Actual source code: ex10.c
1: /*
2: This example implements the model described in
4: Rauenzahn, Mousseau, Knoll. "Temporal accuracy of the nonequilibrium radiation diffusion
5: equations employing a Saha ionization model" 2005.
7: The paper discusses three examples, the first two are nondimensional with a simple
8: ionization model. The third example is fully dimensional and uses the Saha ionization
9: model with realistic parameters.
10: */
12: #include <petscts.h>
13: #include <petscdm.h>
14: #include <petscdmda.h>
16: typedef enum {
17: BC_DIRICHLET,
18: BC_NEUMANN,
19: BC_ROBIN
20: } BCType;
21: static const char *const BCTypes[] = {"DIRICHLET", "NEUMANN", "ROBIN", "BCType", "BC_", 0};
22: typedef enum {
23: JACOBIAN_ANALYTIC,
24: JACOBIAN_MATRIXFREE,
25: JACOBIAN_FD_COLORING,
26: JACOBIAN_FD_FULL
27: } JacobianType;
28: static const char *const JacobianTypes[] = {"ANALYTIC", "MATRIXFREE", "FD_COLORING", "FD_FULL", "JacobianType", "FD_", 0};
29: typedef enum {
30: DISCRETIZATION_FD,
31: DISCRETIZATION_FE
32: } DiscretizationType;
33: static const char *const DiscretizationTypes[] = {"FD", "FE", "DiscretizationType", "DISCRETIZATION_", 0};
34: typedef enum {
35: QUADRATURE_GAUSS1,
36: QUADRATURE_GAUSS2,
37: QUADRATURE_GAUSS3,
38: QUADRATURE_GAUSS4,
39: QUADRATURE_LOBATTO2,
40: QUADRATURE_LOBATTO3
41: } QuadratureType;
42: static const char *const QuadratureTypes[] = {"GAUSS1", "GAUSS2", "GAUSS3", "GAUSS4", "LOBATTO2", "LOBATTO3", "QuadratureType", "QUADRATURE_", 0};
44: typedef struct {
45: PetscScalar E; /* radiation energy */
46: PetscScalar T; /* material temperature */
47: } RDNode;
49: typedef struct {
50: PetscReal meter, kilogram, second, Kelvin; /* Fundamental units */
51: PetscReal Joule, Watt; /* Derived units */
52: } RDUnit;
54: typedef struct _n_RD *RD;
56: struct _n_RD {
57: void (*MaterialEnergy)(RD, const RDNode *, PetscScalar *, RDNode *);
58: DM da;
59: PetscBool monitor_residual;
60: DiscretizationType discretization;
61: QuadratureType quadrature;
62: JacobianType jacobian;
63: PetscInt initial;
64: BCType leftbc;
65: PetscBool view_draw;
66: char view_binary[PETSC_MAX_PATH_LEN];
67: PetscBool test_diff;
68: PetscBool endpoint;
69: PetscBool bclimit;
70: PetscBool bcmidpoint;
71: RDUnit unit;
73: /* model constants, see Table 2 and RDCreate() */
74: PetscReal rho, K_R, K_p, I_H, m_p, m_e, h, k, c, sigma_b, beta, gamma;
76: /* Domain and boundary conditions */
77: PetscReal Eapplied; /* Radiation flux from the left */
78: PetscReal L; /* Length of domain */
79: PetscReal final_time;
80: };
82: static PetscErrorCode RDDestroy(RD *rd)
83: {
84: PetscFunctionBeginUser;
85: PetscCall(DMDestroy(&(*rd)->da));
86: PetscCall(PetscFree(*rd));
87: PetscFunctionReturn(PETSC_SUCCESS);
88: }
90: /* The paper has a time derivative for material energy (Eq 2) which is a dependent variable (computable from temperature
91: * and density through an uninvertible relation). Computing this derivative is trivial for trapezoid rule (used in the
92: * paper), but does not generalize nicely to higher order integrators. Here we use the implicit form which provides
93: * time derivatives of the independent variables (radiation energy and temperature), so we must compute the time
94: * derivative of material energy ourselves (could be done using AD).
95: *
96: * There are multiple ionization models, this interface dispatches to the one currently in use.
97: */
98: static void RDMaterialEnergy(RD rd, const RDNode *n, PetscScalar *Em, RDNode *dEm)
99: {
100: rd->MaterialEnergy(rd, n, Em, dEm);
101: }
103: /* Solves a quadratic equation while propagating tangents */
104: static void QuadraticSolve(PetscScalar a, PetscScalar a_t, PetscScalar b, PetscScalar b_t, PetscScalar c, PetscScalar c_t, PetscScalar *x, PetscScalar *x_t)
105: {
106: PetscScalar disc = b * b - 4. * a * c, disc_t = 2. * b * b_t - 4. * a_t * c - 4. * a * c_t, num = -b + PetscSqrtScalar(disc), /* choose positive sign */
107: num_t = -b_t + 0.5 / PetscSqrtScalar(disc) * disc_t, den = 2. * a, den_t = 2. * a_t;
108: *x = num / den;
109: *x_t = (num_t * den - num * den_t) / PetscSqr(den);
110: }
112: /* The primary model presented in the paper */
113: static void RDMaterialEnergy_Saha(RD rd, const RDNode *n, PetscScalar *inEm, RDNode *dEm)
114: {
115: PetscScalar Em, alpha, alpha_t, T = n->T, T_t = 1., chi = rd->I_H / (rd->k * T), chi_t = -chi / T * T_t, a = 1., a_t = 0, b = 4. * rd->m_p / rd->rho * PetscPowScalarReal(2. * PETSC_PI * rd->m_e * rd->I_H / PetscSqr(rd->h), 1.5) * PetscExpScalar(-chi) * PetscPowScalarReal(chi, 1.5), /* Eq 7 */
116: b_t = -b * chi_t + 1.5 * b / chi * chi_t, c = -b, c_t = -b_t;
117: QuadraticSolve(a, a_t, b, b_t, c, c_t, &alpha, &alpha_t); /* Solve Eq 7 for alpha */
118: Em = rd->k * T / rd->m_p * (1.5 * (1. + alpha) + alpha * chi); /* Eq 6 */
119: if (inEm) *inEm = Em;
120: if (dEm) {
121: dEm->E = 0;
122: dEm->T = Em / T * T_t + rd->k * T / rd->m_p * (1.5 * alpha_t + alpha_t * chi + alpha * chi_t);
123: }
124: }
125: /* Reduced ionization model, Eq 30 */
126: static void RDMaterialEnergy_Reduced(RD rd, const RDNode *n, PetscScalar *Em, RDNode *dEm)
127: {
128: PetscScalar alpha, alpha_t, T = n->T, T_t = 1., chi = -0.3 / T, chi_t = -chi / T * T_t, a = 1., a_t = 0., b = PetscExpScalar(chi), b_t = b * chi_t, c = -b, c_t = -b_t;
129: QuadraticSolve(a, a_t, b, b_t, c, c_t, &alpha, &alpha_t);
130: if (Em) *Em = (1. + alpha) * T + 0.3 * alpha;
131: if (dEm) {
132: dEm->E = 0;
133: dEm->T = alpha_t * T + (1. + alpha) * T_t + 0.3 * alpha_t;
134: }
135: }
