C*************************************************************** C Test program for subroutine HIBEMA by Stephen Kirkup C*************************************************************** C C Copyright 2001- Stephen Kirkup C School of Computing Engineering and Physical Sciences 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/HIBEMA.FOR C C Issued under the GNU General Public License 2007, 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 C This program is a test for the subroutine HIBEMA. The program computes C the solution to an acoustic/Helmholtz problem interior to a sphere C by the boundary element method. C C For the interior problem, the domain lies interior 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 HIBEMA accepts the wavenumber, a description of the C boundary of the domain and the position of the interior 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 interior points. C C The test problems C ----------------- C C In this test the domain is a sphere of side 1 (metre). 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 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 interior 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 MAXNPI : The limit on the number of interior points. C External modules related to the package C --------------------------------------- C subroutine HIBEMA: Subroutine for solving the interior Helmholtz C equation. (file HIBEMA.FOR contains HIBEMA 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 HIBEMAT 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) C Limit on the number of test problems INTEGER MAXTEST PARAMETER (MAXTEST=5) C Limit on the number of points interior to the boundary, where C acoustic properties are sought INTEGER MAXNPI PARAMETER (MAXNPI=6) 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 Wavenumber parameter for HIBEMA COMPLEX*16 K 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 interior to the boundary(ies) where the acoustic C properties are sought and the directional vectors at those points. C [Only necessary if an interior solution is sought.] C Number of interior points and counter INTEGER NPI,IPI C Coordinates of the interior points REAL*8 PINT(MAXNPI,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 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 interior points COMPLEX*16 PPHIIN(MAXTEST,MAXNPI) C Data structures that are used to define each test problem in turn C and are input parameters to HIBEMA. 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(MAXNPI) C Validation and control parameters for HIBEMA 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 weighting parameter COMPLEX*16 MU C Output from subroutine HIBEMA 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 interior points COMPLEX*16 PIPHI(MAXNPI) C Workspace for HIBEMA COMPLEX*16 WKSPC1(MAXNS,MAXNS) COMPLEX*16 WKSPC2(MAXNS,MAXNS) COMPLEX*16 WKSPC3(MAXNPI,MAXNS) COMPLEX*16 WKSPC4(MAXNPI,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 HIBEMA and contain the numerical solution to the test C problems. C At the centres of the elements 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 At the interior points C Velocity potential phi COMPLEX*16 IPHIVAL(MAXTEST,MAXNPI) C Counter through the x,y coordinates INTEGER ICOORD C The coordinates of the centres of the elements REAL*8 SELCNT(MAXNS,2) C Other variables used in specifying the boundary condition REAL*8 Z,ZP,RPQ,R,DRBDN,QA(2),QB(2),NORMQ(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 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 END DO C Set the points in the acoustic domain where the acoustic properties C are sought, PINT. NPI=4 DATA ((PINT(IPI,ICOORD),ICOORD=1,2),IPI=1,4) * / 0.000D0, 0.000D0, * 0.000D0, 0.500D0, * 