C*************************************************************** C Subroutine LEBEM3 by Stephen Kirkup C*************************************************************** C C Copyright 2001- Stephen Kirkup C John Tyndall Nuclear Research Institute C School of Computing Engineering and Physical Sciences C University of Central Lancashire - www.uclan.ac.uk C Westlakes Campus C Samuel Lindow Building C West Lakes Science and Technology Park C Whitehaven C Cumbria CA24 3JY C United Kingdom C smkirkup@uclan.ac.uk C C This open source code can be found at C www.boundary-element-method.com/fortran/LEBEM3.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 This subroutine computes the solution to the three-dimensional C Laplace equation C __ 2 C \/ {\phi} = 0 C C in the domain exterior to a closed boundary. C C The boundary (S) is defined (approximated) by a set of planar C triangular panels. The domain of the equation is within the C boundary. C C The boundary condition may be Dirichlet, Robin or Neumann. It is C 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 solution at the point q on S, v(q) is the C derivative of {\phi} with respect to the outward normal to S at q and C {\alpha}, {\beta} and f are real-valued functions defined on S. The C functions {\alpha} and {\beta} must be specified to define the nature C of the boundary condition. Important examples are {\alpha}=1, C {\beta}=0 which is equivalent to a Dirichlet boundary condition and C {\alpha}=0, {\beta}=1 which is equivalent to a Neumann boundary C condition. The specification of f completes the definition of the C 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 |------->-----| LEBEM3 | : C | (e.g.LEBEM3_t.for) | : | | : C | | : -------------------------- : C ---------------------- : | : C : > : C : | : C : ------------------------ : C Package ---------------->: | subordinate routines | : C : ------------------------ : C : : C : (this file) : C :..................................: C / | | C |_ > > C / | | C ................ ................ ................ C : : : -------- : : -------- : C : (geom3d.for) :---<---: | L3LC | : : | GLS | : C : : : -------- : : -------- : C :..............: : -------------: : -------------: C : |subordinate|: : |subordinate|: C : | routines |: : | routines |: C : -------------: : -------------: C : : : : C : (l3lc.for) : : (gls.for) : C :..............: :..............: C C C The contents of the main program must be linked to LEBEM3.FOR, L3LC.FOR C and GLS.FOR. C C Method of solution C ------------------ C C In the main program, the boundary must be described as a set of C panels. The panels are defined by three indices (integers) which C label a node or vertex on the boundary. The data structure VERTEX C lists and enumerates the coordinates of the vertices, the data C structure SELV defines each panel by indicating the labels for C the three nodes that are its vertices and hence enumerates the C panels. 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 panels. The boundary functions {\alpha} (SALPHA), {\beta} C (SBETA) and f (SF) are also defined by their values at the centres C of the panels. C Normally a solution in the domain is required. By listing the C coordinates of all the exterior points in PENT, the subroutine C returns the value of {\phi} at these points in PEPHI. C C C Format and parameters of the subroutine C --------------------------------------- C C The subroutine has the form C C SUBROUTINE LEBEM3(MAXNV,NV,VERTEX,MAXNSE,NSE,SELV, C * MAXNPE,NPE,PENT, C * SALPHA,SBETA,SF,SIPHI,SIVEL,PDEPHI, C * LSOL,LVALID,EGEOM, C * SPHI,SVEL,PEPHI, C * WKSPC1,WKSPC2,WKSPC3,WKSPC4, C * WKSPC5,WKSPC6,WKSPC7) C The parameters to the subroutine C ================================ 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>=4. C integer NV: The number of vertices on S. 4<=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 C vertex. C integer MAXNSE: The limit on the number of panels describing S. C MAXNSE>=4. C integer NSE: The number of panels describing S. 4<=NSE<=MAXNSE. C integer SELV(MAXNSE,3): The indices of the three vertices defining C each panel. The i-th panel 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 MAXNPE: Limit on the number of points exterior to the C boundary. MAXNPE>=1. C integer NPE: The number of exterior points. 