1 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/tutorials/ex19.c for the steady-state version. 12 13 There used to be a SNES example (src/snes/tutorials/ex27.c) that 14 implemented this algorithm without using TS and was used for the numerical 15 results in the paper 16 17 Todd S. Coffey and C. T. Kelley and David E. Keyes, Pseudotransient 18 Continuation and Differential-Algebraic Equations, 2003. 19 20 That example was removed because it used obsolete interfaces, but the 21 algorithms from the paper can be reproduced using this example. 22 23 Note: The paper describes the algorithm as being linearly implicit but the 24 numerical results were created using nonlinearly implicit Euler. The 25 algorithm as described (linearly implicit) is more efficient and is the 26 default when using TSPSEUDO. If you want to reproduce the numerical 27 results from the paper, you'll have to change the SNES to converge the 28 nonlinear solve (e.g., -snes_type newtonls). The DAE versus ODE variants 29 are controlled using the -parabolic option. 30 31 Comment preserved from snes/tutorials/ex27.c, since removed: 32 33 [H]owever Figure 3.1 was generated with a slightly different algorithm 34 (see targets runex27 and runex27_p) than described in the paper. In 35 particular, the described algorithm is linearly implicit, advancing to 36 the next step after one Newton step, so that the steady state residual 37 is always used, but the figure was generated by converging each step to 38 a relative tolerance of 1.e-3. On the example problem, setting 39 -snes_type ksponly has only minor impact on number of steps, but 40 significantly reduces the required number of linear solves. 41 42 See also https://lists.mcs.anl.gov/pipermail/petsc-dev/2010-March/002362.html 43 */ 44 45 /* ------------------------------------------------------------------------ 46 47 We thank David E. Keyes for contributing the driven cavity discretization 48 within this example code. 49 50 This problem is modeled by the partial differential equation system 51 52 - Lap(U) - Grad_y(Omega) = 0 53 - Lap(V) + Grad_x(Omega) = 0 54 Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0 55 T_t - Lap(T) + PR*Div([U*T,V*T]) = 0 56 57 in the unit square, which is uniformly discretized in each of x and 58 y in this simple encoding. 59 60 No-slip, rigid-wall Dirichlet conditions are used for [U,V]. 61 Dirichlet conditions are used for Omega, based on the definition of 62 vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each 63 constant coordinate boundary, the tangential derivative is zero. 64 Dirichlet conditions are used for T on the left and right walls, 65 and insulation homogeneous Neumann conditions are used for T on 66 the top and bottom walls. 67 68 A finite difference approximation with the usual 5-point stencil 69 is used to discretize the boundary value problem to obtain a 70 nonlinear system of equations. Upwinding is used for the divergence 71 (convective) terms and central for the gradient (source) terms. 72 73 The Jacobian can be either 74 * formed via finite differencing using coloring (the default), or 75 * applied matrix-free via the option -snes_mf 76 (for larger grid problems this variant may not converge 77 without a preconditioner due to ill-conditioning). 78 79 ------------------------------------------------------------------------- */ 80 81 /* 82 Include "petscdmda.h" so that we can use distributed arrays (DMDAs). 