Actual source code: biharmonic2.c

      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  -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: (this fails after a few timesteps 12/17/2017)
     22: ---------------
     23: ./biharmonic2 -ts_monitor -snes_monitor -ts_monitor_draw_solution  -pc_type lu  -draw_pause .1 -snes_converged_reason   -ts_type beuler    -da_refine 6  -draw_fields 1  -kappa .00001 -ts_dt 5.96046e-06 -cahn-hillard

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

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

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

     53:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     54:      Initialize program
     55:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     56:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
     57:   ctx.kappa = 1.0;
     58:   PetscOptionsGetReal(NULL,NULL,"-kappa",&ctx.kappa,NULL);
     59:   ctx.cahnhillard = PETSC_FALSE;

     61:   PetscOptionsGetBool(NULL,NULL,"-cahn-hillard",&ctx.cahnhillard,NULL);
     62:   PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),2,vbounds);
     63:   PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),600,600);
     64:   ctx.energy = 1;
     65:   /*PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);*/
     66:   PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);
     67:   ctx.tol     = 1.0e-8;
     68:   PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL);
     69:   ctx.theta   = .001;
     70:   ctx.theta_c = 1.0;
     71:   PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL);
     72:   PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL);

     74:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     75:      Create distributed array (DMDA) to manage parallel grid and vectors
     76:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     77:   DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10,2,2,NULL,&da);
     78:   DMSetFromOptions(da);
     79:   DMSetUp(da);
     80:   DMDASetFieldName(da,0,"Biharmonic heat equation: w = -kappa*u_xx");
     81:   DMDASetFieldName(da,1,"Biharmonic heat equation: u");
     82:   DMDAGetInfo(da,0,&Mx,0,0,0,0,0,0,0,0,0,0,0);
     83:   dt   = 1.0/(10.*ctx.kappa*Mx*Mx*Mx*Mx);

     85:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     86:      Extract global vectors from DMDA; then duplicate for remaining
     87:      vectors that are the same types
     88:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     89:   DMCreateGlobalVector(da,&x);
     90:   VecDuplicate(x,&r);

     92:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     93:      Create timestepping solver context
     94:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     95:   TSCreate(PETSC_COMM_WORLD,&ts);
     96:   TSSetDM(ts,da);
     97:   TSSetProblemType(ts,TS_NONLINEAR);
     98:   TSSetIFunction(ts,NULL,FormFunction,&ctx);
     99:   TSSetMaxTime(ts,.02);
    100:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_INTERPOLATE);

    102:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    103:      Create matrix data structure; set Jacobian evaluation routine

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

    113:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    114:   TSGetSNES(ts,&snes);
    115:   DMCreateColoring(da,IS_COLORING_GLOBAL,&iscoloring);
    116:   DMSetMatType(da,MATAIJ);
    117:   DMCreateMatrix(da,&J);
    118:   MatFDColoringCreate(J,iscoloring,&matfdcoloring);
    119:   MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))SNESTSFormFunction,ts);
    120:   MatFDColoringSetFromOptions(matfdcoloring);
    121:   MatFDColoringSetUp(J,iscoloring,matfdcoloring);
    122:   ISColoringDestroy(&iscoloring);
    123:   SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,matfdcoloring);

    125:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    126:      Customize nonlinear solver
    127:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    128:   TSSetType(ts,TSBEULER);

    130:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    131:      Set initial conditions
    132:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    133:   FormInitialSolution(da,x,ctx.kappa);
    134:   TSSetTimeStep(ts,dt);
    135:   TSSetSolution(ts,x);

    137:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    138:      Set runtime options
    139:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    140:   TSSetFromOptions(ts);

    142:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    143:      Solve nonlinear system
    144:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    145:   TSSolve(ts,x);
    146:   TSGetStepNumber(ts,&steps);

    148:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    149:      Free work space.  All PETSc objects should be destroyed when they
    150:      are no longer needed.
    151:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    152:   MatDestroy(&J);
    153:   MatFDColoringDestroy(&matfdcoloring);
    154:   VecDestroy(&x);
    155:   VecDestroy(&r);
    156:   TSDestroy(&ts);
    157:   DMDestroy(&da);

    159:   PetscFinalize();
    160:   return ierr;
    161: }

    163: typedef struct {PetscScalar w,u;} Field;
    164: /* ------------------------------------------------------------------- */
    165: /*
    166:    FormFunction - Evaluates nonlinear function, F(x).

