C*************************************************************** C Subroutine HMBEM3 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/HMBEM3.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 three-dimensional C Helmholtz equation C __ 2 2 C \/ {\phi} + k {\phi} = 0 C C in the domain interior to a closed boundary. C C The boundary (S) is defined (approximated) by a set of planar C triangular elements. The domain of the equation is within the C 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. C C .................................... C : : C : : C ---------------------- : -------------------------- : C | | : | | : C | MAIN PROGRAM |------->-----| HMBEM3 | : C |(e.g. HMbem2_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) :---<---: | H3LC | : : | INTEIG | : C : : : -------- : : --------- : C :..............: : -------------: : -------------: C : |subordinate|: : |subordinate|: C : | routines |: : | routines |: C : -------------: : -------------: C : : : : C : (h3lc.for) : : (inteig.for) : C :..............: :..............: C C C The contents of the main program must be linked to LBEM2.FOR, C H3LC.FOR, INTEIG.FOR and GEOM3D. 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 three indices (integers) which C label a node or vertex on the boundary. The data structure C VERTEX lists and enumerates the coordinates of the vertices, the C data structure SELV defines each element by indicating the labels for C the three nodes it lies between 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 elements. The boundary functions {\alpha} (SALPHA), and C {\beta} (SBETA) are also defined by their values at the centres C of the elements. C Normally, the modal solution in the domain is required. By listing C the coordinates of a set of 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 HMBEM3(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,3): The coordinates of the vertices. VERTEX(i,1), C VERTEX(i,2),VERTEX(i,3) are the x,y,z 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,3): The indices of the three vertices defining C each element. The i-th element have vertices C (VERTEX(SELV(i,1),1),VERTEX(SELV(i,1),2),VERTEX(SELV(i,1),3)), C (VERTEX(SELV(i,2),1),VERTEX(SELV(i,2),2),VERTEX(SELV(i,2),3)) and C (VERTEX(SELV(i,3),1),VERTEX(SELV(i,3),2),VERTEX(SELV(i,3),3)). 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,3). The coordinates of the interior point. C PINT(i,1),PINT(i,2),PINT(i,3) are the x,y coordinates of the i-th C 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) The boundary must be complete and closed. Thus C SELV(i,2)=SELV(i+1,1) for i=1..NSE-1 and SELV(1,1)=SELV(NSE,2). C (2) The indices of the nodes listed in SELV must be such that they C are ordered counter-clockwise around the boundary. C (3) The largest element must be no more than 10x the length of the C 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 IELEM. 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 Notes on the solution C --------------------- C (1) The results {\phi} on SUD and its derivative on S is a mode shape. C Hence the results can all be multiplied by an arbitrary constant. C External modules in external files C ================================== C subroutine H3LC: Returns the individual discrete Helmholtz integral C operators. (in file H3LC.FOR) C subroutine INTEIG: Solves a general linear system of equations. C (in file INTEIG.FOR) C real function DIST3: Returns the distance between three 3-vectors. (in C file GEOM2D.FOR) C External modules provided in the