Actual source code: biharmonic3.c
1: static char help[] = "Solves biharmonic equation in 1d.\n";
3: /*
4: Solves the equation biharmonic equation in split form
6: w = -kappa \Delta u
7: u_t = \Delta w
8: -1 <= u <= 1
9: Periodic boundary conditions
11: Evolve the biharmonic heat equation with bounds: (same as biharmonic)
12: ---------------
13: ./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_time_step 9.53674e-9
15: w = -kappa \Delta u + u^3 - u
16: u_t = \Delta w
17: -1 <= u <= 1
18: Periodic boundary conditions
20: Evolve the Cahn-Hillard equations:
21: ---------------
22: ./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_time_step 5.96046e-06 -cahn-hillard
24: */
25: #include <petscdm.h>
26: #include <petscdmda.h>
27: #include <petscts.h>
29: /*
30: User-defined routines
31: */
32: extern PetscErrorCode FormFunction(TS, PetscReal, Vec, Vec, Vec, void *), FormInitialSolution(DM, Vec, PetscReal);
33: typedef struct {
34: PetscBool cahnhillard;
35: PetscReal kappa;
36: PetscInt energy;
37: PetscReal tol;
38: PetscReal theta;
39: PetscReal theta_c;
40: } UserCtx;
42: int main(int argc, char **argv)
43: {
44: TS ts; /* nonlinear solver */
45: Vec x, r; /* solution, residual vectors */
46: Mat J; /* Jacobian matrix */
47: PetscInt steps, Mx;
48: DM da;
49: MatFDColoring matfdcoloring;
50: ISColoring iscoloring;
51: PetscReal dt;
52: PetscReal vbounds[] = {-100000, 100000, -1.1, 1.1};
53: SNES snes;
54: UserCtx ctx;
56: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
57: Initialize program
58: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
59: PetscFunctionBeginUser;
60: PetscCall(PetscInitialize(&argc, &argv, NULL, help));
61: ctx.kappa = 1.0;
62: PetscCall(PetscOptionsGetReal(NULL, NULL, "-kappa", &ctx.kappa, NULL));
63: ctx.cahnhillard = PETSC_FALSE;
64: PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL));
65: PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds));
66: PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600));
67: ctx.energy = 1;
68: PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL));
69: ctx.tol = 1.0e-8;
70: PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL));
71: ctx.theta = .001;
72: ctx.theta_c = 1.0;
73: PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL));
74: PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL));
76: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
77: Create distributed array (DMDA) to manage parallel grid and vectors
78: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
79: PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da));
80: PetscCall(DMSetFromOptions(da));
81: PetscCall(DMSetUp(da));
82: PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx"));
83: PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u"));
84: PetscCall(DMDAGetInfo(da, NULL, &Mx, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
85: dt = 1.0 / (10. * ctx.kappa * Mx * Mx * Mx * Mx);
87: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
88: Extract global vectors from DMDA; then duplicate for remaining
89: vectors that are the same types
90: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
91: PetscCall(DMCreateGlobalVector(da, &x));
92: PetscCall(VecDuplicate(x, &r));
94: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
95: Create timestepping solver context
96: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
97: PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
98: PetscCall(TSSetDM(ts, da));
99: PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
100: PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx));
101: PetscCall(TSSetMaxTime(ts, .02));
102: PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
104: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
105: Create matrix data structure; set Jacobian evaluation routine
107: < Set Jacobian matrix data structure and default Jacobian evaluation
108: routine. User can override with:
109: -snes_mf : matrix-free Newton-Krylov method with no preconditioning
110: (unless user explicitly sets preconditioner)
111: -snes_mf_operator : form matrix used to construct the preconditioner as set by the user,
112: but use matrix-free approx for Jacobian-vector
113: products within Newton-Krylov method
115: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
116: PetscCall(TSGetSNES(ts, &snes));
117: PetscCall(SNESSetType(snes, SNESVINEWTONRSLS));
118: PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring));
119: PetscCall(DMSetMatType(da, MATAIJ));
120: PetscCall(DMCreateMatrix(da, &J));
121: PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring));
122: PetscCall(MatFDColoringSetFunction(matfdcoloring, (MatFDColoringFn *)SNESTSFormFunction, ts));
123: PetscCall(MatFDColoringSetFromOptions(matfdcoloring));
124: PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring));
125: PetscCall(ISColoringDestroy(&iscoloring));
126: PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring));
128: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
129: Customize nonlinear solver
130: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
131: PetscCall(TSSetType(ts, TSBEULER));
133: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
134: Set initial conditions
135: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
136: PetscCall(FormInitialSolution(da, x, ctx.kappa));
137: PetscCall(TSSetTimeStep(ts, dt));
138: PetscCall(TSSetSolution(ts, x));
140: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
141: Set runtime options
142: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
143: PetscCall(TSSetFromOptions(ts));
145: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
146: Solve nonlinear system
147: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148: PetscCall(TSSolve(ts, x));
149: PetscCall(TSGetStepNumber(ts, &steps));
151: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152: Free work space. All PETSc objects should be destroyed when they
153: are no longer needed.
