C*************************************************************** C Subroutine HMBEMA 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/HMBEMA.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 subroutine computes the modal solutions to the axisymmetric C Helmholtz equation C __ 2 2 C \/ {\phi} + k {\phi} = 0 C C in the domain interior to a closed axisymmetric boundary. C C The boundary (S) is defined (approximated) by a set of straight line C elements. The domain of the equation is within the boundary. C C The nature of the boundary condition may be Dirichlet, Robin or C Neumann and it is specified by the boundary functions {\alpha} and C {\beta}. It is assumed to have the following general form C C {\alpha}(q) {\phi}(q) + {\beta}(q) v(q) = 0 C C where {\alpha} and {\beta} are complex-valued functions defined on S. C Important examples are {\alpha}=1, {\beta}=0 which is equivalent to a C Dirichlet boundary condition and {\alpha}=0, {\beta}=1 which is C equivalent to a Neumann boundary condition. C C C How to use the subroutine C ------------------------- C C The following diagram shows how the subroutine is to be used. A main C program is required. C C .................................... C : : C : : C ---------------------- : -------------------------- : C | | : | | : C | MAIN PROGRAM |------->-----| HMBEMA | : C | (e.g. hmbema_t.for)| : | | : C | | : -------------------------- : C ---------------------- : | : C : > : C : | : C : ------------------------ : C Package ---------------->: | subordinate routines | : C : ------------------------ : C : : C : (this file) : C :..................................: C / | | C |_ > > C / | | C ................ ................ ................ C : : : -------- : : -------- : C : (geom2d.for) :---<---: | H3ALC | : : | CGLS | : C : : : -------- : : -------- : C :..............: : -------------: : -------------: C : |subordinate|: : |subordinate|: C ................ : | routines |: : | routines |: C : : : -------------: : -------------: C : (geom3d.for) :---<---: : : : C : : : (h3alc.for) : : (cgls.for) : C :..............: :..............: :..............: C C C The contents of the main program must be linked to HMBEMA.FOR, C H3ALC.FOR, CGLS.FOR, GEOM2D.FOR and GEOM3D.FOR. C C Method of solution C ------------------ C C In the main program, the boundary must be described as a set of C elements. The elements are defined by two indices (integers) which C label a node or vertex on the boundary generator (the (r,z) C coordinate). The data structure VERTEX lists and enumerates the C (r,z) coordinates of the vertices, the data structure SELV defines C each element by indicating the labels for the two edge nodes on the C generator and hence enumerates the elements. C The boundary solution points (the points on the boundary at which C {\phi} (SPHI) and d {\phi}/dn (SVEL) are returned) are at the centres C of the generator of the elements. The boundary functions {\alpha} C (SALPHA), {\beta} (SBETA) and f (SF) are also defined by their values C at the centres of the elements. C Normally a solution in the domain is required. By