Actual source code: biharmonic3.c
2: static char help[] = "Solves biharmonic equation in 1d.\n";
4: /*
5: Solves the equation biharmonic equation in split form
7: w = -kappa \Delta u
8: u_t = \Delta w
9: -1 <= u <= 1
10: Periodic boundary conditions
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
16: w = -kappa \Delta u + u^3 - u
17: u_t = \Delta w
18: -1 <= u <= 1
19: Periodic boundary conditions
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
25: */
26: #include <petscdm.h>
27: #include <petscdmda.h>
28: #include <petscts.h>
29: #include <petscdraw.h>
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;
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;
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));
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);
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));
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));
106: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
107: Create matrix data structure; set Jacobian evaluation routine
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
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));
130: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
131: Customize nonlinear solver
132: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
133: PetscCall(TSSetType(ts, TSBEULER));
135: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136: Set initial conditions
137: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
138: PetscCall(FormInitialSolution(da, x, ctx.kappa));
139: PetscCall(TSSetTimeStep(ts, dt));
140: PetscCall(TSSetSolution(ts, x));
142: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143: Set runtime options
144: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
145: PetscCall(TSSetFromOptions(ts));
147: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
148: Solve nonlinear system
149: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
150: PetscCall(TSSolve(ts, x));
151: PetscCall(TSGetStepNumber(ts, &steps));
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));
164: PetscCall(PetscFinalize());
165: return 0;
166: }
168: typedef struct {
169: PetscScalar w, u;
170: } Field;
171: /* ------------------------------------------------------------------- */
172: /*
173: FormFunction - Evaluates nonlinear function, F(x).
175: Input Parameters:
176: . ts - the TS context
177: . X - input vector
178: . ptr - optional user-defined context, as set by SNESSetFunction()
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;
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));
199: hx = 1.0 / (PetscReal)Mx;
200: sx = 1.0 / (hx * hx);
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));
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));
220: /*
221: Get local grid boundaries
222: */
223: PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
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: }
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: }
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;
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));
280: hx = 1.0 / (PetscReal)Mx;
281: sx = 1.0 / (hx * hx);
283: /*
284: Get pointers to vector data
285: */
286: PetscCall(DMCreateLocalVector(da, &Xg));
287: PetscCall(DMDAVecGetArray(da, Xg, &x));
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));
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;
308: /*
309: Restore vectors
310: */
311: PetscCall(DMDAVecRestoreArray(da, Xg, &x));
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: }
321: /*TEST
323: build:
324: requires: !complex !single
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
330: TEST*/