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 { 36 PetscBool cahnhillard; 37 PetscReal kappa; 38 PetscInt energy; 39 PetscReal tol; 40 PetscReal theta; 41 PetscReal theta_c; 42 } UserCtx; 43 44 int main(int argc, char **argv) 45 { 46 TS ts; /* nonlinear solver */ 47 Vec x, r; /* solution, residual vectors */ 48 Mat J; /* Jacobian matrix */ 49 PetscInt steps, Mx; 50 DM da; 51 MatFDColoring matfdcoloring; 52 ISColoring iscoloring; 53 PetscReal dt; 54 PetscReal vbounds[] = {-100000, 100000, -1.1, 1.1}; 55 SNES snes; 56 UserCtx ctx; 57 58 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 59 Initialize program 60 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 61 PetscFunctionBeginUser; 62 PetscCall(PetscInitialize(&argc, &argv, (char *)0, help)); 63 ctx.kappa = 1.0; 64 PetscCall(PetscOptionsGetReal(NULL, NULL, "-kappa", &ctx.kappa, NULL)); 65 ctx.cahnhillard = PETSC_FALSE; 66 PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL)); 67 PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds)); 68 PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600)); 69 ctx.energy = 1; 70 PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL)); 71 ctx.tol = 1.0e-8; 72 PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL)); 73 ctx.theta = .001; 74 ctx.theta_c = 1.0; 75 PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL)); 76 PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL)); 77 78 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 79 Create distributed array (DMDA) to manage parallel grid and vectors 80 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 81 PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da)); 82 PetscCall(DMSetFromOptions(da)); 83 PetscCall(DMSetUp(da)); 84 PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx")); 85 PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u")); 86 PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)); 87 dt = 1.0 / (10. * ctx.kappa * Mx * Mx * Mx * Mx); 88 89 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 90 Extract global vectors from DMDA; then duplicate for remaining 91 vectors that are the same types 92 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 93 PetscCall(DMCreateGlobalVector(da, &x)); 94 PetscCall(VecDuplicate(x, &r)); 95 96 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 97 Create timestepping solver context 98 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 99 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); 100 PetscCall(TSSetDM(ts, da)); 101 PetscCall(TSSetProblemType(ts, TS_NONLINEAR)); 102 PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx)); 103 PetscCall(TSSetMaxTime(ts, .02)); 104 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER)); 105 106 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 107 Create matrix data structure; set Jacobian evaluation routine 108 109 < Set Jacobian matrix data structure and default Jacobian evaluation 110 routine. User can override with: 111 -snes_mf : matrix-free Newton-Krylov method with no preconditioning 112 (unless user explicitly sets preconditioner) 113 -snes_mf_operator : form preconditioning matrix as set by the user, 114 but use matrix-free approx for Jacobian-vector 115 products within Newton-Krylov method 116 117 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 118 PetscCall(TSGetSNES(ts, &snes)); 119 PetscCall(SNESSetType(snes, SNESVINEWTONRSLS)); 120 PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring)); 121 PetscCall(DMSetMatType(da, MATAIJ)); 122 PetscCall(DMCreateMatrix(da, &J)); 123 PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring)); 124 PetscCall(MatFDColoringSetFunction(matfdcoloring, (PetscErrorCode(*)(void))SNESTSFormFunction, ts)); 125 PetscCall(MatFDColoringSetFromOptions(matfdcoloring)); 126 PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring)); 127 PetscCall(ISColoringDestroy(&iscoloring)); 128 PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring)); 129 130 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 131 Customize nonlinear solver 132 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 133 PetscCall(TSSetType(ts, TSBEULER)); 134 135 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 136 Set initial conditions 137 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 138 PetscCall(FormInitialSolution(da, x, ctx.kappa)); 139 PetscCall(TSSetTimeStep(ts, dt)); 140 PetscCall(TSSetSolution(ts, x)); 141 142 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 143 Set runtime options 144 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 145 PetscCall(TSSetFromOptions(ts)); 146 147 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 148 Solve nonlinear system 149 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 150 PetscCall(TSSolve(ts, x)); 151 PetscCall(TSGetStepNumber(ts, &steps)); 152 153 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 154 Free work space. All PETSc objects should be destroyed when they 155 are no longer needed. 