C ************************** C * PDE2D 9.2 MAIN PROGRAM * C ************************** C *** 2D PROBLEM SOLVED (GALERKIN METHOD) *** C############################################################################## C Is double precision mode to be used? Double precision is recommended # C on 32-bit computers. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If double precision mode is used, variables and functions assigned +# C + names beginning with a letter in the range A-H or O-Z will be DOUBLE +# C + PRECISION, and you should use double precision constants and FORTRAN +# C + expressions throughout; otherwise such variables and functions will +# C + be of type REAL. In either case, variables and functions assigned +# C + names beginning with I,J,K,L,M or N will be of INTEGER type. +# C + +# C + It is possible to convert a single precision PDE2D program to double +# C + precision after it has been created, using an editor. Just change +# C + all occurrences of "real" to "double precision" +# C + " tdp" to "dtdp" (note leading blank) +# C + Any user-written code or routines must be converted "by hand", of +# C + course. To convert from double to single, reverse the changes. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## implicit double precision (a-h,o-z) parameter (neqnmx= 99) parameter (ndelmx= 20) parameter (nxgrid=0,nygrid=0,npgrid=0,nqgrid=0) C############################################################################## C NV0 = number of vertices in initial triangulation # C############################################################################## PARAMETER (NV0 = 7) C############################################################################## C NT0 = number of triangles in initial triangulation # C############################################################################## PARAMETER (NT0 = 7) C############################################################################## C How many differential equations (NEQN) are there in your problem? # C############################################################################## PARAMETER (NEQN = 1) C DIMENSIONS OF WORK ARRAYS C SET TO 1 FOR AUTOMATIC ALLOCATION PARAMETER (IRWK8Z= 1) PARAMETER (IIWK8Z= 1) PARAMETER (NXP8Z=101,NYP8Z=101,KDEG8Z=1,NBPT8Z=51) C############################################################################## C The solution is normally saved on a NX+1 by NY+1 rectangular grid of # C points # C (XA + I*(XB-XA)/NX , YA + J*(YB-YA)/NY) # C I=0,...,NX, J=0,...,NY. Enter values for NX and NY. Suggested values # C are NX=NY=25. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you want to save the solution at an arbitrary user-specified +# C + set of points, set NY=0 and NX+1=number of points. In this case you +# C + can request tabular output of the solution, but you cannot make any +# C + solution plots. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## PARAMETER (NX = 100) PARAMETER (NY = 50) PARAMETER (NSAVE = 1) common/parm8z/ pi,D1 ,D2 dimension vxy(2,nv0+1),iabc(3,nt0+1),iarc(nt0+1),xgrid(nxgrid+1),y &grid(nygrid+1),ixarc(2),iyarc(2),pgrid(npgrid+1),qgrid(nqgrid+1),i &parc(2),iqarc(2),xbd8z(nbpt8z,nt0+4),ybd8z(nbpt8z,nt0+4),xout8z(0: &nx,0:ny),yout8z(0:nx,0:ny),inrg8z(0:nx,0:ny),xcross(100),ycross(10 &0),tout8z(0:nsave),uout8z(0:nx,0:ny,3*neqn,0:nsave),uout(0:nx,0:ny &,3,neqn,0:nsave),xres8z(nxp8z),yres8z(nyp8z),ures8z(neqn,nxp8z,nyp &8z),xd0(ndelmx),yd0(ndelmx) equivalence (uout,uout8z) allocatable iwrk8z(:),rwrk8z(:) C dimension iwrk8z(iiwk8z),rwrk8z(irwk8z) character*40 title logical plot,symm,fdiff,evcmpx,crankn,noupdt,adapt,nodist,fillin,e &con8z,ncon8z,restrt,gridid common/dtdp14/ sint8z(20),bint8z(20),slim8z(20),blim8z(20) common/dtdp15/ evlr8z,ev0r,evli8z,ev0i,evcmpx common/dtdp16/ p8z,evr8z(50),evi8z(50) common/dtdp19/ toler(neqnmx),adapt common/dtdp22/ nxa8z,nya8z,ifgr8z,kd8z,nbp8z common/dtdp23/ work8z(nxp8z*nyp8z+6) common/dtdp30/ econ8z,ncon8z common/dtdp46/ eps8z,cgtl8z,npmx8z,itype common/dtdp63/ amin8z(3*neqnmx),amax8z(3*neqnmx) common/dtdp65/ intri,iotri pi = 4.0*atan(1.d0) nxa8z = nxp8z nya8z = nyp8z kd8z = kdeg8z nbp8z = nbpt8z C############################################################################## C If you don't want to read the FINE PRINT, default NPROB. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you want to solve several similar problems in the same run, set +# C + NPROB equal to the number of problems you want to solve. Then NPROB +# C + loops through the main program will be done, with IPROB=1,...,NPROB, +# C + and you can make the problem parameters vary with IPROB. NPROB +# C + defaults to 1. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## NPROB = 1 do 78755 iprob=1,nprob C############################################################################## C You may now define global parameters, which may be referenced in any # C of the "FORTRAN expressions" you input throughout the rest of this # C interactive session. You will be prompted alternately for parameter # C names and their values; enter a blank name when you are finished. # C # C Parameter names are valid FORTRAN variable names, starting in # C column 1. Thus each name consists of 1 to 6 alphanumeric characters, # C the first of which must be a letter. If the first letter is in the # C range I-N, the parameter must be an integer. # C # C Parameter values are either FORTRAN constants or FORTRAN expressions # C involving only constants and global parameters defined on earlier # C lines. They may also be functions of the problem number IPROB, if # C you are solving several similar problems in one run (NPROB > 1). Note # C that you are defining global CONSTANTS, not functions; e.g., parameter # C values may not reference any of the independent or dependent variables # C of your problem. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you define other parameters here later, using an editor, you must +# C + add them to COMMON block /PARM8Z/ everywhere this block appears, if +# C + they are to be "global" parameters. +# C + +# C + The variable PI is already included as a global parameter, with an +# C + accurate value 3.14159... +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## D1 = & 5 D2 = & 5 C *******INITIAL TRIANGULATION OPTION INTRI = 3 C *******SET IOTRI = 1 TO DUMP FINAL TRIANGULATION TO FILE pde2d.tri IOTRI = 0 C############################################################################## C For a general region, an initial triangulation is constructed # C which generally consists of only as many triangles as needed to define # C the region and to satisfy the following rules (triangles adjacent # C to a curved boundary may be considered to have one curved edge): # C # C 1. The end points of each arc are included as vertices in the # C triangulation. # C # C 2. No vertex of any triangle may touch another in a point which is # C not a vertex of the other triangle. # C # C 3. No triangle may have all three vertices on the boundary. # C # C Now enter the number of vertices in the initial triangulation (NV0) # C and the vertices (VXY(1,i),VXY(2,i)), i=1,...,NV0, when prompted. # C # C The vertices may be numbered in any order, but that order will define # C the vertex numbers referred to in the next list. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If the elements of VXY, IABC and IARC are defaulted, these initial +# C + triangulation arrays will be read from the file 'pde2d.tri'; in this +# C + case the values of NV0 and NT0 entered below must be the same as +# C + those in the file. +# C + +# C + A file 'pde2d.tri' is normally created when the final triangulation +# C + from another program (cases INTRI=1,2 or 3) is dumped. Reset IOTRI +# C + to 1 in the other program, to cause the final triangulation to be +# C + dumped into pde2d.tri. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## call dtdpwx(vxy,2*nv0,0) C VXY(1,1) = & -1 VXY(2,1) = & 0 C VXY(1,2) = & 0 VXY(2,2) = & 0 C VXY(1,3) = & 1 VXY(2,3) = & 0 C VXY(1,4) = & 0 VXY(2,4) = & 1 C VXY(1,5) = & -1 VXY(2,5) = & 1 C VXY(1,6) = & -0.5 VXY(2,6) = & 0.5 C VXY(1,7) = & 0.2 VXY(2,7) = & 0.4 C############################################################################## C Now enter the number of triangles in the initial triangulation (NT0), # C and for each triangle k, give the vertex numbers of vertices a,b,c: # C IABC(1,k) , IABC(2,k) , IABC(3,k) # C where the third vertex (c) is not on the boundary of R (or on a curved # C interface arc), and the number, # C IARC(k) # C of the arc cut off by the base, ab, of the triangle. Put IARC(k)=0 # C if the base of triangle k does not intersect any boundary (or curved # C interface) arc. Recall that negative arc numbers correspond to # C 'fixed' boundary conditions, and positive arc numbers ( < 1000) # C correspond to 'free' boundary conditions. # C # C The order of this list defines the initial triangle numbers. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Any of the partial differential equation coefficients or other +# C + functions of X and Y which you are asked to supply later may be +# C + defined as functions of a variable 'KTRI', which holds the number +# C + of the INITIAL triangle in which (X,Y) lies. This is useful when +# C + the region is a composite of different materials, so that some +# C + material parameters have different values in different subregions. +# C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++# C + If any interfaces between subregions are curved, the interface arcs +# C + may be assigned unique arc numbers of 1000 and above, and treated +# C + like boundary arcs in the initial triangulation definition. (You +# C + will not be allowed to specify boundary conditions on them, however.)+# C + The triangulation refinement will follow the interface arcs, so that +# C + no final triangles will straddle an interface. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C IABC(1,1) = & 1 IABC(2,1) = & 2 IABC(3,1) = & 6 IARC(1) = & 1 C IABC(1,2) = & 2 IABC(2,2) = & 4 IABC(3,2) = & 6 IARC(2) = & 0 C IABC(1,3) = & 4 IABC(2,3) = & 5 IABC(3,3) = & 6 IARC(3) = & 2 C IABC(1,4) = & 5 IABC(2,4) = & 1 IABC(3,4) = & 6 IARC(4) = & -1 C IABC(1,5) = & 2 IABC(2,5) = & 3 IABC(3,5) = & 7 IARC(5) = & 1 C IABC(1,6) = & 3 IABC(2,6) = & 4 IABC(3,6) = & 7 IARC(6) = & 3 C IABC(1,7) = & 4 IABC(2,7) = & 2 IABC(3,7) = & 7 IARC(7) = & 0 call dtdpu(vxy,nv0,iabc,iarc,nt0) C############################################################################## C How many triangles (NTF) are desired for the final triangulation? # C############################################################################## NTF = 2000 C############################################################################## C If you don't want to read the FINE PRINT, enter ISOLVE = 4. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The following linear system solvers are available: +# C + +# C + 1. Band method +# C + The band solver uses a reverse Cuthill-McKee ordering. +# C + 2. Frontal method +# C + This is an out-of-core version of the band solver. +# C + 3. Jacobi bi-conjugate gradient method +# C + This is a preconditioned bi-conjugate gradient, or +# C + Lanczos, iterative method. (This solver is MPI- +# C + enhanced, if MPI is available.) If you want to +# C + override the default convergence tolerance, set a +# C + new relative tolerance CGTL8Z in the main program. +# C + 4. Sparse direct method +# C + This is based on Harwell Library routines MA27/MA37, +# C + developed by AEA Industrial Technology at Harwell +# C + Laboratory, Oxfordshire, OX11 0RA, United Kingdom +# C + (used by permission). +# C + 5. Local solver +# C + Choose this option ONLY if alternative linear system +# C + solvers have been installed locally. See subroutines +# C + (D)TD3M, (D)TD3N in file (d)subs.f for instructions +# C + on how to add local solvers. +# C + 6. MPI-based parallel band solver +# C + This is a parallel solver which runs efficiently on +# C + multiple processor machines, under MPI. It is a +# C + band solver, with the matrix distributed over the +# C + available processors. Choose this option ONLY if the +# C + solver has been activated locally. See subroutine +# C + (D)TD3O in file (d)subs.f for instructions on how to +# C + activate this solver and the MPI-enhancements to the +# C + conjugate gradient solver. +# C + +# C + Enter ISOLVE = 1,2,3,4,5 or 6 to select a linear system solver. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ISOLVE = 4 C############################################################################## C Enter the element degree (1,2,3 or 4) desired. A suggested value is # C IDEG = 3. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + A negative value for IDEG can be entered, and elements of degree +# C + ABS(IDEG) will be used, with a lower order numerical integration +# C + scheme. This results in a slight increase in speed, but negative +# C + values of IDEG are normally not recommended. However, IDEG = -1 +# C + is the only choice which will produce a lumped (diagonal) mass +# C + matrix. +# C + +# C + The spatial discretization error is O(h**2), O(h**3), O(h**4) or +# C + O(h**5) when elements of degree 1,2,3 or 4, respectively, are used, +# C + where h is the maximum triangle diameter, even if the region is +# C + curved. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## IDEG = 3 C *******STEADY-STATE PROBLEM itype = 1 t0 = 0.0 dt = 1.0 crankn = .false. noupdt = .false. C Number of Newton iterations NSTEPS = 1 C############################################################################## C PDE2D solves the equation: # C # C d/dX* A(X,Y,C,Cx,Cy) # C + d/dY* B(X,Y,C,Cx,Cy) # C = F(X,Y,C,Cx,Cy) # C # C with 'fixed' boundary condition: # C # C C = FB(X,Y) # C # C or 'free' boundary condition: # C # C A*nx + B*ny = GB(X,Y,C,Cx,Cy) # C # C where C(X,Y) is the unknown and F,A,B,FB,GB are user-supplied # C functions. # C # C note: # C (nx,ny) = unit outward normal to the boundary # C Cx = d(C)/dX # C Cy = d(C)/dY # C # C Is this problem symmetric? If you don't want to read the FINE PRINT, # C it is safe to enter 'no'. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + This problem is called symmetric if the matrix +# C + +# C + F.C F.Cx F.Cy +# C + A.C A.Cx A.Cy +# C + B.C B.Cx B.Cy +# C + +# C + is always symmetric, where F.C means d(F)/d(C), and similarly +# C + for the other terms. In addition, GB must not depend on Cx,Cy. +# C + +# C + The memory and execution time are halved if the problem is known to +# C + be symmetric. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## SYMM = .TRUE. FDIFF = .FALSE. C############################################################################## C You may calculate one or more integrals (over the entire region) of # C some functions of the solution and its derivatives. How many integrals # C (NINT), if any, do you want to calculate? # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + In the FORTRAN program created by the preprocessor, the computed +# C + values of the integrals will be returned in the vector SINT8Z. If +# C + several iterations or time steps are done, only the last computed +# C + values are saved in SINT8Z (all values are printed). +# C + +# C + A limiting value, SLIM8Z(I), for the I-th integral can be set +# C + below in the main program. The computations will then stop +# C + gracefully whenever SINT8Z(I) > SLIM8Z(I), for any I=1...NINT. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## NINT = 1 C############################################################################## C You may calculate one or more boundary integrals (over the entire # C boundary) of some functions of the solution and its derivatives. How # C many boundary integrals (NBINT), if any, do you want to calculate? # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + In the FORTRAN program created by the preprocessor, the computed +# C + values of the integrals will be returned in the vector BINT8Z. If +# C + several iterations or time steps are done, only the last computed +# C + values are saved in BINT8Z (all values are printed). +# C + +# C + A limiting value, BLIM8Z(I), for the I-th boundary integral can be +# C + set below in the main program. The computations will then stop +# C + gracefully whenever BINT8Z(I) > BLIM8Z(I), for any I=1...NBINT. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## NBINT = 1 ndel = 0 RESTRT = .FALSE. GRIDID = .TRUE. C############################################################################## C The solution is saved on an NX+1 by NY+1 rectangular grid covering the # C rectangle (XA,XB) x (YA,YB). Enter values for XA,XB,YA,YB. These # C variables are usually defaulted. # C # C The default is a rectangle which just covers the entire region. # C############################################################################## C defaults for xa,xb,ya,yb call dtdpv (xgrid,nxgrid,ygrid,nygrid,vxy,nv0,iarc,nt0,xa,xb,ya, &yb) C DEFINE XA,XB,YA,YB HERE: call dtdpx2(nx,ny,xa,xb,ya,yb,hx8z,hy8z,xout8z,yout8z,npts8z) C SOLUTION SAVED EVERY NOUT ITERATIONS NOUT = NSTEPS C *******allocate workspace call dtdpzz(ntf,ideg,isolve,symm,neqn,ii8z,ir8z) if (iiwk8z.gt.1) ii8z = iiwk8z if (irwk8z.gt.1) ir8z = irwk8z allocate (iwrk8z(ii8z),rwrk8z(ir8z)) C *******DRAW TRIANGULATION PLOTS (OVER C *******RECTANGLE (XA,XB) x (YA,YB))? PLOT = .TRUE. C *******call pde solver call dtdp2x(xgrid, nxgrid, ygrid, nygrid, ixarc, iyarc, vxy, nv0, &iabc, nt0, iarc, restrt, gridid, neqn, ntf, ideg, isolve, nsteps, &nout, t0, dt, plot, symm, fdiff, itype, nint, nbint, ndel, xd0, yd &0, crankn, noupdt, xbd8z, ybd8z, nbd8z, xout8z, yout8z, uout, inrg &8z, npts8z, ny, tout8z, nsave, iwrk8z, ii8z, rwrk8z, ir8z) deallocate (iwrk8z,rwrk8z) C *******read from restart file to array ures8z C call dtdpr2(1,xres8z,nxp8z,yres8z,nyp8z,ures8z,neqn) C *******write array ures8z back to restart file C call dtdpr2(2,xres8z,nxp8z,yres8z,nyp8z,ures8z,neqn) C *******call user-written postprocessor call postpr(tout8z,nsave,xout8z,yout8z,inrg8z,nx,ny,uout,neqn) C *******CONTOUR PLOT C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means C (possibly as modified by UPRINT,..) # C 2 A # C 3 B # C############################################################################## IVAR = 1 ISET1 = 1 ISET2 = NSAVE ISINC = 1 C############################################################################## C If you don't want to read the FINE PRINT, enter 'no'. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Do you want to scale the axes on the plot so that the region is +# C + undistorted? Otherwise the axes will be scaled so that the figure +# C + approximately fills the plot space. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## NODIST = .TRUE. C alow = amin8z(ivar) ahigh = amax8z(ivar) C############################################################################## C Enter lower (UMIN) and upper (UMAX) bounds for the contour values. UMIN # C and UMAX are often defaulted. # C # C Labeled contours will be drawn corresponding to the values # C # C UMIN + S*(UMAX-UMIN), for S=0.05,0.15,...0.95. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + By default, UMIN and UMAX are set to the minimum and maximum values +# C + of the variable to be plotted. For a common scaling, you may want +# C + to set UMIN=ALOW, UMAX=AHIGH. ALOW and AHIGH are the minimum and +# C + maximum values over all output points and over all saved time steps +# C + or iterations. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## UMIN = 0.0 UMAX = 0.0 C############################################################################## C Do you want two additional unlabeled contours to be drawn between each # C pair of labeled contours? # C############################################################################## FILLIN = .FALSE. C############################################################################## C Enter a title, WITHOUT quotation marks. A maximum of 40 characters # C are allowed. The default is no title. # C############################################################################## TITLE = ' ' TITLE = 'C ' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78756 is8z=iset1,iset2,isinc call dtdplg(uout8z(0,0,ivar,is8z),nx,ny,xa,ya,hx8z,hy8z,inrg8z,xbd &8z,ybd8z,nbd8z,title,umin,umax,nodist,fillin,tout8z(is8z)) 78756 continue 78755 continue call endgks stop end subroutine tran8z(p,q,x,y) implicit double precision (a-h,o-z) x = p y = q return end subroutine xy8z(i8z,iarc8z,s,x,y,s0,sf) implicit double precision (a-h,o-z) dimension pxy(2,1000) common/parm8z/ pi,D1 ,D2 x = 0.0 y = 0.0 C############################################################################## C Enter the arc number (IARC) of a curved arc. # C############################################################################## IARC = 3 if (iarc8z.eq.iarc) then C############################################################################## C The arc is parameterized as X=X(S),Y=Y(S) where S varies from S0 to SF # C as it traces out the arc, in either direction. # C # C Enter FORTRAN expressions for S0, SF, X(S) and Y(S). S0 and SF default # C to 0 and 1. # C############################################################################## if (i8z.eq.0) then S0 = & 0 SF = & 1 else X = & s Y = & cos(pi/2*s) endif endif return end subroutine dis8z(x,y,ktri,triden,shape) implicit double precision (a-h,o-z) common/parm8z/ pi,D1 ,D2 adapt = dtdplx(2) C############################################################################## C Now enter a FORTRAN expression for TRIDEN(X,Y), which controls the # C grading of the triangulation. TRIDEN should be largest where the # C triangulation is to be most dense. The default is TRIDEN(X,Y)=1.0 # C (a uniform triangulation). # C # C If you want the triangulation to be graded "adaptively", set # C TRIDEN = ADAPT and and make sure there is no "pde2d.adp" file in the # C working directory the first time you run the program, then run the # C program 2 or more times, possibly increasing NTF each time. On the # C first run, a uniform triangulation will be generated, but on each # C subsequent run, information output by the previous run to "pde2d.adp" # C will be used to guide the grading of the new triangulation. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If TRIDEN = ADAPT, after each run a file "pde2d.adp" is written +# C + which tabulates the values of the magnitude of the gradient of the +# C + solution (at the last time step or iteration) at an output NXP8Z by +# C + NYP8Z grid of points (NXP8Z and NYP8Z are set to 101 in a PARAMETER +# C + statement in the main program, so they can be changed if desired). +# C + If NEQN > 1, a normalized average of the gradients of the NEQN +# C + solution components is used. +# C + +# C + You can do all the "runs" in one program, by setting NPROB > 1. +# C + Each pass through the DO loop, PDE2D will read the gradient values +# C + output the previous pass. +# C + +# C + TRIDEN may also be a function of the initial triangle number KTRI. +# C + +# C + Increase the variable EXAG from its default value of 1.5 if you want +# C + to exaggerate the grading of an adaptive triangulation (make it less +# C + uniform). EXAG should normally not be larger than about 2.0. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## TRIDEN = 1.0 EXAG = 1.5 C############################################################################## C If you don't want to read the FINE PRINT, default SHAPE. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Enter a FORTRAN expression for SHAPE(X,Y), which controls the +# C + approximate shape of the triangles. The triangulation refinement +# C + will proceed with the goal of generating triangles with an average +# C + height to width ratio of approximately SHAPE(X,Y) near the point +# C + (X,Y). SHAPE must be positive. The default is SHAPE(X,Y)=1.0. +# C + +# C + SHAPE may also be a function of the initial triangle number KTRI. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## SHAPE = 1.0 if (triden.eq.adapt) triden = dtdpgr(x,y)**exag return end subroutine pdes8z(yd8z,i8z,j8z,kint8z,idel8z,jdel8z,x,y,ktri,t) implicit double precision (a-h,o-z) parameter (neqnmx= 99) parameter (ndelmx= 20) C un8z(1,I),un8z(2,I),un8z(3,I) hold the (rarely used) values C of UI,UIx,UIy from the previous iteration or time step common/dtdp4/un8z(3,neqnmx),uu8z(3,neqnmx) common/dtdp17/normx,normy,iarc double precision normx,normy,delamp(ndelmx,neqnmx) common/parm8z/ pi,D1 ,D2 C = uu8z(1, 1) Cx = uu8z(2, 1) Cy = uu8z(3, 1) Cnorm = Cx*normx + Cy*normy if (i8z.eq.0) then yd8z = 0.0 C############################################################################## C Enter FORTRAN expressions for the functions whose integrals are to be # C calculated and printed. They may be functions of # C # C X,Y,C,Cx,Cy and (if applicable) T # C # C The integrals may also contain references to the initial triangle # C number KTRI. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you only want to integrate a function over part of the region, +# C + define that function to be zero in the rest of the region. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C INTEGRAL DEFINED if (kint8z.eq. 1) yd8z = & 2*pi*y*C C############################################################################## C Enter FORTRAN expressions for the functions whose integrals are to be # C calculated and printed. They may be functions of # C # C X,Y,C,Cx,Cy and (if applicable) T # C # C The components (NORMx,NORMy) of the unit outward normal vector, and the # C initial triangle number KTRI, and the boundary arc number IARC may also # C be referenced. You can also reference the normal derivative Cnorm. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + CURVED interior interface arcs are considered part of the boundary, +# C + for the boundary integral computations, ONLY IF they have arc numbers+# C + in the range 8000-8999. In this case, since an interface arc is +# C + considered to be a boundary for both of the subregions it separates, +# C + the boundary integral will be computed twice on each curved +# C + interface arc, once with (NORMx,NORMy) defined in each direction. +# C + +# C + If you only want to integrate a function over part of the boundary, +# C + define that function to be zero on the rest of the boundary. You +# C + can examine the point (X,Y) to determine if it is on the desired +# C + boundary segment, or the boundary arc number IARC, or the initial +# C + triangle number KTRI (if INTRI=3), +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C BND. INTEGRAL1 DEFINED D = 5 if (kint8z.eq.