Date: 2022-03-31T04:48:49+00:00

Actual source code: biharmonic3.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: ./biharmonic3 -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:
     22: ---------------
     23: ./biharmonic3 -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;
     43:   DM             da;
     44:   MatFDColoring  matfdcoloring;
     45:   ISColoring     iscoloring;
     46:   PetscReal      dt;
     47:   PetscReal      vbounds[] = {-100000,100000,-1.1,1.1};
     48:   SNES           snes;
     49:   UserCtx        ctx;

     51:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     52:      Initialize program
     53:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     54:   PetscInitialize(&argc,&argv,(char*)0,help);
     55:   ctx.kappa       = 1.0;
     56:   PetscOptionsGetReal(NULL,NULL,"-kappa",&ctx.kappa,NULL);
     57:   ctx.cahnhillard = PETSC_FALSE;
     58:   PetscOptionsGetBool(NULL,NULL,"-cahn-hillard",&ctx.cahnhillard,NULL);
     59:   PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),2,vbounds);
     60:   PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),600,600);
     61:   ctx.energy      = 1;
     62:   PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);
     63:   ctx.tol     = 1.0e-8;
     64:   PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL);
     65:   ctx.theta   = .001;
     66:   ctx.theta_c = 1.0;
     67:   PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL);
     68:   PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL);

     70:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     71:      Create distributed array (DMDA) to manage parallel grid and vectors
     72:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     73:   DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10,2,2,NULL,&da);
     74:   DMSetFromOptions(da);
     75:   DMSetUp(da);
     76:   DMDASetFieldName(da,0,"Biharmonic heat equation: w = -kappa*u_xx");
     77:   DMDASetFieldName(da,1,"Biharmonic heat equation: u");
     78:   DMDAGetInfo(da,0,&Mx,0,0,0,0,0,0,0,0,0,0,0);
     79:   dt   = 1.0/(10.*ctx.kappa*Mx*Mx*Mx*Mx);

     81:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     82:      Extract global vectors from DMDA; then duplicate for remaining
     83:      vectors that are the same types
     84:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     85:   DMCreateGlobalVector(da,&x);
     86:   VecDuplicate(x,&r);

     88:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     89:      Create timestepping solver context
     90:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
     91:   TSCreate(PETSC_COMM_WORLD,&ts);
     92:   TSSetDM(ts,da);
     93:   TSSetProblemType(ts,TS_NONLINEAR);
     94:   TSSetIFunction(ts,NULL,FormFunction,&ctx);
     95:   TSSetMaxTime(ts,.02);
     96:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);

     98:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     99:      Create matrix data structure; set Jacobian evaluation routine

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

    109:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    110:   TSGetSNES(ts,&snes);
    111:   SNESSetType(snes,SNESVINEWTONRSLS);
    112:   DMCreateColoring(da,IS_COLORING_GLOBAL,&iscoloring);
    113:   DMSetMatType(da,MATAIJ);
    114:   DMCreateMatrix(da,&J);
    115:   MatFDColoringCreate(J,iscoloring,&matfdcoloring);
    116:   MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))SNESTSFormFunction,ts);
    117:   MatFDColoringSetFromOptions(matfdcoloring);
    118:   MatFDColoringSetUp(J,iscoloring,matfdcoloring);
    119:   ISColoringDestroy(&iscoloring);
    120:   SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,matfdcoloring);

    122:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    123:      Customize nonlinear solver
    124:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    125:   TSSetType(ts,TSBEULER);

    127:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    128:      Set initial conditions
    129:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    130:   FormInitialSolution(da,x,ctx.kappa);
    131:   TSSetTimeStep(ts,dt);
    132:   TSSetSolution(ts,x);

    134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    135:      Set runtime options
    136:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    137:   TSSetFromOptions(ts);

    139:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    140:      Solve nonlinear system
    141:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    142:   TSSolve(ts,x);
    143:   TSGetStepNumber(ts,&steps);

    145:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    146:      Free work space.  All PETSc objects should be destroyed when they
    147:      are no longer needed.
    148:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    149:   MatDestroy(&J);
    150:   MatFDColoringDestroy(&matfdcoloring);
    151:   VecDestroy(&x);
    152:   VecDestroy(&r);
    153:   TSDestroy(&ts);
    154:   DMDestroy(&da);

    156:   PetscFinalize();
    157:   return 0;
    158: }

    160: typedef struct {PetscScalar w,u;} Field;
    161: /* ------------------------------------------------------------------- */
    162: /*
    163:    FormFunction - Evaluates nonlinear function, F(x).

