Actual source code: ex26.c

petsc-3.4.4 2014-03-13
  2: static char help[] = "Transient nonlinear driven cavity in 2d.\n\
  3:   \n\
  4: The 2D driven cavity problem is solved in a velocity-vorticity formulation.\n\
  5: The flow can be driven with the lid or with bouyancy or both:\n\
  6:   -lidvelocity <lid>, where <lid> = dimensionless velocity of lid\n\
  7:   -grashof <gr>, where <gr> = dimensionless temperature gradent\n\
  8:   -prandtl <pr>, where <pr> = dimensionless thermal/momentum diffusity ratio\n\
  9:   -contours : draw contour plots of solution\n\n";
 10: /*
 11:       See src/snes/examples/tutorials/ex19.c for the steady-state version.

 13:       See src/snes/examples/tutorials/ex27.c for a version written specifically
 14:       for pseudo-transient continuation, without using TS.
 15: */

 17: /*T
 18:    Concepts: TS^solving a system of nonlinear equations (parallel multicomponent example);
 19:    Concepts: DMDA^using distributed arrays;
 20:    Concepts: TS^multicomponent
 21:    Concepts: TS^differential-algebraic equation
 22:    Processors: n
 23: T*/
 24: /* ------------------------------------------------------------------------

 26:     We thank David E. Keyes for contributing the driven cavity discretization
 27:     within this example code.

 29:     This problem is modeled by the partial differential equation system

 31:         - Lap(U) - Grad_y(Omega) = 0
 32:         - Lap(V) + Grad_x(Omega) = 0
 33:         Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0
 34:         T_t - Lap(T) + PR*Div([U*T,V*T]) = 0

 36:     in the unit square, which is uniformly discretized in each of x and
 37:     y in this simple encoding.

 39:     No-slip, rigid-wall Dirichlet conditions are used for [U,V].
 40:     Dirichlet conditions are used for Omega, based on the definition of
 41:     vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each
 42:     constant coordinate boundary, the tangential derivative is zero.
 43:     Dirichlet conditions are used for T on the left and right walls,
 44:     and insulation homogeneous Neumann conditions are used for T on
 45:     the top and bottom walls.

 47:     A finite difference approximation with the usual 5-point stencil
 48:     is used to discretize the boundary value problem to obtain a
 49:     nonlinear system of equations.  Upwinding is used for the divergence
 50:     (convective) terms and central for the gradient (source) terms.

 52:     The Jacobian can be either
 53:       * formed via finite differencing using coloring (the default), or
 54:       * applied matrix-free via the option -snes_mf
 55:         (for larger grid problems this variant may not converge
 56:         without a preconditioner due to ill-conditioning).

 58:   ------------------------------------------------------------------------- */

 60: /*
 61:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 62:    Include "petscts.h" so that we can use TS solvers.  Note that this
 63:    file automatically includes:
 64:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 65:      petscmat.h - matrices
 66:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 67:      petscviewer.h - viewers               petscpc.h  - preconditioners
 68:      petscksp.h   - linear solvers         petscsnes.h - nonlinear solvers
 69: */
 70: #include <petscts.h>
 71: #include <petscdmda.h>

 73: /*
 74:    User-defined routines and data structures
 75: */
 76: typedef struct {
 77:   PetscScalar u,v,omega,temp;
 78: } Field;

 80: PetscErrorCode FormIFunctionLocal(DMDALocalInfo*,PetscReal,Field**,Field**,Field**,void*);

 82: typedef struct {
 83:   PassiveReal lidvelocity,prandtl,grashof;   /* physical parameters */
 84:   PetscBool   parabolic;                     /* allow a transient term corresponding roughly to artificial compressibility */
 85:   PetscReal   cfl_initial;                   /* CFL for first time step */
 86: } AppCtx;

 88: PetscErrorCode FormInitialSolution(TS,Vec,AppCtx*);

 92: int main(int argc,char **argv)
 93: {
 94:   AppCtx            user;             /* user-defined work context */
 95:   PetscInt          mx,my,steps;
 96:   PetscErrorCode    ierr;
 97:   TS                ts;
 98:   DM                da;
 99:   Vec               X;
100:   PetscReal         ftime;
101:   TSConvergedReason reason;

103:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return(1);

105:   TSCreate(PETSC_COMM_WORLD,&ts);

107:   DMDACreate2d(PETSC_COMM_WORLD,DMDA_BOUNDARY_NONE,DMDA_BOUNDARY_NONE,DMDA_STENCIL_STAR,-4,-4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);
108:   TSSetDM(ts,(DM)da);

110:   DMDAGetInfo(da,0,&mx,&my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,
111:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
112:   /*
113:      Problem parameters (velocity of lid, prandtl, and grashof numbers)
114:   */
115:   user.lidvelocity = 1.0/(mx*my);
116:   user.prandtl     = 1.0;
117:   user.grashof     = 1.0;
118:   user.parabolic   = PETSC_FALSE;
119:   user.cfl_initial = 50.;

121:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Driven cavity/natural convection options","");
122:   PetscOptionsReal("-lidvelocity","Lid velocity, related to Reynolds number","",user.lidvelocity,&user.lidvelocity,NULL);
123:   PetscOptionsReal("-prandtl","Ratio of viscous to thermal diffusivity","",user.prandtl,&user.prandtl,NULL);
124:   PetscOptionsReal("-grashof","Ratio of bouyant to viscous forces","",user.grashof,&user.grashof,NULL);
125:   PetscOptionsBool("-parabolic","Relax incompressibility to make the system parabolic instead of differential-algebraic","",user.parabolic,&user.parabolic,NULL);
126:   PetscOptionsReal("-cfl_initial","Advective CFL for the first time step","",user.cfl_initial,&user.cfl_initial,NULL);
127:   PetscOptionsEnd();

129:   DMDASetFieldName(da,0,"x-velocity");
130:   DMDASetFieldName(da,1,"y-velocity");
131:   DMDASetFieldName(da,2,"Omega");
132:   DMDASetFieldName(da,3,"temperature");

134:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135:      Create user context, set problem data, create vector data structures.
136:      Also, compute the initial guess.
137:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

139:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
140:      Create time integration context
141:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
142:   DMSetApplicationContext(da,&user);
143:   DMDATSSetIFunctionLocal(da,INSERT_VALUES,(DMDATSIFunctionLocal)FormIFunctionLocal,&user);
144:   TSSetDuration(ts,10000,1e12);
145:   TSSetInitialTimeStep(ts,0.0,user.cfl_initial/(user.lidvelocity*mx));
146:   TSSetFromOptions(ts);

148:   PetscPrintf(PETSC_COMM_WORLD,"%Dx%D grid, lid velocity = %G, prandtl # = %G, grashof # = %G\n",mx,my,user.lidvelocity,user.prandtl,user.grashof);


151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152:      Solve the nonlinear system
153:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

155:   DMCreateGlobalVector(da,&X);
156:   FormInitialSolution(ts,X,&user);

158:   TSSolve(ts,X);
159:   TSGetSolveTime(ts,&ftime);
160:   TSGetTimeStepNumber(ts,&steps);
161:   TSGetConvergedReason(ts,&reason);

163:   PetscPrintf(PETSC_COMM_WORLD,"%s at time %G after %D steps\n",TSConvergedReasons[reason],ftime,steps);

165:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
166:      Free work space.  All PETSc objects should be destroyed when they
167:      are no longer needed.
168:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
169:   VecDestroy(&X);
170:   DMDestroy(&da);
171:   TSDestroy(&ts);

173:   PetscFinalize();
174:   return 0;
175: }

177: /* ------------------------------------------------------------------- */


182: /*
183:    FormInitialSolution - Forms initial approximation.

185:    Input Parameters:
186:    user - user-defined application context
187:    X - vector

189:    Output Parameter:
190:    X - vector
191:  */
192: PetscErrorCode FormInitialSolution(TS ts,Vec X,AppCtx *user)
193: {
194:   DM             da;
195:   PetscInt       i,j,mx,xs,ys,xm,ym;
197:   PetscReal      grashof,dx;
198:   Field          **x;

200:   grashof = user->grashof;
201:   TSGetDM(ts,&da);
202:   DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0);
203:   dx      = 1.0/(mx-1);

205:   /*
206:      Get local grid boundaries (for 2-dimensional DMDA):
207:        xs, ys   - starting grid indices (no ghost points)
208:        xm, ym   - widths of local grid (no ghost points)
209:   */
210:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

212:   /*
213:      Get a pointer to vector data.
214:        - For default PETSc vectors, VecGetArray() returns a pointer to
215:          the data array.  Otherwise, the routine is implementation dependent.
216:        - You MUST call VecRestoreArray() when you no longer need access to
217:          the array.
218:   */
219:   DMDAVecGetArray(da,X,&x);

221:   /*
222:      Compute initial guess over the locally owned part of the grid
223:      Initial condition is motionless fluid and equilibrium temperature
224:   */
225:   for (j=ys; j<ys+ym; j++) {
226:     for (i=xs; i<xs+xm; i++) {
227:       x[j][i].u     = 0.0;
228:       x[j][i].v     = 0.0;
229:       x[j][i].omega = 0.0;
230:       x[j][i].temp  = (grashof>0)*i*dx;
231:     }
232:   }

234:   /*
235:      Restore vector
236:   */
237:   DMDAVecRestoreArray(da,X,&x);
238:   return 0;
239: }

243: PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info,PetscReal ptime,Field **x,Field **xdot,Field **f,void *ptr)
244: {
245:   AppCtx         *user = (AppCtx*)ptr;
247:   PetscInt       xints,xinte,yints,yinte,i,j;
248:   PetscReal      hx,hy,dhx,dhy,hxdhy,hydhx;
249:   PetscReal      grashof,prandtl,lid;
250:   PetscScalar    u,udot,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym;

253:   grashof = user->grashof;
254:   prandtl = user->prandtl;
255:   lid     = user->lidvelocity;

257:   /*
258:      Define mesh intervals ratios for uniform grid.

