Actual source code: ex34.c

  1: static const char help[] = "An elastic wave equation driven by Dieterich-Ruina friction\n";
2: /*
3: This whole derivation comes from Erickson, Birnir, and Lavallee [2010]. The model comes from the continuum limit in Carlson and Langer [1989],

5:   u_{tt}   = c^2 u_{xx} - \tilde\gamma^2 u - (\gamma^2 / \xi) (\theta + \ln(u_t + 1))
6:   \theta_t = -(u_t + 1) (\theta + (1 + \epsilon) \ln(u_t +1))

8: which can be reduced to a first order system,

10:   u_t      = v
11:   v_t      = c^2 u_{xx} - \tilde\gamma^2 u - (\gamma^2 / \xi)(\theta + ln(v + 1)))
12:   \theta_t = -(v + 1) (\theta + (1 + \epsilon) \ln(v+1))
13: */

15: #include <petscdm.h>
16: #include <petscdmda.h>
17: #include <petscts.h>

19: typedef struct {
20:   PetscScalar u, v, th;
21: } Field;

23: typedef struct _User *User;
24: struct _User {
25:   PetscReal epsilon;    /* inverse of seismic ratio, B-A / A */
26:   PetscReal gamma;      /* wave frequency for interblock coupling */
27:   PetscReal gammaTilde; /* wave frequency for coupling to plate */
28:   PetscReal xi;         /* interblock spring constant */
29:   PetscReal c;          /* wavespeed */
30: };

32: static PetscErrorCode FormRHSFunction(TS ts, PetscReal t, Vec U, Vec F, void *ctx)
33: {
34:   User               user = (User)ctx;
35:   DM                 dm, cdm;
36:   DMDALocalInfo      info;
37:   Vec                C;
38:   Field             *f;
39:   const Field       *u;
40:   const PetscScalar *x;
41:   PetscInt           i;

43:   PetscFunctionBeginUser;
44:   PetscCall(TSGetDM(ts, &dm));
45:   PetscCall(DMGetCoordinateDM(dm, &cdm));
46:   PetscCall(DMGetCoordinatesLocal(dm, &C));
47:   PetscCall(DMDAGetLocalInfo(dm, &info));
49:   PetscCall(DMDAVecGetArray(dm, F, &f));
51:   for (i = info.xs; i < info.xs + info.xm; ++i) {
52:     const PetscScalar hx = i + 1 == info.xs + info.xm ? x[i] - x[i - 1] : x[i + 1] - x[i];

54:     f[i].u  = hx * (u[i].v);
55:     f[i].v  = -hx * (PetscSqr(user->gammaTilde) * u[i].u + (PetscSqr(user->gamma) / user->xi) * (u[i].th + PetscLogScalar(u[i].v + 1)));
56:     f[i].th = -hx * (u[i].v + 1) * (u[i].th + (1 + user->epsilon) * PetscLogScalar(u[i].v + 1));
57:   }
59:   PetscCall(DMDAVecRestoreArray(dm, F, &f));
61:   PetscFunctionReturn(PETSC_SUCCESS);
62: }

64: static PetscErrorCode FormIFunction(TS ts, PetscReal t, Vec U, Vec Udot, Vec F, void *ctx)
65: {
66:   User          user = (User)ctx;
67:   DM            dm, cdm;
68:   DMDALocalInfo info;
69:   Vec           Uloc, C;
70:   Field        *u, *udot, *f;
71:   PetscScalar  *x;
72:   PetscInt      i;

74:   PetscFunctionBeginUser;
75:   PetscCall(TSGetDM(ts, &dm));
76:   PetscCall(DMDAGetLocalInfo(dm, &info));
77:   PetscCall(DMGetCoordinateDM(dm, &cdm));
78:   PetscCall(DMGetCoordinatesLocal(dm, &C));
79:   PetscCall(DMGetLocalVector(dm, &Uloc));
80:   PetscCall(DMGlobalToLocalBegin(dm, U, INSERT_VALUES, Uloc));
81:   PetscCall(DMGlobalToLocalEnd(dm, U, INSERT_VALUES, Uloc));
84:   PetscCall(DMDAVecGetArray(dm, F, &f));
86:   for (i = info.xs; i < info.xs + info.xm; ++i) {
87:     if (i == 0) {
88:       const PetscScalar hx = x[i + 1] - x[i];
89:       f[i].u               = hx * udot[i].u;
90:       f[i].v               = hx * udot[i].v - PetscSqr(user->c) * (u[i + 1].u - u[i].u) / hx;
91:       f[i].th              = hx * udot[i].th;
92:     } else if (i == info.mx - 1) {
93:       const PetscScalar hx = x[i] - x[i - 1];
94:       f[i].u               = hx * udot[i].u;
95:       f[i].v               = hx * udot[i].v - PetscSqr(user->c) * (u[i - 1].u - u[i].u) / hx;
96:       f[i].th              = hx * udot[i].th;
97:     } else {
98:       const PetscScalar hx = x[i + 1] - x[i];
99:       f[i].u               = hx * udot[i].u;
100:       f[i].v               = hx * udot[i].v - PetscSqr(user->c) * (u[i - 1].u - 2. * u[i].u + u[i + 1].u) / hx;
101:       f[i].th              = hx * udot[i].th;
102:     }
103:   }
106:   PetscCall(DMDAVecRestoreArray(dm, F, &f));
108:   PetscCall(DMRestoreLocalVector(dm, &Uloc));
109:   PetscFunctionReturn(PETSC_SUCCESS);
110: }