83 Include "petscts.h" so that we can use TS solvers. Note that this 84 file automatically includes: 85 petscsys.h - base PETSc routines petscvec.h - vectors 86 petscmat.h - matrices 87 petscis.h - index sets petscksp.h - Krylov subspace methods 88 petscviewer.h - viewers petscpc.h - preconditioners 89 petscksp.h - linear solvers petscsnes.h - nonlinear solvers 90 */ 91 #include <petscts.h> 92 #include <petscdm.h> 93 #include <petscdmda.h> 94 95 /* 96 User-defined routines and data structures 97 */ 98 typedef struct { 99 PetscScalar u,v,omega,temp; 100 } Field; 101 102 PetscErrorCode FormIFunctionLocal(DMDALocalInfo*,PetscReal,Field**,Field**,Field**,void*); 103 104 typedef struct { 105 PetscReal lidvelocity,prandtl,grashof; /* physical parameters */ 106 PetscBool parabolic; /* allow a transient term corresponding roughly to artificial compressibility */ 107 PetscReal cfl_initial; /* CFL for first time step */ 108 } AppCtx; 109 110 PetscErrorCode FormInitialSolution(TS,Vec,AppCtx*); 111 112 int main(int argc,char **argv) 113 { 114 AppCtx user; /* user-defined work context */ 115 PetscInt mx,my,steps; 116 TS ts; 117 DM da; 118 Vec X; 119 PetscReal ftime; 120 TSConvergedReason reason; 121 122 PetscFunctionBeginUser; 123 PetscCall(PetscInitialize(&argc,&argv,(char*)0,help)); 124 PetscCall(TSCreate(PETSC_COMM_WORLD,&ts)); 125 PetscCall(DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da)); 126 PetscCall(DMSetFromOptions(da)); 127 PetscCall(DMSetUp(da)); 128 PetscCall(TSSetDM(ts,(DM)da)); 129 130 PetscCall(DMDAGetInfo(da,0,&mx,&my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE)); 131 /* 132 Problem parameters (velocity of lid, prandtl, and grashof numbers) 133 */ 134 user.lidvelocity = 1.0/(mx*my); 135 user.prandtl = 1.0; 136 user.grashof = 1.0; 137 user.parabolic = PETSC_FALSE; 138 user.cfl_initial = 50.; 139 140 PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Driven cavity/natural convection options",""); 141 PetscCall(PetscOptionsReal("-lidvelocity","Lid velocity, related to Reynolds number","",user.lidvelocity,&user.lidvelocity,NULL)); 142 PetscCall(PetscOptionsReal("-prandtl","Ratio of viscous to thermal diffusivity","",user.prandtl,&user.prandtl,NULL)); 143 PetscCall(PetscOptionsReal("-grashof","Ratio of bouyant to viscous forces","",user.grashof,&user.grashof,NULL)); 144 PetscCall(PetscOptionsBool("-parabolic","Relax incompressibility to make the system parabolic instead of differential-algebraic","",user.parabolic,&user.parabolic,NULL)); 145 PetscCall(PetscOptionsReal("-cfl_initial","Advective CFL for the first time step","",user.cfl_initial,&user.cfl_initial,NULL)); 146 PetscOptionsEnd(); 147 148 PetscCall(DMDASetFieldName(da,0,"x-velocity")); 149 PetscCall(DMDASetFieldName(da,1,"y-velocity")); 150 PetscCall(DMDASetFieldName(da,2,"Omega")); 151 PetscCall(DMDASetFieldName(da,3,"temperature")); 152 153 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 154 Create user context, set problem data, create vector data structures. 155 Also, compute the initial guess. 156 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 157 158 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 159 Create time integration context 160 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 161 PetscCall(DMSetApplicationContext(da,&user)); 162 PetscCall(DMDATSSetIFunctionLocal(da,INSERT_VALUES,(DMDATSIFunctionLocal)FormIFunctionLocal,&user)); 163 PetscCall(TSSetMaxSteps(ts,10000)); 164 PetscCall(TSSetMaxTime(ts,1e12)); 165 PetscCall(TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER)); 166 PetscCall(TSSetTimeStep(ts,user.cfl_initial/(user.lidvelocity*mx))); 167 PetscCall(TSSetFromOptions(ts)); 168 169 PetscCall(PetscPrintf(PETSC_COMM_WORLD,"%" PetscInt_FMT "x%" PetscInt_FMT " grid, lid velocity = %g, prandtl # = %g, grashof # = %g\n",mx,my,(double)user.lidvelocity,(double)user.prandtl,(double)user.grashof)); 170 171 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 172 Solve the nonlinear system 173 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 174 175 PetscCall(DMCreateGlobalVector(da,&X)); 176 PetscCall(FormInitialSolution(ts,X,&user)); 177 178 PetscCall(TSSolve(ts,X)); 179 PetscCall(TSGetSolveTime(ts,&ftime)); 180 PetscCall(TSGetStepNumber(ts,&steps)); 181 PetscCall(TSGetConvergedReason(ts,&reason)); 182 183 PetscCall(PetscPrintf(PETSC_COMM_WORLD,"%s at time %g after %" PetscInt_FMT " steps\n",TSConvergedReasons[reason],(double)ftime,steps)); 184 185 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 186 Free work space. All PETSc objects should be destroyed when they 187 are no longer needed. 