1 2 static char help[] = "Solves biharmonic equation in 1d.\n"; 3 4 /* 5 Solves the equation biharmonic equation in split form 6 7 w = -kappa \Delta u 8 u_t = \Delta w 9 -1 <= u <= 1 10 Periodic boundary conditions 11 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 15 16 w = -kappa \Delta u + u^3 - u 17 u_t = \Delta w 18 -1 <= u <= 1 19 Periodic boundary conditions 20 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 24 25 */ 26 #include <petscdm.h> 27 #include <petscdmda.h> 28 #include <petscts.h> 29 #include <petscdraw.h> 30 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; 36 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; 50 51 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 52 Initialize program 53 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 54 PetscCall(PetscInitialize(&argc,&argv,(char*)0,help)); 55 ctx.kappa = 1.0; 56 PetscCall(PetscOptionsGetReal(NULL,NULL,"-kappa",&ctx.kappa,NULL)); 57 ctx.cahnhillard = PETSC_FALSE; 58 PetscCall(PetscOptionsGetBool(NULL,NULL,"-cahn-hillard",&ctx.cahnhillard,NULL)); 59 PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),2,vbounds)); 60 PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),600,600)); 61 ctx.energy = 1; 62 PetscCall(PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL)); 63 ctx.tol = 1.0e-8; 64 PetscCall(PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL)); 65 ctx.theta = .001; 66 ctx.theta_c = 1.0; 67 PetscCall(PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL)); 68 PetscCall(PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL)); 69 70 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 71 Create distributed array (DMDA) to manage parallel grid and vectors 72 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 73 PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10,2,2,NULL,&da)); 74 PetscCall(DMSetFromOptions(da)); 75 PetscCall(DMSetUp(da)); 76 PetscCall(DMDASetFieldName(da,0,"Biharmonic heat equation: w = -kappa*u_xx")); 77 PetscCall(DMDASetFieldName(da,1,"Biharmonic heat equation: u")); 78 PetscCall(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); 80 81 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 82 Extract global vectors from DMDA; then duplicate for remaining 83 vectors that are the same types 84 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 85 PetscCall(DMCreateGlobalVector(da,&x)); 86 PetscCall(VecDuplicate(x,&r)); 87 88 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 89 Create timestepping solver context 90 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 91 PetscCall(TSCreate(PETSC_COMM_WORLD,&ts)); 92 PetscCall(TSSetDM(ts,da)); 93 PetscCall(TSSetProblemType(ts,TS_NONLINEAR)); 94 PetscCall(TSSetIFunction(ts,NULL,FormFunction,&ctx)); 95 PetscCall(TSSetMaxTime(ts,.02)); 96 PetscCall(TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER)); 97 98 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 99 Create matrix data structure; set Jacobian evaluation routine 100 101 < 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 108 109 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 110 PetscCall(TSGetSNES(ts,&snes)); 111 PetscCall(SNESSetType(snes,SNESVINEWTONRSLS)); 112 PetscCall(DMCreateColoring(da,IS_COLORING_GLOBAL,&iscoloring)); 113 PetscCall(DMSetMatType(da,MATAIJ)); 114 PetscCall(DMCreateMatrix(da,&J)); 115 PetscCall(MatFDColoringCreate(J,iscoloring,&matfdcoloring)); 116 PetscCall(MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))SNESTSFormFunction,ts)); 117 PetscCall(MatFDColoringSetFromOptions(matfdcoloring)); 118 PetscCall(MatFDColoringSetUp(J,iscoloring,matfdcoloring)); 119 PetscCall(ISColoringDestroy(&iscoloring)); 120 PetscCall(SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,matfdcoloring)); 121 122 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 123 Customize nonlinear solver 124 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 125 PetscCall(TSSetType(ts,TSBEULER)); 126 127 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 128 Set initial conditions 129 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 130 PetscCall(FormInitialSolution(da,x,ctx.kappa)); 131 PetscCall(TSSetTimeStep(ts,dt)); 132 PetscCall(TSSetSolution(ts,x)); 133 134 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 135 Set runtime options 136 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 137 PetscCall(TSSetFromOptions(ts)); 138 139 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 140 Solve nonlinear system 141 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 142 PetscCall(TSSolve(ts,x)); 143 PetscCall(TSGetStepNumber(ts,&steps)); 144 145 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 146 Free work space. All PETSc objects should be destroyed when they 147 are no longer needed. 