0<=NPE<=MAXNPE. C real PENT(MAXNPE,3). The coordinates of the exterior point. C PENT(i,1),PENT(i,2),PENT(i,3) are the x,y,z coordinates of the i-th C point. C The boundary condition ({\alpha} phi + {\beta} v = f) (input) C real SALPHA(MAXNSE): The values of {\alpha} at the centres C of the panels. C real SBETA(MAXNSE): The values of {\beta} at the centres C of the panels. C real SF(MAXNSE): The values of "f" at the centres of the C panels. C Incident field C real SIPHI(MAXNSE): The incident velocity potential at the C centres of the panels C real SIVEL(MAXNSE): The derivative of the incident velocity C centres of the panels C real PDEPHI(MAXNPE): The incident velocity potential at the chosen C exterior points C Validation and control parameters (input) C logical LSOL: A switch to control whether the particular solution is C required. 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 real SPHI(MAXNSE): The velocity potential ({\phi}) at the C centres of the boundary panels. C real SVEL(MAXNSE): The velocity (v or d{\phi}/dn where n is C the outward normal to the boundary) at the centres of the boundary C panels. C real PEPHI(MAXNPE): The velocity potential ({\phi}) at the C exterior points. C Working space C real WKSPC1(MAXNSE,MAXNSE) C real WKSPC2(MAXNSE,MAXNSE) C real WKSPC3(MAXNPE,MAXNSE) C real WKSPC4(MAXNPE,MAXNSE) C real WKSPC5(MAXNSE) C real WKSPC6(MAXNSE) C logical WKSPC7(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. C (3) The indices of the nodes listed in SELV must be such that they C are ordered counter-clockwise around the boundary, when viewed C from just outside the boundary at the panel. C (4) The largest panel must be no more than 10x the area of the C smallest panel. C Notes on the exterior points C ---------------------------- C (1) The points in PENT 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 PEPHI that is near C zero. This property can be if a useful check. 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 L3LC: Returns the individual discrete Laplace integral C operators. (in file L3LC.FOR) C subroutine GLS: Solves a general linear system of equations. C (in file GLS.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 real function FNEXP(X): real X : Returns the natural logarithm of X. C The subroutine SUBROUTINE LEBEM3(MAXNV,NV,VERTEX,MAXNSE,NSE,SELV, * MAXNPE,NPE,PENT, * SALPHA,SBETA,SF,SIPHI,SIVEL,PDEPHI, * LSOL,LVALID,EGEOM, * SPHI,SVEL,PEPHI, * WKSPC1,WKSPC2,WKSPC3,WKSPC4, * WKSPC5,WKSPC6,WKSPC7) PARAMETER (MAXNQ=100) 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 panels describing S INTEGER MAXNSE C The number of panels describing S INTEGER NSE C The indices of the vertices describing each panel INTEGER SELV(MAXNSE,3) C Interior points at which the solution is to be observed C Limit on the number of points exterior to the boundary where C solution is sought INTEGER MAXNPE C The number of exterior points INTEGER NPE C Coordinates of the exterior points REAL*8 PENT(MAXNPE,3) C The boundary condition is such that {\alpha} {\phi} + {\beta} v = f C where alpha, beta and f are real valued functions over S. C The functions are set values at the central points. C function alpha REAL*8 SALPHA(MAXNSE) C function beta REAL*8 SBETA(MAXNSE) C function f REAL*8 SF(MAXNSE) C The incident potential on S REAL*8 SIPHI(MAXNSE) C The derivative of the incident potential on S REAL*8 SIVEL(MAXNSE) C The incident potential at the chosen exterior points REAL*8 PDEPHI(MAXNPE) C Validation and control parameters LOGICAL LSOL LOGICAL LVALID REAL*8 EGEOM C Solution C function phi REAL*8 SPHI(MAXNSE) C function vel REAL*8 SVEL(MAXNSE) C domain solution REAL*8 PEPHI(MAXNPE) C Working space REAL*8 WKSPC1(MAXNSE,MAXNSE) REAL*8 WKSPC2(MAXNSE,MAXNSE) REAL*8 WKSPC3(MAXNPE,MAXNSE) REAL*8 WKSPC4(MAXNPE,MAXNSE) REAL*8 WKSPC5(MAXNSE) REAL*8 WKSPC6(MAXNSE) LOGICAL WKSPC7(MAXNSE) c External function REAL*8 DIST3 REAL*8 AREA C Constants C Real scalars: 0, 1, 2, half, pi REAL*8 ZERO,ONE,TWO,THREE,HALF,THIRD,PI C Geometrical description of the boundary C panels counter INTEGER ISE,JSE C The points exterior to the boundary where the solution is sought INTEGER IPI C Parameters for L3LC 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 panel (LPONEL=.TRUE.) and C one for the case when P does not lie on the panel. 