188 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 189 PetscCall(VecDestroy(&X)); 190 PetscCall(DMDestroy(&da)); 191 PetscCall(TSDestroy(&ts)); 192 193 PetscCall(PetscFinalize()); 194 return 0; 195 } 196 197 /* ------------------------------------------------------------------- */ 198 199 /* 200 FormInitialSolution - Forms initial approximation. 201 202 Input Parameters: 203 user - user-defined application context 204 X - vector 205 206 Output Parameter: 207 X - vector 208 */ 209 PetscErrorCode FormInitialSolution(TS ts,Vec X,AppCtx *user) 210 { 211 DM da; 212 PetscInt i,j,mx,xs,ys,xm,ym; 213 PetscReal grashof,dx; 214 Field **x; 215 216 grashof = user->grashof; 217 PetscCall(TSGetDM(ts,&da)); 218 PetscCall(DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0)); 219 dx = 1.0/(mx-1); 220 221 /* 222 Get local grid boundaries (for 2-dimensional DMDA): 223 xs, ys - starting grid indices (no ghost points) 224 xm, ym - widths of local grid (no ghost points) 225 */ 226 PetscCall(DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL)); 227 228 /* 229 Get a pointer to vector data. 230 - For default PETSc vectors, VecGetArray() returns a pointer to 231 the data array. Otherwise, the routine is implementation dependent. 232 - You MUST call VecRestoreArray() when you no longer need access to 233 the array. 234 */ 235 PetscCall(DMDAVecGetArray(da,X,&x)); 236 237 /* 238 Compute initial guess over the locally owned part of the grid 239 Initial condition is motionless fluid and equilibrium temperature 240 */ 241 for (j=ys; j<ys+ym; j++) { 242 for (i=xs; i<xs+xm; i++) { 243 x[j][i].u = 0.0; 244 x[j][i].v = 0.0; 245 x[j][i].omega = 0.0; 246 x[j][i].temp = (grashof>0)*i*dx; 247 } 248 } 249 250 /* 251 Restore vector 252 */ 253 PetscCall(DMDAVecRestoreArray(da,X,&x)); 254 return 0; 255 } 256 257 PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info,PetscReal ptime,Field **x,Field **xdot,Field **f,void *ptr) 258 { 259 AppCtx *user = (AppCtx*)ptr; 260 PetscInt xints,xinte,yints,yinte,i,j; 261 PetscReal hx,hy,dhx,dhy,hxdhy,hydhx; 262 PetscReal grashof,prandtl,lid; 263 PetscScalar u,udot,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym; 264 265 PetscFunctionBeginUser; 266 grashof = user->grashof; 267 prandtl = user->prandtl; 268 lid = user->lidvelocity; 269 270 /* 271 Define mesh intervals ratios for uniform grid. 272 273 Note: FD formulae below are normalized by multiplying through by 274 local volume element (i.e. hx*hy) to obtain coefficients O(1) in two dimensions. 275 276 */ 277 dhx = (PetscReal)(info->mx-1); dhy = (PetscReal)(info->my-1); 278 hx = 1.0/dhx; hy = 1.0/dhy; 279 hxdhy = hx*dhy; hydhx = hy*dhx; 280 281 xints = info->xs; xinte = info->xs+info->xm; yints = info->ys; yinte = info->ys+info->ym; 282 283 /* Test whether we are on the bottom edge of the global array */ 284 if (yints == 0) { 285 j = 0; 286 yints = yints + 1; 287 /* bottom edge */ 288 for (i=info->xs; i<info->xs+info->xm; i++) { 289 f[j][i].u = x[j][i].u; 290 f[j][i].v = x[j][i].v; 291 f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy; 292 f[j][i].temp = x[j][i].temp-x[j+1][i].temp; 293 } 294 } 295 296 /* Test whether we are on the top edge of the global array */ 297 if (yinte == info->my) { 298 j = info->my - 1; 299 yinte = yinte - 1; 300 /* top edge */ 301 for (i=info->xs; i<info->xs+info->xm; i++) { 302 