNS)
C The incident potential at the exterior points
COMPLEX*16 PPHIIN(MAXTEST,MAXNPE)
C Data structures that are used to define each test problem in turn
C and are input parameters to AEBEMA.
C SALPHA(j) is assigned the value of {\alpha} at the centre of the
C j-th element.
COMPLEX*16 SALPHA(MAXNS)
C SBETA(j) is assigned the value of {\beta} at the centre of the
C j-th element.
COMPLEX*16 SBETA(MAXNS)
C SF(j) is assigned the value of f at the centre of the j-th element.
COMPLEX*16 SF(MAXNS)
C The incident potential at the centres of the elements in each test
C case
COMPLEX*16 SFFPHI(MAXNS)
C The derivative of the incident potential at the centres of the
C elements in each test
COMPLEX*16 SFFVEL(MAXNS)
C The incident potential at the interior points
COMPLEX*16 PFFPHI(MAXNPE)
C Validation and control parameters for AEBEMA
C Switch for particular solution
LOGICAL LSOL
C Validation switch
LOGICAL LVALID
C The maximum absolute error in the parameters that describe the
C geometry of the boundary.
REAL*8 EGEOM
C The parameter
COMPLEX*16 MU
C Output from subroutine AEBEMA
C The velocity potential (phi - the solution) at the centres of the
C elements
COMPLEX*16 SPHI(MAXNS)
C The normal derivative of the velocity potential at the centres of the
C elements
COMPLEX*16 SVEL(MAXNS)
C The velocity potential (phi - the solution) at exterior points
COMPLEX*16 PEPHI(MAXNPE)
C Element areas
REAL*8 ELAREA(MAXNS),AREACN
C Computing radiation ratio
REAL*8 POWER,BPOWER,RADRAT
C Workspace for AEBEMA
COMPLEX*16 WKSPC1(MAXNS,MAXNS)
COMPLEX*16 WKSPC2(MAXNS,MAXNS)
COMPLEX*16 WKSPC3(MAXNPE,MAXNS)
COMPLEX*16 WKSPC4(MAXNPE,MAXNS)
COMPLEX*16 WKSPC5(MAXNS)
COMPLEX*16 WKSPC6(MAXNS)
LOGICAL WKSPC7(MAXNS)
C Acoustic properties. These data structures are appended after each
C execution of AEBEMA and contain the numerical solution to the test
C problems.
C At the centres of the elements
C Sound pressure [standard unit: newtons per sphere metre (or
C pascals) and phase]
COMPLEX*16 SPRESS(MAXTEST,MAXNS)
C Velocity potential phi
COMPLEX*16 SPHIVAL(MAXTEST,MAXNS)
C Velocity (v) [standard unit: metres per second (and phase)]
C At the centres of the elements
COMPLEX*16 SV(MAXTEST,MAXNS)
C Sound intensity [standard unit: watts per sphere metre]
REAL*8 SINTY(MAXTEST,MAXNS)
C At the exterior points
C Sound pressure [standard unit: newtons per sphere metre (or
C pascals) and phase]
COMPLEX*16 IPRESS(MAXTEST,MAXNPE)
C Velocity potential phi
COMPLEX*16 IPHIVAL(MAXTEST,MAXNPE)
C Counter through the x,y coordinates
INTEGER ICOORD
C Local storage of pressure, pressure/velocity
COMPLEX*16 PRESSURE
C The coordinates of the centres of the elements
REAL*8 SELCNT(MAXNS,2)
REAL*8 EPS
C INITIALISATION
C --------------
C Set constants
ZERO=0.0D0
ONE=1.0D0
TWO=2.0D0
FOUR=4.0D0
PI=4.0D0*ATAN(ONE)
CZERO=CMPLX(ZERO,ZERO)
CONE=CMPLX(ONE,ZERO)
CIMAG=CMPLX(ZERO,ONE)
EPS=1.0E-10
C Reference for decibel scales
PREREF=2.0D-05
C Describe the nodes and elements that make up the boundary
C :The circle that generates the sphere is divided into NS=18 uniform
C : elements. VERTEX and SELV are defined anti-clockwise around the
C : boundary so that the normal to the boundary is assumed to be
C : outward
C :Set up nodes
C : Set NS, the number of elements
NS=18
C : Set NV, the number of vertices (equal to the number of elements)
NV=NS+1
C : Set coordinates of the nodes
DATA ((VERTEX(IV,ICOORD),ICOORD=1,2),IV=1,19)
* / 0.000D0, 1.000D0,
* 0.174D0, 0.985D0,
* 0.342D0, 0.940D0,
* 0.500D0, 0.866D0,
* 0.643D0, 0.766D0,
* 0.766D0, 0.643D0,
* 0.866D0, 0.500D0,
* 0.940D0, 0.342D0,
* 0.985D0, 0.174D0,
* 1.000D0, 0.000D0,
* 0.985D0,-0.174D0,
* 0.940D0,-0.342D0,
* 0.866D0,-0.500D0,
* 0.766D0,-0.643D0,
* 0.643D0,-0.766D0,
* 0.500D0,-0.866D0,
* 0.342D0,-0.940D0,
* 0.174D0,-0.985D0,
* 0.000D0,-1.000D0 /
C :Describe the elements that make up the two boundarys
C : Set NS, the number of elements
NS=18
C : Set nodal indices that describe the elements of the boundarys.