listing the (r,z) C coordinates of all the interior points in PINT, the subroutine C returns the value of {\phi} at these points in PIPHI. C C C Format and parameters of the subroutine C --------------------------------------- C C The subroutine has the form C C SUBROUTINE HMBEMA(KA,KB,MAXNK,NK, C * MAXNEIG, C * MAXNV,NV,VERTEX,MAXNSE,NSE,SELV, C * MAXNPI,NPI,PINT, C * SALPHA,SBETA, C * LVALID,EGEOM, C * NEIG,EIGVAL,SPHI,SVEL,PIPHI, C * WKSPC1,WKSPC2,WKSPC3,WKSPC4,WKSPC5,WKSPC6,WKSPC7, C * WKSP00,WKSP01,WKSP02,WKSP03,WKSP04,WKSP05, C * WKSP06,WKSP07,WKSP08,WKSP09,WKSP10,WKSP11, C * WKSP12) C The parameters to the subroutine C ================================ C Wavenumber interpolation information (input) C Lower limit of k-range C REAL*8 KA C Upper limit of k-range C REAL*8 KB C Limit on the number of interpolation points in the k-range C INTEGER MAXNK C Number of interpolation points in the k-range C INTEGER NK C Geometry of the boundary S (input) C integer MAXNV: The limit on the number of vertices of the polygon C that defines (approximates) S. MAXNV>=3. C integer NV: The number of vertices on S. 3<=NV<=MAXNV. C real VERTEX(MAXNV,2): The coordinates of the vertices. VERTEX(i,1), C VERTEX(i,2) are the x,y coordinates of the i-th vertex. C integer MAXNSE: The limit on the number of elements describing S. C MAXNSE>=3. C integer NSE: The number of elements describing S. 3<=NSE<=MAXNSE. C integer SELV(MAXNSE,2): The indices of the two vertices defining C each element. The i-th element have vertices C (VERTEX(SELV(i,1),1),VERTEX(SELV(i,1),2)) and C (VERTEX(SELV(i,2),1),VERTEX(SELV(i,2),2)). C Interior points at which the solution is to be observed (input) C integer MAXNPI: Limit on the number of points interior to the C boundary. MAXNPI>=1. C integer NPI: The number of interior points. 0<=NPI<=MAXNPI. C real PINT(MAXNPI,2). The coordinates of the interior point. C PINT(i,1),PINT(i,2) are the x,y coordinates of the i-th point. C The boundary condition ({\alpha} {\phi} + {\beta} v = 0) (input) C complex SALPHA(MAXNSE): The values of {\alpha} at the centres C of the elements. C complex SBETA(MAXNSE): The values of {\beta} at the centres C of the elements. C Validation and control parameters (input) C logical LVALID: A switch to enable the choice of checking of C subroutine parameters. C real EGEOM: The maximum absolute error in the parameters that C describe the geometry. C Solution (output) C integer NEIG: The number of eigenvalues. C complex EIGVAL(MAXNEIG): The eigenvalues. C complex SPHI(MAXNEIG,MAXNSE): The velocity potential ({\phi}) at C the centres of the boundary elements. C complex SVEL(MAXNEIG,MAXNSE): The velocity (v or d{\phi}/dn C where n is the outward normal to the boundary) at the centres of the C boundary elements. C complex PIPHI(MAXNEIG,MAXNPI): The velocity potential ({\phi}) at C the interior points. C Working space C complex WKSPC1(MAXNK,MAXNSE,MAXNSE) C complex WKSPC2(MAXNK,MAXNSE,MAXNSE) C complex WKSPC3(MAXNK,MAXNSE,MAXNSE) C logical WKSPC4(MAXNSE) C complex WKSP00((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C complex