C*************************************************************** C Test program for 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_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 This program is a test for the subroutine LEBEM3. The program computes C the solution to a Laplace problem exterior to a sphere by the C boundary element method. C C Background C ---------- C C We wish to solve the Laplace equation C C __ 2 C \/ {\phi} = 0 C C C For the exterior problem, the domain lies exterior 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 real-valued functions defined C on S. C C Subroutine LEBEM3 accepts a description of the boundary of the domain C and the position of the exterior points where the solution ({\phi}) C is sought, the boundary condition and returns the solution ({\phi} C and v) on S and the value of {\phi} at the exterior points. C C The test problems C ----------------- C C In this test the domain is a sphere of radius 1 (metre). The solution C to the problem with a Dirichlet boundary condition ({\alpha}=1, C {\beta}=0) on the upper surface (z>=0) and with a Neumann boundary C condition ({\alpha}=0, beta=1) on the lower surface (z<0) are C sought. C Assuming the sphere is centred at the origin the boundary conditions C are specified through taking the solution to be determined by C C C {\phi} = 1/r , C C where r is the distance from a point within the sphere, C which is a solution of the Laplace equation. C C The boundary is described by a set of NS=36 planar triangular elements C of approximately equal size. The boundary solution points are the C centres of the elements. C The solution is sought at the exterior points (0,2), (10,10). 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 MAXNPE : The limit on the number of exterior points. C External modules related to the package C --------------------------------------- C subroutine LEBEM3: Subroutine for solving the exterior Laplace C equation (file LEBEM3.FOR contains the subroutine LEBEM3) C subroutine H3LC: Returns the individual discrete Laplace integral C operators. (file H3LC.FOR contains H3LC and subordinate routines) C subroutine GLS: Solves a general linear system of equations. C (file GLS.FOR contains CGSL and subordinate routines) C file GEOM3D.FOR contains the set of relevant geometric subroutines C The program PROGRAM LBEM3T 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=40) C Limit on the number of vertices (equal to the number of elements) INTEGER MAXNV PARAMETER (MAXNV=30) C Limit on the number of points exterior to the boundary, where C the solutions are sought INTEGER MAXNPE PARAMETER (MAXNPE=6) C Constants C Real scalars: 0, 1, 2, pi REAL*8 ZERO,ONE,TWO,THREE,FOUR,PI 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,3) C The three nodes that define each element on the boundaries INTEGER SELV(MAXNS,3) C The points exterior to the boundary(ies) where the solutions C are sought and the directional vectors at those points. C [Only necessary if an exterior solution is sought.] C Number of exterior points and counter INTEGER NPE,IPE C Coordinates of the exterior points REAL*8 PEXT(MAXNPE,3) C Data structures that are used to define each test problem in turn C and are input parameters to LEBEM3. C SALPHA(j) is assigned the value of {\alpha} at the centre of the C j-th element. REAL*8 SALPHA(MAXNS) C SBETA(j) is assigned the value of {\beta} at the centre of the C j-th element. REAL*8 SBETA(MAXNS) C SF(j) is assigned the value of f at the centre of the j-th element. REAL*8 SF(MAXNS) C Incident field C real SIPHI(MAXNSE): The incident velocity potential at the C centres of the elements REAL*8 SIPHI(MAXNS) C real SIVEL(MAXNSE): The derivative of the incident velocity C centres of the elements REAL*8 SIVEL(MAXNS) C real PIIPHI(MAXNPE): The incident velocity potential at the chosen C interior points REAL*8 PIIPHI(MAXNPE) C Validation and control parameters for LEBEM3 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 Output from