C : The indices refer to the nodes in VERTEX. The order of the
C : nodes in SELV dictates that the normal is outward from the
C : boundary into the acoustic domain.
DATA ((SELV(IS,ICOORD),ICOORD=1,2),IS=1,18)
* / 1, 2,
* 2, 3,
* 3, 4,
* 4, 5,
* 5, 6,
* 6, 7,
* 7, 8,
* 8, 9,
* 9, 10,
* 10, 11,
* 11, 12,
* 12, 13,
* 13, 14,
* 14, 15,
* 15, 16,
* 16, 17,
* 17, 18,
* 18, 19 /
C Set the centres of the elements, the collocation points
DO IS=1,NS
SELCNT(IS,1)=(VERTEX(SELV(IS,1),1)
* +VERTEX(SELV(IS,2),1))/TWO
SELCNT(IS,2)=(VERTEX(SELV(IS,1),2)
* +VERTEX(SELV(IS,2),2))/TWO
ELAREA(IS)=AREACN(VERTEX(SELV(IS,1),1),VERTEX(SELV(IS,1),2),
* VERTEX(SELV(IS,2),1),VERTEX(SELV(IS,2),2))
END DO
C Set the points in the acoustic domain where the acoustic properties
C are sought, PEXT .
NPE=0
C The number of points on the boundary is equal to the number of
C elements
NSP=NS
C Set up test problems
C :Set the number of test problems
NTEST=10
DO 400 ITEST=1,NTEST
C Properties of the acoustic medium. C the propagation velocity
C and RHO the density of the acoustic medium. C>0, RHO>0
C :Acoustic medium is air at 20 celcius and 1 atmosphere.
C [C in metres per second, RHO in kilograms per cubic metre.]
CVAL(ITEST)=344.0D0
RHOVAL(ITEST)=1.205D0
C :Set acoustic frequency value (hertz) in FRVAL
FRVAL(ITEST)=ITEST*100.0D0
C : Set the wavenumber in KVAL
KVAL(ITEST)=TWO*PI*FRVAL(ITEST)/CVAL(ITEST)
C :Set nature of the boundary condition by prescribing the values of
C the boundary functions SALVAL and SBEVAL at the collocation points
C :In this case a Dirichlet (phi-valued) boundary condition
K=KVAL(ITEST)
DO 160 ISP=1,NSP
SALVAL(ITEST,ISP)=CZERO
SBEVAL(ITEST,ISP)=CONE
SFVAL(ITEST,ISP)=1.0D0
160 CONTINUE
DO 180 ISP=1,NSP
SPHIIN(ITEST,ISP)=CZERO
SVELIN(ITEST,ISP)=CZERO
180 CONTINUE
DO 190 IPE=1,NPE
PPHIIN(ITEST,IPE)=CZERO
190 CONTINUE
C Set up validation and control parameters
C :Switch for particular solution
LSOL=.TRUE.
C :Switch on the validation of AEBEMA
LVALID=.TRUE.