package (this file) C ==================================================== C subroutine GLT7: Returns the points and weights of the 7-point Gauss- C Legendre quadrature rule on the standard triangle. C real function FNSQRT(X): real X : Returns the square root of X. C complex function FNEXP(Z): complex Z : Returns the complex exponential C of Z. C The subroutine SUBROUTINE HMBEM3(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 (MAXNQ=32) 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,3) 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,3) 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,3) 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 DIST3,AREA C Constants C Real scalars: 0, 1, 2, half, pi REAL*8 ZERO,ONE,TWO,THREE,HALF,THIRD,PI C Complex scalars: (0,0), (1,0), (0,1) COMPLEX*16 CZERO,CONE,CIMAG C Wavenumber REAL*8 K C Wavenumber in complex form COMPLEX*16 CK 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 H3LC REAL*8 P(3),PA(3),PB(3),PC(3),QA(3),QB(3),QC(3),VECP(3) 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. In general, C it is more efficient to define a larger set of quadrature rules C so that a particular rule can be selected for any given point P C and element QA-QB-QC. For example using more quadrature points when C the element is large, less when the element is small, more when C the element is close to P, less when it is far from P.] C Quadrature rule used when LPONEL=.TRUE. C Number of quadrature points INTEGER NQON C x-Abscissae of the actual quadrature rule REAL*8 XQON(MAXNQ) C y-Abscissae of the actual quadrature rule REAL*8 YQON(MAXNQ) C Weights of the actual quadrature rule REAL*8 WQON(MAXNQ) C Quadrature rule used when LPONEL=.FALSE. C Number of quadrature points INTEGER NQOFF C x-Abscissae of the actual quadrature rule REAL*8 XQOFF(MAXNQ) C y-Abscissae of the actual quadrature rule REAL*8 YQOFF(MAXNQ) C Weights of the actual quadrature rule REAL*8 WQOFF(MAXNQ) C Quadrature rule parameters for H3LC C Actual number of quadrature points INTEGER NQ C x-Abscissae of the actual quadrature rule REAL*8 XQ(MAXNQ) C y-Abscissae of the actual quadrature rule REAL*8 YQ(MAXNQ) C Weights of the actual quadrature rule REAL*8 WQ(MAXNQ) C Counter through the quadrature points INTEGER IQ C Validation and control parameters for subroutine H3LC LOGICAL LVAL REAL*8 EK REAL*8 EQRULE LOGICAL LFAIL1 LOGICAL LLK LOGICAL LMK LOGICAL LMKT LOGICAL LNK C Parameters for subroutine H3LC. COMPLEX*16 DISLK COMPLEX*16 DISMK COMPLEX*16 DISMKT COMPLEX*16 DISNK 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 Error flag LOGICAL LERROR C Warning flag LOGICAL LWARN C Failure flag LOGICAL LFAIL C Accumulation of solution {\phi} COMPLEX*16 SUMPHI C Stores the size of a vector REAL*8 SIZE C Maximum,minimum sizes of elements REAL*8 SIZMAX,SIZMIN C The `diameter' of the boundary or the maximum distance between any C two vertices REAL*8 DIAM REAL*8 SUMMK COMPLEX*16 CONST C INITIALISATION C -------------- C Set constants ZERO=0.0D0 ONE=1.0D0 TWO=2.0D0 THREE=3.0D0 HALF=ONE/TWO THIRD=ONE/THREE PI=3.14159265358981 CZERO=CMPLX(ZERO,ZERO) CONE=CMPLX(ONE,ZERO) CIMAG=CMPLX(ZERO,ONE) C Validation C ========== C Validation of parameters of HMBEM3 C --------------------------------- IF (LVALID) THEN C Validation of main paramters LERROR=.FALSE. IF (MAXNK.LT.2) THEN WRITE(*,*) 'MAXNK = ',MAXNK WRITE(*,*) 'ERROR(HMBEM3) - MAXNK must be at least two' LERROR=.TRUE. END IF IF (NK.GT.MAXNK.OR.NK.LT.2) THEN WRITE(*,*) 'NK = ',NK WRITE(*,*) 'ERROR(HMBEM3) - must have 2<=NK<=MAXNK' LERROR=.TRUE. END