Actual source code: biharmonic2.c
petsc-3.4.2 2013-07-02

      2: static char help[] = "Solves biharmonic equation in 1d.\n";

      4: /*
      5:   Solves the equation biharmonic equation in split form

      7:     w = -kappa \Delta u
      8:     u_t =  \Delta w
      9:     -1  <= u <= 1
     10:     Periodic boundary conditions

     12: Evolve the biharmonic heat equation with bounds:  (same as biharmonic)
     13: ---------------
     14: ./biharmonic2 -ts_monitor -snes_monitor -ts_monitor_draw_solution  -pc_type lu  -draw_pause .1 -snes_converged_reason --wait   -ts_type beuler  -da_refine 5 -draw_fields 1 -ts_dt 9.53674e-9

     16:     w = -kappa \Delta u  + u^3  - u
     17:     u_t =  \Delta w
     18:     -1  <= u <= 1
     19:     Periodic boundary conditions

     21: Evolve the Cahn-Hillard equations:
     22: ---------------
     23: ./biharmonic2 -ts_monitor -snes_monitor -ts_monitor_draw_solution  -pc_type lu  -draw_pause .1 -snes_converged_reason  --wait   -ts_type beuler    -da_refine 6 -vi  -draw_fields 1  -kappa .00001 -ts_dt 5.96046e-06 -cahn-hillard

     26: */
     27: #include <petscdmda.h>
     28: #include <petscts.h>
     29: #include <petscdraw.h>

     31: /*
     32:    User-defined routines
     33: */
     34: extern PetscErrorCode FormFunction(TS,PetscReal,Vec,Vec,Vec,void*),FormInitialSolution(DM,Vec,PetscReal);
     35: typedef struct {PetscBool cahnhillard;PetscReal kappa;PetscInt energy;PetscReal tol;PetscReal theta;PetscReal theta_c;} UserCtx;

     39: int main(int argc,char **argv)
     40: {
     41:   TS             ts;                           /* nonlinear solver */
     42:   Vec            x,r;                          /* solution, residual vectors */
     43:   Mat            J;                            /* Jacobian matrix */
     44:   PetscInt       steps,Mx,maxsteps = 10000000;
     46:   DM             da;
     47:   MatFDColoring  matfdcoloring;
     48:   ISColoring     iscoloring;
     49:   PetscReal      dt;
     50:   PetscReal      vbounds[] = {-100000,100000,-1.1,1.1};
     51:   PetscBool      wait;
     52:   Vec            ul,uh;
     53:   SNES           snes;
     54:   UserCtx        ctx;

     56:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     57:      Initialize program
     58:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     59:   PetscInitialize(&argc,&argv,(char*)0,help);
     60:   ctx.kappa = 1.0;
     61:   PetscOptionsGetReal(NULL,"-kappa",&ctx.kappa,NULL);
     62:   ctx.cahnhillard = PETSC_FALSE;

     64:   PetscOptionsGetBool(NULL,"-cahn-hillard",&ctx.cahnhillard,NULL);
     65:   PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),2,vbounds);
     66:   PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),600,600);
     67:   ctx.energy = 1;
     68:   /*PetscOptionsGetInt(NULL,"-energy",&ctx.energy,NULL);*/
     69:   PetscOptionsInt("-energy","type of energy (1=double well, 2=double obstacle, 3=logarithmic)","",ctx.energy,&ctx.energy,NULL);
     70:   ctx.tol     = 1.0e-8;
     71:   PetscOptionsGetReal(NULL,"-tol",&ctx.tol,NULL);
     72:   ctx.theta   = .001;
     73:   ctx.theta_c = 1.0;
     74:   PetscOptionsGetReal(NULL,"-theta",&ctx.theta,NULL);
     75:   PetscOptionsGetReal(NULL,"-theta_c",&ctx.theta_c,NULL);

     77:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     78:      Create distributed array (DMDA) to manage parallel grid and vectors
     79:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     80:   DMDACreate1d(PETSC_COMM_WORLD, DMDA_BOUNDARY_PERIODIC, -10,2,2,NULL,&da);
     81:   DMDASetFieldName(da,0,"Biharmonic heat equation: w = -kappa*u_xx");
     82:   DMDASetFieldName(da,1,"Biharmonic heat equation: u");
     83:   DMDAGetInfo(da,0,&Mx,0,0,0,0,0,0,0,0,0,0,0);
     84:   dt   = 1.0/(10.*ctx.kappa*Mx*Mx*Mx*Mx);

