C ************************** C * PDE2D 9.2 MAIN PROGRAM * C ************************** C *** 2D PROBLEM SOLVED (COLLOCATION 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) C############################################################################## C NP1GRID = number of P1-grid lines # C############################################################################## C-----------------------------------------> INPUT FROM GUI <------------------- PARAMETER (NP1GRID = 20) C############################################################################## C NP2GRID = number of P2-grid lines # C############################################################################## C-----------------------------------------> INPUT FROM GUI <------------------- PARAMETER (NP2GRID = 50) C############################################################################## C How many differential equations (NEQN) are there in your problem? # C############################################################################## PARAMETER (NEQN = 2) parameter (np3grid = 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,NZP8Z=KDEG8Z+1) C############################################################################## C The solution is normally saved on an NP1+1 by NP2+1 rectangular grid # C of points, # C P1 = P1A + I*(P1B-P1A)/NP1, I = 0,...,NP1 # C P2 = P2A + J*(P2B-P2A)/NP2, J = 0,...,NP2 # C Enter values for NP1 and NP2. Suggested values: NP1=NP2=25. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you want to save the solution at an arbitrary user-specified set +# C + of points, set NP2=0 and NP1+1=number of points. In this case you +# C + can request tabular output, but no plots can be made. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## PARAMETER (NP1 = 5) PARAMETER (NP2 = 20) PARAMETER (NSAVE = 1) common/parm8z/ pi,E ,Rmu ,R1 ,R2 dimension p1grid(np1grid),p2grid(np2grid),p3grid(np3grid),p1out8z( &0:np1,0:np2),p2out8z(0:np1,0:np2),p3out8z(0:np1,0:np2),p1cross(100 &),p2cross(100),tout8z(0:nsave),uout8z(0:np1,0:np2,4*neqn,0:nsave), &uout(0:np1,0:np2,4,neqn,0:nsave),xres8z(nxp8z),yres8z(nyp8z),zres8 &z(nzp8z),ures8z(neqn,nxp8z,nyp8z,nzp8z) equivalence (uout,uout8z) allocatable iwrk8z(:),rwrk8z(:) C dimension iwrk8z(iiwk8z),rwrk8z(irwk8z) character*40 title logical linear,crankn,noupdt,nodist,fillin,evcmpx,adapt,plot,lsqfi &t,fdiff,solid,econ8z,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/dtdp30/ econ8z,ncon8z common/dtdp45/ perdc(neqnmx) common/dtdp46/ eps8z,cgtl8z,npmx8z,itype common/dtdp52/ nxa8z,nya8z,nza8z,kd8z common/dtdp53/ work8z(nxp8z*nyp8z*nzp8z+9) common/dtdp64/ amin8z(4*neqnmx),amax8z(4*neqnmx) pi = 4.0*atan(1.d0) zr8z = 0.0 nxa8z = nxp8z nya8z = nyp8z nza8z = nzp8z kd8z = kdeg8z 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############################################################################## C-----------------------------------------> INPUT FROM GUI <------------------- E = & 100 Rmu = & 0.2 R1 = & 6 R2 = & 10 C############################################################################## C A collocation finite element method is used, with bi-cubic Hermite # C basis functions on the elements (small rectangles) defined by the grid # C points: # C P1GRID(1),...,P1GRID(NP1GRID) # C P2GRID(1),...,P2GRID(NP2GRID) # C You will first be prompted for NP1GRID, the number of P1-grid points, # C then for P1GRID(1),...,P1GRID(NP1GRID). Any points defaulted will be # C uniformly spaced between the