f[j][i].u = x[j][i].u - lid; 303 f[j][i].v = x[j][i].v; 304 f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy; 305 f[j][i].temp = x[j][i].temp-x[j-1][i].temp; 306 } 307 } 308 309 /* Test whether we are on the left edge of the global array */ 310 if (xints == 0) { 311 i = 0; 312 xints = xints + 1; 313 /* left edge */ 314 for (j=info->ys; j<info->ys+info->ym; j++) { 315 f[j][i].u = x[j][i].u; 316 f[j][i].v = x[j][i].v; 317 f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx; 318 f[j][i].temp = x[j][i].temp; 319 } 320 } 321 322 /* Test whether we are on the right edge of the global array */ 323 if (xinte == info->mx) { 324 i = info->mx - 1; 325 xinte = xinte - 1; 326 /* right edge */ 327 for (j=info->ys; j<info->ys+info->ym; j++) { 328 f[j][i].u = x[j][i].u; 329 f[j][i].v = x[j][i].v; 330 f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx; 331 f[j][i].temp = x[j][i].temp - (PetscReal)(grashof>0); 332 } 333 } 334 335 /* Compute over the interior points */ 336 for (j=yints; j<yinte; j++) { 337 for (i=xints; i<xinte; i++) { 338 339 /* 340 convective coefficients for upwinding 341 */ 342 vx = x[j][i].u; avx = PetscAbsScalar(vx); 343 vxp = .5*(vx+avx); vxm = .5*(vx-avx); 344 vy = x[j][i].v; avy = PetscAbsScalar(vy); 345 vyp = .5*(vy+avy); vym = .5*(vy-avy); 346 347 /* U velocity */ 348 u = x[j][i].u; 349 udot = user->parabolic ? xdot[j][i].u : 0.; 350 uxx = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx; 351 uyy = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy; 352 f[j][i].u = udot + uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx; 353 354 /* V velocity */ 355 u = x[j][i].v; 356 udot = user->parabolic ? xdot[j][i].v : 0.; 357 uxx = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx; 358 uyy = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy; 359 f[j][i].v = udot + uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy; 360 361 /* Omega */ 362 u = x[j][i].omega; 363 uxx = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx; 364 uyy = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy; 365 f[j][i].omega = (xdot[j][i].omega + uxx + uyy 366 + (vxp*(u - x[j][i-1].omega) 367 + vxm*(x[j][i+1].omega - u)) * hy 368 + (vyp*(u - x[j-1][i].omega) 369 + vym*(x[j+1][i].omega - u)) * hx 370 - .5 * grashof * (x[j][i+1].temp - x[j][i-1].temp) * hy); 371 372 /* Temperature */ 373 u = x[j][i].temp; 374 uxx = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx; 375 uyy = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy; 376 f[j][i].temp = (xdot[j][i].temp + uxx + uyy 377 + prandtl * ((vxp*(u - x[j][i-1].temp) 378 + vxm*(x[j][i+1].temp - u)) * hy 379 + (vyp*(u - x[j-1][i].temp) 380 + vym*(x[j+1][i].temp - u)) * hx)); 381 } 382 } 383 384 /* 385 Flop count (multiply-adds are counted as 2 operations) 386 */ 387 PetscCall(PetscLogFlops(84.0*info->ym*info->xm)); 388 PetscFunctionReturn(0); 389 } 390 391 /*TEST 392 393 test: 394 args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03d.vts' 395 requires: !complex !single 396 397 test: 398 suffix: 2 399 nsize: 4 400 args: -da_grid_x 20 -da_grid_y 20 -lidvelocity 100 -grashof 1e3 -ts_max_steps 100 -ts_rtol 1e-3 -ts_atol 1e-3 -ts_type rosw -ts_rosw_type ra3pw -ts_monitor -ts_monitor_solution_vtk 'foo-%03d.vts' 401 requires: !complex !single 402 403 test: 404 suffix: 3 405 nsize: 4 406 args: -da_refine 2 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 -pc_type none -ts_type beuler -ts_monitor -snes_monitor_short -snes_type aspin -da_overlap 4 407 requires: !complex !single 408 409 test: 410 suffix: 4 411 nsize: 2 412 args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 413 requires: !complex !single 414 415 test: 416 suffix: asm 417 nsize: 4 418 args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 419 requires: !complex !single 420 421 TEST*/ 422