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 panel QA-QB-QC. For example using more quadrature points when C the panel is large, less when the panel is small, more when C the panel 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 L3LC C Actual number of quadrature points INTEGER NQ C Abscissae of the actual quadrature rule REAL*8 XQ(MAXNQ) C 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 L3LC LOGICAL LVAL REAL*8 EQRULE LOGICAL LFAIL1 LOGICAL LL LOGICAL LM LOGICAL LMT LOGICAL LN C Parameters for subroutine L3LC. REAL*8 DISL REAL*8 DISM REAL*8 DISMT REAL*8 DISN C Other variables C Error flag LOGICAL LERROR C Failure flag LOGICAL LFAIL C Accumulation of solution {\phi} REAL*8 SUMPHI C Maximum,minimum sizes of panels REAL*8 SIZMAX,SIZMIN,SIZE C The `diameter' of the boundary or the maximum distance between any C three vertices REAL*8 DIAM REAL*8 SUMM 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=4.0D0*ATAN(ONE) C Validation C ========== C Validation of parameters of LEBEM3 C --------------------------------- IF (LVALID) THEN C Validation of main paramters LERROR=.FALSE. IF (MAXNV.LT.3) THEN WRITE(*,*) 'MAXNV = ',MAXNV WRITE(*,*) 'ERROR(LEBEM3) - must have MAXNV>=4' LERROR=.TRUE. END IF IF (NV.LT.3.OR.NV.GT.MAXNV) THEN WRITE(*,*) 'NV = ',NV WRITE(*,*) 'ERROR(LEBEM3) - must have 4<=NV<=MAXNV' LERROR=.TRUE. END IF IF (MAXNSE.LT.3) THEN WRITE(*,*) 'MAXNSE = ',MAXNSE WRITE(*,*) 'ERROR(LEBEM3) - must have MAXNSE>=4' LERROR=.TRUE. END IF IF (NSE.LT.3.OR.NSE.GT.MAXNSE) THEN WRITE(*,*) 'NSE = ',NSE WRITE(*,*) 'ERROR(LEBEM3) - must have 4<=NSE<=MAXNSE' LERROR=.TRUE. END IF IF (MAXNPE.LT.1) THEN WRITE(*,*) 'MAXNPE = ',MAXNPE WRITE(*,*) 'ERROR(LEBEM3) - must have MAXNPE>=1' LERROR=.TRUE. END IF IF (NPE.LT.0.OR.NPE.GT.MAXNPE) THEN WRITE(*,*) 'NPE = ',NPE WRITE(*,*) 'ERROR(LEBEM3) - must have 3<=NPE<=MAXNPE' LERROR=.TRUE. END IF IF (EGEOM.LE.ZERO) THEN WRITE(*,*) 'NPE = ',NPE WRITE(*,*) 'ERROR(LEBEM3) - EGEOM must be positive' LERROR=.TRUE. END IF IF (LERROR) THEN LFAIL=.TRUE. WRITE(*,*) WRITE(*,*) 'Error(s) found in the main parameters of LEBEM3' WRITE(*,*) 'Execution terminated' STOP 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 the boundary defined by SELV is complete and closed DO 120 ISE=1,NSE 120 CONTINUE C Check that EGEOM is not too large IF (EGEOM.GT.DIAM/100.0D0) THEN WRITE(*,*) 'EGEOM = ',EGEOM WRITE(*,*) 'ERROR(LEBEM3) - 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 LERROR=.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' LERROR=.TRUE. END IF END IF END IF 140 CONTINUE 130 CONTINUE IF (LERROR) THEN WRITE(*,*) WRITE(*,*) 'ERROR(LEBEM3) - Vertices (see above) coincide' WRITE(*,*) 'Execution terminated' STOP END IF END IF C Check that the panels 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(LEBEM3) - panels of disproportionate' WRITE(*,*) ' sizes' END IF END IF C Validation of the surface functions IF (LVALID.AND.LSOL) 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(LEBEM3) - 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 up validation and control parameters C Switch off the validation of L3LC LVAL=.FALSE. 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 panel. 