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 /*T 46 Concepts: TS^solving a system of nonlinear equations (parallel multicomponent example); 47 Concepts: DMDA^using distributed arrays; 48 Concepts: TS^multicomponent 49 Concepts: TS^differential-algebraic equation 50 Processors: n 51 T*/ 52 /* ------------------------------------------------------------------------ 53 54 We thank David E. Keyes for contributing the driven cavity discretization 55 within this example code. 56 57 This problem is modeled by the partial differential equation system 58 59 - Lap(U) - Grad_y(Omega) = 0 60 - Lap(V) + Grad_x(Omega) = 0 61 Omega_t - Lap(Omega) + Div([U*Omega,V*Omega]) - GR*Grad_x(T) = 0 62 T_t - Lap(T) + PR*Div([U*T,V*T]) = 0 63 64 in the unit square, which is uniformly discretized in each of x and 65 y in this simple encoding. 66 67 No-slip, rigid-wall Dirichlet conditions are used for [U,V]. 68 Dirichlet conditions are used for Omega, based on the definition of 69 vorticity: Omega = - Grad_y(U) + Grad_x(V), where along each 70 constant coordinate boundary, the tangential derivative is zero. 71 Dirichlet conditions are used for T on the left and right walls, 72 and insulation homogeneous Neumann conditions are used for T on 73 the top and bottom walls. 74 75 A finite difference approximation with the usual 5-point stencil 76 is used to discretize the boundary value problem to obtain a 77 nonlinear system of equations. Upwinding is used for the divergence 78 (convective) terms and central for the gradient (source) terms. 79 80 The Jacobian can be either 81 * formed via finite differencing using coloring (the default), or 82 * applied matrix-free via the option -snes_mf 83 (for larger grid problems this variant may not converge 84 without a preconditioner due to ill-conditioning). 85 86 ------------------------------------------------------------------------- */ 87 88 /* 89 Include "petscdmda.h" so that we can use distributed arrays (DMDAs). 90 Include "petscts.h" so that we can use TS solvers. Note that this 91 file automatically includes: 92 petscsys.h - base PETSc routines petscvec.h - vectors 93 petscmat.h - matrices 94 petscis.h - index sets petscksp.h - Krylov subspace methods 95 petscviewer.h - viewers petscpc.h - preconditioners 96 petscksp.h - linear solvers petscsnes.h - nonlinear solvers 97 */ 98 #include <petscts.h> 99 #include <petscdm.h> 100 #include <petscdmda.h> 101 102 /* 103 User-defined routines and data structures 104 */ 105 typedef struct { 106 PetscScalar u,v,omega,temp; 107 } Field; 108 109 PetscErrorCode FormIFunctionLocal(DMDALocalInfo*,PetscReal,Field**,Field**,Field**,void*); 110 111 typedef struct { 112 PetscReal lidvelocity,prandtl,grashof; /* physical parameters */ 113 PetscBool parabolic; /* allow a transient term corresponding roughly to artificial compressibility */ 114 PetscReal cfl_initial; /* CFL for first time step */ 115 } AppCtx; 116 117 PetscErrorCode FormInitialSolution(TS,Vec,AppCtx*); 118 119 int main(int argc,char **argv) 120 { 121 AppCtx user; /* user-defined work context */ 122 PetscInt mx,my,steps; 123 PetscErrorCode ierr; 124 TS ts; 125 DM da; 126 Vec X; 127 PetscReal ftime; 128 TSConvergedReason reason; 129 130 ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr; 131 ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr); 132 ierr = DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,4,1,0,0,&da);CHKERRQ(ierr); 133 ierr = DMSetFromOptions(da);CHKERRQ(ierr); 134 