points you define; the first and last # C points cannot be defaulted. Then you will be prompted similarly # C for the number and values of the P2-grid points. The limits on the # C parameters are then: # C P1GRID(1) < P1 < P1GRID(NP1GRID) # C P2GRID(1) < P2 < P2GRID(NP2GRID) # C # C############################################################################## call dtdpwx(p1grid,np1grid,0) call dtdpwx(p2grid,np2grid,0) C P1GRID DEFINED C-----------------------------------------> INPUT FROM GUI <------------------- P1GRID(1) = & R1 C-----------------------------------------> INPUT FROM GUI <------------------- P1GRID(NP1GRID) = & R2 C P2GRID DEFINED C-----------------------------------------> INPUT FROM GUI <------------------- P2GRID(1) = & 0 C-----------------------------------------> INPUT FROM GUI <------------------- P2GRID(NP2GRID) = & pi C p3grid(1) = 0 call dtdpwx(p1grid,np1grid,1) call dtdpwx(p2grid,np2grid,1) C############################################################################## C If you don't want to read the FINE PRINT, enter ISOLVE = 1. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The following linear system solvers are available: +# C + +# C + 1. Sparse direct method +# C + Harwell Library routine MA27 (used by permission) is +# C + used to solve the (positive definite) "normal" +# C + equations A**T*A*x = A**T*b. The normal equations, +# C + which are essentially the equations which would result +# C + if a least squares finite element method were used +# C + instead of a collocation method, are substantially +# C + more ill-conditioned than the original system Ax = b, +# C + so it may be important to use high precision if this +# C + option is chosen. +# C + 2. Frontal method +# C + This is an out-of-core band solver. If you want to +# C + override the default number of rows in the buffer (11),+# C + set a new value for NPMX8Z in the main program. +# C + 3. Jacobi conjugate gradient iterative method +# C + A preconditioned conjugate gradient iterative method +# C + is used to solve the (positive definite) normal +# C + equations. High precision is also important if this +# C + option is chosen. (This solver is MPI-enhanced, if +# C + MPI is available.) If you want to override the +# C + default convergence tolerance, set a new relative +# C + tolerance CGTL8Z in the main program. +# C + 4. Local solver (normal equations) +# C + 5. Local solver (original equations) +# C + Choose these options ONLY if alterative 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 = 1 C *******STEADY-STATE PROBLEM itype = 1 t0 = 0.0 dt = 1.0 crankn = .false. noupdt = .false. C############################################################################## C Is this a linear problem? ("linear" means all differential equations # C and all boundary conditions are linear) # C############################################################################## LINEAR = .TRUE. C Number of Newton iterations NSTEPS = 1 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############################################################################## C-----------------------------------------> INPUT FROM GUI <------------------- 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 = 0 lsqfit = .false. RESTRT = .FALSE. GRIDID = .TRUE. C############################################################################## C If you do not have any periodic boundary conditions, enter IPERDC=0. # C # C Enter IPERDC=1 for periodic