C*************************************************************** C Test program for subroutine HIBEM2 by Stephen Kirkup C *************************************************************** C C Copyright 2001- 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/HIBEM2_T.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 HIBEM2. The program computes C the solution to a Helmholtz problem interior to a square 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 HIBEM2 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 square of side 0.1 (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. C Assuming the square has vertices (0,0), (0,0.1), (0.1,0.1) and (0.1,0) C in the x-y plane, the boundary conditions are specified through C taking the solution to be determined by C C {\phi} = sin(r x) sin( r y), C C where r = k/(sqrt(2)), which is clearly a solution of the Helmholtz C equation. C C The form of the test problems are illustrated in the following C diagram. C C ^ C y | C | ^n C | C 0.1 -------------------------------------- C |B C| C | | C | * * | C | | C <- | {\phi} = sin(rx) sin(ry) | -> C n | | n C | * | C | | C | | C | | C | * * | C | | C |A D| C 0.0 -------------------------------------- C | n C 0.0 \ / 0.1 ---> x C C Note that on side CD v = d {\phi}/dx, on AB v = - d {\phi}/dx, C on BC v = d {\phi}/dy and on DA v = - d{\phi}/dy: the sign C is negative if the normal is in the negative x or y direction. 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 HIBEM2: Subroutine for solving the interior Helmholtz C equation. (file HIBEM2.FOR contains HIBEM2 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 IBEM2T 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 potentials are sought INTEGER MAXNPI PARAMETER (MAXNPI=6) C Constants C Real scalars: 0, 1, 2, pi REAL*8 ZERO,ONE,TWO,PI C Complex scalars: (0,0), (1,0), (0,1) COMPLEX*16 CZERO,CONE,CIMAG 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 potentials C 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 HIBEM2. 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 HIBEM2 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 HIBEM2 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 HIBEM2 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 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 C COMPLEX*16 SV(MAXTEST,MAXNS) 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 X,Y C Other variables REAL*8 CONST REAL*8 EPS COMPLEX*16 K COMPLEX*16 H(0:1) COMPLEX*16 KR REAL*8 RR(2),PX,PY,R REAL*8 SIZE2 REAL*8 RK C INITIALISATION C -------------- C Set constants ZERO=0.0D0 ONE=1.0D0 TWO=2.0D0 PI=4.0D0*ATAN(ONE) CZERO=CMPLX(ZERO,ZERO) CONE=CMPLX(ONE,ZERO) CIMAG=CMPLX(ZERO,ONE) EPS=1.0E-10 C Describe the nodes and elements that make up the boundary C :The square with vertices (0,0), (0.1,0), (0.1,0.1), (0,0.1) is divided C : into NS=32 uniform elements. VERTEX and SELV are defined C : anti-clockwise around the boundary so that the normal to the C : boundary is assumed to be outward C :Set up nodes C : Set NS, the number of elements NS=32 C : Set NV, the number of vertices (equal to the number of elements) NV=NS C : Set coordinates of the nodes DATA ((VERTEX(IV,ICOORD),ICOORD=1,2),IV=1,32) * / 0.00000000000D0, 0.00000000000D0, * 0.00000000000D0, 0.01250000000D0, * 0.00000000000D0, 0.02500000000D0, * 0.00000000000D0, 0.03750000000D0, * 0.00000000000D0, 0.05000000000D0, * 0.00000000000D0, 0.06250000000D0, * 0.00000000000D0, 0.07500000000D0, * 0.00000000000D0, 0.08750000000D0, * 0.00000000000D0, 0.10000000000D0, * 0.01250000000D0, 0.10000000000D0, * 0.02500000000D0, 0.10000000000D0, * 0.03750000000D0, 0.10000000000D0, * 0.05000000000D0, 0.10000000000D0, * 0.06250000000D0, 0.10000000000D0, * 0.07500000000D0, 0.10000000000D0, * 0.08750000000D0, 0.10000000000D0, * 0.10000000000D0, 0.10000000000D0, * 0.10000000000D0, 0.08750000000D0, * 0.10000000000D0, 0.07500000000D0, * 0.10000000000D0, 0.06250000000D0, * 0.10000000000D0, 0.05000000000D0, * 0.10000000000D0, 0.03750000000D0, * 0.10000000000D0, 0.02500000000D0, * 0.10000000000D0, 0.01250000000D0, * 0.10000000000D0, 0.00000000000D0, * 0.08750000000D0, 0.00000000000D0, * 0.07500000000D0, 0.00000000000D0, * 0.06250000000D0, 0.00000000000D0, * 0.05000000000D0, 0.00000000000D0, * 0.03750000000D0, 0.00000000000D0, * 0.02500000000D0, 0.00000000000D0, * 0.01250000000D0, 0.00000000000D0 / C :Describe the elements that make up the two boundarys C : Set NS, the number of elements NS=32 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 domain. DATA ((SELV(IS,ICOORD),ICOORD=1,2),IS=1,32) * / 