     86:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     87:      Extract global vectors from DMDA; then duplicate for remaining
     88:      vectors that are the same types
     89:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     90:   DMCreateGlobalVector(da,&x);
     91:   VecDuplicate(x,&r);

     93:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     94:      Create timestepping solver context
     95:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     96:   TSCreate(PETSC_COMM_WORLD,&ts);
     97:   TSSetDM(ts,da);
     98:   TSSetProblemType(ts,TS_NONLINEAR);
     99:   TSSetIFunction(ts,NULL,FormFunction,&ctx);
    100:   TSSetDuration(ts,maxsteps,.02);
    101:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_INTERPOLATE);

    103:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    104:      Create matrix data structure; set Jacobian evaluation routine

    106: <     Set Jacobian matrix data structure and default Jacobian evaluation
    107:      routine. User can override with:
    108:      -snes_mf : matrix-free Newton-Krylov method with no preconditioning
    109:                 (unless user explicitly sets preconditioner)
    110:      -snes_mf_operator : form preconditioning matrix as set by the user,
    111:                          but use matrix-free approx for Jacobian-vector
    112:                          products within Newton-Krylov method

    114:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    115:   TSGetSNES(ts,&snes);
    116:   DMCreateColoring(da,IS_COLORING_GLOBAL,MATAIJ,&iscoloring);
    117:   DMCreateMatrix(da,MATAIJ,&J);
    118:   MatFDColoringCreate(J,iscoloring,&matfdcoloring);
    119:   ISColoringDestroy(&iscoloring);
    120:   MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))SNESTSFormFunction,ts);
    121:   MatFDColoringSetFromOptions(matfdcoloring);
    122:   SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,matfdcoloring);

    124:   {
    125:     VecDuplicate(x,&ul);
    126:     VecDuplicate(x,&uh);
    127:     VecStrideSet(ul,0,SNES_VI_NINF);
    128:     VecStrideSet(ul,1,-1.0);
    129:     VecStrideSet(uh,0,SNES_VI_INF);
    130:     VecStrideSet(uh,1,1.0);
    131:     TSVISetVariableBounds(ts,ul,uh);
    132:   }

    134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    135:      Customize nonlinear solver
    136:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    137:   TSSetType(ts,TSBEULER);

    139:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    140:      Set initial conditions
    141:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    142:   FormInitialSolution(da,x,ctx.kappa);
    143:   TSSetInitialTimeStep(ts,0.0,dt);
    144:   TSSetSolution(ts,x);

    146:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    147:      Set runtime options
    148:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    149:   TSSetFromOptions(ts);

    151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    152:      Solve nonlinear system
    153:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    154:   TSSolve(ts,x);
    155:   wait = PETSC_FALSE;
    156:   PetscOptionsGetBool(NULL,"-wait",&wait,NULL);
    157:   if (wait) {
    158:     PetscSleep(-1);
    159:   }
    160:   TSGetTimeStepNumber(ts,&steps);

    162:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    163:      Free work space.  All PETSc objects should be destroyed when they
    164:      are no longer needed.
    165:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    166:   {
    167:     VecDestroy(&ul);
    168:     VecDestroy(&uh);
    169:   }
    170:   MatDestroy(&J);
    171:   MatFDColoringDestroy(&matfdcoloring);
    172:   VecDestroy(&x);
    173:   VecDestroy(&r);
    174:   TSDestroy(&ts);
    175:   DMDestroy(&da);

    177:   PetscFinalize();
    178:   return(0);
    179: }

    181: typedef struct {PetscScalar w,u;} Field;
    182: /* ------------------------------------------------------------------- */
    185: /*
    186:    FormFunction - Evaluates nonlinear function, F(x).