conditions at P1 = P1GRID(1),P1GRID(NP1GRID)# C IPERDC=2 for periodic conditions at P2 = P2GRID(1),P2GRID(NP2GRID)# C IPERDC=4 for periodic conditions on both P1 and P2 # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + When periodic boundary conditions are selected, they apply to all +# C + variables by default. To turn off periodic boundary conditions on +# C + the I-th variable, set PERDC(I) to 0 (or another appropriate value +# C + of IPERDC) below in the main program and set the desired boundary +# C + conditions in subroutine GB8Z, "by hand". +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C-----------------------------------------> INPUT FROM GUI <------------------- IPERDC = 0 C############################################################################## C The solution is saved on an NP1+1 by NP2+1 rectangular grid covering # C the rectangle (P1A,P1B) x (P2A,P2B). Enter values for P1A,P1B,P2A,P2B. # C These variables are usually defaulted. # C # C The defaults are P1A = P1GRID(1), P1B = P1GRID(NP1GRID) # C P2A = P2GRID(1), P2B = P2GRID(NP2GRID) # C # C############################################################################## C defaults for p1a,p1b,p2a,p2b p1a = p1grid(1) p1b = p1grid(np1grid) p2a = p2grid(1) p2b = p2grid(np2grid) C DEFINE P1A,P1B,P2A,P2B HERE: call dtdpx3(np1,np2,0,p1a,p1b,p2a,p2b,zr8z,zr8z,hp18z,hp28z,hp38z, &p1out8z,p2out8z,p3out8z,npts8z) C SOLUTION SAVED EVERY NOUT ITERATIONS NOUT = NSTEPS C *******allocate workspace call dtdpqx(np1grid,np2grid,np3grid,isolve,neqn,ii8z,ir8z,iperdc) if (iiwk8z.gt.1) ii8z = iiwk8z if (irwk8z.gt.1) ir8z = irwk8z allocate (iwrk8z(ii8z),rwrk8z(ir8z)) C *******DRAW GRID LINES? PLOT = .FALSE. C *******call pde solver call dtdp3x(p1grid, p2grid, p3grid, np1grid,np2grid, -1, neqn, p1o &ut8z, p2out8z, p3out8z, uout, tout8z, npts8z, t0, dt, nsteps, nout &, nsave, crankn, noupdt, itype, linear, isolve, rwrk8z, ir8z, iwrk &8z, ii8z, iperdc, plot, lsqfit, fdiff, nint, nbint, restrt, gridid &) deallocate (iwrk8z,rwrk8z) C *******read from restart file to array ures8z C call dtdpr3(1,xres8z,nxp8z,yres8z,nyp8z,zres8z,nzp8z,ures8z,neqn) C *******write array ures8z back to restart file C call dtdpr3(2,xres8z,nxp8z,yres8z,nyp8z,zres8z,nzp8z,ures8z,neqn) C *******call user-written postprocessor call postpr(tout8z,nsave,p1out8z,p2out8z,np1,np2,uout,neqn) C *******VECTOR PLOTS C############################################################################## C Enter values for IVAR1, IVAR2 to select the components Vr1 and Vr2 # C of the vector to be plotted. # C IVAR1,IVAR2 = 1 means U (possibly as modified by UPRINT,...) # C 2 Ux # C 3 Uy # C 4 V # C 5 Vx # C 6 Vy # C # C Vr1 and Vr2 are assumed to be the components of the vector in # C Cartesian coordinates. # C############################################################################## IVAR1 = 1 IVAR2 = 4 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 ivr18z = ivar1 + (ivar1-1)/3 ivr28z = ivar2 + (ivar2-1)/3 a1mag = max(abs(amin8z(ivr18z)),abs(amax8z(ivr18z))) a2mag = max(abs(amin8z(ivr28z)),abs(amax8z(ivr28z))) C############################################################################## C For the purpose of scaling the arrows, the ranges of the two components # C of the vector are assumed to be (-VR1MAG,VR1MAG) and (-VR2MAG,VR2MAG). # C Enter values for VR1MAG and VR2MAG. VR1MAG and VR2MAG are often # C defaulted. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + By default, VR1MAG and VR2MAG are the maxima of the absolute values +# C + of the first and second components. For a common scaling, you may +# C + want to set VR1MAG=A1MAG, VR2MAG=A2MAG. A1MAG, A2MAG are the +# C + maxima of the absolute values over all output points and over all +# C + saved time steps or iterations. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## VR1MAG = 0.0 VR2MAG = 0.0 C############################################################################## C Enter a title, WITHOUT quotation marks. A maximum of 40 characters # C are allowed. The default is no title. # C############################################################################## TITLE = ' ' TITLE = 'Displacements (U,V) ' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78760 is8z=iset1,iset2,isinc call dtdplq(uout8z(0,0,ivr18z,is8z),uout8z(0,0,ivr28z,is8z),uout8z &(0,0,4,is8z),np1,np2,0,p1a,p1b,p2a,p2b,zr8z,zr8z,3,ix8z,jy8z,0,tit &le,vr1mag,vr2mag,zr8z,zr8z,nodist,tout8z(is8z),zr8z,zr8z,zr8z,zr8z &,2,ical8z) 78760 continue 78755 continue call endgks stop end subroutine tran8z(itrans,p1,p2,p38z) implicit double precision (a-h,o-z) common /dtdp41/x,y,z8z,x1,x2,x3,y1,y2,y3,z1,z2,z3,x11,x21,x31,x12, &x22,x32,x13,x23,x33,y11,y21,y31,y12,y22,y32,y13,y23,y33,z11,z21,z3 &1,z12,z22,z32,z13,z23,z33 common/parm8z/ pi,E ,Rmu ,R1 ,R2 C############################################################################## C You can solve problems in your region only if you can describe it by # C X = X(P1,P2) # C Y = Y(P1,P2) # C with constant limits on the parameters P1,P2. If your region is # C rectangular, enter ITRANS=0 and the trivial parameterization # C X = P1 # C Y = P2 # C will be used. Otherwise, you need to read the FINE PRINT below. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If P1,P2 represent polar or other non-Cartesian coordinates, you can +# C + reference the Cartesian coordinates X,Y and derivatives of your +# C + unknowns with respect to these coordinates, when you define your +# C + PDE coefficients, boundary conditions, and volume and boundary +# C + integrals, if you enter ITRANS .NE. 0. Enter: +# C + ITRANS = 1, if P1,P2 are polar coordinates, that is, if +# C + P1=R, P2=Theta, where X = R*cos(Theta) +# C + Y = R*sin(Theta) +# C + ITRANS = -1, same as ITRANS=1, but P1=Theta, P2=R +# C + ITRANS = 3, to define your own coordinate transformation. In +# C + this case, you will be prompted to define X,Y and +# C + their first and second derivatives in terms of P1,P2. +# C + Because of symmetry, you will not be prompted for all +# C + of the second derivatives. If you make a mistake in +# C + computing any of these derivatives, PDE2D will usually +# C + be able to issue a warning message. (X1 = dX/dP1, etc) +# C + ITRANS = -3, same as ITRANS=3, but you will only be prompted to +# C + define X,Y; their first and second derivatives will +# C + be approximated using finite differences. +# C + When ITRANS = -3 or 3, the first derivatives of X,Y must all be +# C + continuous. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ITRANS = 1 C z8z = p38z z3 = 1 return end subroutine pdes8z(yd8z,i8z,j8z,kint8z,p1,p2,p38z,t,uu8z) implicit double precision (a-h,o-z) parameter (neqnmx= 99) C un8z(1,I),un8z(2,I),... hold the (rarely used) values C of UI,UI1,... from the previous iteration or time step common /dtdp5x/un8z(10,neqnmx) common /dtdp18/norm1,norm2,n38z double precision norm1,norm2,n38z,normx,normy,nz8z dimension uu8z(10,neqnmx) common/parm8z/ pi,E ,Rmu ,R1 ,R2 zr8z = 0.0 U = uu8z(1, 1) U1 = uu8z(2, 1) U2 = uu8z(3, 1) U11= uu8z(5, 1) U22= uu8z(6, 1) U12= uu8z(8, 1) U21= uu8z(8, 1) V = uu8z(1, 2) V1 = uu8z(2, 2) V2 = uu8z(3, 2) V11= uu8z(5, 2) V22= uu8z(6, 2) V12= uu8z(8, 2) V21= uu8z(8, 2) call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,norm1,norm2,n38z,x,y,z8z,normx,normy,nz8z,3 &) call dtdpcc(p1,p2,p38z, & U1,U2,zr8z,U11,U22,zr8z,U12,zr8z,zr8z, & x,y,z8z,Ux,Uy,uz8z,Uxx,Uyy,uzz8z,Uxy,uxz8z,uyz8z, & Uyx,uzx8z,uzy8z,dvol,darea) Unorm = Ux*normx + Uy*normy call dtdpcc(p1,p2,p38z, & V1,V2,zr8z,V11,V22,zr8z,V12,zr8z,zr8z, & x,y,z8z,Vx,Vy,uz8z,Vxx,Vyy,uzz8z,Vxy,uxz8z,uyz8z, & Vyx,uzx8z,uzy8z,dvol,darea) Vnorm = Vx*normx + Vy*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,U,Ux,Uy,Uxx,Uyy,Uxy # C V,Vx,Vy,Vxx,Vyy,Vxy and (if applicable) T # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced (U1 = dU/dP1, etc): # C U1,U2,U11,U22,U12 # C V1,V2,V11,V22,V12 # 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 INTEGRAL1 DEFINED if (kint8z.eq.1) yd8z = & V 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,U,Ux,Uy,Uxx,Uyy,Uxy # C V,Vx,Vy,Vxx,Vyy,Vxy and (if applicable) T # C # C The components (NORMx,NORMy) of the unit outward normal vector # C may also be referenced. # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced: # C U1,U2,U11,U22,U12 # C V1,V2,V11,V22,V12 # C You can also reference the normal derivatives Unorm,Vnorm. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# 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. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C BND. INTEGRAL1 DEFINED C if (kint8z.eq.-1) yd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] if (kint8z.gt.0) yd8z = yd8z*dvol if (kint8z.lt.0) yd8z = yd8z*darea else C############################################################################## C Now enter FORTRAN expressions to define the PDE coefficients, which # C may be functions of # C # C X,Y,U,Ux,Uy,Uxx,Uyy,Uxy # C V,Vx,Vy,Vxx,Vyy,Vxy # C # C and, in some cases, of the parameter T. # C # C Recall that the PDEs have the form # C # C F1 = 0 # C F2 = 0 # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced (U1 = dU/dP1, etc): # C U1,U2,U11,U22,U12 # C V1,V2,V11,V22,V12 # C############################################################################## if (j8z.eq.0) then yd8z = 0.0 C-----------------------------------------> INPUT FROM GUI <------------------- C F1 DEFINED if (i8z.eq. 1) yd8z = & E/2/(1+Rmu)*(Uxx+Uyy+(Uxx+Vyx)/(1-2*Rmu)) C-----------------------------------------> INPUT FROM GUI <------------------- C F2 DEFINED if (i8z.eq. 2) yd8z = & E/2/(1+Rmu)*(Vxx+Vyy+(Uxy+Vyy)/(1-2*Rmu)) - 10 else endif endif return end function u8z(i8z,p1,p2,p38z,t0) implicit double precision (a-h,o-z) common/parm8z/ pi,E ,Rmu ,R1 ,R2 call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,z18z,z28z,z38z,x,y,z8z,d18z,d28z,d38z,1) u8z = 0.0 return end subroutine gb8z(gd8z,ifac8z,i8z,j8z,p1,p2,p38z,t,uu8z) implicit double precision (a-h,o-z) parameter (neqnmx= 99) dimension uu8z(10,neqnmx) C un8z(1,I),un8z(2,I),... hold the (rarely used) values C of UI,UI1,... from the previous iteration or time step common /dtdp5x/ un8z(10,neqnmx) common /dtdp18/norm1,norm2,n38z double precision none,norm1,norm2,n38z,normx,normy,nz8z common/parm8z/ pi,E ,Rmu ,R1 ,R2 none = dtdplx(2) zr8z = 0.0 U = uu8z(1, 1) U1 = uu8z(2, 1) U2 = uu8z(3, 1) V = uu8z(1, 2) V1 = uu8z(2, 2) V2 = uu8z(3, 