137: /* Eq 5 */
138: static void RDSigma_R(RD rd, RDNode *n, PetscScalar *sigma_R, RDNode *dsigma_R)
139: {
140: *sigma_R = rd->K_R * rd->rho * PetscPowScalar(n->T, -rd->gamma);
141: dsigma_R->E = 0;
142: dsigma_R->T = -rd->gamma * (*sigma_R) / n->T;
143: }
145: /* Eq 4 */
146: static void RDDiffusionCoefficient(RD rd, PetscBool limit, RDNode *n, RDNode *nx, PetscScalar *D_R, RDNode *dD_R, RDNode *dxD_R)
147: {
148: PetscScalar sigma_R, denom;
149: RDNode dsigma_R, ddenom, dxdenom;
151: RDSigma_R(rd, n, &sigma_R, &dsigma_R);
152: denom = 3. * rd->rho * sigma_R + (int)limit * PetscAbsScalar(nx->E) / n->E;
153: ddenom.E = -(int)limit * PetscAbsScalar(nx->E) / PetscSqr(n->E);
154: ddenom.T = 3. * rd->rho * dsigma_R.T;
155: dxdenom.E = (int)limit * (PetscRealPart(nx->E) < 0 ? -1. : 1.) / n->E;
156: dxdenom.T = 0;
157: *D_R = rd->c / denom;
158: if (dD_R) {
159: dD_R->E = -rd->c / PetscSqr(denom) * ddenom.E;
160: dD_R->T = -rd->c / PetscSqr(denom) * ddenom.T;
161: }
162: if (dxD_R) {
163: dxD_R->E = -rd->c / PetscSqr(denom) * dxdenom.E;
164: dxD_R->T = -rd->c / PetscSqr(denom) * dxdenom.T;
165: }
166: }
168: static PetscErrorCode RDStateView(RD rd, Vec X, Vec Xdot, Vec F)
169: {
170: DMDALocalInfo info;
171: PetscInt i;
172: const RDNode *x, *xdot, *f;
173: MPI_Comm comm;
175: PetscFunctionBeginUser;
176: PetscCall(PetscObjectGetComm((PetscObject)rd->da, &comm));
177: PetscCall(DMDAGetLocalInfo(rd->da, &info));
178: PetscCall(DMDAVecGetArrayRead(rd->da, X, (void *)&x));
179: PetscCall(DMDAVecGetArrayRead(rd->da, Xdot, (void *)&xdot));
180: PetscCall(DMDAVecGetArrayRead(rd->da, F, (void *)&f));
181: for (i = info.xs; i < info.xs + info.xm; i++) {
182: PetscCall(PetscSynchronizedPrintf(comm, "x[%" PetscInt_FMT "] (%10.2G,%10.2G) (%10.2G,%10.2G) (%10.2G,%10.2G)\n", i, (double)PetscRealPart(x[i].E), (double)PetscRealPart(x[i].T), (double)PetscRealPart(xdot[i].E), (double)PetscRealPart(xdot[i].T),
183: (double)PetscRealPart(f[i].E), (double)PetscRealPart(f[i].T)));
184: }
185: PetscCall(DMDAVecRestoreArrayRead(rd->da, X, (void *)&x));
186: PetscCall(DMDAVecRestoreArrayRead(rd->da, Xdot, (void *)&xdot));
187: PetscCall(DMDAVecRestoreArrayRead(rd->da, F, (void *)&f));
188: PetscCall(PetscSynchronizedFlush(comm, PETSC_STDOUT));
189: PetscFunctionReturn(PETSC_SUCCESS);
190: }
192: static PetscScalar RDRadiation(RD rd, const RDNode *n, RDNode *dn)
193: {
194: PetscScalar sigma_p = rd->K_p * rd->rho * PetscPowScalar(n->T, -rd->beta), sigma_p_T = -rd->beta * sigma_p / n->T, tmp = 4. * rd->sigma_b * PetscSqr(PetscSqr(n->T)) / rd->c - n->E, tmp_E = -1., tmp_T = 4. * rd->sigma_b * 4 * n->T * (PetscSqr(n->T)) / rd->c, rad = sigma_p * rd->c * rd->rho * tmp, rad_E = sigma_p * rd->c * rd->rho * tmp_E, rad_T = rd->c * rd->rho * (sigma_p_T * tmp + sigma_p * tmp_T);
195: if (dn) {
196: dn->E = rad_E;
197: dn->T = rad_T;
198: }
199: return rad;
200: }
202: static PetscScalar RDDiffusion(RD rd, PetscReal hx, const RDNode x[], PetscInt i, RDNode d[])
203: {
204: PetscReal ihx = 1. / hx;
205: RDNode n_L, nx_L, n_R, nx_R, dD_L, dxD_L, dD_R, dxD_R, dfluxL[2], dfluxR[2];
206: PetscScalar D_L, D_R, fluxL, fluxR;
208: n_L.E = 0.5 * (x[i - 1].E + x[i].E);
209: n_L.T = 0.5 * (x[i - 1].T + x[i].T);
210: nx_L.E = (x[i].E - x[i - 1].E) / hx;
211: nx_L.T = (x[i].T - x[i - 1].T) / hx;
212: RDDiffusionCoefficient(rd, PETSC_TRUE, &n_L, &nx_L, &D_L, &dD_L, &dxD_L);
213: fluxL = D_L * nx_L.E;
214: dfluxL[0].E = -ihx * D_L + (0.5 * dD_L.E - ihx * dxD_L.E) * nx_L.E;
215: dfluxL[1].E = +ihx * D_L + (0.5 * dD_L.E + ihx * dxD_L.E) * nx_L.E;
216: dfluxL[0].T = (0.5 * dD_L.T - ihx * dxD_L.T) * nx_L.E;
217: dfluxL[1].T = (0.5 * dD_L.T + ihx * dxD_L.T) * nx_L.E;
219: n_R.E = 0.5 * (x[i].E + x[i + 1].E);
220: n_R.T = 0.5 * (x[i].T + x[i + 1].T);
221: nx_R.E = (x[i + 1].E - x[i].E) / hx;
222: nx_R.T = (x[i + 1].T - x[i].T) / hx;
223: RDDiffusionCoefficient(rd, PETSC_TRUE, &n_R, &nx_R, &D_R, &dD_R, &dxD_R);
224: fluxR = D_R * nx_R.E;
225: dfluxR[0].E = -ihx * D_R + (0.5 * dD_R.E - ihx * dxD_R.E) * nx_R.E;
226: dfluxR[1].E = +ihx * D_R + (0.5 * dD_R.E + ihx * dxD_R.E) * nx_R.E;
227: dfluxR[0].T = (0.5 * dD_R.T - ihx * dxD_R.T) * nx_R.E;
228: dfluxR[1].T = (0.5 * dD_R.T + ihx * dxD_R.T) * nx_R.E;
230: if (d) {
231: d[0].E = -ihx * dfluxL[0].E;
232: d[0].T = -ihx * dfluxL[0].T;
233: d[1].E = ihx * (dfluxR[0].E - dfluxL[1].E);
234: d[1].T = ihx * (dfluxR[0].T - dfluxL[1].T);
235: d[2].E = ihx * dfluxR[1].E;
236: d[2].T = ihx * dfluxR[1].T;
237: }
238: return ihx * (fluxR - fluxL);
239: }
241: static PetscErrorCode RDGetLocalArrays(RD rd, TS ts, Vec X, Vec Xdot, PetscReal *Theta, PetscReal *dt, Vec *X0loc, RDNode **x0, Vec *Xloc, RDNode **x, Vec *Xloc_t, RDNode **xdot)
242: {
243: PetscBool istheta;
245: PetscFunctionBeginUser;
246: PetscCall(DMGetLocalVector(rd->da, X0loc));
247: PetscCall(DMGetLocalVector(rd->da, Xloc));
248: PetscCall(DMGetLocalVector(rd->da, Xloc_t));
250: PetscCall(DMGlobalToLocalBegin(rd->da, X, INSERT_VALUES, *Xloc));
251: PetscCall(DMGlobalToLocalEnd(rd->da, X, INSERT_VALUES, *Xloc));
252: PetscCall(DMGlobalToLocalBegin(rd->da, Xdot, INSERT_VALUES, *Xloc_t));
253: PetscCall(DMGlobalToLocalEnd(rd->da, Xdot, INSERT_VALUES, *Xloc_t));
255: /*
256: The following is a hack to subvert TSTHETA which is like an implicit midpoint method to behave more like a trapezoid
257: rule. These methods have equivalent linear stability, but the nonlinear stability is somewhat different. The
258: radiation system is inconvenient to write in explicit form because the ionization model is "on the left".