0.000D0, -0.500D0, * 0.500D0, 0.000D0 / 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=3 C TEST PROBLEM 1 C ============== C : Set the wavenumber in KVAL KVAL(1)=CONE 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 DO 160 ISP=1,NSP SALVAL(1,ISP)=CONE SBEVAL(1,ISP)=CZERO 160 CONTINUE C :The test problem is devised so that C {\phi}=sin((k z) C :Set K, the wavenumber K=KVAL(1) DO 170 ISP=1,NSP Z=SELCNT(ISP,2) SFVAL(1,ISP)=SIN(K*Z) 170 CONTINUE DO 180 ISP=1,NSP SPHIIN(1,ISP)=0.0D0 SVELIN(1,ISP)=0.0D0 180 CONTINUE DO 190 IPI=1,NPI PPHIIN(1,IPI)=0.0D0 190 CONTINUE C TEST PROBLEM 2 C ============== C : Set the wavenumber in KVAL KVAL(2)=CONE 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 Neumann (v-valued) boundary condition DO 200 ISP=1,NSP SALVAL(2,ISP)=CZERO SBEVAL(2,ISP)=CONE 200 CONTINUE C :The test problem is devised so that C {\phi}=sin((k/sqrt(2))x) sin((k/sqrt(2))y) C :Set K, the wavenumber K=KVAL(2) C Differentiate with respect to x,y to obtain outward normal C derivative. DO 210 ISP=1,NSP Z=SELCNT(ISP,2) QA(1)=VERTEX(SELV(ISP,1),1) QA(2)=VERTEX(SELV(ISP,1),2) QB(1)=VERTEX(SELV(ISP,2),1) QB(2)=VERTEX(SELV(ISP,2),2) CALL NORM2(QA,QB,NORMQ) SFVAL(2,ISP)=K*COS(K*Z)*NORMQ(2) 210 CONTINUE DO 220 ISP=1,NSP SPHIIN(2,ISP)=0.0D0 SVELIN(2,ISP)=0.0D0 220 CONTINUE DO 230 IPI=1,NPI PPHIIN(2,IPI)=0.0D0 230 CONTINUE C TEST PROBLEM 3 C ============== C : Set the wavenumber in KVAL KVAL(3)=CONE 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 DO 240 ISP=1,NSP SALVAL(3,ISP)=CONE SBEVAL(3,ISP)=CZERO 240 CONTINUE C :The test problem is devised so that C {\phi}=sin((k/sqrt(2))x) sin((k/sqrt(2))y) C :Set K, the wavenumber K=KVAL(3) C Differentiate with respect to x,y to obtain outward normal C derivative. ZP=0.25D0 DO 250 ISP=1,NSP R=SELCNT(ISP,1) Z=SELCNT(ISP,2) RPQ=SQRT(R*R+(Z-ZP)*(Z-ZP)) SFVAL(3,ISP)=EXP(CIMAG*K*RPQ)/FOUR/PI/RPQ 250 CONTINUE DO 260 ISP=1,NSP R=SELCNT(ISP,1) Z=SELCNT(ISP,2) RPQ=SQRT(R*R+(ZP-Z)*(ZP-Z)) SPHIIN(3,ISP)=EXP(CIMAG*K*RPQ)/FOUR/PI/RPQ QA(1)=VERTEX(SELV(ISP,1),1) QA(2)=VERTEX(SELV(ISP,1),2) QB(1)=VERTEX(SELV(ISP,2),1) QB(2)=VERTEX(SELV(ISP,2),2) CALL NORM2(QA,QB,NORMQ) DRBDN=(R*NORMQ(1)+(Z-ZP)*NORMQ(2))/RPQ SVELIN(3,ISP)=DRBDN* * EXP(CIMAG*K*RPQ)*(CIMAG*K*RPQ-CONE)/FOUR/PI/RPQ/RPQ 260 CONTINUE DO 270 IPI=1,NPI R=PINT(IPI,1) Z=PINT(IPI,2) RPQ=SQRT(R*R+(ZP-Z)*(ZP-Z)) PPHIIN(3,IPI)=EXP(CIMAG*K*RPQ)/FOUR/PI/RPQ 270 CONTINUE C Set up validation and control parameters C :Switch for particular solution LSOL=.TRUE. C :Switch on the validation of HIBEMA LVALID=.TRUE. C :Set EGEOM EGEOM=1.0D-6 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) 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 IPI=1,NPI PFFPHI(IPI)=PPHIIN(ITEST,IPI) 650 CONTINUE C Set the parameter IF(DIMAG(K).GT.DREAL(K)) THEN MU=CZERO ELSE MU=CIMAG/(DREAL(K)-DIMAG(K)+ONE) END IF CALL HIBEMA(K, * MAXNV,NV,VERTEX,MAXNS,NS,SELV, * MAXNPI,NPI,PINT, * SALPHA,SBETA,SF,SFFPHI,SFFVEL,PFFPHI, * LSOL,LVALID,EGEOM,MU, * SPHI,SVEL,PIPHI, * WKSPC1,WKSPC2,WKSPC3,WKSPC4, * WKSPC5,WKSPC6,WKSPC7) C Save results DO 690 ISP=1,NSP SPHIVAL(ITEST,ISP)=SPHI(ISP) SV(ITEST,ISP)=SVEL(ISP) 690 CONTINUE DO 695 ISP=1,NPI IPHIVAL(ITEST,ISP)=PIPHI(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='HIBEMA.OUT') C Formats for output 2810 FORMAT(1X,'Wavenumber = ',F8.2/) 2860 FORMAT(4X,I4,2X,E10.4,'+ ',E10.4,' i ') WRITE(20,*) 'HIBEMA: Computed solution to the acoustic properties' WRITE(20,*) C Loop(ITEST) through the test problems. DO 2000 ITEST=1,NTEST C Output the acoustic frequency WRITE(20,*) WRITE(20,*) WRITE(20,2810) KVAL(ITEST) WRITE(20,*) C Loop(IPI) through the points in the interior DO 2040 IPI=1,NPI WRITE(20,2860) IPI, IPHIVAL(ITEST,IPI) 2040 CONTINUE WRITE(20,*) 2000 CONTINUE CLOSE(20) END