WKSP01((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C complex WKSP02((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C real WKSP03((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C real WKSP04((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C real WKSP05((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C real WKSP06((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C real WKSP07((MAXNK-1)*MAXNSE) C real WKSP08((MAXNK-1)*MAXNSE) C real WKSP09(MAXNK) C integer WKSP10((MAXNK-1)*MAXNSE) C complex WKSP11((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) C complex WKSP12((MAXNK-1)*MAXNSE) C Notes on the geometric parameters C --------------------------------- C (1) Each of the vertices listed in VERTEX must be distinct points C with respect to EGEOM. C (2) The boundary must be complete and closed. Thus C SELV(i,2)=SELV(i+1,1) for i=1..NSE-1 and VERTEX(SELV(1,1),1)=0 C and VERTEX(SELV(NSE,2),1)=0. C (3) The indices of the nodes listed in SELV must be such that they C are ordered counter-clockwise around the generator of the boundary. C (4) The generator of the largest element must be no more than 10x C the length of the generator of the smallest element. C Notes on the interior points C ---------------------------- C (1) The points in PINT should lie within the boundary, as defined C by the parameters VERTEX and SELV. Any point lying outside the C boundary will return a corresponding value in PIPHI that is near C zero. C Notes on the boundary condition C ------------------------------- C (1) For each i=1..NSE, it must not be the case that both of SALPHA(i) C and SBETA(i) are zero C External modules in external files C ================================== C subroutine H3ALC: Returns the individual discrete Helmholtz integral C operators. (in file H3ALC.FOR) C subroutine INTEIG: Solves a general linear system of equations. C (in file INTEIG.FOR) C real function DIST2: Returns the distance between two 2-vectors. (in C file GEOM2D.FOR) C External modules provided in the package (this file) C ==================================================== C subroutine GL8: Returns the points and weights of the 8-point Gauss- C Legendre quadrature rule. C real function FNSQRT(X): real X : Returns the square root of X. C real function FNEXP(Z): real X : Returns the complex exponential of Z. C The subroutine SUBROUTINE HMBEMA(KA,KB,MAXNK,NK, * MAXNEIG, * MAXNV,NV,VERTEX,MAXNSE,NSE,SELV, * MAXNPI,NPI,PINT, * SALPHA,SBETA, * LVALID,EGEOM, * NEIG,EIGVAL,SPHI,SVEL,PIPHI, * WKSPC1,WKSPC2,WKSPC3,WKSPC4,WKSPC5,WKSPC6,WKSPC7, * WKSP00,WKSP01,WKSP02,WKSP03,WKSP04,WKSP05, * WKSP06,WKSP07,WKSP08,WKSP09,WKSP10,WKSP11, * WKSP12) PARAMETER (MAXNGQ=32) PARAMETER (MAXNTQ=10000) C Wavenumber interpolation information C Lower limit of k-range REAL*8 KA C Upper limit of k-range REAL*8 KB C Limit on the number of interpolation points in the k-range INTEGER MAXNK C Number of interpolation points in the k-range INTEGER NK C Eigenvalue information C Limit on the number of eigenvalues determined in the k-range INTEGER MAXNEIG C Boundary geometry C Limit on the number of vertices on S