259: */
260: PetscCall(PetscObjectTypeCompare((PetscObject)ts, TSTHETA, &istheta));
261: if (istheta && rd->endpoint) PetscCall(TSThetaGetTheta(ts, Theta));
262: else *Theta = 1.;
264: PetscCall(TSGetTimeStep(ts, dt));
265: PetscCall(VecWAXPY(*X0loc, -(*Theta) * (*dt), *Xloc_t, *Xloc)); /* back out the value at the start of this step */
266: if (rd->endpoint) PetscCall(VecWAXPY(*Xloc, *dt, *Xloc_t, *X0loc)); /* move the abscissa to the end of the step */
268: PetscCall(DMDAVecGetArray(rd->da, *X0loc, x0));
269: PetscCall(DMDAVecGetArray(rd->da, *Xloc, x));
270: PetscCall(DMDAVecGetArray(rd->da, *Xloc_t, xdot));
271: PetscFunctionReturn(PETSC_SUCCESS);
272: }
274: static PetscErrorCode RDRestoreLocalArrays(RD rd, Vec *X0loc, RDNode **x0, Vec *Xloc, RDNode **x, Vec *Xloc_t, RDNode **xdot)
275: {
276: PetscFunctionBeginUser;
277: PetscCall(DMDAVecRestoreArray(rd->da, *X0loc, x0));
278: PetscCall(DMDAVecRestoreArray(rd->da, *Xloc, x));
279: PetscCall(DMDAVecRestoreArray(rd->da, *Xloc_t, xdot));
280: PetscCall(DMRestoreLocalVector(rd->da, X0loc));
281: PetscCall(DMRestoreLocalVector(rd->da, Xloc));
282: PetscCall(DMRestoreLocalVector(rd->da, Xloc_t));
283: PetscFunctionReturn(PETSC_SUCCESS);
284: }
286: static PetscErrorCode PETSC_UNUSED RDCheckDomain_Private(RD rd, TS ts, Vec X, PetscBool *in)
287: {
288: PetscInt minloc;
289: PetscReal min;
291: PetscFunctionBeginUser;
292: PetscCall(VecMin(X, &minloc, &min));
293: if (min < 0) {
294: SNES snes;
295: *in = PETSC_FALSE;
296: PetscCall(TSGetSNES(ts, &snes));
297: PetscCall(SNESSetFunctionDomainError(snes));
298: PetscCall(PetscInfo(ts, "Domain violation at %" PetscInt_FMT " field %" PetscInt_FMT " value %g\n", minloc / 2, minloc % 2, (double)min));
299: } else *in = PETSC_TRUE;
300: PetscFunctionReturn(PETSC_SUCCESS);
301: }
303: /* Energy and temperature must remain positive */
304: #define RDCheckDomain(rd, ts, X) \
305: do { \
306: PetscBool _in; \
307: PetscCall(RDCheckDomain_Private(rd, ts, X, &_in)); \
308: if (!_in) PetscFunctionReturn(PETSC_SUCCESS); \
309: } while (0)
311: static PetscErrorCode RDIFunction_FD(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
312: {
313: RD rd = (RD)ctx;
314: RDNode *x, *x0, *xdot, *f;
315: Vec X0loc, Xloc, Xloc_t;
316: PetscReal hx, Theta, dt;
317: DMDALocalInfo info;
318: PetscInt i;
320: PetscFunctionBeginUser;
321: PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
322: PetscCall(DMDAVecGetArray(rd->da, F, &f));
323: PetscCall(DMDAGetLocalInfo(rd->da, &info));
324: PetscCall(VecZeroEntries(F));
326: hx = rd->L / (info.mx - 1);
328: for (i = info.xs; i < info.xs + info.xm; i++) {
329: PetscReal rho = rd->rho;
330: PetscScalar Em_t, rad;
332: rad = (1. - Theta) * RDRadiation(rd, &x0[i], 0) + Theta * RDRadiation(rd, &x[i], 0);
333: if (rd->endpoint) {
334: PetscScalar Em0, Em1;
335: RDMaterialEnergy(rd, &x0[i], &Em0, NULL);
336: RDMaterialEnergy(rd, &x[i], &Em1, NULL);
337: Em_t = (Em1 - Em0) / dt;
338: } else {
339: RDNode dEm;
340: RDMaterialEnergy(rd, &x[i], NULL, &dEm);
341: Em_t = dEm.E * xdot[i].E + dEm.T * xdot[i].T;
342: }
343: /* Residuals are multiplied by the volume element (hx). */
344: /* The temperature equation does not have boundary conditions */
345: f[i].T = hx * (rho * Em_t + rad);
347: if (i == 0) { /* Left boundary condition */
348: PetscScalar D_R, bcTheta = rd->bcmidpoint ? Theta : 1.;
349: RDNode n, nx;
351: n.E = (1. - bcTheta) * x0[0].E + bcTheta * x[0].E;
352: n.T = (1. - bcTheta) * x0[0].T + bcTheta * x[0].T;
353: nx.E = ((1. - bcTheta) * (x0[1].E - x0[0].E) + bcTheta * (x[1].E - x[0].E)) / hx;
354: nx.T = ((1. - bcTheta) * (x0[1].T - x0[0].T) + bcTheta * (x[1].T - x[0].T)) / hx;
355: switch (rd->leftbc) {
356: case BC_ROBIN:
357: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D_R, 0, 0);
358: f[0].E = hx * (n.E - 2. * D_R * nx.E - rd->Eapplied);
359: break;
360: case BC_NEUMANN:
361: f[0].E = x[1].E - x[0].E;
362: break;
363: default:
364: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
365: }
366: } else if (i == info.mx - 1) { /* Right boundary */
367: f[i].E = x[i].E - x[i - 1].E; /* Homogeneous Neumann */
368: } else {
369: PetscScalar diff = (1. - Theta) * RDDiffusion(rd, hx, x0, i, 0) + Theta * RDDiffusion(rd, hx, x, i, 0);
370: f[i].E = hx * (xdot[i].E - diff - rad);
371: }
372: }
373: PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
374: PetscCall(DMDAVecRestoreArray(rd->da, F, &f));
375: if (rd->monitor_residual) PetscCall(RDStateView(rd, X, Xdot, F));
376: PetscFunctionReturn(PETSC_SUCCESS);
377: }
379: static PetscErrorCode RDIJacobian_FD(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx)
380: {
381: RD rd = (RD)ctx;
382: RDNode *x, *x0, *xdot;
383: Vec X0loc, Xloc, Xloc_t;
384: PetscReal hx, Theta, dt;
385: DMDALocalInfo info;
386: PetscInt i;
388: PetscFunctionBeginUser;
389: PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
390: PetscCall(DMDAGetLocalInfo(rd->da, &info));
391: hx = rd->L / (info.mx - 1);
392: PetscCall(MatZeroEntries(B));
394: for (i = info.xs; i < info.xs + info.xm; i++) {
395: PetscInt col[3];
396: PetscReal rho = rd->rho;
397: PetscScalar /*Em_t,rad,*/ K[2][6];
398: RDNode dEm_t, drad;
400: /*rad = (1.-Theta)* */ RDRadiation(rd, &x0[i], 0); /* + Theta* */
401: RDRadiation(rd, &x[i], &drad);
403: if (rd->endpoint) {
404: PetscScalar Em0, Em1;
405: RDNode dEm1;
406: RDMaterialEnergy(rd, &x0[i], &Em0, NULL);
407: RDMaterialEnergy(rd, &x[i], &Em1, &dEm1);
408: /*Em_t = (Em1 - Em0) / (Theta*dt);*/
409: dEm_t.E = dEm1.E / (Theta * dt);
410: dEm_t.T = dEm1.T / (Theta * dt);
411: } else {
412: const PetscScalar epsilon = x[i].T * PETSC_SQRT_MACHINE_EPSILON;
413: RDNode n1;
414: RDNode dEm, dEm1;
415: PetscScalar Em_TT;
417: n1.E = x[i].E;
418: n1.T = x[i].T + epsilon;
419: RDMaterialEnergy(rd, &x[i], NULL, &dEm);
420: RDMaterialEnergy(rd, &n1, NULL, &dEm1);
421: /* The Jacobian needs another derivative. We finite difference here instead of