2) call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,norm1,norm2,n38z,x,y,z8z,normx,normy,nz8z,3 &) call dtdpcb( & p1,p2,p38z,U1,U2,zr8z,x,y,z8z,Ux,Uy,uz8z,2) Unorm = Ux*normx + Uy*normy call dtdpcb( & p1,p2,p38z,V1,V2,zr8z,x,y,z8z,Vx,Vy,uz8z,2) Vnorm = Vx*normx + Vy*normy if (j8z.eq.0) gd8z = 0.0 C############################################################################## C Enter FORTRAN expressions to define the boundary condition functions, # C which may be functions of # C # C X,Y,U,Ux,Uy, # C V,Vx,Vy and (if applicable) T # C # C Recall that the boundary conditions have the form # C # C G1 = 0 # C G2 = 0 # C # C Enter NONE to indicate "no" boundary condition. # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced (U1 = dU/dP1, etc): # C U1,U2 # C V1,V2 # C The components (NORMx,NORMy) of the unit outward normal vector # C may also be referenced, as well as the normal derivatives Unorm, # C Vnorm. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If "no" boundary condition is specified, the corresponding PDE is +# C + enforced at points just inside the boundary (exactly on the +# C + boundary, if EPS8Z is set to 0 in the main program). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## Sigxx = E*((1-Rmu)*Ux+Rmu*Vy)/(1-2*Rmu)/(1+Rmu) Sigxy = E*(Uy+Vx)/2./(1+Rmu) Sigyy = E*(Rmu*Ux+(1-Rmu)*Vy)/(1-2*Rmu)/(1+Rmu) if (ifac8z.eq. 1) then C############################################################################## C # C First define the boundary conditions on the face P1 = P1GRID(1). # C############################################################################## if (j8z.eq.0) then C-----------------------------------------> INPUT FROM GUI <------------------- C G1 DEFINED if (i8z.eq. 1) gd8z = & Sigxx*NORMx + Sigxy*NORMy C-----------------------------------------> INPUT FROM GUI <------------------- C G2 DEFINED if (i8z.eq. 2) gd8z = & Sigxy*NORMx + Sigyy*NORMy else endif endif if (ifac8z.eq. 2) then C############################################################################## C # C Now define the boundary conditions on the face P1 = P1GRID(NP1GRID). # C############################################################################## if (j8z.eq.0) then C-----------------------------------------> INPUT FROM GUI <------------------- C G1 DEFINED if (i8z.eq. 1) gd8z = & Sigxx*NORMx + Sigxy*NORMy C-----------------------------------------> INPUT FROM GUI <------------------- C G2 DEFINED if (i8z.eq. 2) gd8z = & Sigxy*NORMx + Sigyy*NORMy else endif endif if (ifac8z.eq. 3) then C############################################################################## C # C Now define the boundary conditions on the face P2 = P2GRID(1). # C############################################################################## if (j8z.eq.0) then C-----------------------------------------> INPUT FROM GUI <------------------- C G1 DEFINED if (i8z.eq. 1) gd8z = & U C-----------------------------------------> INPUT FROM GUI <------------------- C G2 DEFINED if (i8z.eq. 2) gd8z = & V else endif endif if (ifac8z.eq. 4) then C############################################################################## C # C Now define the boundary conditions on the face P2 = P2GRID(NP2GRID). # C############################################################################## if (j8z.eq.0) then C-----------------------------------------> INPUT FROM GUI <------------------- C G1 DEFINED if (i8z.eq. 1) gd8z = & U C-----------------------------------------> INPUT FROM