260:      Note: FD formulae below are normalized by multiplying through by
261:      local volume element (i.e. hx*hy) to obtain coefficients O(1) in two dimensions.


264:   */
265:   dhx   = (PetscReal)(info->mx-1);  dhy = (PetscReal)(info->my-1);
266:   hx    = 1.0/dhx;                   hy = 1.0/dhy;
267:   hxdhy = hx*dhy;                 hydhx = hy*dhx;

269:   xints = info->xs; xinte = info->xs+info->xm; yints = info->ys; yinte = info->ys+info->ym;

271:   /* Test whether we are on the bottom edge of the global array */
272:   if (yints == 0) {
273:     j     = 0;
274:     yints = yints + 1;
275:     /* bottom edge */
276:     for (i=info->xs; i<info->xs+info->xm; i++) {
277:       f[j][i].u     = x[j][i].u;
278:       f[j][i].v     = x[j][i].v;
279:       f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy;
280:       f[j][i].temp  = x[j][i].temp-x[j+1][i].temp;
281:     }
282:   }

284:   /* Test whether we are on the top edge of the global array */
285:   if (yinte == info->my) {
286:     j     = info->my - 1;
287:     yinte = yinte - 1;
288:     /* top edge */
289:     for (i=info->xs; i<info->xs+info->xm; i++) {
290:       f[j][i].u     = x[j][i].u - lid;
291:       f[j][i].v     = x[j][i].v;
292:       f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy;
293:       f[j][i].temp  = x[j][i].temp-x[j-1][i].temp;
294:     }
295:   }

297:   /* Test whether we are on the left edge of the global array */
298:   if (xints == 0) {
299:     i     = 0;
300:     xints = xints + 1;
301:     /* left edge */
302:     for (j=info->ys; j<info->ys+info->ym; j++) {
303:       f[j][i].u     = x[j][i].u;
304:       f[j][i].v     = x[j][i].v;
305:       f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx;
306:       f[j][i].temp  = x[j][i].temp;
307:     }
308:   }

310:   /* Test whether we are on the right edge of the global array */
311:   if (xinte == info->mx) {
312:     i     = info->mx - 1;
313:     xinte = xinte - 1;
314:     /* right edge */
315:     for (j=info->ys; j<info->ys+info->ym; j++) {
316:       f[j][i].u     = x[j][i].u;
317:       f[j][i].v     = x[j][i].v;
318:       f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx;
319:       f[j][i].temp  = x[j][i].temp - (PetscReal)(grashof>0);
320:     }
321:   }

323:   /* Compute over the interior points */
324:   for (j=yints; j<yinte; j++) {
325:     for (i=xints; i<xinte; i++) {

327:       /*
328:         convective coefficients for upwinding
329:       */
330:       vx  = x[j][i].u; avx = PetscAbsScalar(vx);
331:       vxp = .5*(vx+avx); vxm = .5*(vx-avx);
332:       vy  = x[j][i].v; avy = PetscAbsScalar(vy);
333:       vyp = .5*(vy+avy); vym = .5*(vy-avy);

335:       /* U velocity */
336:       u         = x[j][i].u;
337:       udot      = user->parabolic ? xdot[j][i].u : 0.;
338:       uxx       = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx;
339:       uyy       = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy;
340:       f[j][i].u = udot + uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx;

342:       /* V velocity */
343:       u         = x[j][i].v;
344:       udot      = user->parabolic ? xdot[j][i].v : 0.;
345:       uxx       = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx;
346:       uyy       = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy;
347:       f[j][i].v = udot + uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy;

349:       /* Omega */
350:       u             = x[j][i].omega;
351:       uxx           = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx;
352:       uyy           = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy;
353:       f[j][i].omega = (xdot[j][i].omega + uxx + uyy
354:                        + (vxp*(u - x[j][i-1].omega)
355:                           + vxm*(x[j][i+1].omega - u)) * hy
356:                        + (vyp*(u - x[j-1][i].omega)
357:                           + vym*(x[j+1][i].omega - u)) * hx
358:                        - .5 * grashof * (x[j][i+1].temp - x[j][i-1].temp) * hy);

360:       /* Temperature */
361:       u            = x[j][i].temp;
362:       uxx          = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx;
363:       uyy          = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy;
364:       f[j][i].temp =  (xdot[j][i].temp + uxx + uyy
365:                        + prandtl * ((vxp*(u - x[j][i-1].temp)
366:                                      + vxm*(x[j][i+1].temp - u)) * hy
367:                                     + (vyp*(u - x[j-1][i].temp)
368:                                        + vym*(x[j+1][i].temp - u)) * hx));
369:     }
370:   }

372:   /*
373:      Flop count (multiply-adds are counted as 2 operations)
374:   */
375:   PetscLogFlops(84.0*info->ym*info->xm);
376:   return(0);
377: }