ierr = DMSetUp(da);CHKERRQ(ierr); 135 ierr = TSSetDM(ts,(DM)da);CHKERRQ(ierr); 136 137 ierr = DMDAGetInfo(da,0,&mx,&my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE, 138 PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);CHKERRQ(ierr); 139 /* 140 Problem parameters (velocity of lid, prandtl, and grashof numbers) 141 */ 142 user.lidvelocity = 1.0/(mx*my); 143 user.prandtl = 1.0; 144 user.grashof = 1.0; 145 user.parabolic = PETSC_FALSE; 146 user.cfl_initial = 50.; 147 148 ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Driven cavity/natural convection options","");CHKERRQ(ierr); 149 ierr = PetscOptionsReal("-lidvelocity","Lid velocity, related to Reynolds number","",user.lidvelocity,&user.lidvelocity,NULL);CHKERRQ(ierr); 150 ierr = PetscOptionsReal("-prandtl","Ratio of viscous to thermal diffusivity","",user.prandtl,&user.prandtl,NULL);CHKERRQ(ierr); 151 ierr = PetscOptionsReal("-grashof","Ratio of bouyant to viscous forces","",user.grashof,&user.grashof,NULL);CHKERRQ(ierr); 152 ierr = PetscOptionsBool("-parabolic","Relax incompressibility to make the system parabolic instead of differential-algebraic","",user.parabolic,&user.parabolic,NULL);CHKERRQ(ierr); 153 ierr = PetscOptionsReal("-cfl_initial","Advective CFL for the first time step","",user.cfl_initial,&user.cfl_initial,NULL);CHKERRQ(ierr); 154 ierr = PetscOptionsEnd();CHKERRQ(ierr); 155 156 ierr = DMDASetFieldName(da,0,"x-velocity");CHKERRQ(ierr); 157 ierr = DMDASetFieldName(da,1,"y-velocity");CHKERRQ(ierr); 158 ierr = DMDASetFieldName(da,2,"Omega");CHKERRQ(ierr); 159 ierr = DMDASetFieldName(da,3,"temperature");CHKERRQ(ierr); 160 161 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 162 Create user context, set problem data, create vector data structures. 163 Also, compute the initial guess. 164 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 165 166 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 167 Create time integration context 168 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 169 ierr = DMSetApplicationContext(da,&user);CHKERRQ(ierr); 170 ierr = DMDATSSetIFunctionLocal(da,INSERT_VALUES,(DMDATSIFunctionLocal)FormIFunctionLocal,&user);CHKERRQ(ierr); 171 ierr = TSSetMaxSteps(ts,10000);CHKERRQ(ierr); 172 ierr = TSSetMaxTime(ts,1e12);CHKERRQ(ierr); 173 ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); 174 ierr = TSSetTimeStep(ts,user.cfl_initial/(user.lidvelocity*mx));CHKERRQ(ierr); 175 ierr = TSSetFromOptions(ts);CHKERRQ(ierr); 176 177 ierr = PetscPrintf(PETSC_COMM_WORLD,"%Dx%D grid, lid velocity = %g, prandtl # = %g, grashof # = %g\n",mx,my,(double)user.lidvelocity,(double)user.prandtl,(double)user.grashof);CHKERRQ(ierr); 178 179 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 180 Solve the nonlinear system 181 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 182 183 ierr = DMCreateGlobalVector(da,&X);CHKERRQ(ierr); 184 ierr = FormInitialSolution(ts,X,&user);CHKERRQ(ierr); 185 186 ierr = TSSolve(ts,X);CHKERRQ(ierr); 187 ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr); 188 ierr = TSGetStepNumber(ts,&steps);CHKERRQ(ierr); 189 ierr = TSGetConvergedReason(ts,&reason);CHKERRQ(ierr); 190 191 ierr = PetscPrintf(PETSC_COMM_WORLD,"%s at time %g after %D steps\n",TSConvergedReasons[reason],(double)ftime,steps);CHKERRQ(ierr); 192 193 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 194 Free work space. All PETSc objects should be destroyed when they 195 are no longer needed. 