112: /* IJacobian - Compute IJacobian = dF/dU + a dF/dUdot */
113: PetscErrorCode FormIJacobian(TS ts, PetscReal t, Vec U, Vec Udot, PetscReal a, Mat J, Mat Jpre, void *ctx)
114: {
115:   User          user = (User)ctx;
116:   DM            dm, cdm;
117:   DMDALocalInfo info;
118:   Vec           C;
119:   Field        *u, *udot;
120:   PetscScalar  *x;
121:   PetscInt      i;

123:   PetscFunctionBeginUser;
124:   PetscCall(TSGetDM(ts, &dm));
125:   PetscCall(DMDAGetLocalInfo(dm, &info));
126:   PetscCall(DMGetCoordinateDM(dm, &cdm));
127:   PetscCall(DMGetCoordinatesLocal(dm, &C));
131:   for (i = info.xs; i < info.xs + info.xm; ++i) {
132:     if (i == 0) {
133:       const PetscScalar hx  = x[i + 1] - x[i];
134:       const PetscInt    row = i, col[] = {i, i + 1};
135:       const PetscScalar dxx0 = PetscSqr(user->c) / hx, dxxR = -PetscSqr(user->c) / hx;
136:       const PetscScalar vals[3][2][3] = {
137:         {{a * hx, 0, 0},        {0, 0, 0}   },
138:         {{0, a * hx + dxx0, 0}, {0, dxxR, 0}},
139:         {{0, 0, a * hx},        {0, 0, 0}   }
140:       };

142:       PetscCall(MatSetValuesBlocked(Jpre, 1, &row, 2, col, &vals[0][0][0], INSERT_VALUES));
143:     } else if (i == info.mx - 1) {
144:       const PetscScalar hx  = x[i + 1] - x[i];
145:       const PetscInt    row = i, col[] = {i - 1, i};
146:       const PetscScalar dxxL = -PetscSqr(user->c) / hx, dxx0 = PetscSqr(user->c) / hx;
147:       const PetscScalar vals[3][2][3] = {
148:         {{0, 0, 0},    {a * hx, 0, 0}       },
149:         {{0, dxxL, 0}, {0, a * hx + dxx0, 0}},
150:         {{0, 0, 0},    {0, 0, a * hx}       }
151:       };

153:       PetscCall(MatSetValuesBlocked(Jpre, 1, &row, 2, col, &vals[0][0][0], INSERT_VALUES));
154:     } else {
155:       const PetscScalar hx  = x[i + 1] - x[i];
156:       const PetscInt    row = i, col[] = {i - 1, i, i + 1};
157:       const PetscScalar dxxL = -PetscSqr(user->c) / hx, dxx0 = 2. * PetscSqr(user->c) / hx, dxxR = -PetscSqr(user->c) / hx;
158:       const PetscScalar vals[3][3][3] = {
159:         {{0, 0, 0},    {a * hx, 0, 0},        {0, 0, 0}   },
160:         {{0, dxxL, 0}, {0, a * hx + dxx0, 0}, {0, dxxR, 0}},
161:         {{0, 0, 0},    {0, 0, a * hx},        {0, 0, 0}   }
162:       };

164:       PetscCall(MatSetValuesBlocked(Jpre, 1, &row, 3, col, &vals[0][0][0], INSERT_VALUES));
165:     }
166:   }
170:   PetscCall(MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY));
171:   PetscCall(MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY));
172:   if (J != Jpre) {
173:     PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
174:     PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
175:   }
176:   PetscFunctionReturn(PETSC_SUCCESS);
177: }

179: PetscErrorCode FormInitialSolution(TS ts, Vec U, void *ctx)
180: {
181:   /* User            user = (User) ctx; */
182:   DM              dm, cdm;
183:   DMDALocalInfo   info;
184:   Vec             C;
185:   Field          *u;
186:   PetscScalar    *x;
187:   const PetscReal sigma = 1.0;
188:   PetscInt        i;

