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], 4 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)) 7 8 which can be reduced to a first order system, 9 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 */ 14 15 #include <petscdm.h> 16 #include <petscdmda.h> 17 #include <petscts.h> 18 19 typedef struct { 20 PetscScalar u, v, th; 21 } Field; 22 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 }; 31 32 static PetscErrorCode FormRHSFunction(TS ts, PetscReal t, Vec U, Vec F, PetscCtx 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; 42 43 PetscFunctionBeginUser; 44 PetscCall(TSGetDM(ts, &dm)); 45 PetscCall(DMGetCoordinateDM(dm, &cdm)); 46 PetscCall(DMGetCoordinatesLocal(dm, &C)); 47 PetscCall(DMDAGetLocalInfo(dm, &info)); 48 PetscCall(DMDAVecGetArrayRead(dm, U, (void *)&u)); 49 PetscCall(DMDAVecGetArray(dm, F, &f)); 50 PetscCall(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]; 53 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 PetscCall(DMDAVecRestoreArrayRead(dm, U, (void *)&u)); 59 PetscCall(DMDAVecRestoreArray(dm, F, &f)); 60 PetscCall(DMDAVecRestoreArrayRead(cdm, C, (void *)&x)); 61 PetscFunctionReturn(PETSC_SUCCESS); 62 } 63 64 static PetscErrorCode FormIFunction(TS ts, PetscReal t, Vec U, Vec Udot, Vec F, PetscCtx 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; 73 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)); 82 PetscCall(DMDAVecGetArrayRead(dm, Uloc, &u)); 83 PetscCall(DMDAVecGetArrayRead(dm, Udot, &udot)); 84 PetscCall(DMDAVecGetArray(dm, F, &f)); 85 PetscCall(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 PetscCall(DMDAVecRestoreArrayRead(dm, Uloc, &u)); 105 PetscCall(DMDAVecRestoreArrayRead(dm, Udot, &udot)); 106 PetscCall(DMDAVecRestoreArray(dm, F, &f)); 107 PetscCall(DMDAVecRestoreArrayRead(cdm, C, &x)); 108 PetscCall(DMRestoreLocalVector(dm, &Uloc)); 109 PetscFunctionReturn(PETSC_SUCCESS); 110 } 111 112 /* 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, PetscCtx 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; 122 123 PetscFunctionBeginUser; 124 PetscCall(TSGetDM(ts, &dm)); 125 PetscCall(DMDAGetLocalInfo(dm, &info)); 126 PetscCall(DMGetCoordinateDM(dm, &cdm)); 127 PetscCall(DMGetCoordinatesLocal(dm, &C)); 128 PetscCall(DMDAVecGetArrayRead(dm, U, &u)); 129 PetscCall(DMDAVecGetArrayRead(dm, Udot, &udot)); 130 PetscCall(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] = { 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 }; 141 142 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 }; 152 153 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 }; 163 164 PetscCall(MatSetValuesBlocked(Jpre, 1, &row, 3, col, &vals[0][0][0], INSERT_VALUES)); 165 } 166 } 167 PetscCall(DMDAVecRestoreArrayRead(dm, U, &u)); 168 PetscCall(DMDAVecRestoreArrayRead(dm, Udot, &udot)); 169 PetscCall(DMDAVecRestoreArrayRead(cdm, C, &x)); 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 } 178 179 PetscErrorCode FormInitialSolution(TS ts, Vec U, PetscCtx 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; 189 190 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)); 196 PetscCall(DMDAVecGetArrayRead(cdm, C, &x)); 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)); 203 PetscCall(DMDAVecRestoreArrayRead(cdm, C, &x)); 204 PetscFunctionReturn(PETSC_SUCCESS); 205 } 206 207 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; 217 218 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)); 225 226 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(); 240 241 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)); 248 249 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)); 259 260 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)); 265 266 PetscCall(MatDestroy(&J)); 267 PetscCall(VecDestroy(&X)); 268 PetscCall(TSDestroy(&ts)); 269 PetscCall(DMDestroy(&dm)); 270 PetscCall(PetscFinalize()); 271 return 0; 272 } 273 274 /*TEST 275 276 build: 277 requires: !single !complex 278 279 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 281 282 TEST*/ 283