156 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 157 PetscCall(MatDestroy(&J)); 158 PetscCall(MatFDColoringDestroy(&matfdcoloring)); 159 PetscCall(VecDestroy(&x)); 160 PetscCall(VecDestroy(&r)); 161 PetscCall(TSDestroy(&ts)); 162 PetscCall(DMDestroy(&da)); 163 164 PetscCall(PetscFinalize()); 165 return 0; 166 } 167 168 typedef struct { 169 PetscScalar w, u; 170 } Field; 171 /* ------------------------------------------------------------------- */ 172 /* 173 FormFunction - Evaluates nonlinear function, F(x). 174 175 Input Parameters: 176 . ts - the TS context 177 . X - input vector 178 . ptr - optional user-defined context, as set by SNESSetFunction() 179 180 Output Parameter: 181 . F - function vector 182 */ 183 PetscErrorCode FormFunction(TS ts, PetscReal ftime, Vec X, Vec Xdot, Vec F, void *ptr) 184 { 185 DM da; 186 PetscInt i, Mx, xs, xm; 187 PetscReal hx, sx; 188 PetscScalar r, l; 189 Field *x, *xdot, *f; 190 Vec localX, localXdot; 191 UserCtx *ctx = (UserCtx *)ptr; 192 193 PetscFunctionBegin; 194 PetscCall(TSGetDM(ts, &da)); 195 PetscCall(DMGetLocalVector(da, &localX)); 196 PetscCall(DMGetLocalVector(da, &localXdot)); 197 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)); 198 199 hx = 1.0 / (PetscReal)Mx; 200 sx = 1.0 / (hx * hx); 201 202 /* 203 Scatter ghost points to local vector,using the 2-step process 204 DMGlobalToLocalBegin(),DMGlobalToLocalEnd(). 205 By placing code between these two statements, computations can be 206 done while messages are in transition. 207 */ 208 PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX)); 209 PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX)); 210 PetscCall(DMGlobalToLocalBegin(da, Xdot, INSERT_VALUES, localXdot)); 211 PetscCall(DMGlobalToLocalEnd(da, Xdot, INSERT_VALUES, localXdot)); 212 213 /* 214 Get pointers to vector data 215 */ 216 PetscCall(DMDAVecGetArrayRead(da, localX, &x)); 217 PetscCall(DMDAVecGetArrayRead(da, localXdot, &xdot)); 218 PetscCall(DMDAVecGetArray(da, F, &f)); 219 220 /* 221 Get local grid boundaries 222 */ 223 PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL)); 224 225 /* 226 Compute function over the locally owned part of the grid 227 */ 228 for (i = xs; i < xs + xm; i++) { 229 f[i].w = x[i].w + ctx->kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx; 230 if (ctx->cahnhillard) { 231 switch (ctx->energy) { 232 case 1: /* double well */ 233 f[i].w += -x[i].u * x[i].u * x[i].u + x[i].u; 234 break; 235 case 2: /* double obstacle */ 236 f[i].w += x[i].u; 237 break; 238 case 3: /* logarithmic */ 239 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; 240 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; 241 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; 242 break; 243 case 4: 244 break; 245 } 246 } 247 f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx; 248 if (ctx->energy == 4) { 249 f[i].u = xdot[i].u; 250 /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */ 251 r = (1.0 - x[i + 1].u * x[i + 1].u) * (x[i + 2].w - x[i].w) * .5 / hx; 252 l = (1.0 - x[i - 1].u * x[i - 1].u) * (x[i].w - x[i - 2].w) * .5 / hx; 253 f[i].u -= (r - l) * .5 / hx; 254 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; 255 } 256 } 257 258 /* 259 Restore vectors 260 */ 261 PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot)); 262 PetscCall(DMDAVecRestoreArrayRead(da, localX, &x)); 263 PetscCall(DMDAVecRestoreArray(da, F, &f)); 264 PetscCall(DMRestoreLocalVector(da, &localX)); 265 PetscCall(DMRestoreLocalVector(da, &localXdot)); 266 PetscFunctionReturn(PETSC_SUCCESS); 267 } 268 269 /* ------------------------------------------------------------------- */ 270 PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa) 271 { 272 PetscInt i, xs, xm, Mx, xgs, xgm; 273 Field *x; 274 PetscReal hx, xx, r, sx; 275 Vec Xg; 276 277 PetscFunctionBegin; 278 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)); 279 280 hx = 1.0 / (PetscReal)Mx; 281 sx = 1.0 / (hx * hx); 282 283 /* 284 Get pointers to vector data 285 */ 286 PetscCall(DMCreateLocalVector(da, &Xg)); 287 PetscCall(DMDAVecGetArray(da, Xg, &x)); 288 289 /* 290 Get local grid boundaries 291 */ 292 PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL)); 293 PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL)); 294 295 /* 296 Compute u function over the locally owned part of the grid including ghost points 297 */ 298 for (i = xgs; i < xgs + xgm; i++) { 299 xx = i * hx; 300 r = PetscSqrtReal((xx - .5) * (xx - .5)); 301 if (r < .125) x[i].u = 1.0; 302 else x[i].u = -.50; 303 /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */ 304 x[i].w = 0; 305 } 306 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; 307 308 /* 309 Restore vectors 310 */ 311 PetscCall(DMDAVecRestoreArray(da, Xg, &x)); 312 313 /* Grab only the global part of the vector */ 314 PetscCall(VecSet(X, 0)); 315 PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X)); 316 PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X)); 317 PetscCall(VecDestroy(&Xg)); 318 PetscFunctionReturn(PETSC_SUCCESS); 319 } 320 321 /*TEST 322 323 build: 324 requires: !complex !single 325 326 test: 327 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 50 328 requires: x 329 330 TEST*/ 331