148 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 149 PetscCall(MatDestroy(&J)); 150 PetscCall(MatFDColoringDestroy(&matfdcoloring)); 151 PetscCall(VecDestroy(&x)); 152 PetscCall(VecDestroy(&r)); 153 PetscCall(TSDestroy(&ts)); 154 PetscCall(DMDestroy(&da)); 155 156 PetscCall(PetscFinalize()); 157 return 0; 158 } 159 160 typedef struct {PetscScalar w,u;} Field; 161 /* ------------------------------------------------------------------- */ 162 /* 163 FormFunction - Evaluates nonlinear function, F(x). 164 165 Input Parameters: 166 . ts - the TS context 167 . X - input vector 168 . ptr - optional user-defined context, as set by SNESSetFunction() 169 170 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; 182 183 PetscFunctionBegin; 184 PetscCall(TSGetDM(ts,&da)); 185 PetscCall(DMGetLocalVector(da,&localX)); 186 PetscCall(DMGetLocalVector(da,&localXdot)); 187 PetscCall(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)); 188 189 hx = 1.0/(PetscReal)Mx; sx = 1.0/(hx*hx); 190 191 /* 192 Scatter ghost points to local vector,using the 2-step process 193 DMGlobalToLocalBegin(),DMGlobalToLocalEnd(). 194 By placing code between these two statements, computations can be 195 done while messages are in transition. 196 */ 197 PetscCall(DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX)); 198 PetscCall(DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX)); 199 PetscCall(DMGlobalToLocalBegin(da,Xdot,INSERT_VALUES,localXdot)); 200 PetscCall(DMGlobalToLocalEnd(da,Xdot,INSERT_VALUES,localXdot)); 201 202 /* 203 Get pointers to vector data 204 */ 205 PetscCall(DMDAVecGetArrayRead(da,localX,&x)); 206 PetscCall(DMDAVecGetArrayRead(da,localXdot,&xdot)); 207 PetscCall(DMDAVecGetArray(da,F,&f)); 208 209 /* 210 Get local grid boundaries 211 */ 212 PetscCall(DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL)); 213 214 /* 215 Compute function over the locally owned part of the grid 216 */ 217 for (i=xs; i<xs+xm; i++) { 218 f[i].w = x[i].w + ctx->kappa*(x[i-1].u + x[i+1].u - 2.0*x[i].u)*sx; 219 if (ctx->cahnhillard) { 220 switch (ctx->energy) { 221 case 1: /* double well */ 222 f[i].w += -x[i].u*x[i].u*x[i].u + x[i].u; 223 break; 224 case 2: /* double obstacle */ 225 f[i].w += x[i].u; 226 break; 227 case 3: /* logarithmic */ 228 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; 229 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; 230 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; 231 break; 232 case 4: 233 break; 234 } 235 } 236 f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx; 237 if (ctx->energy==4) { 238 f[i].u = xdot[i].u; 239 /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */ 240 r = (1.0 - x[i+1].u*x[i+1].u)*(x[i+2].w-x[i].w)*.5/hx; 241 l = (1.0 - x[i-1].u*x[i-1].u)*(x[i].w-x[i-2].w)*.5/hx; 242 f[i].u -= (r - l)*.5/hx; 243 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; 244 } 245 } 246 247 /* 248 Restore vectors 249 */ 250 PetscCall(DMDAVecRestoreArrayRead(da,localXdot,&xdot)); 251 PetscCall(DMDAVecRestoreArrayRead(da,localX,&x)); 252 PetscCall(DMDAVecRestoreArray(da,F,&f)); 253 PetscCall(DMRestoreLocalVector(da,&localX)); 254 PetscCall(DMRestoreLocalVector(da,&localXdot)); 255 PetscFunctionReturn(0); 256 } 257 258 /* ------------------------------------------------------------------- */ 259 PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa) 260 { 261 PetscInt i,xs,xm,Mx,xgs,xgm; 262 Field *x; 263 PetscReal hx,xx,r,sx; 264 Vec Xg; 265 266 PetscFunctionBegin; 267 PetscCall(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)); 268 269 hx = 1.0/(PetscReal)Mx; 270 sx = 1.0/(hx*hx); 271 272 /* 273 Get pointers to vector data 274 */ 275 PetscCall(DMCreateLocalVector(da,&Xg)); 276 PetscCall(DMDAVecGetArray(da,Xg,&x)); 277 278 /* 279 Get local grid boundaries 280 */ 281 PetscCall(DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL)); 282 PetscCall(DMDAGetGhostCorners(da,&xgs,NULL,NULL,&xgm,NULL,NULL)); 283 284 /* 285 Compute u function over the locally owned part of the grid including ghost points 286 */ 287 for (i=xgs; i<xgs+xgm; i++) { 288 xx = i*hx; 289 r = PetscSqrtReal((xx-.5)*(xx-.5)); 290 if (r < .125) x[i].u = 1.0; 291 else x[i].u = -.50; 292 /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */ 293 x[i].w = 0; 294 } 295 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; 296 297 /* 298 Restore vectors 299 */ 300 PetscCall(DMDAVecRestoreArray(da,Xg,&x)); 301 302 /* Grab only the global part of the vector */ 303 PetscCall(VecSet(X,0)); 304 PetscCall(DMLocalToGlobalBegin(da,Xg,ADD_VALUES,X)); 305 PetscCall(DMLocalToGlobalEnd(da,Xg,ADD_VALUES,X)); 306 PetscCall(VecDestroy(&Xg)); 307 PetscFunctionReturn(0); 308 } 309 310 /*TEST 311 312 build: 313 requires: !complex !single 314 315 test: 316 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 50 317 requires: x 318 319 TEST*/ 320