196 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 197 ierr = VecDestroy(&X);CHKERRQ(ierr); 198 ierr = DMDestroy(&da);CHKERRQ(ierr); 199 ierr = TSDestroy(&ts);CHKERRQ(ierr); 200 201 ierr = PetscFinalize(); 202 return ierr; 203 } 204 205 /* ------------------------------------------------------------------- */ 206 207 /* 208 FormInitialSolution - Forms initial approximation. 209 210 Input Parameters: 211 user - user-defined application context 212 X - vector 213 214 Output Parameter: 215 X - vector 216 */ 217 PetscErrorCode FormInitialSolution(TS ts,Vec X,AppCtx *user) 218 { 219 DM da; 220 PetscInt i,j,mx,xs,ys,xm,ym; 221 PetscErrorCode ierr; 222 PetscReal grashof,dx; 223 Field **x; 224 225 grashof = user->grashof; 226 ierr = TSGetDM(ts,&da);CHKERRQ(ierr); 227 ierr = DMDAGetInfo(da,0,&mx,0,0,0,0,0,0,0,0,0,0,0);CHKERRQ(ierr); 228 dx = 1.0/(mx-1); 229 230 /* 231 Get local grid boundaries (for 2-dimensional DMDA): 232 xs, ys - starting grid indices (no ghost points) 233 xm, ym - widths of local grid (no ghost points) 234 */ 235 ierr = DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);CHKERRQ(ierr); 236 237 /* 238 Get a pointer to vector data. 239 - For default PETSc vectors, VecGetArray() returns a pointer to 240 the data array. Otherwise, the routine is implementation dependent. 241 - You MUST call VecRestoreArray() when you no longer need access to 242 the array. 243 */ 244 ierr = DMDAVecGetArray(da,X,&x);CHKERRQ(ierr); 245 246 /* 247 Compute initial guess over the locally owned part of the grid 248 Initial condition is motionless fluid and equilibrium temperature 249 */ 250 for (j=ys; j<ys+ym; j++) { 251 for (i=xs; i<xs+xm; i++) { 252 x[j][i].u = 0.0; 253 x[j][i].v = 0.0; 254 x[j][i].omega = 0.0; 255 x[j][i].temp = (grashof>0)*i*dx; 256 } 257 } 258 259 /* 260 Restore vector 261 */ 262 ierr = DMDAVecRestoreArray(da,X,&x);CHKERRQ(ierr); 263 return 0; 264 } 265 266 PetscErrorCode FormIFunctionLocal(DMDALocalInfo *info,PetscReal ptime,Field **x,Field **xdot,Field **f,void *ptr) 267 { 268 AppCtx *user = (AppCtx*)ptr; 269 PetscErrorCode ierr; 270 PetscInt xints,xinte,yints,yinte,i,j; 271 PetscReal hx,hy,dhx,dhy,hxdhy,hydhx; 272 PetscReal grashof,prandtl,lid; 273 PetscScalar u,udot,uxx,uyy,vx,vy,avx,avy,vxp,vxm,vyp,vym; 274 275 PetscFunctionBeginUser; 276 grashof = user->grashof; 277 prandtl = user->prandtl; 278 lid = user->lidvelocity; 279 280 /* 281 Define mesh intervals ratios for uniform grid. 282 283 Note: FD formulae below are normalized by multiplying through by 284 local volume element (i.e. hx*hy) to obtain coefficients O(1) in two dimensions. 285 286 */ 287 dhx = (PetscReal)(info->mx-1); dhy = (PetscReal)(info->my-1); 288 hx = 1.0/dhx; hy = 1.0/dhy; 289 hxdhy = hx*dhy; hydhx = hy*dhx; 290 291 xints = info->xs; xinte = info->xs+info->xm; yints = info->ys; yinte = info->ys+info->ym; 292 293 /* Test whether we are on the bottom edge of the global array */ 294 if (yints == 0) { 295 j = 0; 296 yints = yints + 1; 297 /* bottom edge */ 298 for (i=info->xs; i<info->xs+info->xm; i++) { 299 f[j][i].u = x[j][i].u; 300 f[j][i].v = x[j][i].v; 301 f[j][i].omega = x[j][i].omega + (x[j+1][i].u - x[j][i].u)*dhy; 302 f[j][i].temp = x[j][i].temp-x[j+1][i].temp; 303 } 304 } 305 306 /* Test whether we are on the top edge of the global array */ 307 if (yinte == info->my) { 308 j = info->my - 1; 309 yinte = yinte - 1; 310 /* top edge */ 311 for (i=info->xs; i<info->xs+info->xm; i++) { 312 f[j][i].u = x[j][i].u - lid; 313 f[j][i].v = x[j][i].v; 