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_dt 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_dt 5.96046e-06 -cahn-hillard
24: */
25: #include <petscdm.h>
26: #include <petscdmda.h>
27: #include <petscts.h>
28: #include <petscdraw.h>
30: /*
31: User-defined routines
32: */
33: extern PetscErrorCode FormFunction(TS, PetscReal, Vec, Vec, Vec, void *), FormInitialSolution(DM, Vec, PetscReal);
34: typedef struct {
35: PetscBool cahnhillard;
36: PetscReal kappa;
37: PetscInt energy;
38: PetscReal tol;
39: PetscReal theta;
40: PetscReal theta_c;
41: } UserCtx;
43: int main(int argc, char **argv)
44: {
45: TS ts; /* nonlinear solver */
46: Vec x, r; /* solution, residual vectors */
47: Mat J; /* Jacobian matrix */
48: PetscInt steps, Mx;
49: DM da;
50: MatFDColoring matfdcoloring;
51: ISColoring iscoloring;
52: PetscReal dt;
53: PetscReal vbounds[] = {-100000, 100000, -1.1, 1.1};
54: SNES snes;
55: UserCtx ctx;
57: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
58: Initialize program
59: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
60: PetscFunctionBeginUser;
61: PetscCall(PetscInitialize(&argc, &argv, NULL, help));
62: ctx.kappa = 1.0;
63: PetscCall(PetscOptionsGetReal(NULL, NULL, "-kappa", &ctx.kappa, NULL));
64: ctx.cahnhillard = PETSC_FALSE;
65: PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL));
66: PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds));
67: PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600));
68: ctx.energy = 1;
69: PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL));
70: ctx.tol = 1.0e-8;
71: PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL));
72: ctx.theta = .001;
73: ctx.theta_c = 1.0;
74: PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL));
75: PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL));
77: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
78: Create distributed array (DMDA) to manage parallel grid and vectors
79: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
80: PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da));
81: PetscCall(DMSetFromOptions(da));
82: PetscCall(DMSetUp(da));
83: PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx"));
84: PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u"));
85: PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
86: dt = 1.0 / (10. * ctx.kappa * Mx * Mx * Mx * Mx);
88: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
89: Extract global vectors from DMDA; then duplicate for remaining
90: vectors that are the same types
91: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
92: PetscCall(DMCreateGlobalVector(da, &x));
93: PetscCall(VecDuplicate(x, &r));
95: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
96: Create timestepping solver context
97: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
98: PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
99: PetscCall(TSSetDM(ts, da));
100: PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
101: PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx));
102: PetscCall(TSSetMaxTime(ts, .02));
103: PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
105: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106: Create matrix data structure; set Jacobian evaluation routine
108: < Set Jacobian matrix data structure and default Jacobian evaluation
109: routine. User can override with:
110: -snes_mf : matrix-free Newton-Krylov method with no preconditioning
111: (unless user explicitly sets preconditioner)
112: -snes_mf_operator : form preconditioning matrix as set by the user,
113: but use matrix-free approx for Jacobian-vector
114: products within Newton-Krylov method
116: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
117: PetscCall(TSGetSNES(ts, &snes));
118: PetscCall(SNESSetType(snes, SNESVINEWTONRSLS));
119: PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring));
120: PetscCall(DMSetMatType(da, MATAIJ));
121: PetscCall(DMCreateMatrix(da, &J));
122: PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring));
123: PetscCall(MatFDColoringSetFunction(matfdcoloring, (PetscErrorCode (*)(void))SNESTSFormFunction, ts));
124: PetscCall(MatFDColoringSetFromOptions(matfdcoloring));
125: PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring));
126: PetscCall(ISColoringDestroy(&iscoloring));
127: PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring));
129: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
130: Customize nonlinear solver
131: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
132: PetscCall(TSSetType(ts, TSBEULER));
134: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135: Set initial conditions
136: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
137: PetscCall(FormInitialSolution(da, x, ctx.kappa));
138: PetscCall(TSSetTimeStep(ts, dt));
139: PetscCall(TSSetSolution(ts, x));
141: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
142: Set runtime options
143: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
144: PetscCall(TSSetFromOptions(ts));
146: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
147: Solve nonlinear system
148: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
149: PetscCall(TSSolve(ts, x));
150: PetscCall(TSGetStepNumber(ts, &steps));
152: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153: Free work space. All PETSc objects should be destroyed when they
154: are no longer needed.
155: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
156: PetscCall(MatDestroy(&J));
157: PetscCall(MatFDColoringDestroy(&matfdcoloring));
158: PetscCall(VecDestroy(&x));
159: PetscCall(VecDestroy(&r));
160: PetscCall(TSDestroy(&ts));
161: PetscCall(DMDestroy(&da));
163: PetscCall(PetscFinalize());
164: return 0;
165: }
167: typedef struct {
168: PetscScalar w, u;
169: } Field;
170: /* ------------------------------------------------------------------- */
171: /*
172: FormFunction - Evaluates nonlinear function, F(x).