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 panel. C Set up quadrature rule data. If LPONEL is false then use the standard C Gaussian quadrature rule above. If LPONEL is true then 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 SUMM=0.0D0 DO 180 JSE=1,NSE C Set QA and QB, the coordinates of the edges of the JSEth panel 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 M operator is required. Set LM true, C LL,LMT,LN false. LL=.FALSE. LM=.TRUE. LMT=.FALSE. LN=.FALSE. C Call L3LC. CALL L3LC(P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQON,XQON,YQON,WQON, * LVAL,EGEOM,EQRULE,LFAIL1, * LL,LM,LMT,LN,DISL,DISM,DISMT,DISN) SUMM=SUMM+DISM 180 CONTINUE IF (ABS(SUMM-0.5D0).LT.0.1) THEN WRITE(*,*) WRITE(*,*) 'ERROR(LEBEM3) - 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(SUMM+0.5D0).GT.0.1) THEN WRITE(*,*) WRITE(*,*) 'WARNING(LEBEM3) - in geometry' WRITE(*,*) ' The boundary panels may be arranged incorrectly' END IF END IF C Validation that the points in PENT are exterior points IF (LVALID) THEN DO IPI=1,NPE P(1)=PENT(IPI,1) P(2)=PENT(IPI,2) P(3)=PENT(IPI,3) VECP(1)=0.0D0 VECP(2)=0.0D0 VECP(3)=1.0D0 SUMM=0.0D0 DO 210 JSE=1,NSE C Set QA and QB, the coordinates of the edges of the JSEth panel 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 M operator is required. Set LM true, C LL,LMT,LN false. LL=.FALSE. LM=.TRUE. LMT=.FALSE. LN=.FALSE. C Call L3LC. CALL L3LC(P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQON,XQON,YQON,WQON, * LVAL,EGEOM,EQRULE,LFAIL1, * LL,LM,LMT,LN,DISL,DISM,DISMT,DISN) SUMM=SUMM+DISM 210 CONTINUE IF (ABS(SUMM).GT.0.25) THEN WRITE(*,*) WRITE(*,*) 'WARNING(IBEM2) - The observation point' WRITE(*,*) ' (',P(1),',',P(2),',',P(3),')' WRITE(*,*) ' may not be exterior to the boundary' END IF END DO END IF C Compute the discrete L, M, Mt and N 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 panel at P CALL NORM3(PA,PB,PC,VECP) C Loop(ISE) through the panels DO 520 JSE=1,NSE C Set QA and QB, the coordinates of the edges of the JSEth panel 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 L3LC C : Select the quadrature rule XQON-WQON in the case when the C : point p lies on the panel, 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 All operators are required LL=.TRUE. LM=.TRUE. LMT=.TRUE. LN=.TRUE. C Call L3LC. CALL L3LC(P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQ,XQ,YQ,WQ, * LVAL,EGEOM,EQRULE,LFAIL1, * LL,LM,LMT,LN,DISL,DISM,DISMT,DISN) WKSPC1(ISE,JSE)=DISL+DISMT WKSPC2(ISE,JSE)=DISM+DISN C Close loop(JSE) 520 CONTINUE WKSPC1(ISE,ISE)=WKSPC1(ISE,ISE)+HALF WKSPC2(ISE,ISE)=WKSPC2(ISE,ISE)-HALF IF (LSOL) WKSPC6(ISE)=SIPHI(ISE)+SIVEL(ISE) C Close loop(ISE) 510 CONTINUE IF (LSOL) THEN CALL GLS(MAXNSE,NSE,WKSPC2,WKSPC1,WKSPC6,SALPHA,SBETA,SF, * SPHI,SVEL,LFAIL,WKSPC5,WKSPC7) END IF C SOLUTION IN THE DOMAIN C Compute sound pressures at the selected exterior points. C Loop through the the points in the exterior region DO 800 IPI=1,NPE C Set P P(1)=PENT(IPI,1) P(2)=PENT(IPI,2) P(3)=PENT(IPI,3) C Set VECP, this is arbitrary as the derivative of the potential C at P is not sought. VECP(1)=ONE VECP(2)=ZERO VECP(3)=ZERO C Initialise SUMPHI to the incident potential SUMPHI=PDEPHI(IPI) C Loop(ISE) through the panels DO 850 JSE=1,NSE C Compute the discrete L and M integral operators. C Set QA and QB, the coordinates of the edges of the JSEth panel 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 L, M operators are required. Set LL,LM true, C LMT,LN false. LL=.TRUE. LM=.TRUE. LMT=.FALSE. LN=.FALSE. C Call L3LC. CALL L3LC(P,VECP,QA,QB,QC,LPONEL, * MAXNQ,NQOFF,XQOFF,YQOFF,WQOFF, * LVAL,EGEOM,EQRULE,LFAIL, * LL,LM,LMT,LN,DISL,DISM,DISMT,DISN) C Accumulate phi IF (LSOL) SUMPHI=SUMPHI-DISL*SVEL(JSE)+DISM*SPHI(JSE) IF (.NOT.LSOL) THEN WKSPC3(IPI,JSE)=DISL WKSPC4(IPI,JSE)=DISM END IF C Close loop (JSE) through the panels 850 CONTINUE PEPHI(IPI)=SUMPHI C Close loop(IPI) through the exterior points 800 CONTINUE END C ---------------------------------------------------------------------- C Subordinate routines for LEBEM3 C ============================== C Set up standard Gaussian Quadrature rule. Stores the values C for the 7 point Gaussian quadrature over the standard triangle. C Taken from "Some criteria for numerically integrated matrices and C quadrature formulas for triangles" by M.E.Laursen and M.Gellert in C International Journal for numerical methods in fluids Vol 12 67-76 C 1978. 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 L3LC (not in file L3LC.FOR) C Subroutine for returning the square root. REAL*8 FUNCTION FNSQRT(X) REAL*8 X FNSQRT=SQRT(X) END