422: * propagating second derivatives through the ionization model. */
423: Em_TT = (dEm1.T - dEm.T) / epsilon;
424: /*Em_t = dEm.E * xdot[i].E + dEm.T * xdot[i].T;*/
425: dEm_t.E = dEm.E * a;
426: dEm_t.T = dEm.T * a + Em_TT * xdot[i].T;
427: }
429: PetscCall(PetscMemzero(K, sizeof(K)));
430: /* Residuals are multiplied by the volume element (hx). */
431: if (i == 0) {
432: PetscScalar D, bcTheta = rd->bcmidpoint ? Theta : 1.;
433: RDNode n, nx;
434: RDNode dD, dxD;
436: n.E = (1. - bcTheta) * x0[0].E + bcTheta * x[0].E;
437: n.T = (1. - bcTheta) * x0[0].T + bcTheta * x[0].T;
438: nx.E = ((1. - bcTheta) * (x0[1].E - x0[0].E) + bcTheta * (x[1].E - x[0].E)) / hx;
439: nx.T = ((1. - bcTheta) * (x0[1].T - x0[0].T) + bcTheta * (x[1].T - x[0].T)) / hx;
440: switch (rd->leftbc) {
441: case BC_ROBIN:
442: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, &dD, &dxD);
443: K[0][1 * 2 + 0] = (bcTheta / Theta) * hx * (1. - 2. * D * (-1. / hx) - 2. * nx.E * dD.E + 2. * nx.E * dxD.E / hx);
444: K[0][1 * 2 + 1] = (bcTheta / Theta) * hx * (-2. * nx.E * dD.T);
445: K[0][2 * 2 + 0] = (bcTheta / Theta) * hx * (-2. * D * (1. / hx) - 2. * nx.E * dD.E - 2. * nx.E * dxD.E / hx);
446: break;
447: case BC_NEUMANN:
448: K[0][1 * 2 + 0] = -1. / Theta;
449: K[0][2 * 2 + 0] = 1. / Theta;
450: break;
451: default:
452: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
453: }
454: } else if (i == info.mx - 1) {
455: K[0][0 * 2 + 0] = -1. / Theta;
456: K[0][1 * 2 + 0] = 1. / Theta;
457: } else {
458: /*PetscScalar diff;*/
459: RDNode ddiff[3];
460: /*diff = (1.-Theta)*RDDiffusion(rd,hx,x0,i,0) + Theta* */ RDDiffusion(rd, hx, x, i, ddiff);
461: K[0][0 * 2 + 0] = -hx * ddiff[0].E;
462: K[0][0 * 2 + 1] = -hx * ddiff[0].T;
463: K[0][1 * 2 + 0] = hx * (a - ddiff[1].E - drad.E);
464: K[0][1 * 2 + 1] = hx * (-ddiff[1].T - drad.T);
465: K[0][2 * 2 + 0] = -hx * ddiff[2].E;
466: K[0][2 * 2 + 1] = -hx * ddiff[2].T;
467: }
469: K[1][1 * 2 + 0] = hx * (rho * dEm_t.E + drad.E);
470: K[1][1 * 2 + 1] = hx * (rho * dEm_t.T + drad.T);
472: col[0] = i - 1;
473: col[1] = i;
474: col[2] = i + 1 < info.mx ? i + 1 : -1;
475: PetscCall(MatSetValuesBlocked(B, 1, &i, 3, col, &K[0][0], INSERT_VALUES));
476: }
477: PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
478: PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
479: PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
480: if (A != B) {
481: PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
482: PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
483: }
484: PetscFunctionReturn(PETSC_SUCCESS);
485: }
487: /* Evaluate interpolants and derivatives at a select quadrature point */
488: static void RDEvaluate(PetscReal interp[][2], PetscReal deriv[][2], PetscInt q, const RDNode x[], PetscInt i, RDNode *n, RDNode *nx)
489: {
490: PetscInt j;
491: n->E = 0;
492: n->T = 0;
493: nx->E = 0;
494: nx->T = 0;
495: for (j = 0; j < 2; j++) {
496: n->E += interp[q][j] * x[i + j].E;
497: n->T += interp[q][j] * x[i + j].T;
498: nx->E += deriv[q][j] * x[i + j].E;
499: nx->T += deriv[q][j] * x[i + j].T;
500: }
501: }
503: /*
504: Various quadrature rules. The nonlinear terms are non-polynomial so no standard quadrature will be exact.
505: */
506: static PetscErrorCode RDGetQuadrature(RD rd, PetscReal hx, PetscInt *nq, PetscReal weight[], PetscReal interp[][2], PetscReal deriv[][2])
507: {
508: PetscInt q, j;
509: const PetscReal *refweight, (*refinterp)[2], (*refderiv)[2];
511: PetscFunctionBeginUser;
512: switch (rd->quadrature) {
513: case QUADRATURE_GAUSS1: {
514: static const PetscReal ww[1] = {1.};
515: static const PetscReal ii[1][2] = {
516: {0.5, 0.5}
517: };
518: static const PetscReal dd[1][2] = {
519: {-1., 1.}
520: };
521: *nq = 1;
522: refweight = ww;
523: refinterp = ii;
524: refderiv = dd;
525: } break;
526: case QUADRATURE_GAUSS2: {
527: static const PetscReal ii[2][2] = {
528: {0.78867513459481287, 0.21132486540518713},
529: {0.21132486540518713, 0.78867513459481287}
530: };
531: static const PetscReal dd[2][2] = {
532: {-1., 1.},
533: {-1., 1.}
534: };
535: static const PetscReal ww[2] = {0.5, 0.5};
536: *nq = 2;
537: refweight = ww;
538: refinterp = ii;
539: refderiv = dd;
540: } break;
541: case QUADRATURE_GAUSS3: {
542: static const PetscReal ii[3][2] = {
543: {0.8872983346207417, 0.1127016653792583},
544: {0.5, 0.5 },
545: {0.1127016653792583, 0.8872983346207417}
546: };
547: static const PetscReal dd[3][2] = {
548: {-1, 1},
549: {-1, 1},
550: {-1, 1}
551: };
552: static const PetscReal ww[3] = {5. / 18, 8. / 18, 5. / 18};
553: *nq = 3;
554: refweight = ww;
555: refinterp = ii;
556: refderiv = dd;
557: } break;
558: case QUADRATURE_GAUSS4: {
559: static const PetscReal ii[][2] = {
560: {0.93056815579702623, 0.069431844202973658},
561: {0.66999052179242813, 0.33000947820757187 },
562: {0.33000947820757187, 0.66999052179242813 },
563: {0.069431844202973658, 0.93056815579702623 }
564: };
565: static const PetscReal dd[][2] = {
566: {-1, 1},
567: {-1, 1},
568: {-1, 1},
569: {-1, 1}
570: };
571: static const PetscReal ww[] = {0.17392742256872692, 0.3260725774312731, 0.3260725774312731, 0.17392742256872692};
573: *nq = 4;
574: refweight = ww;
575: refinterp = ii;
576: refderiv = dd;
577: } break;
578: case QUADRATURE_LOBATTO2: {
579: static const PetscReal ii[2][2] = {
580: {1., 0.},
581: {0., 1.}
582: };
583: static const PetscReal dd[2][2] = {
584: {-1., 1.},
585: {-1., 1.}
586: };
587: static const PetscReal ww[2] = {0.5, 0.5};
588: *nq = 2;
589: refweight = ww;
590: refinterp = ii;
591: refderiv = dd;
592: } break;
593: case QUADRATURE_LOBATTO3: {
594: static const PetscReal ii[3][2] = {
595: {1, 0 },
596: {0.5, 0.5},
597: {0, 1 }
598: };
599: static const PetscReal dd[3][2] = {
600: {-1, 1},
601: {-1, 1},
602: {-1, 1}
603: };
604: static const PetscReal ww[3] = {1. / 6, 4. / 6, 1. / 6};
605: *nq = 3;
606: refweight = ww;
607: refinterp = ii;
608: refderiv = dd;
609: } break;