    188:    Input Parameters:
    189: .  ts - the TS context
    190: .  X - input vector
    191: .  ptr - optional user-defined context, as set by SNESSetFunction()

    193:    Output Parameter:
    194: .  F - function vector
    195:  */
    196: PetscErrorCode FormFunction(TS ts,PetscReal ftime,Vec X,Vec Xdot,Vec F,void *ptr)
    197: {
    198:   DM             da;
    200:   PetscInt       i,Mx,xs,xm;
    201:   PetscReal      hx,sx;
    202:   Field          *x,*xdot,*f;
    203:   Vec            localX,localXdot;
    204:   UserCtx        *ctx = (UserCtx*)ptr;

    207:   TSGetDM(ts,&da);
    208:   DMGetLocalVector(da,&localX);
    209:   DMGetLocalVector(da,&localXdot);
    210:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
    211:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

    213:   hx = 1.0/(PetscReal)Mx; sx = 1.0/(hx*hx);

    215:   /*
    216:      Scatter ghost points to local vector,using the 2-step process
    217:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    218:      By placing code between these two statements, computations can be
    219:      done while messages are in transition.
    220:   */
    221:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    222:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    223:   DMGlobalToLocalBegin(da,Xdot,INSERT_VALUES,localXdot);
    224:   DMGlobalToLocalEnd(da,Xdot,INSERT_VALUES,localXdot);

    226:   /*
    227:      Get pointers to vector data
    228:   */
    229:   DMDAVecGetArray(da,localX,&x);
    230:   DMDAVecGetArray(da,localXdot,&xdot);
    231:   DMDAVecGetArray(da,F,&f);

    233:   /*
    234:      Get local grid boundaries
    235:   */
    236:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    238:   /*
    239:      Compute function over the locally owned part of the grid
    240:   */
    241:   for (i=xs; i<xs+xm; i++) {
    242:     f[i].w =  x[i].w + ctx->kappa*(x[i-1].u + x[i+1].u - 2.0*x[i].u)*sx;
    243:     if (ctx->cahnhillard) {
    244:       switch (ctx->energy) {
    245:       case 1: /* double well */
    246:         f[i].w += -x[i].u*x[i].u*x[i].u + x[i].u;
    247:         break;
    248:       case 2: /* double obstacle */
    249:         f[i].w += x[i].u;
    250:         break;
    251:       case 3: /* logarithmic */
    252:         if (PetscRealPart(x[i].u) < -1.0 + 2.0*ctx->tol)     f[i].w += .5*ctx->theta*(-log(ctx->tol) + log((1.0-x[i].u)/2.0)) + ctx->theta_c*x[i].u;
    253:         else if (PetscRealPart(x[i].u) > 1.0 - 2.0*ctx->tol) f[i].w += .5*ctx->theta*(-log((1.0+x[i].u)/2.0) + log(ctx->tol)) + ctx->theta_c*x[i].u;
    254:         else                                                 f[i].w += .5*ctx->theta*(-log((1.0+x[i].u)/2.0) + log((1.0-x[i].u)/2.0)) + ctx->theta_c*x[i].u;
    255:         break;
    256:       }
    257:     }
    258:     f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx;
    259:   }

    261:   /*
    262:      Restore vectors
    263:   */
    264:   DMDAVecRestoreArray(da,localXdot,&xdot);
    265:   DMDAVecRestoreArray(da,localX,&x);
    266:   DMDAVecRestoreArray(da,F,&f);
    267:   DMRestoreLocalVector(da,&localX);
    268:   DMRestoreLocalVector(da,&localXdot);
    269:   return(0);
    270: }

    272: /* ------------------------------------------------------------------- */
    275: PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa)
    276: {
    278:   PetscInt       i,xs,xm,Mx;
    279:   Field          *x;
    280:   PetscReal      hx,xx,r,sx;

    283:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
    284:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

    286:   hx = 1.0/(PetscReal)Mx;
    287:   sx = 1.0/(hx*hx);

    289:   /*
    290:      Get pointers to vector data
    291:   */
    292:   DMDAVecGetArray(da,X,&x);

    294:   /*
    295:      Get local grid boundaries
    296:   */
    297:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    299:   /*
    300:      Compute function over the locally owned part of the grid
    301:   */
    302:   for (i=xs; i<xs+xm; i++) {
    303:     xx = i*hx;
    304:     r = PetscSqrtScalar((xx-.5)*(xx-.5));
    305:     if (r < .125) x[i].u = 1.0;
    306:     else          x[i].u = -.50;
    307:     /*  u[i] = PetscPowScalar(x - .5,4.0); */
    308:   }
    309:   for (i=xs; i<xs+xm; i++) x[i].w = -kappa*(x[i-1].u + x[i+1].u - 2.0*x[i].u)*sx;

    311:   /*
    312:      Restore vectors
    313:   */
    314:   DMDAVecRestoreArray(da,X,&x);
    315:   return(0);
    316: }