INTEGER MAXNV C The number of vertices on S INTEGER NV C The coordinates of the vertices on S REAL*8 VERTEX(MAXNV,2) C Limit on the number of elements describing S INTEGER MAXNSE C The number of elements describing S INTEGER NSE C The indices of the vertices describing each element INTEGER SELV(MAXNSE,2) C Interior points at which the solution is to be observed C Limit on the number of points interior to the boundary where C solution is sought INTEGER MAXNPI C The number of interior points INTEGER NPI C Coordinates of the interior points REAL*8 PINT(MAXNPI,2) C The boundary condition is such that {\alpha} {\phi} + {\beta} v = 0 C where alpha, beta are complex valued functions over S. C The functions are set values at the collocation points. C function alpha COMPLEX*16 SALPHA(MAXNSE) C function beta COMPLEX*16 SBETA(MAXNSE) C Validation and control parameters LOGICAL LVALID REAL*8 EGEOM C Solution C Number of eigenvalues determined in the k-range INTEGER NEIG C eigenvalues COMPLEX*16 EIGVAL(MAXNEIG) C function phi COMPLEX*16 SPHI(MAXNEIG,MAXNSE) C function vel COMPLEX*16 SVEL(MAXNEIG,MAXNSE) C domain solution COMPLEX*16 PIPHI(MAXNEIG,MAXNPI) C Working space COMPLEX*16 WKSPC1(MAXNK,MAXNSE,MAXNSE) COMPLEX*16 WKSPC2(MAXNK,MAXNSE,MAXNSE) COMPLEX*16 WKSPC3(MAXNK,MAXNSE,MAXNSE) LOGICAL WKSPC4(MAXNSE) C External function REAL*8 DIST2 REAL*8 K COMPLEX*16 CK C Constants C Real scalars: 0, 1, 2, half, pi REAL*8 ZERO,ONE,TWO,HALF,PI C Complex scalars: (0,0), (1,0), (0,1) COMPLEX*16 CZERO,CONE,CIMAG C Geometrical description of the boundary C Elements counter INTEGER ISE,JSE C The points interior to the boundary where the solution is sought INTEGER IIP C Parameters for H2LC REAL*8 P(2),PA(2),PB(2),QA(2),QB(2),VECP(2) LOGICAL LPONEL C Quadrature rule information C [Note that in this program two quadrature rules are used: one for C the case when the point P lies on the element (LPONEL=.TRUE.) and C one for the case when P does not lie on the element.] C Quadrature rule used when LPONEL=.TRUE. C Number of quadrature points INTEGER NGQON C Abscissae of the actual quadrature rule REAL*8 AGQON(MAXNGQ) C Weights of the actual quadrature rule REAL*8 WGQON(MAXNGQ) C Quadrature rule used when LPONEL=.FALSE. C Number of quadrature points INTEGER NGQOFF C Abscissae of the actual quadrature rule REAL*8 AGQOFF(MAXNGQ) C Weights of the actual quadrature rule REAL*8 WGQOFF(MAXNGQ) C Quadrature rule parameters for H3ALC C Counter through the quadrature points INTEGER IGQ C Abscissae of the actual quadrature rule in the theta direction REAL*8 AGQ(MAXNGQ) C Weights of the actual quadrature rule REAL*8 WGQ(MAXNGQ) C Abscissae of the actual quadrature rule in the theta direction REAL*8 ATQ(MAXNTQ) C Weights of the actual quadrature rule REAL*8 WTQ(MAXNTQ) C Validation and control parameters for subroutine H2LC LOGICAL LVAL REAL*8 EK REAL*8 EQRULE LOGICAL LLK LOGICAL LMK LOGICAL LMKT LOGICAL LNK C Parameters for subroutine H2LC. COMPLEX*16 DISLK COMPLEX*16 DISMK COMPLEX*16 DISMKT COMPLEX*16 DISNK REAL*8 WKSPCE(2*MAXNTQ+MAXNGQ) C Parameters for subroutine INTEIG C The k-coordiate of the interpolation