C :Set EGEOM
EGEOM=1.0D-6
400 CONTINUE
C Loop(ITEST) through the test problems
DO 500 ITEST=1,NTEST
C Set OMEGA, the angular frequency omega and K, the wavenumber
K=KVAL(ITEST)
OMEGA=2.0D0*PI*FRVAL(ITEST)
C Set up particular alpha and beta functions for this wavenumber
C and type of boundary condition
DO 640 ISP=1,NSP
SALPHA(ISP)=SALVAL(ITEST,ISP)
SBETA(ISP)=SBEVAL(ITEST,ISP)
SF(ISP)=SFVAL(ITEST,ISP)
SFFPHI(ISP)=SPHIIN(ITEST,ISP)
SFFVEL(ISP)=SVELIN(ITEST,ISP)
640 CONTINUE
DO 650 IPE=1,NPE
PFFPHI(IPE)=PPHIIN(ITEST,IPE)
650 CONTINUE
C Set the parameter
MU=CIMAG/(K+ONE)
CALL AEBEMA(K,
* MAXNV,NV,VERTEX,MAXNS,NS,SELV,
* MAXNPE,NPE,PEXT ,
* SALPHA,SBETA,SF,SFFPHI,SFFVEL,PFFPHI,
* LSOL,LVALID,EGEOM,MU,
* SPHI,SVEL,PEPHI,
* WKSPC1,WKSPC2,WKSPC3,WKSPC4,
* WKSPC5,WKSPC6,WKSPC7)
C Compute the sound pressure at the exterior points. Also compute
C the velocity and intensity at the points for each type of boundary
C condition and each related input function f and at each point.
DO 690 ISP=1,NSP
SPHIVAL(ITEST,ISP)=SPHI(ISP)
PRESSURE=CIMAG*RHOVAL(ITEST)*OMEGA*SPHI(ISP)
SPRESS(ITEST,ISP)=PRESSURE
SV(ITEST,ISP)=SVEL(ISP)
SINTY(ITEST,ISP)=
* DBLE(CONJG(PRESSURE)*SVEL(ISP))/TWO
690 CONTINUE
DO 695 ISP=1,NPE
IPHIVAL(ITEST,ISP)=PEPHI(ISP)
IPRESS(ITEST,ISP)=CIMAG*RHOVAL(ITEST)*OMEGA*PEPHI(ISP)
695 CONTINUE
C Close loop(ITEST) through the test problems
500 CONTINUE
C Output the solutions
C Open file for the output data
OPEN(UNIT=20,FILE='AEBEMA.OUT')
C Formats for output
999 FORMAT(3F16.4)
WRITE(20,*) 'AEBEMA: Computed solution to the acoustic properties'
WRITE(20,*)
WRITE(20,*) 'The radiation ratio of a pulsating sphere'
WRITE(20,*)
WRITE(20,*)
WRITE(20,*) ' Frequency Wavenumber '//
*'Radiation ratio'
C Loop(ITEST) through the test problems.
DO 2000 ITEST=1,NTEST
C Output the acoustic frequency
C Loop(ISP) through the points on the boundary
POWER=0.0D0
BPOWER=0.0D0
DO 2030 ISP=1,NSP
C Output the sound pressure, velocity and intensity at each point
POWER=POWER+SINTY(ITEST,ISP)*ELAREA(ISP)
BPOWER=BPOWER+RHOVAL(ITEST)*CVAL(ITEST)*
* ABS(SV(ITEST,ISP))**2*ELAREA(ISP)/2
2030 CONTINUE
RADRAT=POWER/BPOWER
WRITE(20,999) FRVAL(ITEST),KVAL(ITEST),RADRAT
2000 CONTINUE
CLOSE(20)
END
REAL*8 FUNCTION AREACN(R1,Z1,R2,Z2)
REAL*8 R1,Z1,R2,Z2
REAL*8 A,B
REAL*8 PI
PI=4.0D0*ATAN(1.0D0)
IF (ABS(Z2-Z1).GT.0.000001) THEN
A=R1-(R2-R1)*Z1/(Z2-Z1)
B=(R2-R1)/(Z2-Z1)
AREACN=
* ABS(2.0D0*PI*SQRT(1.0D0+B*B)*(Z2-Z1)*(A+B*(Z1+Z2)/2.0D0))
ELSE
AREA=PI*ABS(R1*R1-R2*R2)
END IF
END