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;

 44:   TSGetDM(ts, &dm);
 45:   DMGetCoordinateDM(dm, &cdm);
 46:   DMGetCoordinatesLocal(dm, &C);
 47:   DMDAGetLocalInfo(dm, &info);
 48:   DMDAVecGetArrayRead(dm,  U, (void*)&u);
 49:   DMDAVecGetArray(dm,  F, &f);
 50:   DMDAVecGetArrayRead(cdm, C, (void*)&x);
 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:   }
 58:   DMDAVecRestoreArrayRead(dm,  U, (void*)&u);
 59:   DMDAVecRestoreArray(dm,  F, &f);
 60:   DMDAVecRestoreArrayRead(cdm, C, (void*)&x);
 61:   return 0;
 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;

 75:   TSGetDM(ts, &dm);
 76:   DMDAGetLocalInfo(dm, &info);
 77:   DMGetCoordinateDM(dm, &cdm);
 78:   DMGetCoordinatesLocal(dm, &C);
 79:   DMGetLocalVector(dm, &Uloc);
 80:   DMGlobalToLocalBegin(dm, U, INSERT_VALUES, Uloc);
 81:   DMGlobalToLocalEnd(dm, U, INSERT_VALUES, Uloc);
 82:   DMDAVecGetArrayRead(dm,  Uloc, &u);
 83:   DMDAVecGetArrayRead(dm,  Udot, &udot);
 84:   DMDAVecGetArray(dm,  F,    &f);
 85:   DMDAVecGetArrayRead(cdm, C,    &x);
 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:   }
104:   DMDAVecRestoreArrayRead(dm,  Uloc, &u);
105:   DMDAVecRestoreArrayRead(dm,  Udot, &udot);
106:   DMDAVecRestoreArray(dm,  F,    &f);
107:   DMDAVecRestoreArrayRead(cdm, C,    &x);
108:   DMRestoreLocalVector(dm, &Uloc);
109:   return 0;
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;

124:   TSGetDM(ts, &dm);
125:   DMDAGetLocalInfo(dm, &info);
126:   DMGetCoordinateDM(dm, &cdm);
127:   DMGetCoordinatesLocal(dm, &C);
128:   DMDAVecGetArrayRead(dm,  U,    &u);
129:   DMDAVecGetArrayRead(dm,  Udot, &udot);
130:   DMDAVecGetArrayRead(cdm, C,    &x);
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] = {{{a*hx,     0,0},{0,0,   0}},
137:                                          {{0,a*hx+dxx0,0},{0,dxxR,0}},
138:                                          {{0,0,     a*hx},{0,0,   0}}};

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

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

158:       MatSetValuesBlocked(Jpre, 1, &row, 3, col, &vals[0][0][0], INSERT_VALUES);
159:     }
160:   }
161:   DMDAVecRestoreArrayRead(dm,  U,    &u);
162:   DMDAVecRestoreArrayRead(dm,  Udot, &udot);
163:   DMDAVecRestoreArrayRead(cdm, C,    &x);
164:   MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY);
165:   MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY);
166:   if (J != Jpre) {
167:     MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY);
168:     MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY);
169:   }
170:   return 0;
171: }

173: PetscErrorCode FormInitialSolution(TS ts, Vec U, void *ctx)
174: {
175:   /* User            user = (User) ctx; */
176:   DM              dm, cdm;
177:   DMDALocalInfo   info;
178:   Vec             C;
179:   Field          *u;
180:   PetscScalar    *x;
181:   const PetscReal sigma = 1.0;
182:   PetscInt        i;

185:   TSGetDM(ts, &dm);
186:   DMGetCoordinateDM(dm, &cdm);
187:   DMGetCoordinatesLocal(dm, &C);
188:   DMDAGetLocalInfo(dm, &info);
189:   DMDAVecGetArray(dm,  U, &u);
190:   DMDAVecGetArrayRead(cdm, C, &x);
191:   for (i = info.xs; i < info.xs+info.xm; ++i) {
192:     u[i].u  = 1.5 * PetscExpScalar(-PetscSqr(x[i] - 10)/PetscSqr(sigma));
193:     u[i].v  = 0.0;
194:     u[i].th = 0.0;
195:   }
196:   DMDAVecRestoreArray(dm,  U, &u);
197:   DMDAVecRestoreArrayRead(cdm, C, &x);
198:   return 0;
199: }

201: int main(int argc, char **argv)
202: {
203:   DM                dm;
204:   TS                ts;
205:   Vec               X;
206:   Mat               J;
207:   PetscInt          steps, mx;
208:   PetscReal         ftime, hx, dt;
209:   TSConvergedReason reason;
210:   struct _User      user;
211:   PetscErrorCode    ierr;

213:   PetscInitialize(&argc, &argv, NULL,help);
214:   DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, 11, 3, 1, NULL, &dm);
215:   DMSetFromOptions(dm);
216:   DMSetUp(dm);
217:   DMDASetUniformCoordinates(dm, 0.0, 20.0, 0.0, 0.0, 0.0, 0.0);
218:   DMCreateGlobalVector(dm, &X);

220:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Dynamic Friction Options", "");
221:   {
222:     user.epsilon    = 0.1;
223:     user.gamma      = 0.5;
224:     user.gammaTilde = 0.5;
225:     user.xi         = 0.5;
226:     user.c          = 0.5;
227:     PetscOptionsReal("-epsilon", "Inverse of seismic ratio", "", user.epsilon, &user.epsilon, NULL);
228:     PetscOptionsReal("-gamma", "Wave frequency for interblock coupling", "", user.gamma, &user.gamma, NULL);
229:     PetscOptionsReal("-gamma_tilde", "Wave frequency for plate coupling", "", user.gammaTilde, &user.gammaTilde, NULL);
230:     PetscOptionsReal("-xi", "Interblock spring constant", "", user.xi, &user.xi, NULL);
231:     PetscOptionsReal("-c", "Wavespeed", "", user.c, &user.c, NULL);
232:   }
233:   PetscOptionsEnd();

235:   TSCreate(PETSC_COMM_WORLD, &ts);
236:   TSSetDM(ts, dm);
237:   TSSetRHSFunction(ts, NULL, FormRHSFunction, &user);
238:   TSSetIFunction(ts, NULL, FormIFunction, &user);
239:   DMSetMatType(dm, MATAIJ);
240:   DMCreateMatrix(dm, &J);
241:   TSSetIJacobian(ts, J, J, FormIJacobian, &user);

243:   ftime = 800.0;
244:   TSSetMaxTime(ts,ftime);
245:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);
246:   FormInitialSolution(ts, X, &user);
247:   TSSetSolution(ts, X);
248:   VecGetSize(X, &mx);
249:   hx   = 20.0/(PetscReal)(mx-1);
250:   dt   = 0.4 * PetscSqr(hx) / PetscSqr(user.c); /* Diffusive stability limit */
251:   TSSetTimeStep(ts,dt);
252:   TSSetFromOptions(ts);

254:   TSSolve(ts, X);
255:   TSGetSolveTime(ts, &ftime);
256:   TSGetStepNumber(ts, &steps);
257:   TSGetConvergedReason(ts, &reason);
258:   PetscPrintf(PETSC_COMM_WORLD, "%s at time %g after %D steps\n", TSConvergedReasons[reason], (double)ftime, steps);

260:   MatDestroy(&J);
261:   VecDestroy(&X);
262:   TSDestroy(&ts);
263:   DMDestroy(&dm);
264:   PetscFinalize();
265:   return 0;
266: }

268: /*TEST

270:     build:
271:       requires: !single  !complex

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

276: TEST*/