610: default:
611: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Unknown quadrature %d", (int)rd->quadrature);
612: }
614: for (q = 0; q < *nq; q++) {
615: weight[q] = refweight[q] * hx;
616: for (j = 0; j < 2; j++) {
617: interp[q][j] = refinterp[q][j];
618: deriv[q][j] = refderiv[q][j] / hx;
619: }
620: }
621: PetscFunctionReturn(PETSC_SUCCESS);
622: }
624: /*
625: Finite element version
626: */
627: static PetscErrorCode RDIFunction_FE(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
628: {
629: RD rd = (RD)ctx;
630: RDNode *x, *x0, *xdot, *f;
631: Vec X0loc, Xloc, Xloc_t, Floc;
632: PetscReal hx, Theta, dt, weight[5], interp[5][2], deriv[5][2];
633: DMDALocalInfo info;
634: PetscInt i, j, q, nq;
636: PetscFunctionBeginUser;
637: PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
639: PetscCall(DMGetLocalVector(rd->da, &Floc));
640: PetscCall(VecZeroEntries(Floc));
641: PetscCall(DMDAVecGetArray(rd->da, Floc, &f));
642: PetscCall(DMDAGetLocalInfo(rd->da, &info));
644: /* Set up shape functions and quadrature for elements (assumes a uniform grid) */
645: hx = rd->L / (info.mx - 1);
646: PetscCall(RDGetQuadrature(rd, hx, &nq, weight, interp, deriv));
648: for (i = info.xs; i < PetscMin(info.xs + info.xm, info.mx - 1); i++) {
649: for (q = 0; q < nq; q++) {
650: PetscReal rho = rd->rho;
651: PetscScalar Em_t, rad, D_R, D0_R;
652: RDNode n, n0, nx, n0x, nt, ntx;
653: RDEvaluate(interp, deriv, q, x, i, &n, &nx);
654: RDEvaluate(interp, deriv, q, x0, i, &n0, &n0x);
655: RDEvaluate(interp, deriv, q, xdot, i, &nt, &ntx);
657: rad = (1. - Theta) * RDRadiation(rd, &n0, 0) + Theta * RDRadiation(rd, &n, 0);
658: if (rd->endpoint) {
659: PetscScalar Em0, Em1;
660: RDMaterialEnergy(rd, &n0, &Em0, NULL);
661: RDMaterialEnergy(rd, &n, &Em1, NULL);
662: Em_t = (Em1 - Em0) / dt;
663: } else {
664: RDNode dEm;
665: RDMaterialEnergy(rd, &n, NULL, &dEm);
666: Em_t = dEm.E * nt.E + dEm.T * nt.T;
667: }
668: RDDiffusionCoefficient(rd, PETSC_TRUE, &n0, &n0x, &D0_R, 0, 0);
669: RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, 0, 0);
670: for (j = 0; j < 2; j++) {
671: f[i + j].E += (deriv[q][j] * weight[q] * ((1. - Theta) * D0_R * n0x.E + Theta * D_R * nx.E) + interp[q][j] * weight[q] * (nt.E - rad));
672: f[i + j].T += interp[q][j] * weight[q] * (rho * Em_t + rad);
673: }
674: }
675: }
676: if (info.xs == 0) {
677: switch (rd->leftbc) {
678: case BC_ROBIN: {
679: PetscScalar D_R, D_R_bc;
680: PetscReal ratio, bcTheta = rd->bcmidpoint ? Theta : 1.;
681: RDNode n, nx;
683: n.E = (1 - bcTheta) * x0[0].E + bcTheta * x[0].E;
684: n.T = (1 - bcTheta) * x0[0].T + bcTheta * x[0].T;
685: nx.E = (x[1].E - x[0].E) / hx;
686: nx.T = (x[1].T - x[0].T) / hx;
687: RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, 0, 0);
688: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D_R_bc, 0, 0);
689: ratio = PetscRealPart(D_R / D_R_bc);
690: PetscCheck(ratio <= 1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Limited diffusivity is greater than unlimited");
691: PetscCheck(ratio >= 1e-3, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Heavily limited diffusivity");
692: f[0].E += -ratio * 0.5 * (rd->Eapplied - n.E);
693: } break;
694: case BC_NEUMANN:
695: /* homogeneous Neumann is the natural condition */
696: break;
697: default:
698: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
699: }
700: }
702: PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
703: PetscCall(DMDAVecRestoreArray(rd->da, Floc, &f));
704: PetscCall(VecZeroEntries(F));
705: PetscCall(DMLocalToGlobalBegin(rd->da, Floc, ADD_VALUES, F));
706: PetscCall(DMLocalToGlobalEnd(rd->da, Floc, ADD_VALUES, F));
707: PetscCall(DMRestoreLocalVector(rd->da, &Floc));
709: if (rd->monitor_residual) PetscCall(RDStateView(rd, X, Xdot, F));
710: PetscFunctionReturn(PETSC_SUCCESS);
711: }
713: static PetscErrorCode RDIJacobian_FE(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx)
714: {
715: RD rd = (RD)ctx;
716: RDNode *x, *x0, *xdot;
717: Vec X0loc, Xloc, Xloc_t;
718: PetscReal hx, Theta, dt, weight[5], interp[5][2], deriv[5][2];
719: DMDALocalInfo info;
720: PetscInt i, j, k, q, nq;
721: PetscScalar K[4][4];
723: PetscFunctionBeginUser;
724: PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
725: PetscCall(DMDAGetLocalInfo(rd->da, &info));
726: hx = rd->L / (info.mx - 1);
727: PetscCall(RDGetQuadrature(rd, hx, &nq, weight, interp, deriv));
728: PetscCall(MatZeroEntries(B));
729: for (i = info.xs; i < PetscMin(info.xs + info.xm, info.mx - 1); i++) {
730: PetscInt rc[2];
732: rc[0] = i;
733: rc[1] = i + 1;
734: PetscCall(PetscMemzero(K, sizeof(K)));
735: for (q = 0; q < nq; q++) {
736: PetscScalar D_R;
737: PETSC_UNUSED PetscScalar rad;
738: RDNode n, nx, nt, ntx, drad, dD_R, dxD_R, dEm;
739: RDEvaluate(interp, deriv, q, x, i, &n, &nx);
740: RDEvaluate(interp, deriv, q, xdot, i, &nt, &ntx);
741: rad = RDRadiation(rd, &n, &drad);
742: RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, &dD_R, &dxD_R);
743: RDMaterialEnergy(rd, &n, NULL, &dEm);
744: for (j = 0; j < 2; j++) {
745: for (k = 0; k < 2; k++) {
746: K[j * 2 + 0][k * 2 + 0] += (+interp[q][j] * weight[q] * (a - drad.E) * interp[q][k] + deriv[q][j] * weight[q] * ((D_R + dxD_R.E * nx.E) * deriv[q][k] + dD_R.E * nx.E * interp[q][k]));
747: K[j * 2 + 0][k * 2 + 1] += (+interp[q][j] * weight[q] * (-drad.T * interp[q][k]) + deriv[q][j] * weight[q] * (dxD_R.T * deriv[q][k] + dD_R.T * interp[q][k]) * nx.E);
748: K[j * 2 + 1][k * 2 + 0] += interp[q][j] * weight[q] * drad.E * interp[q][k];
749: K[j * 2 + 1][k * 2 + 1] += interp[q][j] * weight[q] * (a * rd->rho * dEm.T + drad.T) * interp[q][k];
750: }
751: }
752: }
753: PetscCall(MatSetValuesBlocked(B, 2, rc, 2, rc, &K[0][0], ADD_VALUES));
754: }
755: if (info.xs == 0) {
756: switch (rd->leftbc) {
757: case BC_ROBIN: {
758: PetscScalar D_R, D_R_bc;
759: PetscReal ratio;
760: RDNode n, nx;
762: n.E = (1 - Theta) * x0[0].E + Theta * x[0].E;