154: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: PetscCall(MatDestroy(&J));
156: PetscCall(MatFDColoringDestroy(&matfdcoloring));
157: PetscCall(VecDestroy(&x));
158: PetscCall(VecDestroy(&r));
159: PetscCall(TSDestroy(&ts));
160: PetscCall(DMDestroy(&da));
162: PetscCall(PetscFinalize());
163: return 0;
164: }
166: typedef struct {
167: PetscScalar w, u;
168: } Field;
169: /* ------------------------------------------------------------------- */
170: /*
171: FormFunction - Evaluates nonlinear function, F(x).
173: Input Parameters:
174: . ts - the TS context
175: . X - input vector
176: . ptr - optional user-defined context, as set by SNESSetFunction()
178: Output Parameter:
179: . F - function vector
180: */
181: PetscErrorCode FormFunction(TS ts, PetscReal ftime, Vec X, Vec Xdot, Vec F, void *ptr)
182: {
183: DM da;
184: PetscInt i, Mx, xs, xm;
185: PetscReal hx, sx;
186: PetscScalar r, l;
187: Field *x, *xdot, *f;
188: Vec localX, localXdot;
189: UserCtx *ctx = (UserCtx *)ptr;
191: PetscFunctionBegin;
192: PetscCall(TSGetDM(ts, &da));
193: PetscCall(DMGetLocalVector(da, &localX));
194: PetscCall(DMGetLocalVector(da, &localXdot));
195: 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));
197: hx = 1.0 / (PetscReal)Mx;
198: sx = 1.0 / (hx * hx);
200: /*
201: Scatter ghost points to local vector,using the 2-step process
202: DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
203: By placing code between these two statements, computations can be
204: done while messages are in transition.
205: */
206: PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
207: PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
208: PetscCall(DMGlobalToLocalBegin(da, Xdot, INSERT_VALUES, localXdot));
209: PetscCall(DMGlobalToLocalEnd(da, Xdot, INSERT_VALUES, localXdot));
211: /*
212: Get pointers to vector data
213: */
214: PetscCall(DMDAVecGetArrayRead(da, localX, &x));
215: PetscCall(DMDAVecGetArrayRead(da, localXdot, &xdot));
216: PetscCall(DMDAVecGetArray(da, F, &f));
218: /*
219: Get local grid boundaries
220: */
221: PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
223: /*
224: Compute function over the locally owned part of the grid
225: */
226: for (i = xs; i < xs + xm; i++) {
227: f[i].w = x[i].w + ctx->kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;
228: if (ctx->cahnhillard) {
229: switch (ctx->energy) {
230: case 1: /* double well */
231: f[i].w += -x[i].u * x[i].u * x[i].u + x[i].u;
232: break;
233: case 2: /* double obstacle */
234: f[i].w += x[i].u;
235: break;
236: case 3: /* logarithmic */
237: 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;
238: 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;
239: 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;
240: break;
241: case 4:
242: break;
243: }
244: }
245: f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx;
246: if (ctx->energy == 4) {
247: f[i].u = xdot[i].u;
248: /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */
249: r = (1.0 - x[i + 1].u * x[i + 1].u) * (x[i + 2].w - x[i].w) * .5 / hx;
250: l = (1.0 - x[i - 1].u * x[i - 1].u) * (x[i].w - x[i - 2].w) * .5 / hx;
251: f[i].u -= (r - l) * .5 / hx;
252: 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;
253: }
254: }
256: /*
257: Restore vectors
258: */
259: PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot));
260: PetscCall(DMDAVecRestoreArrayRead(da, localX, &x));
261: PetscCall(DMDAVecRestoreArray(da, F, &f));
262: PetscCall(DMRestoreLocalVector(da, &localX));
263: PetscCall(DMRestoreLocalVector(da, &localXdot));
264: PetscFunctionReturn(PETSC_SUCCESS);
265: }
267: /* ------------------------------------------------------------------- */
268: PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa)
269: {
270: PetscInt i, xs, xm, Mx, xgs, xgm;
271: Field *x;
272: PetscReal hx, xx, r, sx;
273: Vec Xg;
275: PetscFunctionBegin;
276: 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));
278: hx = 1.0 / (PetscReal)Mx;
279: sx = 1.0 / (hx * hx);
281: /*
282: Get pointers to vector data
283: */
284: PetscCall(DMCreateLocalVector(da, &Xg));
285: PetscCall(DMDAVecGetArray(da, Xg, &x));
287: /*
288: Get local grid boundaries
289: */
290: PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
291: PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL));
293: /*
294: Compute u function over the locally owned part of the grid including ghost points
295: */
296: for (i = xgs; i < xgs + xgm; i++) {
297: xx = i * hx;
298: r = PetscSqrtReal((xx - .5) * (xx - .5));
299: if (r < .125) x[i].u = 1.0;
300: else x[i].u = -.50;
301: /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
302: x[i].w = 0;
303: }
304: 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;
306: /*
307: Restore vectors
308: */
309: PetscCall(DMDAVecRestoreArray(da, Xg, &x));
311: /* Grab only the global part of the vector */
312: PetscCall(VecSet(X, 0));
313: PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X));
314: PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X));
315: PetscCall(VecDestroy(&Xg));
316: PetscFunctionReturn(PETSC_SUCCESS);
317: }
319: /*TEST
321: build:
322: requires: !complex !single
324: test:
325: args: -ts_monitor -snes_monitor -pc_type lu -snes_converged_reason -ts_type beuler -da_refine 5 -ts_time_step 9.53674e-9 -ts_max_steps 50
326: requires: x
328: TEST*/