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