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, * 19, 20, * 20, 21, * 21, 22, * 22, 23, * 23, 24, * 24, 25, * 25, 26, * 26, 27, * 27, 28, * 28, 29, * 29, 30, * 30, 31, * 31, 32, * 32, 1 / C Set the centres of the elements, the collocation points DO 100 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 100 CONTINUE C Set the points in the domain where the potentials are sought, PINT. C : Let them be the points (0.025,0.025), (0.075,0.025), (0.025,0.075), C : (0.075,0.075) and (0.5,0.5). NPI=5 DATA ((PINT(IPI,ICOORD),ICOORD=1,2),IPI=1,5) * / 0.0250D0, 0.0250D0, * 0.0750D0, 0.0250D0, * 0.0250D0, 0.0750D0, * 0.0750D0, 0.0750D0, * 0.0500D0, 0.0500D0 / 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 ============== 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/sqrt(2))x) sin((k/sqrt(2))y) C :Set K, the wavenumber K=KVAL(1) DO 170 ISP=1,NSP X=SELCNT(ISP,1) Y=SELCNT(ISP,2) SFVAL(1,ISP)=SIN(K*X/SQRT(TWO))*SIN(K*Y/SQRT(TWO)) 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 (d phi/ dn -valued) boundary condition DO 205 ISP=1,NSP SALVAL(2,ISP)=CZERO SBEVAL(2,ISP)=CONE 205 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. C Dphi/Dx = (k/sqrt(2)) * cos((k/sqrt(2))x) sin((k/sqrt(2))y) C Dphi/Dy = (k/sqrt(2)) * sin((k/sqrt(2))x) cos((k/sqrt(2))y) CONST=K/SQRT(TWO) DO 210 ISP=1,NSP X=SELCNT(ISP,1) Y=SELCNT(ISP,2) C Face X=0. Note the normal is in the direction of -x IF (X.LT.EPS) THEN RK=DBLE(K) SFVAL(2,ISP)=-CONST*COS(RK*X/SQRT(TWO))*SIN(RK*Y/SQRT(TWO)) C Face X=1. Note that the normal is in the direction of +x ELSE IF (X.GT.0.1D0-EPS) THEN SFVAL(2,ISP)=CONST*COS(RK*X/SQRT(TWO))*SIN(RK*Y/SQRT(TWO)) C Face Y=0. Note the normal is in the direction of -y ELSE IF (Y.LT.EPS) THEN SFVAL(2,ISP)=-CONST*SIN(RK*X/SQRT(TWO))*COS(RK*Y/SQRT(TWO)) ELSE C Face Y=1. Note the normal is in the direction of +y SFVAL(2,ISP)=CONST*SIN(RK*X/SQRT(TWO))*COS(RK*Y/SQRT(TWO)) END IF 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 :The test problem computes the field produced by a unit source at C the point (0.5,0.25) within the square with a rigid boundary. C :Set nature of the boundary condition by prescribing the values of C the boundary functions SALVAL and SBEVAL at the collocation points C :The rigid boundary implies the bondary condition v=0. DO 240 ISP=1,NSP SALVAL(3,ISP)=CZERO SBEVAL(3,ISP)=CONE SFVAL(3,ISP)=CZERO 240 CONTINUE C The incident velocity potential is given by {\phi}_inc=i*h0(kr)/4 C where r is the distance from the point (0.5,0.25) PX=0.05D0 PY=0.025D0 C :Set K, the wavenumber K=KVAL(3) DO 260 ISP=1,NSP X=SELCNT(ISP,1) Y=SELCNT(ISP,2) RR(1)=X-PX RR(2)=Y-PY R=SIZE2(RR) KR=K*R CALL FNHANK(KR,H) SPHIIN(3,ISP)=CIMAG*H(0)/4.0D0 C Face X=0. Note the normal is in the direction of -x IF (X.LT.EPS) THEN SVELIN(3,ISP)=(-CIMAG*K*H(1)/4.0D0)*(-RR(1)/R) C Face X=1. Note that the normal is in the direction of +x ELSE IF (X.GT.0.1D0-EPS) THEN SVELIN(3,ISP)=(-CIMAG*K*H(1)/4.0D0)*(RR(1)/R) C Face Y=0. Note the normal is in the direction of -y ELSE IF (Y.LT.EPS) THEN SVELIN(3,ISP)=(-CIMAG*K*H(1)/4.0D0)*(-RR(2)/R) C Face Y=1. Note the normal is in the direction of +y ELSE SVELIN(3,ISP)=(-CIMAG*K*H(1)/4.0D0)*(RR(2)/R) END IF 260 CONTINUE DO 270 IPI=1,NPI X=PINT(IPI,1) Y=PINT(IPI,2) RR(1)=X-PX RR(2)=Y-PY R=SIZE2(RR) KR=K*R CALL FNHANK(KR,H) PPHIIN(3,IPI)=CIMAG*H(0)/4.0D0 270 CONTINUE C Set up validation and control parameters C :Switch for particular solution LSOL=.TRUE. C :Switch on the validation of HIBEM2 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 IF(DIMAG(K).GT.DREAL(K)) THEN MU=CZERO ELSE MU=CIMAG/(DREAL(K)-DIMAG(K)+ONE) END IF CALL HIBEM2(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 Compute the sound pressure at the interior 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) 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='HIBEM2.OUT') C Formats for output 2810 FORMAT(1X,'Wavenumber = ',F8.2/) 2860 FORMAT(4X,I4,2X,E10.4,'+ ',E10.4,' i ', * E10.4, '+ ',E10.4,' i ',4X, * E10.4, '+ ',E10.4,7X,F10.4) WRITE(20,*) 'HIBEM2: Computed solution to the Helmholtz Equation' WRITE(20,*) C Loop(ITEST) through the test problems. DO 2000 ITEST=1,NTEST C Output the wavenumber WRITE(20,*) WRITE(20,*) WRITE(20,2810) KVAL(ITEST) WRITE(20,*) WRITE(20,*) C Loop(ISP) through the points on the boundary WRITE(20,*) C Loop(IPE) through the points in the exterior DO 2040 IPI=1,NPI C Output the sound pressure, its magnitude(dB) and phase WRITE(20,2860) IPI,DBLE(IPHIVAL(ITEST,IPI)), * AIMAG(IPHIVAL(ITEST,IPI)) 2040 CONTINUE WRITE(20,*) 2000 CONTINUE CLOSE(20) END