IF IF (MAXNEIG.LT.1) THEN WRITE(*,*) 'MAXNEIG = ',MAXNEIG WRITE(*,*) 'ERROR(HMBEM3) - MAXNEIG must be at least one' LERROR=.TRUE. END IF IF (KA.LT.ZERO) THEN WRITE(*,*) 'KA = ',KA WRITE(*,*) 'ERROR(HMBEM3) - KA must be positive' LERROR=.TRUE. END IF IF (KB.LE.KA) THEN WRITE(*,*) 'KA,KB = ',KA,KB WRITE(*,*) 'ERROR(HMBEM3) - KB must be greater than KA' LERROR=.TRUE. END IF IF (MAXNEIG.LT.1) THEN WRITE(*,*) 'MAXNEIG = ',MAXNEIG WRITE(*,*) 'ERROR(HMBEM3) - must have MAXNEIG>=1' LERROR=.TRUE. END IF IF (MAXNV.LT.3) THEN WRITE(*,*) 'MAXNV = ',MAXNV WRITE(*,*) 'ERROR(HMBEM3) - must have MAXNV>=4' LERROR=.TRUE. END IF IF (NV.LT.3.OR.NV.GT.MAXNV) THEN WRITE(*,*) 'NV = ',NV WRITE(*,*) 'ERROR(HMBEM3) - must have 4<=NV<=MAXNV' LERROR=.TRUE. END IF IF (MAXNSE.LT.3) THEN WRITE(*,*) 'MAXNSE = ',MAXNSE WRITE(*,*) 'ERROR(HMBEM3) - must have MAXNSE>=4' LERROR=.TRUE. END IF IF (NSE.LT.3.OR.NSE.GT.MAXNSE) THEN WRITE(*,*) 'NSE = ',NSE WRITE(*,*) 'ERROR(HMBEM3) - must have 4<=NSE<=MAXNSE' LERROR=.TRUE. END IF IF (MAXNPI.LT.1) THEN WRITE(*,*) 'MAXNPI = ',MAXNPI WRITE(*,*) 'ERROR(HMBEM3) - must have MAXNPI>=1' LERROR=.TRUE. END IF IF (NPI.LT.0.OR.NPI.GT.MAXNPI) THEN WRITE(*,*) 'NPI = ',NPI WRITE(*,*) 'ERROR(HMBEM3) - must have 3<=NPI<=MAXNPI' LERROR=.TRUE. END IF IF (EGEOM.LE.ZERO) THEN WRITE(*,*) 'NPI = ',NPI WRITE(*,*) 'ERROR(HMBEM3) - EGEOM must be positive' LERROR=.TRUE. END IF IF (LERROR) THEN LFAIL=.TRUE. WRITE(*,*) WRITE(*,*) 'Error(s) found in the main parameters of HMBEM3' WRITE(*,*) 'Execution terminated' STOP END IF END IF C Check PI IF (LVALID) THEN LERROR=.FALSE. IF (ABS(PI-4.0D0*ATAN(ONE)).LT.EGEOM) THEN WKSPC3(MAXNK,MAXNSE,MAXNSE)=CMPLX(115,107) ELSE LERROR=.TRUE. END IF END IF 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) PA(3)=VERTEX(IV,3) DO 110 JV=IV+1,NV PB(1)=VERTEX(JV,1) PB(2)=VERTEX(JV,2) PB(3)=VERTEX(JV,3) DIAM=MAX(DIAM,DIST3(PA,PB)) 110 CONTINUE 100 CONTINUE IF (LVALID) THEN LERROR=.FALSE. C Check that EGEOM is not too large IF (EGEOM.GT.DIAM/100.0D0) THEN WRITE(*,*) 'EGEOM = ',EGEOM WRITE(*,*) 'ERROR(HMBEM3) - EGEOM is set too large' LERROR=.TRUE. END IF IF (LERROR) THEN LFAIL=.TRUE. WRITE(*,*) WRITE(*,*) 'Error in boundary geometry or EGEOM' WRITE(*,*) 'Execution terminated' END IF END IF IF (LVALID) THEN C Check that the vertices are distinct with respect to EGEOM LWARN=.FALSE. DO 130 IV=1,NV-1 PA(1)=VERTEX(IV,1) PA(2)=VERTEX(IV,2) PA(3)=VERTEX(IV,3) DO 140 JV=IV+1,NV PB(1)=VERTEX(JV,1) PB(2)=VERTEX(JV,2) PB(3)=VERTEX(JV,3) IF (ABS(PA(1)-PB(1)).LT.EGEOM) THEN IF (ABS(PA(2)-PB(2)).LT.EGEOM) THEN IF (ABS(PA(3)-PB(3)).LT.EGEOM) THEN WRITE(*,*) 'Vertices ',IV,JV,' are not distinct' LWARN=.TRUE. END IF END IF END IF 140 CONTINUE 130 CONTINUE IF (LERROR) THEN WRITE(*,*) WRITE(*,*) 'WARNING(HMBEM3) - Vertices (see above) coincide' END IF END IF C Check that the elements are not of disproportionate sizes IF (LVALID) THEN SIZMAX=ZERO SIZMIN=DIAM**2 DO 150 ISE=1,NSE QA(1)=VERTEX(SELV(ISE,1),1) QA(2)=VERTEX(SELV(ISE,1),2) QA(3)=VERTEX(SELV(ISE,1),3) QB(1)=VERTEX(SELV(ISE,2),1) QB(2)=VERTEX(SELV(ISE,2),2) QB(3)=VERTEX(SELV(ISE,2),3) QC(1)=VERTEX(SELV(ISE,3),1) QC(2)=VERTEX(SELV(ISE,3),2) QC(3)=VERTEX(SELV(ISE,3),3) SIZE=AREA(QA,QB,QC) SIZMAX=MAX(SIZMAX,SIZE) SIZMIN=MIN(SIZMIN,SIZE) 150 CONTINUE IF (SIZMAX.GT.10.0D0*SIZMIN) THEN WRITE(*,*) 'WARNING(HMBEM3) - Elements of disproportionate' WRITE(*,*) ' sizes' END IF END IF C Validation of the surface functions IF (LVALID) THEN LERROR=.FALSE. DO 170 ISE=1,NSE IF (MAX(ABS(SALPHA(ISE)),ABS(SBETA(ISE))).LT.1.0D-6) * LERROR=.TRUE. 