Date: 2022-03-04T17:39:41+00:00

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

     25: */
     26: #include <petscdm.h>
     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;

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

     52:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     53:      Initialize program
     54:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     55:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
     56:   ctx.kappa = 1.0;
     57:   PetscOptionsGetReal(NULL,NULL,"-kappa",&ctx.kappa,NULL);
     58:   ctx.cahnhillard = PETSC_FALSE;

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

     73:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     74:      Create distributed array (DMDA) to manage parallel grid and vectors
     75:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     76:   DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10,2,2,NULL,&da);
     77:   DMSetFromOptions(da);
     78:   DMSetUp(da);
     79:   DMDASetFieldName(da,0,"Biharmonic heat equation: w = -kappa*u_xx");
     80:   DMDASetFieldName(da,1,"Biharmonic heat equation: u");
     81:   DMDAGetInfo(da,0,&Mx,0,0,0,0,0,0,0,0,0,0,0);
     82:   dt   = 1.0/(10.*ctx.kappa*Mx*Mx*Mx*Mx);

     84:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     85:      Extract global vectors from DMDA; then duplicate for remaining
     86:      vectors that are the same types
     87:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     88:   DMCreateGlobalVector(da,&x);
     89:   VecDuplicate(x,&r);

     91:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     92:      Create timestepping solver context
     93:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     94:   TSCreate(PETSC_COMM_WORLD,&ts);
     95:   TSSetDM(ts,da);
     96:   TSSetProblemType(ts,TS_NONLINEAR);
     97:   TSSetIFunction(ts,NULL,FormFunction,&ctx);
     98:   TSSetMaxTime(ts,.02);
     99:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_INTERPOLATE);

    101:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    102:      Create matrix data structure; set Jacobian evaluation routine

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

    112:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    113:   TSGetSNES(ts,&snes);
    114:   DMCreateColoring(da,IS_COLORING_GLOBAL,&iscoloring);
    115:   DMSetMatType(da,MATAIJ);
    116:   DMCreateMatrix(da,&J);
    117:   MatFDColoringCreate(J,iscoloring,&matfdcoloring);
    118:   MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))SNESTSFormFunction,ts);
    119:   MatFDColoringSetFromOptions(matfdcoloring);
    120:   MatFDColoringSetUp(J,iscoloring,matfdcoloring);
    121:   ISColoringDestroy(&iscoloring);
    122:   SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,matfdcoloring);

    124:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    125:      Customize nonlinear solver
    126:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    127:   TSSetType(ts,TSBEULER);

    129:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    130:      Set initial conditions
    131:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    132:   FormInitialSolution(da,x,ctx.kappa);
    133:   TSSetTimeStep(ts,dt);
    134:   TSSetSolution(ts,x);

    136:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    137:      Set runtime options
    138:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    139:   TSSetFromOptions(ts);

    141:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    142:      Solve nonlinear system
    143:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    144:   TSSolve(ts,x);
    145:   TSGetStepNumber(ts,&steps);

    147:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    148:      Free work space.  All PETSc objects should be destroyed when they
    149:      are no longer needed.
    150:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    151:   MatDestroy(&J);
    152:   MatFDColoringDestroy(&matfdcoloring);
    153:   VecDestroy(&x);
    154:   VecDestroy(&r);
    155:   TSDestroy(&ts);
    156:   DMDestroy(&da);

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

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

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

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

    186:   TSGetDM(ts,&da);
    187:   DMGetLocalVector(da,&localX);
    188:   DMGetLocalVector(da,&localXdot);
    189:   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);

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

    193:   /*
    194:      Scatter ghost points to local vector,using the 2-step process
    195:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    196:      By placing code between these two statements, computations can be
    197:      done while messages are in transition.
    198:   */
    199:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    200:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    201:   DMGlobalToLocalBegin(da,Xdot,INSERT_VALUES,localXdot);
    202:   DMGlobalToLocalEnd(da,Xdot,INSERT_VALUES,localXdot);

    204:   /*
    205:      Get pointers to vector data
    206:   */
    207:   DMDAVecGetArrayRead(da,localX,&x);
    208:   DMDAVecGetArrayRead(da,localXdot,&xdot);
    209:   DMDAVecGetArray(da,F,&f);

    211:   /*
    212:      Get local grid boundaries
    213:   */
    214:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    216:   /*
    217:      Compute function over the locally owned part of the grid
    218:   */
    219:   for (i=xs; i<xs+xm; i++) {
    220:     f[i].w =  x[i].w + ctx->kappa*(x[i-1].u + x[i+1].u - 2.0*x[i].u)*sx;
    221:     if (ctx->cahnhillard) {
    222:       switch (ctx->energy) {
    223:       case 1: /* double well */
    224:         f[i].w += -x[i].u*x[i].u*x[i].u + x[i].u;
    225:         break;
    226:       case 2: /* double obstacle */
    227:         f[i].w += x[i].u;
    228:         break;
    229:       case 3: /* logarithmic */
    230:         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;
    231:         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;
    232:         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;
    233:         break;
    234:       }
    235:     }
    236:     f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx;
    237:   }

    239:   /*
    240:      Restore vectors
    241:   */
    242:   DMDAVecRestoreArrayRead(da,localXdot,&xdot);
    243:   DMDAVecRestoreArrayRead(da,localX,&x);
    244:   DMDAVecRestoreArray(da,F,&f);
    245:   DMRestoreLocalVector(da,&localX);
    246:   DMRestoreLocalVector(da,&localXdot);
    247:   return(0);
    248: }

    250: /* ------------------------------------------------------------------- */
    251: PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa)
    252: {
    254:   PetscInt       i,xs,xm,Mx,xgs,xgm;
    255:   Field          *x;
    256:   PetscReal      hx,xx,r,sx;
    257:   Vec            Xg;

    260:   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);

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

    265:   /*
    266:      Get pointers to vector data
    267:   */
    268:   DMCreateLocalVector(da,&Xg);
    269:   DMDAVecGetArray(da,Xg,&x);

    271:   /*
    272:      Get local grid boundaries
    273:   */
    274:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
    275:   DMDAGetGhostCorners(da,&xgs,NULL,NULL,&xgm,NULL,NULL);

    277:   /*
    278:      Compute u function over the locally owned part of the grid including ghost points
    279:   */
    280:   for (i=xgs; i<xgs+xgm; i++) {
    281:     xx = i*hx;
    282:     r = PetscSqrtReal((xx-.5)*(xx-.5));
    283:     if (r < .125) x[i].u = 1.0;
    284:     else          x[i].u = -.50;
    285:     /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
    286:     x[i].w = 0;
    287:   }
    288:   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;

    290:   /*
    291:      Restore vectors
    292:   */
    293:   DMDAVecRestoreArray(da,Xg,&x);

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

    303: /*TEST

    305:    build:
    306:      requires: !complex !single

    308:    test:
    309:      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
    310:      requires: x

    312: TEST*/