174: Input Parameters:
175: . ts - the TS context
176: . X - input vector
177: . ptr - optional user-defined context, as set by SNESSetFunction()
179: Output Parameter:
180: . F - function vector
181: */
182: PetscErrorCode FormFunction(TS ts, PetscReal ftime, Vec X, Vec Xdot, Vec F, void *ptr)
183: {
184: DM da;
185: PetscInt i, Mx, xs, xm;
186: PetscReal hx, sx;
187: PetscScalar r, l;
188: Field *x, *xdot, *f;
189: Vec localX, localXdot;
190: UserCtx *ctx = (UserCtx *)ptr;
192: PetscFunctionBegin;
193: PetscCall(TSGetDM(ts, &da));
194: PetscCall(DMGetLocalVector(da, &localX));
195: PetscCall(DMGetLocalVector(da, &localXdot));
196: 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: hx = 1.0 / (PetscReal)Mx;
199: sx = 1.0 / (hx * hx);
201: /*
202: Scatter ghost points to local vector,using the 2-step process
203: DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
204: By placing code between these two statements, computations can be
205: done while messages are in transition.
206: */
207: PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
208: PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
209: PetscCall(DMGlobalToLocalBegin(da, Xdot, INSERT_VALUES, localXdot));
210: PetscCall(DMGlobalToLocalEnd(da, Xdot, INSERT_VALUES, localXdot));
212: /*
213: Get pointers to vector data
214: */
215: PetscCall(DMDAVecGetArrayRead(da, localX, &x));
216: PetscCall(DMDAVecGetArrayRead(da, localXdot, &xdot));
217: PetscCall(DMDAVecGetArray(da, F, &f));
219: /*
220: Get local grid boundaries
221: */
222: PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
224: /*
225: Compute function over the locally owned part of the grid
226: */
227: for (i = xs; i < xs + xm; i++) {
228: f[i].w = x[i].w + ctx->kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;
229: if (ctx->cahnhillard) {
230: switch (ctx->energy) {
231: case 1: /* double well */
232: f[i].w += -x[i].u * x[i].u * x[i].u + x[i].u;
233: break;
234: case 2: /* double obstacle */
235: f[i].w += x[i].u;
236: break;
237: case 3: /* logarithmic */
238: 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;
239: 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;
240: 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;
241: break;
242: case 4:
243: break;
244: }
245: }
246: f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx;
247: if (ctx->energy == 4) {
248: f[i].u = xdot[i].u;
249: /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */
250: r = (1.0 - x[i + 1].u * x[i + 1].u) * (x[i + 2].w - x[i].w) * .5 / hx;
251: l = (1.0 - x[i - 1].u * x[i - 1].u) * (x[i].w - x[i - 2].w) * .5 / hx;
252: f[i].u -= (r - l) * .5 / hx;
253: 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;
254: }
255: }
257: /*
258: Restore vectors
259: */
260: PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot));
261: PetscCall(DMDAVecRestoreArrayRead(da, localX, &x));
262: PetscCall(DMDAVecRestoreArray(da, F, &f));
263: PetscCall(DMRestoreLocalVector(da, &localX));
264: PetscCall(DMRestoreLocalVector(da, &localXdot));
265: PetscFunctionReturn(PETSC_SUCCESS);
266: }
268: /* ------------------------------------------------------------------- */
269: PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa)
270: {
271: PetscInt i, xs, xm, Mx, xgs, xgm;
272: Field *x;
273: PetscReal hx, xx, r, sx;
274: Vec Xg;
276: PetscFunctionBegin;
277: 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: hx = 1.0 / (PetscReal)Mx;
280: sx = 1.0 / (hx * hx);
282: /*
283: Get pointers to vector data
284: */
285: PetscCall(DMCreateLocalVector(da, &Xg));
286: PetscCall(DMDAVecGetArray(da, Xg, &x));
288: /*
289: Get local grid boundaries
290: */
291: PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
292: PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL));
294: /*
295: Compute u function over the locally owned part of the grid including ghost points
296: */
297: for (i = xgs; i < xgs + xgm; i++) {
298: xx = i * hx;
299: r = PetscSqrtReal((xx - .5) * (xx - .5));
300: if (r < .125) x[i].u = 1.0;
301: else x[i].u = -.50;
302: /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
303: x[i].w = 0;
304: }
305: 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: Restore vectors
309: */
310: PetscCall(DMDAVecRestoreArray(da, Xg, &x));
312: /* Grab only the global part of the vector */
313: PetscCall(VecSet(X, 0));
314: PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X));
315: PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X));
316: PetscCall(VecDestroy(&Xg));
317: PetscFunctionReturn(PETSC_SUCCESS);
318: }
320: /*TEST
322: build:
323: requires: !complex !single
325: test:
326: 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
327: requires: x
329: TEST*/