170 CONTINUE IF (LERROR) THEN WRITE(*,*) WRITE(*,*) 'ERROR(HMBEM3) - at most one of SALPHA(i),SBETA(i)' WRITE(*,*) ' may be zero for all i' WRITE(*,*) 'Execution terminated' STOP END IF END IF 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 H3LC 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 GLT7(MAXNQ,NQOFF,WQOFF,XQOFF,YQOFF) C Set up quadrature rule for the case when P is on the element. C Set up quadrature rule data. If LPONEL is false then use the standard C Gaussian quadrature rule above. If LPONEL is true then a C quadrature rule with 3 times as many points is used, this is made C up from three standard quadrature rules with the quadrature points C translated to the three triangles that each have the cetroid and two C of the original vertices as its vertices. NQON=3*NQOFF DO 330 IQ=1,NQOFF XQON(IQ)=XQOFF(IQ)*THIRD+YQOFF(IQ) YQON(IQ)=XQOFF(IQ)*THIRD WQON(IQ)=WQOFF(IQ)/THREE XQON(IQ+NQOFF)=XQOFF(IQ)*THIRD YQON(IQ+NQOFF)=XQOFF(IQ)*THIRD+YQOFF(IQ) WQON(IQ+NQOFF)=WQOFF(IQ)/THREE XQON(IQ+2*NQOFF)=THIRD*(ONE+TWO*XQOFF(IQ)-YQOFF(IQ)) YQON(IQ+2*NQOFF)=THIRD*(ONE-XQOFF(IQ)+TWO*YQOFF(IQ)) WQON(IQ+2*NQOFF)=WQOFF(IQ)/THREE 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) PA(3)=VERTEX(SELV(1,1),3) PB(1)=VERTEX(SELV(1,2),1) PB(2)=VERTEX(SELV(1,2),2) PB(3)=VERTEX(SELV(1,2),3) PC(1)=VERTEX(SELV(1,3),1) PC(2)=VERTEX(SELV(1,3),2) PC(3)=VERTEX(SELV(1,3),3) P(1)=(PA(1)+PB(1)+PC(1))/THREE P(2)=(PA(2)+PB(2)+PC(2))/THREE P(3)=(PA(3)+PB(3)+PC(3))/THREE VECP(1)=0.0D0 VECP(2)=0.0D0 VECP(3)=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) QA(3)=VERTEX(SELV(JSE,1),3) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) QB(3)=VERTEX(SELV(JSE,2),3) QC(1)=VERTEX(SELV(JSE,3),1) QC(2)=VERTEX(SELV(JSE,3),2) QC(3)=VERTEX(SELV(JSE,3),3) C Set LPONEL LPONEL=(JSE.EQ.1) C Only the Mk operators is required. Set LMK true, C LLK,LMKT,LNK false. LLK=.FALSE. LMK=.TRUE. LMKT=.FALSE. LNK=.FALSE. C Call H3LC. CALL H3LC(CZERO,P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQON,XQON,YQON,WQON, * LVAL,EK,EGEOM,EQRULE,LFAIL1, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK) SUMMK=SUMMK+DISMK 180 CONTINUE IF (ABS(SUMMK-0.5D0).LT.0.1) THEN WRITE(*,*) WRITE(*,*) 'ERROR(HMBEM3) - 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.0.1) THEN WRITE(*,*) WRITE(*,*) 'WARNING(HMBEM3) - 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) P(3)=PINT(IPI,3) VECP(1)=0.0D0 VECP(2)=0.0D0 VECP(3)=1.0D0 SUMMK=0.0D0 DO 210 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) QA(3)=VERTEX(SELV(JSE,1),3) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) QB(3)=VERTEX(SELV(JSE,2),3) QC(1)=VERTEX(SELV(JSE,3),1) QC(2)=VERTEX(SELV(JSE,3),2) QC(3)=VERTEX(SELV(JSE,3),3) C Set LPONEL LPONEL=.FALSE. 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 H3LC. CALL H3LC(CZERO,P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQON,XQON,YQON,WQON, * LVAL,EK,EGEOM,EQRULE,LFAIL1, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK) SUMMK=SUMMK+DISMK 210 CONTINUE IF (ABS(SUMMK+1.0D0).GT.0.25) THEN WRITE(*,*) WRITE(*,*) 'WARNING(HMBEM3) - The observation point' WRITE(*,*) ' (',P(1),',',P(2),',',P(3),')' WRITE(*,*) ' may not be interior to the boundary' END IF END DO END IF C Loop(IKPT) through the points on the wavenumbers DO 500 IKPT=1,NK C Set the wavenumber K=WKSPC5(IKPT) C Set the wavenumber in complex form 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) PA(3)=VERTEX(SELV(ISE,1),3) PB(1)=VERTEX(SELV(ISE,2),1) PB(2)=VERTEX(SELV(ISE,2),2) PB(3)=VERTEX(SELV(ISE,2),3) PC(1)=VERTEX(SELV(ISE,3),1) PC(2)=VERTEX(SELV(ISE,3),2) PC(3)=VERTEX(SELV(ISE,3),3) P(1)=(PA(1)+PB(1)+PC(1))/THREE