matrices REAL*8 WKSPC5(MAXNK) C The list of eigenvectors. EIGVCT(i,j) gives the value of the C j-th component of the i-th eigenvector. The eigenvectors are C normalised COMPLEX*16 WKSPC6((MAXNK-1)*MAXNSE,MAXNSE) LOGICAL WKSPC7(MAXNSE) C Work space C ---------- COMPLEX*16 WKSP00((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) COMPLEX*16 WKSP01((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) COMPLEX*16 WKSP02((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) REAL*8 WKSP03((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) REAL*8 WKSP04((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) REAL*8 WKSP05((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) REAL*8 WKSP06((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) REAL*8 WKSP07((MAXNK-1)*MAXNSE) REAL*8 WKSP08((MAXNK-1)*MAXNSE) REAL*8 WKSP09(MAXNK) INTEGER WKSP10((MAXNK-1)*MAXNSE) COMPLEX*16 WKSP11((MAXNK-1)*MAXNSE,(MAXNK-1)*MAXNSE) COMPLEX*16 WKSP12((MAXNK-1)*MAXNSE) C Other variables C Failure flag LOGICAL LFAIL C Accumulation of solution {\phi} COMPLEX*16 SUMPHI C The `diameter' of the boundary or the maximum distance between any C two vertices REAL*8 DIAM REAL*8 RADMID,SGLEN,GLEN,CIRMID,TDIV INTEGER NDIV REAL*8 SUMMK C INITIALISATION C -------------- C Set constants ZERO=0.0D0 ONE=1.0D0 TWO=2.0D0 HALF=ONE/TWO PI=4.0D0*ATAN(ONE) CZERO=CMPLX(ZERO,ZERO) CONE=CMPLX(ONE,ZERO) CIMAG=CMPLX(ZERO,ONE) C Validation C ========== C Validation of parameters of HMBEMA C --------------------------------- C Find the diameter DIAM of the boundary DIAM=0.0 DO 100 IV=1,NV-1 PA(1)=VERTEX(IV,1) PA(2)=VERTEX(IV,2) DO 110 JV=IV+1,NV PB(1)=VERTEX(JV,1) PB(2)=VERTEX(JV,2) DIAM=MAX(DIAM,DIST2(PA,PB)) 110 CONTINUE 100 CONTINUE C Set the k-values of the interpolation points DO 200 IKPT=1,NK WKSPC5(IKPT)=(KA+KB)/2.0D0- * COS((2*IKPT-1)*PI/DFLOAT(2*NK))*(KB-KA)/2.0D0 200 CONTINUE C Set up validation and control parameters C Switch off the validation of H3ALC LVAL=.FALSE. C Set EK EK=1.0D-6 C Set EQRULE EQRULE=1.0D-6 C Set up the quadrature rule(s). C Set up quadrature rule for the case when P is not on the element. C Set up 8 point Gauss-Legendre rules CALL GL8(MAXNGQ,NGQOFF,WGQOFF,AGQOFF) C Set up quadrature rule for the case when P is on the element. C This is done by splitting the quadrature rule at the centre. NGQON=2*NGQOFF DO 330 IGQ=1,NGQOFF AGQON(IGQ)=AGQOFF(IGQ)/TWO AGQON(NGQOFF+IGQ)=0.5D0+AGQOFF(IGQ)/TWO WGQON(IGQ)=WGQOFF(IGQ)/TWO WGQON(NGQOFF+IGQ)=WGQOFF(IGQ)/TWO 330 CONTINUE C Validation that the surface is closed IF (LVALID) THEN PA(1)=VERTEX(SELV(1,1),1) PA(2)=VERTEX(SELV(1,1),2) PB(1)=VERTEX(SELV(1,2),1) PB(2)=VERTEX(SELV(1,2),2) P(1)=(PA(1)+PB(1))/TWO P(2)=(PA(2)+PB(2))/TWO VECP(1)=0.0D0 VECP(2)=1.0D0 SUMMK=0.0D0 DO 180 JSE=1,NSE C Set QA and QB, the coordinates of the edges of the JSEth element QA(1)=VERTEX(SELV(JSE,1),1) QA(2)=VERTEX(SELV(JSE,1),2) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) C Set LPONEL LPONEL=(JSE.EQ.1) C Quadrature rule in the theta direction is constructed out of individual C Gauss rules so that the length of each is approximately equal to the C length