-1) yd8z = & 2*pi*y*D*Cnorm C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] else C############################################################################## C Now enter FORTRAN expressions to define the PDE coefficients, which # C may be functions of # C # C X,Y,C,Cx,Cy # C # C They may also be functions of the initial triangle number KTRI # C and, in some cases, of the parameter T. # C # C Recall that the PDE has the form # C # C d/dX*A + d/dY*B = F # C # C############################################################################## if (x.le.0.0) then Q = 0 D = D1 else Q = 1 D = D2 endif if (j8z.eq.0) then yd8z = 0.0 C F DEFINED if (i8z.eq. 1) yd8z = & -y*Q C A DEFINED if (i8z.eq. 2) yd8z = & y*D*Cx C B DEFINED if (i8z.eq. 3) yd8z = & y*D*Cy else endif endif return end function u8z(i8z,x,y,ktri,t0) implicit double precision (a-h,o-z) common/parm8z/ pi,D1 ,D2 u8z = 0.0 return end function fb8z(i8z,iarc8z,ktri,s,x,y,t) implicit double precision (a-h,o-z) common/parm8z/ pi,D1 ,D2 fb8z = 0.0 C NO BOUNDARY CONDITIONS DEFINED ON NEGATIVE ARCS. C TO ADD BCs FOR NEGATIVE ARCS, USE BLOCK BELOW AS MODEL IARC = 0 if (iarc8z.eq.iarc) then C############################################################################## C Enter a FORTRAN expression to define FB on this arc. It may be a # C function of X,Y and (if applicable) T. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + This function may also reference the initial triangle number KTRI, +# C + and the arc parameter S (S = P or Q when INTRI=2). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C # C Recall that fixed boundary conditions have the form # C # C C = FB # C # C############################################################################## C FB DEFINED C if (i8z.eq. 1) fb8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] endif return end subroutine gb8z(gd8z,i8z,j8z,iarc8z,ktri,s,x,y,t) implicit double precision (a-h,o-z) parameter (neqnmx= 99) C un8z(1,I),un8z(2,I),un8z(3,I) hold the (rarely used) values C of UI,UIx,UIy from the previous iteration or time step. C (normx,normy) is the (rarely used) unit outward normal vector common/dtdp4/un8z(3,neqnmx),uu8z(neqnmx,3) common/dtdp17/normx,normy,ibarc8z common/dtdp49/bign8z double precision normx,normy common/parm8z/ pi,D1 ,D2 zero(f8z) = bign8z*f8z C = uu8z( 1,1) Cx = uu8z( 1,2) Cy = uu8z( 1,3) if (j8z.eq.0) gd8z = 0.0 C NO BOUNDARY CONDITIONS DEFINED ON POSITIVE ARCS. C TO ADD BCs FOR POSITIVE ARCS, USE BLOCK BELOW AS MODEL IARC = 0 if (iarc8z.eq.iarc) then C############################################################################## C Enter FORTRAN expressions to define the following free boundary # C condition functions on this arc. They may be functions of # C # C X,Y,C,Cx,Cy and (if applicable) T # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + These functions may also reference the components (NORMx,NORMy) of +# C + the unit outward normal vector, the initial triangle number KTRI, +# C + and the arc parameter S (S = P or Q when INTRI=2). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C # C Recall that free boundary conditions have the form # C # C A*nx+B*ny = GB # C # C############################################################################## if (j8z.eq.0) then C GB DEFINED C if (i8z.eq. 1) gd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] else endif endif return end subroutine pmod8z(x,y,ktri,t,a,b) implicit double precision (a-h,o-z) parameter (neqnmx= 99) common/dtdp4/un8z(3,neqnmx),uu8z(3,neqnmx) common/dtdp6/upr8z(3,neqnmx),uab8z(3,neqnmx) common/dtdp9/uprint(neqnmx),aprint(neqnmx),bprint(neqnmx) common/dtdp14/sint(20),bint(20),slim8z(20),blim8z(20) common/parm8z/ pi,D1 ,D2 C = uu8z(1, 1) Cx = uu8z(2, 1) Cy = uu8z(3, 1) a1 = upr8z(2, 1) b1 = upr8z(3, 1) C############################################################################## C If you don't want to read the FINE PRINT, default all of the following # C variables. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Normally, PDE2D saves the values of C,A,B at the output points. +# C + If different variables are to be saved (for later printing or +# C + plotting) the following functions can be used to re-define the +# C + output variables: +# C + define UPRINT(1) to replace C +# C + APRINT(1) A +# C + BPRINT(1) B +# C + Each function may be a function of +# C + +# C + X,Y,C,Cx,Cy,A,B and (if applicable) T +# C + +# C + Each may also be a function of the initial triangle number KTRI and +# C + the integral estimates SINT(1),...,BINT(1),... +# C + +# C + The default for each variable is no change, for example, UPRINT(1) +# C + defaults to C. Enter FORTRAN expressions for each of the +# C + following functions (or default). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C DEFINE UPRINT(*),APRINT(*),BPRINT(*) HERE: return end C dummy routines function axis8z(i8z,x,y,z,ical8z) implicit double precision (a-h,o-z) axis8z = 0 return end subroutine postpr(tout,nsave,xout,yout,inrg,nx,ny,uout,neqn) implicit double precision (a-h,o-z) dimension xout(0:nx,0:ny),yout(0:nx,0:ny),tout(0:nsave) dimension inrg(0:nx,0:ny),uout(0:nx,0:ny,3,neqn,0:nsave) common/parm8z/ pi,D1 ,D2 common /dtdp27/ itask,npes,icomm common /dtdp46/ eps8z,cgtl8z,npmx8z,itype data lun,lud/0,47/ if (itask.gt.0) return C UOUT(I,J,IDER,IEQ,L) = U-sub-IEQ, if IDER=1 C A-sub-IEQ, if IDER=2 C B-sub-IEQ, if IDER=3 C (possibly as modified by UPRINT,..) C at the point (XOUT(I,J) , YOUT(I,J)) C at time/iteration TOUT(L). C INRG(I,J) = 1 if this point is in R C = 0 otherwise C ******* ADD POSTPROCESSING CODE HERE: C IN THE EXAMPLE BELOW, MATLAB PLOTFILES pde2d.m, C pde2d.rdm CREATED (REMOVE COMMENTS TO ACTIVATE) if (lun.eq.0) then lun = 46 open (lun,file='pde2d.m') open (lud,file='pde2d.rdm') endif do 78753 l=0,nsave if (tout(l).ne.dtdplx(2)) nsave0 = l 78753 continue write (lud,78754) nsave0 write (lud,78754) neqn write (lud,78754) nx write (lud,78754) ny 78754 format (i8) do 78756 i=0,nx do 78755 j=0,ny write (lud,78762) xout(i,j),yout(i,j) 78755 continue 78756 continue do 78761 l=0,nsave0 write (lud,78762) tout(l) do 78760 ieq=1,neqn do 78759 ider=1,3 do 78758 i=0,nx do 78757 j=0,ny if (inrg(i,j).eq.1) then write (lud,78762) uout(i,j,ider,ieq,l) else write (lud,78763) endif 78757 continue 78758 continue 78759 continue 78760 continue 78761 continue 78762 format (e16.8) 78763 format ('NaN') write (lun,*) '% Read solution from pde2d.rdm' write (lun,*) 'fid = fopen(''pde2d.rdm'');' write (lun,*) 'NSAVE = fscanf(fid,''%g'',1);' write (lun,*) 'NEQN = fscanf(fid,''%g'',1);' write (lun,*) 'NX = fscanf(fid,''%g'',1);' write (lun,*) 'NY = fscanf(fid,''%g'',1);' if (itype.eq.2) then write (lun,*) 'L0 = 0;' else write (lun,*) 'L0 = 1;' endif write (lun,*) 'T = zeros(NSAVE+1,1);' write (lun,*) 'X = zeros(NY+1,NX+1);' write (lun,*) 'Y = zeros(NY+1,NX+1);' write (lun,*) 'U = zeros(NY+1,NX+1,NSAVE+1,3,NEQN);' write (lun,*) 'for i=0:NX' write (lun,*) 'for j=0:NY' write (lun,*) ' X(j+1,i+1) = fscanf(fid,''%g'',1);' write (lun,*) ' Y(j+1,i+1) = fscanf(fid,''%g'',1);' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'for l=0:NSAVE' write (lun,*) 'T(l+1) = fscanf(fid,''%g'',1);' write (lun,*) 'for ieq=1:NEQN' write (lun,*) 'for ider=1:3' write (lun,*) 'for i=0:NX' write (lun,*) 'for j=0:NY' write (lun,*) & ' U(j+1,i+1,l+1,ider,ieq) = fscanf(fid,''%g'',1);' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'xmin = min(min(X(:,:)));' write (lun,*) 'xmax = max(max(X(:,:)));' write (lun,*) 'ymin = min(min(Y(:,:)));' write (lun,*) 'ymax = max(max(Y(:,:)));' write (lun,*) 'hx = 0.1*(xmax-xmin);' write (lun,*) 'hy = 0.1*(ymax-ymin);' write (lun,*) '% Surface plots of each variable' write (lun,*) 'for IEQ=1:NEQN' write (lun,*) 'IDER = 1;' write (lun,*) & 'umin = min(min(min(U(:,:,L0+1:NSAVE+1,IDER,IEQ))));' write (lun,*) & 'umax = max(max(max(U(:,:,L0+1:NSAVE+1,IDER,IEQ))));' write (lun,*) 'for L=L0:NSAVE' write (lun,*) ' figure' write (lun,*) ' surf(X,Y,U(:,:,L+1,IDER,IEQ))' write (lun,*) ' axis([xmin xmax ymin ymax umin umax])' write (lun,*) ' xlabel(''X'')' write (lun,*) ' ylabel(''Y'')' write (lun,*) ' zlabel([''U'',num2str(IEQ)])' write (lun,*) ' title(['' T = '',num2str(T(L+1))])' write (lun,*) ' view(-37.5,30.0)' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) '% ******* Choose variables for vector plots' write (lun,*) '% (see comments in POSTPR)' write (lun,*) 'IEQ1 = 1;' write (lun,*) 'IDER1 = 2;' write (lun,*) 'IEQ2 = 1;' write (lun,*) 'IDER2 = 3;' write (lun,*) 'for L=L0:NSAVE' write (lun,*) ' figure' write (lun,*) ' quiver(X,Y,U(:,:,L+1,IDER1,IEQ1), ...' write (lun,*) ' U(:,:,L+1,IDER2,IEQ2))' write (lun,*) ' axis([xmin-hx xmax+hx ymin-hy ymax+hy])' write (lun,*) ' xlabel(''X'')' write (lun,*) ' ylabel(''Y'')' write (lun,*) ' title(['' T = '',num2str(T(L+1))])' write (lun,*) 'end' return end