GUI <------------------- C G2 DEFINED if (i8z.eq. 2) gd8z = & V else endif endif return end subroutine pmod8z(p1,p2,p38z,t,uu8z,uprint,uxprnt,uyprnt,uzpr8z) implicit double precision (a-h,o-z) dimension uu8z(10,*),uprint(*),uxprnt(*),uyprnt(*),uzpr8z(*) common/dtdp14/sint(20),bint(20),slim8z(20),blim8z(20) common/parm8z/ pi,E ,Rmu ,R1 ,R2 zr8z = 0.0 U = uu8z(1, 1) U1 = uu8z(2, 1) U2 = uu8z(3, 1) U11= uu8z(5, 1) U22= uu8z(6, 1) U12= uu8z(8, 1) U21= uu8z(8, 1) V = uu8z(1, 2) V1 = uu8z(2, 2) V2 = uu8z(3, 2) V11= uu8z(5, 2) V22= uu8z(6, 2) V12= uu8z(8, 2) V21= uu8z(8, 2) call dtdpcd(p1,p2,p38z) call dtdpcc(p1,p2,p38z, & U1,U2,zr8z,U11,U22,zr8z,U12,zr8z,zr8z, & x,y,z8z,Ux,Uy,uz8z,Uxx,Uyy,uzz8z,Uxy,uxz8z,uyz8z, & Uyx,uzx8z,uzy8z,dvol8z,dare8z) uxprnt( 1) = Ux uyprnt( 1) = Uy call dtdpcc(p1,p2,p38z, & V1,V2,zr8z,V11,V22,zr8z,V12,zr8z,zr8z, & x,y,z8z,Vx,Vy,uz8z,Vxx,Vyy,uzz8z,Vxy,uxz8z,uyz8z, & Vyx,uzx8z,uzy8z,dvol8z,dare8z) uxprnt( 2) = Vx uyprnt( 2) = Vy 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 U,Ux,Uy,V,Vx,Vy at +# C + the output points. If different variables are to be saved (for +# C + later printing or plotting) the following functions can be used to +# C + re-define the output variables: +# C + define UPRINT(1) to replace U +# C + UXPRNT(1) Ux +# C + UYPRNT(1) Uy +# C + UPRINT(2) V +# C + UXPRNT(2) Vx +# C + UYPRNT(2) Vy +# C + Each function may be a function of +# C + +# C + X,Y,U,Ux,Uy,Uxx,Uyy,Uxy +# C + V,Vx,Vy,Vxx,Vyy,Vxy and (if applicable) T +# C + +# C + Each may also be a function of the integral estimates SINT(1),..., +# C + BINT(1),... +# C + +# C + The parameters P1,P2 and derivatives with respect to these may also +# C + be referenced (U1 = dU/dP1, etc): +# C + U1,U2,U11,U22,U12 +# C + V1,V2,V11,V22,V12 +# C + +# C + The default for each variable is no change, for example, UPRINT(1) +# C + defaults to U. Enter FORTRAN expressions for each of the +# C + following functions (or default). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C DEFINE UPRINT(*),UXPRNT(*),UYPRNT(*) HERE: return end function axis8z(i8z,p1,p2,p38z,ical8z) implicit double precision (a-h,o-z) call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,z18z,z28z,z38z,x,y,z8z,d18z,d28z,d38z,1) if (i8z.eq.1) axis8z = x if (i8z.eq.2) axis8z = y return end C dummy routines subroutine xy8z(i8z,iarc8z,s,x,y,s0,sf) implicit double precision (a-h,o-z) return end subroutine dis8z(x,y,ktri,triden,shape) implicit double precision (a-h,o-z) return end function fb8z(i8z,iarc8z,ktri,s,x,y,t) implicit double precision (a-h,o-z) fb8z = 0 return end subroutine postpr(tout,nsave,p1out,p2out,np1,np2,uout,neqn) implicit double precision (a-h,o-z) dimension p1out(0:np1,0:np2),p2out(0:np1,0:np2),tout(0:nsave) dimension uout(0:np1,0:np2,4,neqn,0:nsave) common/parm8z/ pi,E ,Rmu ,R1 ,R2 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 Ux-sub-IEQ, if IDER=2 C Uy-sub-IEQ, if IDER=3 C (possibly as modified by UPRINT,..) C at the point (P1OUT(I,J) , P2OUT(I,J)) C at time/iteration TOUT(L). C ******* ADD POSTPROCESSING CODE HERE: C IN THE EXAMPLE BELOW, MATLAB PLOTFILES pde2d.m, C pde2d.rdm CREATED (REMOVE C! COMMENTS TO ACTIVATE) C! if (lun.eq.0) then C! lun = 46 C! open (lun,file='pde2d.m') C! open (lud,file='pde2d.rdm') C! endif C! do 78753 l=0,nsave C! if (tout(l).ne.dtdplx(2)) nsave0 = l C!78753 continue C! write (lud,78754) nsave0 C! write (lud,78754) neqn C! write (lud,78754) np1 C! write (lud,78754) np2 C!78754 format (i8) C! do 78756 