of the element at the generator. NGQ=NGQON RADMID=(QA(1)+QB(1))/TWO SGLEN=(QA(1)-QB(1))*(QA(1)-QB(1))+ * (QA(2)-QB(2))*(QA(2)-QB(2)) GLEN=SQRT(SGLEN) CIRMID=PI*RADMID NDIV=1+CIRMID/GLEN TDIV=ONE/DBLE(NDIV) NTQ=NDIV*NGQ IF (NTQ.GT.MAXNTQ) THEN WRITE(*,*) 'ERROR(IBEM3A) - MAXNTQ is set too small' STOP END IF DO 146 IDIV=1,NDIV DO 156 IGQ=1,NGQ WTQ((IDIV-1)*NGQ+IGQ)=WGQON(IGQ)/DBLE(NDIV) ATQ((IDIV-1)*NGQ+IGQ)=AGQON(IGQ)/DBLE(NDIV)+ * TDIV*DBLE(IDIV-1) 156 CONTINUE 146 CONTINUE C Only the Mk operator is required. Set LMK true, C LLK,LMKT,LNK false. LLK=.FALSE. LMK=.TRUE. LMKT=.FALSE. LNK=.FALSE. C Call H3ALC. CALL H3ALC(CZERO,P,VECP,QA,QB,LPONEL, * MAXNGQ,NGQON,AGQON,WGQON,MAXNTQ,NTQ,ATQ,WTQ, * LVAL,EK,EGEOM,EQRULE,LFAIL, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK, * WKSPCE) SUMMK=SUMMK+DISMK 180 CONTINUE IF (ABS(SUMMK-0.5D0).LT.0.01) THEN WRITE(*,*) WRITE(*,*) 'ERROR(IBEM3A) - in geometry' WRITE(*,*) ' The boundary could be oriented wrongly' WRITE(*,*) ' On the outer boundary arrange panels' * // ' in clockwise order' WRITE(*,*) ' If there are inner boundaries arrange the' * // ' panels in anticlockwise order' STOP END IF IF (ABS(SUMMK+0.5D0).GT.0.01) THEN WRITE(*,*) WRITE(*,*) 'WARNING(IBEM3A) - in geometry' WRITE(*,*) ' The boundary panels may be arranged incorrectly' END IF END IF C Validation that the points in PINT are interior points IF (LVALID) THEN DO IPI=1,NPI P(1)=PINT(IPI,1) P(2)=PINT(IPI,2) VECP(1)=0.0D0 VECP(2)=1.0D0 SUMMK=0.0D0 DO 210 JSE=1,NSE C Set QA and QB, the coordinates of the edges of the ISEth element QA(1)=VERTEX(SELV(JSE,1),1) QA(2)=VERTEX(SELV(JSE,1),2) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) C Set LPONEL LPONEL=.FALSE. C Quadrature rule in the theta direction is constructed out of individual C Gauss rules so that the length of each is approximately equal to the C length of the element at the generator. NGQ=NGQON RADMID=(QA(1)+QB(1))/TWO SGLEN=(QA(1)-QB(1))*(QA(1)-QB(1))+ * (QA(2)-QB(2))*(QA(2)-QB(2)) GLEN=SQRT(SGLEN) CIRMID=PI*RADMID NDIV=1+CIRMID/GLEN TDIV=ONE/DBLE(NDIV) NTQ=NDIV*NGQ IF (NTQ.GT.MAXNTQ) THEN WRITE(*,*) 'ERROR(IBEM3A) - MAXNTQ is set too small' STOP END IF DO 147 IDIV=1,NDIV DO 157 IGQ=1,NGQ WTQ((IDIV-1)*NGQ+IGQ)=WGQON(IGQ)/DBLE(NDIV) ATQ((IDIV-1)*NGQ+IGQ)=AGQON(IGQ)/DBLE(NDIV)+ * TDIV*DBLE(IDIV-1) 157 CONTINUE 147 CONTINUE C Only the Mk operator is required. Set LMK true, C LLK,LMKT,LNK false. LLK=.FALSE. LMK=.TRUE. LMKT=.FALSE. LNK=.FALSE. C Call H3ALC. CALL H3ALC(CZERO,P,VECP,QA,QB,LPONEL, * MAXNGQ,NGQON,AGQON,WGQON,MAXNTQ,NTQ,ATQ,WTQ, * LVAL,EK,EGEOM,EQRULE,LFAIL, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK, * WKSPCE) SUMMK=SUMMK+DISMK 210 CONTINUE IF (ABS(SUMMK+1.0D0).GT.0.01) THEN WRITE(*,*) WRITE(*,*) 'WARNING(IBEM3A) - The observation point' WRITE(*,*) ' (',P(1),',',P(2),')' WRITE(*,*) ' is may not be interior to the boundary' END IF END DO END IF DO 500 IKPT=1,NK K=WKSPC5(IKPT) CK=CMPLX(K,ZERO) C Compute the discrete Lk, Mk, Mkt and Nk matrices C Loop(ISE) through the points on the boundary DO 510 ISE=1,NSE C Set P