763: n.T = (1 - Theta) * x0[0].T + Theta * x[0].T;
764: nx.E = (x[1].E - x[0].E) / hx;
765: nx.T = (x[1].T - x[0].T) / hx;
766: RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, 0, 0);
767: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D_R_bc, 0, 0);
768: ratio = PetscRealPart(D_R / D_R_bc);
769: PetscCall(MatSetValue(B, 0, 0, ratio * 0.5, ADD_VALUES));
770: } break;
771: case BC_NEUMANN:
772: /* homogeneous Neumann is the natural condition */
773: break;
774: default:
775: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
776: }
777: }
779: PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
780: PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
781: PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
782: if (A != B) {
783: PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
784: PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
785: }
786: PetscFunctionReturn(PETSC_SUCCESS);
787: }
789: /* Temperature that is in equilibrium with the radiation density */
790: static PetscScalar RDRadiationTemperature(RD rd, PetscScalar E)
791: {
792: return PetscPowScalar(E * rd->c / (4. * rd->sigma_b), 0.25);
793: }
795: static PetscErrorCode RDInitialState(RD rd, Vec X)
796: {
797: DMDALocalInfo info;
798: PetscInt i;
799: RDNode *x;
801: PetscFunctionBeginUser;
802: PetscCall(DMDAGetLocalInfo(rd->da, &info));
803: PetscCall(DMDAVecGetArray(rd->da, X, &x));
804: for (i = info.xs; i < info.xs + info.xm; i++) {
805: PetscReal coord = i * rd->L / (info.mx - 1);
806: switch (rd->initial) {
807: case 1:
808: x[i].E = 0.001;
809: x[i].T = RDRadiationTemperature(rd, x[i].E);
810: break;
811: case 2:
812: x[i].E = 0.001 + 100. * PetscExpReal(-PetscSqr(coord / 0.1));
813: x[i].T = RDRadiationTemperature(rd, x[i].E);
814: break;
815: case 3:
816: x[i].E = 7.56e-2 * rd->unit.Joule / PetscPowScalarInt(rd->unit.meter, 3);
817: x[i].T = RDRadiationTemperature(rd, x[i].E);
818: break;
819: default:
820: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "No initial state %" PetscInt_FMT, rd->initial);
821: }
822: }
823: PetscCall(DMDAVecRestoreArray(rd->da, X, &x));
824: PetscFunctionReturn(PETSC_SUCCESS);
825: }
827: static PetscErrorCode RDView(RD rd, Vec X, PetscViewer viewer)
828: {
829: Vec Y;
830: const RDNode *x;
831: PetscScalar *y;
832: PetscInt i, m, M;
833: const PetscInt *lx;
834: DM da;
835: MPI_Comm comm;
837: PetscFunctionBeginUser;
838: /*
839: Create a DMDA (one dof per node, zero stencil width, same layout) to hold Trad
840: (radiation temperature). It is not necessary to create a DMDA for this, but this way
841: output and visualization will have meaningful variable names and correct scales.
842: */
843: PetscCall(DMDAGetInfo(rd->da, 0, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
844: PetscCall(DMDAGetOwnershipRanges(rd->da, &lx, 0, 0));
845: PetscCall(PetscObjectGetComm((PetscObject)rd->da, &comm));
846: PetscCall(DMDACreate1d(comm, DM_BOUNDARY_NONE, M, 1, 0, lx, &da));
847: PetscCall(DMSetFromOptions(da));
848: PetscCall(DMSetUp(da));
849: PetscCall(DMDASetUniformCoordinates(da, 0., rd->L, 0., 0., 0., 0.));
850: PetscCall(DMDASetFieldName(da, 0, "T_rad"));
851: PetscCall(DMCreateGlobalVector(da, &Y));
853: /* Compute the radiation temperature from the solution at each node */
854: PetscCall(VecGetLocalSize(Y, &m));
855: PetscCall(VecGetArrayRead(X, (const PetscScalar **)&x));
856: PetscCall(VecGetArray(Y, &y));
857: for (i = 0; i < m; i++) y[i] = RDRadiationTemperature(rd, x[i].E);
858: PetscCall(VecRestoreArrayRead(X, (const PetscScalar **)&x));
859: PetscCall(VecRestoreArray(Y, &y));
861: PetscCall(VecView(Y, viewer));
862: PetscCall(VecDestroy(&Y));
863: PetscCall(DMDestroy(&da));
864: PetscFunctionReturn(PETSC_SUCCESS);
865: }
867: static PetscErrorCode RDTestDifferentiation(RD rd)
868: {
869: MPI_Comm comm;
870: RDNode n, nx;
871: PetscScalar epsilon;
873: PetscFunctionBeginUser;
874: PetscCall(PetscObjectGetComm((PetscObject)rd->da, &comm));
875: epsilon = 1e-8;
876: {
877: RDNode dEm, fdEm;
878: PetscScalar T0 = 1000., T1 = T0 * (1. + epsilon), Em0, Em1;
879: n.E = 1.;
880: n.T = T0;
881: rd->MaterialEnergy(rd, &n, &Em0, &dEm);
882: n.E = 1. + epsilon;
883: n.T = T0;
884: rd->MaterialEnergy(rd, &n, &Em1, 0);
885: fdEm.E = (Em1 - Em0) / epsilon;
886: n.E = 1.;
887: n.T = T1;
888: rd->MaterialEnergy(rd, &n, &Em1, 0);
889: fdEm.T = (Em1 - Em0) / (T0 * epsilon);
890: PetscCall(PetscPrintf(comm, "dEm {%g,%g}, fdEm {%g,%g}, diff {%g,%g}\n", (double)PetscRealPart(dEm.E), (double)PetscRealPart(dEm.T), (double)PetscRealPart(fdEm.E), (double)PetscRealPart(fdEm.T), (double)PetscRealPart(dEm.E - fdEm.E),
891: (double)PetscRealPart(dEm.T - fdEm.T)));
892: }
893: {
894: PetscScalar D0, D;
895: RDNode dD, dxD, fdD, fdxD;
896: n.E = 1.;
897: n.T = 1.;
898: nx.E = 1.;
899: n.T = 1.;
900: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D0, &dD, &dxD);
901: n.E = 1. + epsilon;
902: n.T = 1.;
903: nx.E = 1.;
904: n.T = 1.;
905: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
906: fdD.E = (D - D0) / epsilon;
907: n.E = 1;
908: n.T = 1. + epsilon;
909: nx.E = 1.;
910: n.T = 1.;
911: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
912: fdD.T = (D - D0) / epsilon;
913: n.E = 1;
914: n.T = 1.;
915: nx.E = 1. + epsilon;
916: n.T = 1.;
917: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
918: fdxD.E = (D - D0) / epsilon;
919: n.E = 1;
920: n.T = 1.;
921: nx.E = 1.;
922: n.T = 1. + epsilon;
923: RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
924: fdxD.T = (D - D0) / epsilon;
925: PetscCall(PetscPrintf(comm, "dD {%g,%g}, fdD {%g,%g}, diff {%g,%g}\n", (double)PetscRealPart(dD.E), (double)PetscRealPart(dD.T), (double)PetscRealPart(fdD.E), (double)PetscRealPart(fdD.T), (double)PetscRealPart(dD.E - fdD.E),
926: (double)PetscRealPart(dD.T - fdD.T)));
927: PetscCall(PetscPrintf(comm, "dxD {%g,%g}, fdxD {%g,%g}, diffx {%g,%g}\n", (double)PetscRealPart(dxD.E), (double)PetscRealPart(dxD.T), (double)PetscRealPart(fdxD.E), (double)PetscRealPart(fdxD.T), (double)PetscRealPart(dxD.E - fdxD.E),