subroutine LEBEM3 C The velocity potential (phi - the solution) at the centres of the C elements REAL*8 SPHI(MAXNS) C The normal derivative of the velocity potential at the centres of the C elements REAL*8 SVEL(MAXNS) C The velocity potential (phi - the solution) at exterior points REAL*8 PEIPHI(MAXNPE) C Workspace for LEBEM3 REAL*8 WKSPC1(MAXNS,MAXNS) REAL*8 WKSPC2(MAXNS,MAXNS) REAL*8 WKSPC3(MAXNPE,MAXNS) REAL*8 WKSPC4(MAXNPE,MAXNS) REAL*8 WKSPC5(MAXNS) REAL*8 WKSPC6(MAXNS) LOGICAL WKSPC7(MAXNS) REAL*8 PHI REAL*8 SIZE3 REAL*8 P(3),NP(3),SIZEP C Counter through the x,y coordinates INTEGER ICOORD C The coordinates of the centres of the elements REAL*8 SELCNT(MAXNS,3) REAL*8 EPS C INITIALISATION C -------------- C Set constants ZERO=0.0D0 ONE=1.0D0 TWO=2.0D0 THREE=3.0D0 FOUR=4.0D0 PI=4.0D0*ATAN(ONE) EPS=1.0E-10 C Describe the nodes and elements that make up the boundary C :The unit sphere, centred at the origin is divided C : into NS=36 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=36 C : Set NV, the number of vertices (equal to the number of elements) NV=NS C : Set coordinates of the nodes C Set up VERTEX, the coordinates of the vertices of the elements NV=20 DATA ((VERTEX(IV,ICOORD),ICOORD=1,3),IV=1,20) * / 0.000D0, 0.000D0, 1.000D0, * 0.000D0, 0.745D0, 0.667D0, * 0.645D0, 0.372D0, 0.667D0, * 0.645D0,-0.372D0, 0.667D0, * 0.000D0,-0.745D0, 0.667D0, * -0.645D0,-0.372D0, 0.667D0, * -0.645D0, 0.372D0, 0.667D0, * 0.500D0, 0.866D0, 0.000D0, * 1.000D0, 0.000D0, 0.000D0, * 0.500D0,-0.866D0, 0.000D0, * -0.500D0,-0.866D0, 0.000D0, * -1.000D0, 0.000D0, 0.000D0, * -0.500D0, 0.866D0, 0.000D0, * 0.000D0, 0.745D0,-0.667D0, * 0.645D0, 0.372D0,-0.667D0, * 0.645D0,-0.372D0,-0.667D0, * 0.000D0,-0.745D0,-0.667D0, * -0.645D0,-0.372D0,-0.667D0, * -0.645D0, 0.372D0,-0.667D0, * 0.000D0, 0.000D0,-1.000D0/ 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=1,3),IS=1,36) * / 1, 3, 2, 1, 4, 3, 1, 5, 4, 1, 6, 5, * 1, 7, 6, 1, 2, 7, 2, 3, 8, 3, 9, 8, * 3, 4, 9, 4,10, 9, 4, 5,10, 5,11,10, * 5, 6,11, 6,12,11, 6, 7,12, 7,13,12, * 7, 2,13, 2, 8,13, 8,15,14, 8, 9,15, * 9,16,15, 9,10,16, 10,17,16, 10,11,17, * 11,18,17, 11,12,18, 12,19,18, 12,13,19, * 13,14,19, 13, 8,14, 14,15,20, 15,16,20, * 16,17,20, 17,18,20, 18,19,20, 19,14,20/ C Set the centres of the elements, the collocation points DO IS=1,NS SELCNT(IS,1)=(VERTEX(SELV(IS,1),1) * +VERTEX(SELV(IS,2),1)+VERTEX(SELV(IS,3),1))/THREE SELCNT(IS,2)=(VERTEX(SELV(IS,1),2) * +VERTEX(SELV(IS,2),2)+VERTEX(SELV(IS,3),2))/THREE SELCNT(IS,3)=(VERTEX(SELV(IS,1),3) * +VERTEX(SELV(IS,2),3)+VERTEX(SELV(IS,3),3))/THREE END DO C Set the points in the domain where the solutions are sought, PEXT. C : Let them be the points (0,0,2), (10,10,10). NPE=2 DATA ((PEXT(IPE,ICOORD),ICOORD=1,3),IPE=1,2) * / 0.000D0, 0.000D0, 4.000D0, * 10.000D0, 10.000D0, 10.000D0 / C The number of points on the boundary is equal to the number of C elements NSP=NS C :Switch for particular solution LSOL=.TRUE. C :Switch on the validation of LEBEM3 LVALID=.TRUE. C :Set EGEOM EGEOM=1.0D-6 C Set up particular alpha and beta functions for this wavenumber C and type of boundary condition DO 640 ISP=1,NSP P(1)=SELCNT(ISP,1) P(2)=SELCNT(ISP,2) P(3)=SELCNT(ISP,3) SIZEP=SIZE3(P) NP(1)=P(1)/SIZEP NP(2)=P(2)/SIZEP NP(3)=P(3)/SIZEP SALPHA(ISP)=ONE SBETA(ISP)=ZERO SF(ISP)=PHI(P) 640 CONTINUE DO 660 ISP=1,NSP SIPHI(ISP)=ZERO SIVEL(ISP)=ZERO 660 CONTINUE DO 670 IPE=1,NPE PIIPHI(IPE)=ZERO 670 CONTINUE CALL LEBEM3(MAXNV,NV,VERTEX,MAXNS,NS,SELV, * MAXNPE,NPE,PEXT, * SALPHA,SBETA,SF,SIPHI,SIVEL,PIIPHI, * LSOL,LVALID,EGEOM, * SPHI,SVEL,PEIPHI, * WKSPC1,WKSPC2,WKSPC3,WKSPC4, * WKSPC5,WKSPC6,WKSPC7) C Open file for the output data OPEN(UNIT=20,FILE='LEBEM3.OUT') WRITE(20,*) 'Solutions at the exterior points' WRITE(20,*) ' P(1) P(2) P(3) Computed Exact' DO 800 IPE=1,NPE P(1)=PEXT(IPE,1) P(2)=PEXT(IPE,2) P(3)=PEXT(IPE,3) WRITE(20,999) P(1),P(2),P(3),PEIPHI(IPE),PHI(P) 800 CONTINUE 999 FORMAT(5F10.4) CLOSE(20) END REAL*8 FUNCTION PHI(P) REAL*8 P(3) REAL*8 SIZE3 PHI=1/SIZE3(P) END