    168:    Input Parameters:
    169: .  ts - the TS context
    170: .  X - input vector
    171: .  ptr - optional user-defined context, as set by SNESSetFunction()

    173:    Output Parameter:
    174: .  F - function vector
    175:  */
    176: PetscErrorCode FormFunction(TS ts,PetscReal ftime,Vec X,Vec Xdot,Vec F,void *ptr)
    177: {
    178:   DM             da;
    180:   PetscInt       i,Mx,xs,xm;
    181:   PetscReal      hx,sx;
    182:   Field          *x,*xdot,*f;
    183:   Vec            localX,localXdot;
    184:   UserCtx        *ctx = (UserCtx*)ptr;

    187:   TSGetDM(ts,&da);
    188:   DMGetLocalVector(da,&localX);
    189:   DMGetLocalVector(da,&localXdot);
    190:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

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

    194:   /*
    195:      Scatter ghost points to local vector,using the 2-step process
    196:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    197:      By placing code between these two statements, computations can be
    198:      done while messages are in transition.
    199:   */
    200:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    201:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    202:   DMGlobalToLocalBegin(da,Xdot,INSERT_VALUES,localXdot);
    203:   DMGlobalToLocalEnd(da,Xdot,INSERT_VALUES,localXdot);

    205:   /*
    206:      Get pointers to vector data
    207:   */
    208:   DMDAVecGetArrayRead(da,localX,&x);
    209:   DMDAVecGetArrayRead(da,localXdot,&xdot);
    210:   DMDAVecGetArray(da,F,&f);

    212:   /*
    213:      Get local grid boundaries
    214:   */
    215:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    217:   /*
    218:      Compute function over the locally owned part of the grid
    219:   */
    220:   for (i=xs; i<xs+xm; i++) {
    221:     f[i].w =  x[i].w + ctx->kappa*(x[i-1].u + x[i+1].u - 2.0*x[i].u)*sx;
    222:     if (ctx->cahnhillard) {
    223:       switch (ctx->energy) {
    224:       case 1: /* double well */
    225:         f[i].w += -x[i].u*x[i].u*x[i].u + x[i].u;
    226:         break;
    227:       case 2: /* double obstacle */
    228:         f[i].w += x[i].u;
    229:         break;
    230:       case 3: /* logarithmic */
    231:         if (PetscRealPart(x[i].u) < -1.0 + 2.0*ctx->tol)     f[i].w += .5*ctx->theta*(-PetscLogReal(ctx->tol) + PetscLogScalar((1.0-x[i].u)/2.0)) + ctx->theta_c*x[i].u;
    232:         else if (PetscRealPart(x[i].u) > 1.0 - 2.0*ctx->tol) f[i].w += .5*ctx->theta*(-PetscLogScalar((1.0+x[i].u)/2.0) + PetscLogReal(ctx->tol)) + ctx->theta_c*x[i].u;
    233:         else                                                 f[i].w += .5*ctx->theta*(-PetscLogScalar((1.0+x[i].u)/2.0) + PetscLogScalar((1.0-x[i].u)/2.0)) + ctx->theta_c*x[i].u;
    234:         break;
    235:       }
    236:     }
    237:     f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx;
    238:   }

    240:   /*
    241:      Restore vectors
    242:   */
    243:   DMDAVecRestoreArrayRead(da,localXdot,&xdot);
    244:   DMDAVecRestoreArrayRead(da,localX,&x);
    245:   DMDAVecRestoreArray(da,F,&f);
    246:   DMRestoreLocalVector(da,&localX);
    247:   DMRestoreLocalVector(da,&localXdot);
    248:   return(0);
    249: }

    251: /* ------------------------------------------------------------------- */
    252: PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa)
    253: {
    255:   PetscInt       i,xs,xm,Mx,xgs,xgm;
    256:   Field          *x;
    257:   PetscReal      hx,xx,r,sx;
    258:   Vec            Xg;

    261:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

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

    266:   /*
    267:      Get pointers to vector data
    268:   */
    269:   DMCreateLocalVector(da,&Xg);
    270:   DMDAVecGetArray(da,Xg,&x);

    272:   /*
    273:      Get local grid boundaries
    274:   */
    275:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
    276:   DMDAGetGhostCorners(da,&xgs,NULL,NULL,&xgm,NULL,NULL);

    278:   /*
    279:      Compute u function over the locally owned part of the grid including ghost points
    280:   */
    281:   for (i=xgs; i<xgs+xgm; i++) {
    282:     xx = i*hx;
    283:     r = PetscSqrtReal((xx-.5)*(xx-.5));
    284:     if (r < .125) x[i].u = 1.0;
    285:     else          x[i].u = -.50;
    286:     /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
    287:     x[i].w = 0;
    288:   }
    289:   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;

    291:   /*
    292:      Restore vectors
    293:   */
    294:   DMDAVecRestoreArray(da,Xg,&x);

    296:   /* Grab only the global part of the vector */
    297:   VecSet(X,0);
    298:   DMLocalToGlobalBegin(da,Xg,ADD_VALUES,X);
    299:   DMLocalToGlobalEnd(da,Xg,ADD_VALUES,X);
    300:   VecDestroy(&Xg);
    301:   return(0);
    302: }

    304: /*TEST

    306:    build:
    307:      requires: !complex !single

    309:    test:
    310:      args: -ts_monitor -snes_monitor  -pc_type lu   -snes_converged_reason  -ts_type beuler  -da_refine 5 -ts_dt 9.53674e-9 -ts_max_steps 50
    311:      requires: x

    313: TEST*/