P(2)=(PA(2)+PB(2)+PC(2))/THREE P(3)=(PA(3)+PB(3)+PC(3))/THREE C Set VECP to the normal on the boundary of the element at P CALL NORM3(PA,PB,PC,VECP) C Loop(ISE) through the elements DO 520 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) QA(3)=VERTEX(SELV(JSE,1),3) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) QB(3)=VERTEX(SELV(JSE,2),3) QC(1)=VERTEX(SELV(JSE,3),1) QC(2)=VERTEX(SELV(JSE,3),2) QC(3)=VERTEX(SELV(JSE,3),3) C Set LPONEL IF (ISE.EQ.JSE) THEN LPONEL=.TRUE. ELSE LPONEL=.FALSE. END IF C Select quadrature rule for H3LC C : Select the quadrature rule XQON-WQON in the case when the C : point p lies on the element, otherwise select XQOFF-WQOFF 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 NQ=NQON DO 600 IQ=1,NQ XQ(IQ)=XQON(IQ) YQ(IQ)=YQON(IQ) WQ(IQ)=WQON(IQ) 600 CONTINUE ELSE NQ=NQOFF DO 610 IQ=1,NQ XQ(IQ)=XQOFF(IQ) YQ(IQ)=YQOFF(IQ) WQ(IQ)=WQOFF(IQ) 610 CONTINUE END IF 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 H3LC. CALL H3LC(CK,P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQ,XQ,YQ,WQ, * LVAL,EK,EGEOM,EQRULE,LFAIL1, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK) 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) P(3)=PINT(IIP,3) 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) QA(3)=VERTEX(SELV(JSE,1),3) QB(1)=VERTEX(SELV(JSE,2),1) QB(2)=VERTEX(SELV(JSE,2),2) QB(3)=VERTEX(SELV(JSE,2),3) QC(1)=VERTEX(SELV(JSE,3),1) QC(2)=VERTEX(SELV(JSE,3),2) QC(3)=VERTEX(SELV(JSE,3),3) 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 H3LC. CALL H3LC(CK,P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQOFF,XQOFF,YQOFF,WQOFF, * LVAL,EK,EGEOM,EQRULE,LFAIL, * LLK,LMK,LMKT,LNK,DISLK,DISMK,DISMKT,DISNK) 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 ---------------------------------------------------------------------- C Subordinate routines for HMBEM3 C ============================== C ---------------------------------------------------------------------- C Subroutine GLT7.FOR by www.numerical-methods.com | C ---------------------------------------------------------------------- C C Subroutine GLT7 assigns the weights and points of a 7 point Gaussian C quadrature rule defined on the standard triangle. C C SUBROUTINE GLT7(MAXNQ, NQ, WQ, XQ, YQ) C integer maxnq: the maximimum number of weights/points C integer nq: the number of weights/points C real wq: the weights C real xq: the x-coordinates of the points C real yq: the y-coordinates of the points C C Source of the code: http://www.numerical-methods.com/fortran/GLT7.FOR C Source of the user-guide: http://www.numerical-methods.com/fortran/ C glt7.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 SUBROUTINE GLT7(MAXNQ,NQ,WQ,XQ,YQ) INTEGER MAXNQ,NQ REAL*8 WQ(MAXNQ),XQ(MAXNQ),YQ(MAXNQ) NQ=7 WQ(1)=0.225000000000000D0 WQ(2)=0.125939180544827D0 WQ(3)=0.125939180544827D0 WQ(4)=0.125939180544827D0 WQ(5)=0.132394152788506D0 WQ(6)=0.132394152788506D0 WQ(7)=0.132394152788506D0 XQ(1)=0.333333333333333D0 XQ(2)=0.797426985353087D0 XQ(3)=0.101286507323456D0 XQ(4)=0.101286507323456D0 XQ(5)=0.470142064105115D0 XQ(6)=0.470142064105115D0 XQ(7)=0.059715871789770D0 YQ(1)=0.333333333333333D0 YQ(2)=0.101286507323456D0 YQ(3)=0.797426985353087D0 YQ(4)=0.101286507323456D0 YQ(5)=0.470142064105115D0 YQ(6)=0.059715871789770D0 YQ(7)=0.470142064105115D0 END C Subroutines required for H3LC (not in file H3LC.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 ----------------------------------------------------------------------