PA(1)=VERTEX(SELV(ISE,1),1) PA(2)=VERTEX(SELV(ISE,1),2) PB(1)=VERTEX(SELV(ISE,2),1) PB(2)=VERTEX(SELV(ISE,2),2) P(1)=(PA(1)+PB(1))/TWO P(2)=(PA(2)+PB(2))/TWO C Set VECP to the normal on the boundary of the element at P CALL NORM2(PA,PB,VECP) C Loop(ISE) through the elements DO 520 JSE=1,NSE C Set QA and QB, the coordinates of the edges of the ISEth element QA(1)=VERTEX(SELV(JSE,1),1) QA(2)=VERTEX(SELV(JSE,1),2) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) C Set LPONEL IF (ISE.EQ.JSE) THEN LPONEL=.TRUE. ELSE LPONEL=.FALSE. END IF C Select quadrature rule for H3ALC C : Select the quadrature rule AGQON-WGQON in the case when the C : point p lies on the element, otherwise select AGQOFF-WGQOFF C [Note that the overall method would benefit from selecting from C a wider set of quadrature rules, and an appropriate method C of selection] IF (LPONEL) THEN NGQ=NGQON DO 600 IGQ=1,NGQ AGQ(IGQ)=AGQON(IGQ) WGQ(IGQ)=WGQON(IGQ) 600 CONTINUE ELSE NGQ=NGQOFF DO 610 IGQ=1,NGQ AGQ(IGQ)=AGQOFF(IGQ) WGQ(IGQ)=WGQOFF(IGQ) 610 CONTINUE END IF C Quadrature rule in the theta direction is constructed out of individual C Gauss rules so that the length of each is approximately equal to the C length of the element at the generator. RADMID=(QA(1)+QB(1))/TWO SGLEN=(QA(1)-QB(1))*(QA(1)-QB(1))+ * (QA(2)-QB(2))*(QA(2)-QB(2)) GLEN=SQRT(SGLEN) CIRMID=PI*RADMID NDIV=1+CIRMID/GLEN TDIV=ONE/DBLE(NDIV) NTQ=NDIV*NGQ IF (NTQ.GT.MAXNTQ) THEN WRITE(*,*) 'ERROR(HMBEMA) - MAXNTQ is set too small' STOP END IF DO 145 IDIV=1,NDIV DO 155 IGQ=1,NGQ WTQ((IDIV-1)*NGQ+IGQ)=WGQ(IGQ)/DBLE(NDIV) ATQ((IDIV-1)*NGQ+IGQ)=AGQ(IGQ)/DBLE(NDIV)+ * TDIV*DBLE(IDIV-1) 155 CONTINUE 145 CONTINUE C Only Lk and Mk operators are required. Set LLK, LMK true, C LMKT,LNK false. LLK=.TRUE. LMK=.TRUE. LMKT=.FALSE. LNK=.FALSE. C Call of H3ALC routine to compute [Lk], [Mk] CALL H3ALC(CK,P,VECP,QA,QB,LPONEL, * MAXNGQ,NGQ,AGQ,WGQ,MAXNTQ,NTQ,ATQ,WTQ, * LVAL,EK,EGEOM,EQRULE,LFAIL, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK, * WKSPCE) WKSPC1(IKPT,ISE,JSE)=DISLK WKSPC2(IKPT,ISE,JSE)=DISMK C Close loop(JSE) 520 CONTINUE WKSPC2(IKPT,ISE,ISE)=WKSPC2(IKPT,ISE,ISE)+0.5D0 C Close loop(ISE) 510 CONTINUE 500 CONTINUE DO 750 JSE=1,NSE IF (ABS(SALPHA(JSE)).GT.ABS(SBETA(JSE))) THEN WKSPC7(JSE)=.TRUE. ELSE WKSPC7(JSE)=.FALSE. END IF 750 CONTINUE DO 700 IKPT=1,NK DO 710 ISE=1,NSE DO 720 JSE=1,NSE IF (WKSPC7(JSE)) THEN CONST=SBETA(JSE)/SALPHA(JSE) WKSPC3(IKPT,ISE,JSE)= * -CONST*WKSPC2(IKPT,ISE,JSE)-WKSPC1(IKPT,ISE,JSE) ELSE CONST=SALPHA(JSE)/SBETA(JSE) WKSPC3(IKPT,ISE,JSE)= * WKSPC2(IKPT,ISE,JSE)+CONST*WKSPC1(IKPT,ISE,JSE) END IF 720 CONTINUE 710 CONTINUE 700 CONTINUE CALL INTEIG(MAXNSE,NSE,MAXNK,NK,WKSPC5,WKSPC3,NEIG,EIGVAL,WKSPC6, * WKSP00,WKSP01,WKSP02,WKSP03,WKSP04,WKSP05,WKSP06,WKSP07, * WKSP08,WKSP09,WKSP10,WKSP11,WKSP12) DO 810 IEIG=1,NEIG C Set k to the eigenfrequency CK=CMPLX(DBLE(EIGVAL(IEIG)),ZERO) DO 820 ISE=1,NSE IF (WKSPC7(ISE)) THEN SVEL(IEIG,ISE)=WKSPC6(IEIG,ISE) SPHI(IEIG,ISE)=-SBETA(ISE)*SVEL(IEIG,ISE)/SALPHA(ISE) ELSE SPHI(IEIG,ISE)=WKSPC6(IEIG,ISE) SVEL(IEIG,ISE)=-SALPHA(ISE)*SPHI(IEIG,ISE)/SBETA(ISE) END IF 820 CONTINUE C SOLUTION IN THE DOMAIN C Compute sound pressures at the selected