Actual source code: biharmonic3.c
petsc-3.13.6 2020-09-29
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      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

     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;
     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:   ctx.tol     = 1.0e-8;
     66:   PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL);
     67:   ctx.theta   = .001;
     68:   ctx.theta_c = 1.0;
     69:   PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL);
     70:   PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL);

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

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

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

    100:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    101:      Create matrix data structure; set Jacobian evaluation routine

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

    111:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    112:   TSGetSNES(ts,&snes);
    113:   SNESSetType(snes,SNESVINEWTONRSLS);
    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:   PetscScalar    r,l;
    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 (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;
    232:         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;
    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:       case 4:
    236:         break;
    237:       }
    238:     }
    239:     f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx;
    240:     if (ctx->energy==4) {
    241:       f[i].u = xdot[i].u;
    242:       /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */
    243:       r       = (1.0 - x[i+1].u*x[i+1].u)*(x[i+2].w-x[i].w)*.5/hx;
    244:       l       = (1.0 - x[i-1].u*x[i-1].u)*(x[i].w-x[i-2].w)*.5/hx;
    245:       f[i].u -= (r - l)*.5/hx;
    246:       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;
    247:     }
    248:   }

    250:   /*
    251:      Restore vectors
    252:   */
    253:   DMDAVecRestoreArrayRead(da,localXdot,&xdot);
    254:   DMDAVecRestoreArrayRead(da,localX,&x);
    255:   DMDAVecRestoreArray(da,F,&f);
    256:   DMRestoreLocalVector(da,&localX);
    257:   DMRestoreLocalVector(da,&localXdot);
    258:   return(0);
    259: }

    262: /* ------------------------------------------------------------------- */
    263: PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa)
    264: {
    266:   PetscInt       i,xs,xm,Mx,xgs,xgm;
    267:   Field          *x;
    268:   PetscReal      hx,xx,r,sx;
    269:   Vec            Xg;

    272:   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);

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

    277:   /*
    278:      Get pointers to vector data
    279:   */
    280:   DMCreateLocalVector(da,&Xg);
    281:   DMDAVecGetArray(da,Xg,&x);

    283:   /*
    284:      Get local grid boundaries
    285:   */
    286:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
    287:   DMDAGetGhostCorners(da,&xgs,NULL,NULL,&xgm,NULL,NULL);

    289:   /*
    290:      Compute u function over the locally owned part of the grid including ghost points
    291:   */
    292:   for (i=xgs; i<xgs+xgm; i++) {
    293:     xx = i*hx;
    294:     r = PetscSqrtReal((xx-.5)*(xx-.5));
    295:     if (r < .125) x[i].u = 1.0;
    296:     else          x[i].u = -.50;
    297:     /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
    298:     x[i].w = 0;
    299:   }
    300:   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;

    302:   /*
    303:      Restore vectors
    304:   */
    305:   DMDAVecRestoreArray(da,Xg,&x);

    307:   /* Grab only the global part of the vector */
    308:   VecSet(X,0);
    309:   DMLocalToGlobalBegin(da,Xg,ADD_VALUES,X);
    310:   DMLocalToGlobalEnd(da,Xg,ADD_VALUES,X);
    311:   VecDestroy(&Xg);
    312:   return(0);
    313: }

    315: /*TEST

    317:    build:
    318:      requires: !complex !single

    320:    test:
    321:      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
    322:      requires: x

    324: TEST*/