928: (double)PetscRealPart(dxD.T - fdxD.T)));
929: }
930: {
931: PetscInt i;
932: PetscReal hx = 1.;
933: PetscScalar a0;
934: RDNode n0[3], n1[3], d[3], fd[3];
936: n0[0].E = 1.;
937: n0[0].T = 1.;
938: n0[1].E = 5.;
939: n0[1].T = 3.;
940: n0[2].E = 4.;
941: n0[2].T = 2.;
942: a0 = RDDiffusion(rd, hx, n0, 1, d);
943: for (i = 0; i < 3; i++) {
944: PetscCall(PetscMemcpy(n1, n0, sizeof(n0)));
945: n1[i].E += epsilon;
946: fd[i].E = (RDDiffusion(rd, hx, n1, 1, 0) - a0) / epsilon;
947: PetscCall(PetscMemcpy(n1, n0, sizeof(n0)));
948: n1[i].T += epsilon;
949: fd[i].T = (RDDiffusion(rd, hx, n1, 1, 0) - a0) / epsilon;
950: PetscCall(PetscPrintf(comm, "ddiff[%" PetscInt_FMT "] {%g,%g}, fd {%g %g}, diff {%g,%g}\n", i, (double)PetscRealPart(d[i].E), (double)PetscRealPart(d[i].T), (double)PetscRealPart(fd[i].E), (double)PetscRealPart(fd[i].T),
951: (double)PetscRealPart(d[i].E - fd[i].E), (double)PetscRealPart(d[i].T - fd[i].T)));
952: }
953: }
954: {
955: PetscScalar rad0, rad;
956: RDNode drad, fdrad;
957: n.E = 1.;
958: n.T = 1.;
959: rad0 = RDRadiation(rd, &n, &drad);
960: n.E = 1. + epsilon;
961: n.T = 1.;
962: rad = RDRadiation(rd, &n, 0);
963: fdrad.E = (rad - rad0) / epsilon;
964: n.E = 1.;
965: n.T = 1. + epsilon;
966: rad = RDRadiation(rd, &n, 0);
967: fdrad.T = (rad - rad0) / epsilon;
968: PetscCall(PetscPrintf(comm, "drad {%g,%g}, fdrad {%g,%g}, diff {%g,%g}\n", (double)PetscRealPart(drad.E), (double)PetscRealPart(drad.T), (double)PetscRealPart(fdrad.E), (double)PetscRealPart(fdrad.T), (double)PetscRealPart(drad.E - drad.E),
969: (double)PetscRealPart(drad.T - fdrad.T)));
970: }
971: PetscFunctionReturn(PETSC_SUCCESS);
972: }
974: static PetscErrorCode RDCreate(MPI_Comm comm, RD *inrd)
975: {
976: RD rd;
977: PetscReal meter = 0, kilogram = 0, second = 0, Kelvin = 0, Joule = 0, Watt = 0;
979: PetscFunctionBeginUser;
980: *inrd = 0;
981: PetscCall(PetscNew(&rd));
983: PetscOptionsBegin(comm, NULL, "Options for nonequilibrium radiation-diffusion with RD ionization", NULL);
984: {
985: rd->initial = 1;
986: PetscCall(PetscOptionsInt("-rd_initial", "Initial condition (1=Marshak, 2=Blast, 3=Marshak+)", "", rd->initial, &rd->initial, 0));
987: switch (rd->initial) {
988: case 1:
989: case 2:
990: rd->unit.kilogram = 1.;
991: rd->unit.meter = 1.;
992: rd->unit.second = 1.;
993: rd->unit.Kelvin = 1.;
994: break;
995: case 3:
996: rd->unit.kilogram = 1.e12;
997: rd->unit.meter = 1.;
998: rd->unit.second = 1.e9;
999: rd->unit.Kelvin = 1.;
1000: break;
1001: default:
1002: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Unknown initial condition %" PetscInt_FMT, rd->initial);
1003: }
1004: /* Fundamental units */
1005: PetscCall(PetscOptionsReal("-rd_unit_meter", "Length of 1 meter in nondimensional units", "", rd->unit.meter, &rd->unit.meter, 0));
1006: PetscCall(PetscOptionsReal("-rd_unit_kilogram", "Mass of 1 kilogram in nondimensional units", "", rd->unit.kilogram, &rd->unit.kilogram, 0));
1007: PetscCall(PetscOptionsReal("-rd_unit_second", "Time of a second in nondimensional units", "", rd->unit.second, &rd->unit.second, 0));
1008: PetscCall(PetscOptionsReal("-rd_unit_Kelvin", "Temperature of a Kelvin in nondimensional units", "", rd->unit.Kelvin, &rd->unit.Kelvin, 0));
1009: /* Derived units */
1010: rd->unit.Joule = rd->unit.kilogram * PetscSqr(rd->unit.meter / rd->unit.second);
1011: rd->unit.Watt = rd->unit.Joule / rd->unit.second;
1012: /* Local aliases */
1013: meter = rd->unit.meter;
1014: kilogram = rd->unit.kilogram;
1015: second = rd->unit.second;
1016: Kelvin = rd->unit.Kelvin;
1017: Joule = rd->unit.Joule;
1018: Watt = rd->unit.Watt;
1020: PetscCall(PetscOptionsBool("-rd_monitor_residual", "Display residuals every time they are evaluated", "", rd->monitor_residual, &rd->monitor_residual, NULL));
1021: PetscCall(PetscOptionsEnum("-rd_discretization", "Discretization type", "", DiscretizationTypes, (PetscEnum)rd->discretization, (PetscEnum *)&rd->discretization, NULL));
1022: if (rd->discretization == DISCRETIZATION_FE) {
1023: rd->quadrature = QUADRATURE_GAUSS2;
1024: PetscCall(PetscOptionsEnum("-rd_quadrature", "Finite element quadrature", "", QuadratureTypes, (PetscEnum)rd->quadrature, (PetscEnum *)&rd->quadrature, NULL));
1025: }
1026: PetscCall(PetscOptionsEnum("-rd_jacobian", "Type of finite difference Jacobian", "", JacobianTypes, (PetscEnum)rd->jacobian, (PetscEnum *)&rd->jacobian, NULL));
1027: switch (rd->initial) {
1028: case 1:
1029: rd->leftbc = BC_ROBIN;
1030: rd->Eapplied = 4 * rd->unit.Joule / PetscPowRealInt(rd->unit.meter, 3);
1031: rd->L = 1. * rd->unit.meter;
1032: rd->beta = 3.0;
1033: rd->gamma = 3.0;
1034: rd->final_time = 3 * second;
1035: break;
1036: case 2:
1037: rd->leftbc = BC_NEUMANN;
1038: rd->Eapplied = 0.;
1039: rd->L = 1. * rd->unit.meter;
1040: rd->beta = 3.0;
1041: rd->gamma = 3.0;
1042: rd->final_time = 1 * second;
1043: break;
1044: case 3:
1045: rd->leftbc = BC_ROBIN;
1046: rd->Eapplied = 7.503e6 * rd->unit.Joule / PetscPowRealInt(rd->unit.meter, 3);
1047: rd->L = 5. * rd->unit.meter;
1048: rd->beta = 3.5;
1049: rd->gamma = 3.5;
1050: rd->final_time = 20e-9 * second;
1051: break;
1052: default:
1053: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Initial %" PetscInt_FMT, rd->initial);
1054: }
1055: PetscCall(PetscOptionsEnum("-rd_leftbc", "Left boundary condition", "", BCTypes, (PetscEnum)rd->leftbc, (PetscEnum *)&rd->leftbc, NULL));
1056: PetscCall(PetscOptionsReal("-rd_E_applied", "Radiation flux at left end of domain", "", rd->Eapplied, &rd->Eapplied, NULL));
1057: PetscCall(PetscOptionsReal("-rd_beta", "Thermal exponent for photon absorption", "", rd->beta, &rd->beta, NULL));
1058: PetscCall(PetscOptionsReal("-rd_gamma", "Thermal exponent for diffusion coefficient", "", rd->gamma, &rd->gamma, NULL));