i=0,np1 C! do 78755 j=0,np2 C! p1 = p1out(i,j) C! p2 = p2out(i,j) C! p38z = 0.0 C! call dtdpcd(p1,p2,p38z) C! call dtdpcb(p1,p2,p38z,z18z,z28z,z38z,x,y,z8z, C! & d18z,d28z,d38z,1) C! write (lud,78762) p1,p2,x,y C!78755 continue C!78756 continue C! do 78761 l=0,nsave0 C! write (lud,78762) tout(l) C! do 78760 ieq=1,neqn C! do 78759 ider=1,3 C! do 78758 i=0,np1 C! do 78757 j=0,np2 C! write (lud,78762) uout(i,j,ider,ieq,l) C!78757 continue C!78758 continue C!78759 continue C!78760 continue C!78761 continue C!78762 format (e16.8) C! write (lun,*) '% Read solution from pde2d.rdm' C! write (lun,*) 'fid = fopen(''pde2d.rdm'');' C! write (lun,*) 'NSAVE = fscanf(fid,''%g'',1);' C! write (lun,*) 'NEQN = fscanf(fid,''%g'',1);' C! write (lun,*) 'NP1 = fscanf(fid,''%g'',1);' C! write (lun,*) 'NP2 = fscanf(fid,''%g'',1);' C! if (itype.eq.2) then C! write (lun,*) 'L0 = 0;' C! else C! write (lun,*) 'L0 = 1;' C! endif C! write (lun,*) 'T = zeros(NSAVE+1,1);' C! write (lun,*) 'P1 = zeros(NP2+1,NP1+1);' C! write (lun,*) 'P2 = zeros(NP2+1,NP1+1);' C! write (lun,*) 'X = zeros(NP2+1,NP1+1);' C! write (lun,*) 'Y = zeros(NP2+1,NP1+1);' C! write (lun,*) 'U = zeros(NP2+1,NP1+1,NSAVE+1,3,NEQN);' C! write (lun,*) 'for i=0:NP1' C! write (lun,*) 'for j=0:NP2' C! write (lun,*) ' P1(j+1,i+1) = fscanf(fid,''%g'',1);' C! write (lun,*) ' P2(j+1,i+1) = fscanf(fid,''%g'',1);' C! write (lun,*) ' X(j+1,i+1) = fscanf(fid,''%g'',1);' C! write (lun,*) ' Y(j+1,i+1) = fscanf(fid,''%g'',1);' C! write (lun,*) 'end' C! write (lun,*) 'end' C! write (lun,*) 'for l=0:NSAVE' C! write (lun,*) 'T(l+1) = fscanf(fid,''%g'',1);' C! write (lun,*) 'for ieq=1:NEQN' C! write (lun,*) 'for ider=1:3' C! write (lun,*) 'for i=0:NP1' C! write (lun,*) 'for j=0:NP2' C! write (lun,*) C! & ' U(j+1,i+1,l+1,ider,ieq) = fscanf(fid,''%g'',1);' C! write (lun,*) 'end' C! write (lun,*) 'end' C! write (lun,*) 'end' C! write (lun,*) 'end' C! write (lun,*) 'end' C! write (lun,*) 'xmin = min(min(X(:,:)));' C! write (lun,*) 'xmax = max(max(X(:,:)));' C! write (lun,*) 'ymin = min(min(Y(:,:)));' C! write (lun,*) 'ymax = max(max(Y(:,:)));' C! write (lun,*) 'hx = 0.1*(xmax-xmin);' C! write (lun,*) 'hy = 0.1*(ymax-ymin);' C! write (lun,*) '% Surface plots of each variable' C! write (lun,*) 'for IEQ=1:NEQN' C! write (lun,*) 'IDER = 1;' C! write (lun,*) C! & 'umin = min(min(min(U(:,:,L0+1:NSAVE+1,IDER,IEQ))));' C! write (lun,*) C! & 'umax = max(max(max(U(:,:,L0+1:NSAVE+1,IDER,IEQ))));' C! write (lun,*) 'for L=L0:NSAVE' C! write (lun,*) ' figure' C! write (lun,*) ' surf(X,Y,U(:,:,L+1,IDER,IEQ))' C! write (lun,*) ' axis([xmin xmax ymin ymax umin umax])' C! write (lun,*) ' xlabel(''X'')' C! write (lun,*) ' ylabel(''Y'')' C! write (lun,*) ' zlabel([''U'',num2str(IEQ)])' C! write (lun,*) ' title(['' T = '',num2str(T(L+1))])' C! write (lun,*) ' view(-37.5,30.0)' C! write (lun,*) 'end' C! write (lun,*) 'end' C! write (lun,*) '% ******* Choose variables for vector plots' C! write (lun,*) '% (see comments in POSTPR)' C! write (lun,*) 'IEQ1 = 1;' C! write (lun,*) 'IDER1 = 2;' C! write (lun,*) 'IEQ2 = 1;' C! write (lun,*) 'IDER2 = 3;' C! write (lun,*) 'for L=L0:NSAVE' C! write (lun,*) ' figure' C! write (lun,*) ' quiver(X,Y,U(:,:,L+1,IDER1,IEQ1), ...' C! write (lun,*) ' U(:,:,L+1,IDER2,IEQ2))' C! write (lun,*) ' axis([xmin-hx xmax+hx ymin-hy ymax+hy])' C! write (lun,*) ' xlabel(''X'')' C! write (lun,*) ' ylabel(''Y'')' C! write (lun,*) ' title(['' T = '',num2str(T(L+1))])' C! write (lun,*) 'end' return end