314 f[j][i].omega = x[j][i].omega + (x[j][i].u - x[j-1][i].u)*dhy; 315 f[j][i].temp = x[j][i].temp-x[j-1][i].temp; 316 } 317 } 318 319 /* Test whether we are on the left edge of the global array */ 320 if (xints == 0) { 321 i = 0; 322 xints = xints + 1; 323 /* left edge */ 324 for (j=info->ys; j<info->ys+info->ym; j++) { 325 f[j][i].u = x[j][i].u; 326 f[j][i].v = x[j][i].v; 327 f[j][i].omega = x[j][i].omega - (x[j][i+1].v - x[j][i].v)*dhx; 328 f[j][i].temp = x[j][i].temp; 329 } 330 } 331 332 /* Test whether we are on the right edge of the global array */ 333 if (xinte == info->mx) { 334 i = info->mx - 1; 335 xinte = xinte - 1; 336 /* right edge */ 337 for (j=info->ys; j<info->ys+info->ym; j++) { 338 f[j][i].u = x[j][i].u; 339 f[j][i].v = x[j][i].v; 340 f[j][i].omega = x[j][i].omega - (x[j][i].v - x[j][i-1].v)*dhx; 341 f[j][i].temp = x[j][i].temp - (PetscReal)(grashof>0); 342 } 343 } 344 345 /* Compute over the interior points */ 346 for (j=yints; j<yinte; j++) { 347 for (i=xints; i<xinte; i++) { 348 349 /* 350 convective coefficients for upwinding 351 */ 352 vx = x[j][i].u; avx = PetscAbsScalar(vx); 353 vxp = .5*(vx+avx); vxm = .5*(vx-avx); 354 vy = x[j][i].v; avy = PetscAbsScalar(vy); 355 vyp = .5*(vy+avy); vym = .5*(vy-avy); 356 357 /* U velocity */ 358 u = x[j][i].u; 359 udot = user->parabolic ? xdot[j][i].u : 0.; 360 uxx = (2.0*u - x[j][i-1].u - x[j][i+1].u)*hydhx; 361 uyy = (2.0*u - x[j-1][i].u - x[j+1][i].u)*hxdhy; 362 f[j][i].u = udot + uxx + uyy - .5*(x[j+1][i].omega-x[j-1][i].omega)*hx; 363 364 /* V velocity */ 365 u = x[j][i].v; 366 udot = user->parabolic ? xdot[j][i].v : 0.; 367 uxx = (2.0*u - x[j][i-1].v - x[j][i+1].v)*hydhx; 368 uyy = (2.0*u - x[j-1][i].v - x[j+1][i].v)*hxdhy; 369 f[j][i].v = udot + uxx + uyy + .5*(x[j][i+1].omega-x[j][i-1].omega)*hy; 370 371 /* Omega */ 372 u = x[j][i].omega; 373 uxx = (2.0*u - x[j][i-1].omega - x[j][i+1].omega)*hydhx; 374 uyy = (2.0*u - x[j-1][i].omega - x[j+1][i].omega)*hxdhy; 375 f[j][i].omega = (xdot[j][i].omega + uxx + uyy 376 + (vxp*(u - x[j][i-1].omega) 377 + vxm*(x[j][i+1].omega - u)) * hy 378 + (vyp*(u - x[j-1][i].omega) 379 + vym*(x[j+1][i].omega - u)) * hx 380 - .5 * grashof * (x[j][i+1].temp - x[j][i-1].temp) * hy); 381 382 /* Temperature */ 383 u = x[j][i].temp; 384 uxx = (2.0*u - x[j][i-1].temp - x[j][i+1].temp)*hydhx; 385 uyy = (2.0*u - x[j-1][i].temp - x[j+1][i].temp)*hxdhy; 386 f[j][i].temp = (xdot[j][i].temp + uxx + uyy 387 + prandtl * ((vxp*(u - x[j][i-1].temp) 388 + vxm*(x[j][i+1].temp - u)) * hy 389 + (vyp*(u - x[j-1][i].temp) 390 + vym*(x[j+1][i].temp - u)) * hx)); 391 } 392 } 393 394 /* 395 Flop count (multiply-adds are counted as 2 operations) 396 */ 397 ierr = PetscLogFlops(84.0*info->ym*info->xm);CHKERRQ(ierr); 398 PetscFunctionReturn(0); 399 } 400 401 /*TEST 402 403 test: 404 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' 405 requires: !complex !single 406 407 test: 408 suffix: 2 409 nsize: 4 410 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' 411 requires: !complex !single 412 413 test: 414 suffix: 3 415 nsize: 4 416 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 4 417 requires: !complex !single 418 419 test: 420 suffix: 4 421 nsize: 2 422 args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 423 requires: !complex !single 424 425 test: 426 suffix: asm 427 nsize: 4 428 args: -da_refine 1 -lidvelocity 100 -grashof 1e3 -ts_max_steps 10 -ts_rtol 1e-3 -ts_atol 1e-3 429 requires: !complex !single 430 431 TEST*/ 432