190:   PetscFunctionBeginUser;
191:   PetscCall(TSGetDM(ts, &dm));
192:   PetscCall(DMGetCoordinateDM(dm, &cdm));
193:   PetscCall(DMGetCoordinatesLocal(dm, &C));
194:   PetscCall(DMDAGetLocalInfo(dm, &info));
195:   PetscCall(DMDAVecGetArray(dm, U, &u));
197:   for (i = info.xs; i < info.xs + info.xm; ++i) {
198:     u[i].u  = 1.5 * PetscExpScalar(-PetscSqr(x[i] - 10) / PetscSqr(sigma));
199:     u[i].v  = 0.0;
200:     u[i].th = 0.0;
201:   }
202:   PetscCall(DMDAVecRestoreArray(dm, U, &u));
204:   PetscFunctionReturn(PETSC_SUCCESS);
205: }

207: int main(int argc, char **argv)
208: {
209:   DM                dm;
210:   TS                ts;
211:   Vec               X;
212:   Mat               J;
213:   PetscInt          steps, mx;
214:   PetscReal         ftime, hx, dt;
215:   TSConvergedReason reason;
216:   struct _User      user;

218:   PetscFunctionBeginUser;
219:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
220:   PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, 11, 3, 1, NULL, &dm));
221:   PetscCall(DMSetFromOptions(dm));
222:   PetscCall(DMSetUp(dm));
223:   PetscCall(DMDASetUniformCoordinates(dm, 0.0, 20.0, 0.0, 0.0, 0.0, 0.0));
224:   PetscCall(DMCreateGlobalVector(dm, &X));

226:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Dynamic Friction Options", "");
227:   {
228:     user.epsilon    = 0.1;
229:     user.gamma      = 0.5;
230:     user.gammaTilde = 0.5;
231:     user.xi         = 0.5;
232:     user.c          = 0.5;
233:     PetscCall(PetscOptionsReal("-epsilon", "Inverse of seismic ratio", "", user.epsilon, &user.epsilon, NULL));
234:     PetscCall(PetscOptionsReal("-gamma", "Wave frequency for interblock coupling", "", user.gamma, &user.gamma, NULL));
235:     PetscCall(PetscOptionsReal("-gamma_tilde", "Wave frequency for plate coupling", "", user.gammaTilde, &user.gammaTilde, NULL));
236:     PetscCall(PetscOptionsReal("-xi", "Interblock spring constant", "", user.xi, &user.xi, NULL));
237:     PetscCall(PetscOptionsReal("-c", "Wavespeed", "", user.c, &user.c, NULL));
238:   }
239:   PetscOptionsEnd();

241:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
242:   PetscCall(TSSetDM(ts, dm));
243:   PetscCall(TSSetRHSFunction(ts, NULL, FormRHSFunction, &user));
244:   PetscCall(TSSetIFunction(ts, NULL, FormIFunction, &user));
245:   PetscCall(DMSetMatType(dm, MATAIJ));
246:   PetscCall(DMCreateMatrix(dm, &J));
247:   PetscCall(TSSetIJacobian(ts, J, J, FormIJacobian, &user));

249:   ftime = 800.0;
250:   PetscCall(TSSetMaxTime(ts, ftime));
251:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
252:   PetscCall(FormInitialSolution(ts, X, &user));
253:   PetscCall(TSSetSolution(ts, X));
254:   PetscCall(VecGetSize(X, &mx));
255:   hx = 20.0 / (PetscReal)(mx - 1);
256:   dt = 0.4 * PetscSqr(hx) / PetscSqr(user.c); /* Diffusive stability limit */
257:   PetscCall(TSSetTimeStep(ts, dt));
258:   PetscCall(TSSetFromOptions(ts));

260:   PetscCall(TSSolve(ts, X));
261:   PetscCall(TSGetSolveTime(ts, &ftime));
262:   PetscCall(TSGetStepNumber(ts, &steps));
263:   PetscCall(TSGetConvergedReason(ts, &reason));
264:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%s at time %g after %" PetscInt_FMT " steps\n", TSConvergedReasons[reason], (double)ftime, steps));

266:   PetscCall(MatDestroy(&J));
267:   PetscCall(VecDestroy(&X));
268:   PetscCall(TSDestroy(&ts));
269:   PetscCall(DMDestroy(&dm));
270:   PetscCall(PetscFinalize());
271:   return 0;
272: }

274: /*TEST

276:     build:
277:       requires: !single !complex

279:     test:
280:       TODO: broken, was not nightly tested, SNES solve eventually fails, -snes_test_jacobian indicates Jacobian is wrong, but even -snes_mf_operator fails

282: TEST*/