interior points. C Loop through the the points in the interior region DO 800 IIP=1,NPI C Set P P(1)=PINT(IIP,1) P(2)=PINT(IIP,2) C Set VECP, this is arbitrary as the velocity/intensity at P C is not sought. VECP(1)=ONE VECP(2)=ZERO C Initialise SUMPHI to zero SUMPHI=CZERO C Loop(ISE) through the elements DO 900 JSE=1,NSE C Compute the discrete Lk and Mk integral operators. C Set QA and QB, the coordinates of the edges of the JSEth element QA(1)=VERTEX(SELV(JSE,1),1) QA(2)=VERTEX(SELV(JSE,1),2) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) C All the points do not lie on the boundary hence LPONEL=.FALSE. LPONEL=.FALSE. C Only Lk, Mk operators are required. Set LLK,LMK true, C LMKT,LNK false. LLK=.TRUE. LMK=.TRUE. LMKT=.FALSE. LNK=.FALSE. C Call of H3ALC routine to compute [Lk], [Mk] CALL H3ALC(CK,P,VECP,QA,QB,LPONEL, * MAXNGQ,NGQ,AGQ,WGQ,MAXNTQ,NTQ,ATQ,WTQ, * LVAL,EK,EGEOM,EQRULE,LFAIL, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK, * WKSPCE) C Accumulate phi SUMPHI=SUMPHI+DISLK*SVEL(IEIG,JSE)-DISMK*SPHI(IEIG,JSE) C Close loop (ISE) through the elements 900 CONTINUE PIPHI(IEIG,IIP)=SUMPHI C Close loop(IIP) through the interior points 800 CONTINUE C Close loop(IEIG) through the eigenfrequencies 810 CONTINUE END C Subordinate routines for HMBEMA C ============================== C ---------------------------------------------------------------------- C Subroutine GL8.FOR by www.numerical-methods.com | C ---------------------------------------------------------------------- C C Subroutine GL8 assigns the weights and points of a 8 point Gaussian C (Gauss-Legendre) quadrature rule defined on the interval [0,1]. C C SUBROUTINE GL8(MAXN, N, WTS, PTS) C integer maxn: the maximimum number of weights/points C integer n: the number of weights/pointsr C real wts: the weights C real pts: the points C C Source of the code: http://www.numerical-methods.com/fortran/GL8.FOR C Source of the user-guide: http://www.numerical-methods.com/fortran/ C gl8.htm C C Licence: This is 'open source'; the software may be used and applied C within other systems as long as its provenance is appropriately C acknowledged. See the GNU Licence http://www.gnu.org/licenses/lgpl.txt C for more information or contact webmaster@numerical-methods.com C Original code 1998. Documentation enhanced 2014 SUBROUTINE GL8(MAXN,N,WTS,PTS) INTEGER MAXN INTEGER N REAL*8 WTS(MAXN) REAL*8 PTS(MAXN) N=8 WTS(1)= 5.061426814519E-02 PTS(1)= 0.980144928249 WTS(2)= 0.111190517227 PTS(2)= 0.898333238707 WTS(3)= 0.156853322939 PTS(3)= 0.762766204958 WTS(4)= 0.181341891689 PTS(4)= 0.591717321248 WTS(5)= 0.181341891689 PTS(5)= 0.408282678752 WTS(6)= 0.156853322939 PTS(6)= 0.237233795042 WTS(7)= 0.111190517227 PTS(7)= 0.101666761293 WTS(8)= 5.061426814519E-02 PTS(8)= 1.985507175123E-02 END C Subroutines required for H3ALC (not in file H3ALC.FOR) C Subroutine for returning the square root. REAL*8 FUNCTION FNSQRT(X) REAL*8 X FNSQRT=SQRT(X) END C Subroutine for returning the exponential. COMPLEX*16 FUNCTION FNEXP(Z) COMPLEX*16 Z FNEXP=EXP(Z) END C ----------------------------------------------------------------------