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

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

    183:   TSGetDM(ts,&da);
    184:   DMGetLocalVector(da,&localX);
    185:   DMGetLocalVector(da,&localXdot);
    186:   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);

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

    190:   /*
    191:      Scatter ghost points to local vector,using the 2-step process
    192:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    193:      By placing code between these two statements, computations can be
    194:      done while messages are in transition.
    195:   */
    196:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    197:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    198:   DMGlobalToLocalBegin(da,Xdot,INSERT_VALUES,localXdot);
    199:   DMGlobalToLocalEnd(da,Xdot,INSERT_VALUES,localXdot);

    201:   /*
    202:      Get pointers to vector data
    203:   */
    204:   DMDAVecGetArrayRead(da,localX,&x);
    205:   DMDAVecGetArrayRead(da,localXdot,&xdot);
    206:   DMDAVecGetArray(da,F,&f);

    208:   /*
    209:      Get local grid boundaries
    210:   */
    211:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

    213:   /*
    214:      Compute function over the locally owned part of the grid
    215:   */
    216:   for (i=xs; i<xs+xm; i++) {
    217:     f[i].w =  x[i].w + ctx->kappa*(x[i-1].u + x[i+1].u - 2.0*x[i].u)*sx;
    218:     if (ctx->cahnhillard) {
    219:       switch (ctx->energy) {
    220:       case 1: /* double well */
    221:         f[i].w += -x[i].u*x[i].u*x[i].u + x[i].u;
    222:         break;
    223:       case 2: /* double obstacle */
    224:         f[i].w += x[i].u;
    225:         break;
    226:       case 3: /* logarithmic */
    227:         if (x[i].u < -1.0 + 2.0*ctx->tol)      f[i].w += .5*ctx->theta*(-PetscLogScalar(ctx->tol) + PetscLogScalar((1.0-x[i].u)/2.0)) + ctx->theta_c*x[i].u;
    228:         else if (x[i].u > 1.0 - 2.0*ctx->tol)  f[i].w += .5*ctx->theta*(-PetscLogScalar((1.0+x[i].u)/2.0) + PetscLogScalar(ctx->tol)) + ctx->theta_c*x[i].u;
    229:         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;
    230:         break;
    231:       case 4:
    232:         break;
    233:       }
    234:     }
    235:     f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx;
    236:     if (ctx->energy==4) {
    237:       f[i].u = xdot[i].u;
    238:       /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */
    239:       r       = (1.0 - x[i+1].u*x[i+1].u)*(x[i+2].w-x[i].w)*.5/hx;
    240:       l       = (1.0 - x[i-1].u*x[i-1].u)*(x[i].w-x[i-2].w)*.5/hx;
    241:       f[i].u -= (r - l)*.5/hx;
    242:       f[i].u += 2.0*ctx->theta_c*x[i].u*(x[i+1].u-x[i-1].u)*(x[i+1].u-x[i-1].u)*.25*sx - (ctx->theta - ctx->theta_c*(1-x[i].u*x[i].u))*(x[i+1].u + x[i-1].u - 2.0*x[i].u)*sx;
    243:     }
    244:   }

    246:   /*
    247:      Restore vectors
    248:   */
    249:   DMDAVecRestoreArrayRead(da,localXdot,&xdot);
    250:   DMDAVecRestoreArrayRead(da,localX,&x);
    251:   DMDAVecRestoreArray(da,F,&f);
    252:   DMRestoreLocalVector(da,&localX);
    253:   DMRestoreLocalVector(da,&localXdot);
    254:   return 0;
    255: }

    257: /* ------------------------------------------------------------------- */
    258: PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa)
    259: {
    260:   PetscInt       i,xs,xm,Mx,xgs,xgm;
    261:   Field          *x;
    262:   PetscReal      hx,xx,r,sx;
    263:   Vec            Xg;

    265:   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);

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

    270:   /*
    271:      Get pointers to vector data
    272:   */
    273:   DMCreateLocalVector(da,&Xg);
    274:   DMDAVecGetArray(da,Xg,&x);

    276:   /*
    277:      Get local grid boundaries
    278:   */
    279:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
    280:   DMDAGetGhostCorners(da,&xgs,NULL,NULL,&xgm,NULL,NULL);

    282:   /*
    283:      Compute u function over the locally owned part of the grid including ghost points
    284:   */
    285:   for (i=xgs; i<xgs+xgm; i++) {
    286:     xx = i*hx;
    287:     r = PetscSqrtReal((xx-.5)*(xx-.5));
    288:     if (r < .125) x[i].u = 1.0;
    289:     else          x[i].u = -.50;
    290:     /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
    291:     x[i].w = 0;
    292:   }
    293:   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;

    295:   /*
    296:      Restore vectors
    297:   */
    298:   DMDAVecRestoreArray(da,Xg,&x);

    300:   /* Grab only the global part of the vector */
    301:   VecSet(X,0);
    302:   DMLocalToGlobalBegin(da,Xg,ADD_VALUES,X);
    303:   DMLocalToGlobalEnd(da,Xg,ADD_VALUES,X);
    304:   VecDestroy(&Xg);
    305:   return 0;
    306: }

    308: /*TEST

    310:    build:
    311:      requires: !complex !single

    313:    test:
    314:      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
    315:      requires: x

    317: TEST*/