1059: PetscCall(PetscOptionsBool("-rd_view_draw", "Draw final solution", "", rd->view_draw, &rd->view_draw, NULL));
1060: PetscCall(PetscOptionsBool("-rd_endpoint", "Discretize using endpoints (like trapezoid rule) instead of midpoint", "", rd->endpoint, &rd->endpoint, NULL));
1061: PetscCall(PetscOptionsBool("-rd_bcmidpoint", "Impose the boundary condition at the midpoint (Theta) of the interval", "", rd->bcmidpoint, &rd->bcmidpoint, NULL));
1062: PetscCall(PetscOptionsBool("-rd_bclimit", "Limit diffusion coefficient in definition of Robin boundary condition", "", rd->bclimit, &rd->bclimit, NULL));
1063: PetscCall(PetscOptionsBool("-rd_test_diff", "Test differentiation in constitutive relations", "", rd->test_diff, &rd->test_diff, NULL));
1064: PetscCall(PetscOptionsString("-rd_view_binary", "File name to hold final solution", "", rd->view_binary, rd->view_binary, sizeof(rd->view_binary), NULL));
1065: }
1066: PetscOptionsEnd();
1068: switch (rd->initial) {
1069: case 1:
1070: case 2:
1071: rd->rho = 1.;
1072: rd->c = 1.;
1073: rd->K_R = 1.;
1074: rd->K_p = 1.;
1075: rd->sigma_b = 0.25;
1076: rd->MaterialEnergy = RDMaterialEnergy_Reduced;
1077: break;
1078: case 3:
1079: /* Table 2 */
1080: rd->rho = 1.17e-3 * kilogram / (meter * meter * meter); /* density */
1081: rd->K_R = 7.44e18 * PetscPowRealInt(meter, 5) * PetscPowReal(Kelvin, 3.5) * PetscPowRealInt(kilogram, -2);
1082: rd->K_p = 2.33e20 * PetscPowRealInt(meter, 5) * PetscPowReal(Kelvin, 3.5) * PetscPowRealInt(kilogram, -2);
1083: rd->I_H = 2.179e-18 * Joule; /* Hydrogen ionization potential */
1084: rd->m_p = 1.673e-27 * kilogram; /* proton mass */
1085: rd->m_e = 9.109e-31 * kilogram; /* electron mass */
1086: rd->h = 6.626e-34 * Joule * second; /* Planck's constant */
1087: rd->k = 1.381e-23 * Joule / Kelvin; /* Boltzman constant */
1088: rd->c = 3.00e8 * meter / second; /* speed of light */
1089: rd->sigma_b = 5.67e-8 * Watt * PetscPowRealInt(meter, -2) * PetscPowRealInt(Kelvin, -4); /* Stefan-Boltzman constant */
1090: rd->MaterialEnergy = RDMaterialEnergy_Saha;
1091: break;
1092: }
1094: PetscCall(DMDACreate1d(comm, DM_BOUNDARY_NONE, 20, sizeof(RDNode) / sizeof(PetscScalar), 1, NULL, &rd->da));
1095: PetscCall(DMSetFromOptions(rd->da));
1096: PetscCall(DMSetUp(rd->da));
1097: PetscCall(DMDASetFieldName(rd->da, 0, "E"));
1098: PetscCall(DMDASetFieldName(rd->da, 1, "T"));
1099: PetscCall(DMDASetUniformCoordinates(rd->da, 0., 1., 0., 0., 0., 0.));
1101: *inrd = rd;
1102: PetscFunctionReturn(PETSC_SUCCESS);
1103: }
1105: int main(int argc, char *argv[])
1106: {
1107: RD rd;
1108: TS ts;
1109: SNES snes;
1110: Vec X;
1111: Mat A, B;
1112: PetscInt steps;
1113: PetscReal ftime;
1115: PetscFunctionBeginUser;
1116: PetscCall(PetscInitialize(&argc, &argv, 0, NULL));
1117: PetscCall(RDCreate(PETSC_COMM_WORLD, &rd));
1118: PetscCall(DMCreateGlobalVector(rd->da, &X));
1119: PetscCall(DMSetMatType(rd->da, MATAIJ));
1120: PetscCall(DMCreateMatrix(rd->da, &B));
1121: PetscCall(RDInitialState(rd, X));
1123: PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
1124: PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
1125: PetscCall(TSSetType(ts, TSTHETA));
1126: PetscCall(TSSetDM(ts, rd->da));
1127: switch (rd->discretization) {
1128: case DISCRETIZATION_FD:
1129: PetscCall(TSSetIFunction(ts, NULL, RDIFunction_FD, rd));
1130: if (rd->jacobian == JACOBIAN_ANALYTIC) PetscCall(TSSetIJacobian(ts, B, B, RDIJacobian_FD, rd));
1131: break;
1132: case DISCRETIZATION_FE:
1133: PetscCall(TSSetIFunction(ts, NULL, RDIFunction_FE, rd));
1134: if (rd->jacobian == JACOBIAN_ANALYTIC) PetscCall(TSSetIJacobian(ts, B, B, RDIJacobian_FE, rd));
1135: break;
1136: }
1137: PetscCall(TSSetMaxTime(ts, rd->final_time));
1138: PetscCall(TSSetTimeStep(ts, 1e-3));
1139: PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
1140: PetscCall(TSSetFromOptions(ts));
1142: A = B;
1143: PetscCall(TSGetSNES(ts, &snes));
1144: switch (rd->jacobian) {
1145: case JACOBIAN_ANALYTIC:
1146: break;
1147: case JACOBIAN_MATRIXFREE:
1148: break;
1149: case JACOBIAN_FD_COLORING: {
1150: PetscCall(SNESSetJacobian(snes, A, B, SNESComputeJacobianDefaultColor, 0));
1151: } break;
1152: case JACOBIAN_FD_FULL:
1153: PetscCall(SNESSetJacobian(snes, A, B, SNESComputeJacobianDefault, ts));
1154: break;
1155: }
1157: if (rd->test_diff) PetscCall(RDTestDifferentiation(rd));
1158: PetscCall(TSSolve(ts, X));
1159: PetscCall(TSGetSolveTime(ts, &ftime));
1160: PetscCall(TSGetStepNumber(ts, &steps));
1161: PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Steps %" PetscInt_FMT " final time %g\n", steps, (double)ftime));
1162: if (rd->view_draw) PetscCall(RDView(rd, X, PETSC_VIEWER_DRAW_WORLD));
1163: if (rd->view_binary[0]) {
1164: PetscViewer viewer;
1165: PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD, rd->view_binary, FILE_MODE_WRITE, &viewer));
1166: PetscCall(RDView(rd, X, viewer));
1167: PetscCall(PetscViewerDestroy(&viewer));
1168: }
1169: PetscCall(VecDestroy(&X));
1170: PetscCall(MatDestroy(&B));
1171: PetscCall(RDDestroy(&rd));
1172: PetscCall(TSDestroy(&ts));
1173: PetscCall(PetscFinalize());
1174: return 0;
1175: }
1176: /*TEST
1178: test:
1179: args: -da_grid_x 20 -rd_initial 1 -rd_discretization fd -rd_jacobian fd_coloring -rd_endpoint -ts_max_time 1 -ts_dt 2e-1 -ts_theta_initial_guess_extrapolate 0 -ts_monitor -snes_monitor_short -ksp_monitor_short
1180: requires: !single
1182: test:
1183: suffix: 2
1184: args: -da_grid_x 20 -rd_initial 1 -rd_discretization fe -rd_quadrature lobatto2 -rd_jacobian fd_coloring -rd_endpoint -ts_max_time 1 -ts_dt 2e-1 -ts_theta_initial_guess_extrapolate 0 -ts_monitor -snes_monitor_short -ksp_monitor_short
1185: requires: !single
1187: test:
1188: suffix: 3
1189: nsize: 2
1190: args: -da_grid_x 20 -rd_initial 1 -rd_discretization fd -rd_jacobian analytic -rd_endpoint -ts_max_time 3 -ts_dt 1e-1 -ts_theta_initial_guess_extrapolate 0 -ts_monitor -snes_monitor_short -ksp_monitor_short
1191: requires: !single
1193: TEST*/