Actual source code: ex46.c

      1: static char help[] = "Time dependent Navier-Stokes problem in 2d and 3d with finite elements.\n\
      2: We solve the Navier-Stokes in a rectangular\n\
      3: domain, using a parallel unstructured mesh (DMPLEX) to discretize it.\n\
      4: This example supports discretized auxiliary fields (Re) as well as\n\
      5: multilevel nonlinear solvers.\n\
      6: Contributed by: Julian Andrej <juan@tf.uni-kiel.de>\n\n\n";

      8: #include <petscdmplex.h>
      9: #include <petscsnes.h>
     10: #include <petscts.h>
     11: #include <petscds.h>

     13: /*
     14:   Navier-Stokes equation:

     16:   du/dt + u . grad u - \Delta u - grad p = f
     17:   div u  = 0
     18: */

     20: typedef struct {
     21:   PetscInt mms;
     22: } AppCtx;

     24: #define REYN 400.0

     26: /* MMS1

     28:   u = t + x^2 + y^2;
     29:   v = t + 2*x^2 - 2*x*y;
     30:   p = x + y - 1;

     32:   f_x = -2*t*(x + y) + 2*x*y^2 - 4*x^2*y - 2*x^3 + 4.0/Re - 1.0
     33:   f_y = -2*t*x       + 2*y^3 - 4*x*y^2 - 2*x^2*y + 4.0/Re - 1.0

     35:   so that

     37:     u_t + u \cdot \nabla u - 1/Re \Delta u + \nabla p + f = <1, 1> + <t (2x + 2y) + 2x^3 + 4x^2y - 2xy^2, t 2x + 2x^2y + 4xy^2 - 2y^3> - 1/Re <4, 4> + <1, 1>
     38:                                                     + <-t (2x + 2y) + 2xy^2 - 4x^2y - 2x^3 + 4/Re - 1, -2xt + 2y^3 - 4xy^2 - 2x^2y + 4/Re - 1> = 0
     39:     \nabla \cdot u                                  = 2x - 2x = 0

     41:   where

     43:     <u, v> . <<u_x, v_x>, <u_y, v_y>> = <u u_x + v u_y, u v_x + v v_y>
     44: */
     45: PetscErrorCode mms1_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
     46: {
     47:   u[0] = time + x[0]*x[0] + x[1]*x[1];
     48:   u[1] = time + 2.0*x[0]*x[0] - 2.0*x[0]*x[1];
     49:   return 0;
     50: }

     52: PetscErrorCode mms1_p_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *p, void *ctx)
     53: {
     54:   *p = x[0] + x[1] - 1.0;
     55:   return 0;
     56: }

     58: /* MMS 2*/

     60: static PetscErrorCode mms2_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
     61: {
     62:   u[0] = PetscSinReal(time + x[0])*PetscSinReal(time + x[1]);
     63:   u[1] = PetscCosReal(time + x[0])*PetscCosReal(time + x[1]);
     64:   return 0;
     65: }

     67: static PetscErrorCode mms2_p_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *p, void *ctx)
     68: {
     69:   *p = PetscSinReal(time + x[0] - x[1]);
     70:   return 0;
     71: }

     73: static void f0_mms1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     74:                       const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     75:                       const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     76:                       PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
     77: {
     78:   const PetscReal Re    = REYN;
     79:   const PetscInt  Ncomp = dim;
     80:   PetscInt        c, d;

     82:   for (c = 0; c < Ncomp; ++c) {
     83:     for (d = 0; d < dim; ++d) {
     84:       f0[c] += u[d] * u_x[c*dim+d];
     85:     }
     86:   }
     87:   f0[0] += u_t[0];
     88:   f0[1] += u_t[1];

     90:   f0[0] += -2.0*t*(x[0] + x[1]) + 2.0*x[0]*x[1]*x[1] - 4.0*x[0]*x[0]*x[1] - 2.0*x[0]*x[0]*x[0] + 4.0/Re - 1.0;
     91:   f0[1] += -2.0*t*x[0]          + 2.0*x[1]*x[1]*x[1] - 4.0*x[0]*x[1]*x[1] - 2.0*x[0]*x[0]*x[1] + 4.0/Re - 1.0;
     92: }

     94: static void f0_mms2_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     95:                       const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     96:                       const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     97:                       PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
     98: {
     99:   const PetscReal Re    = REYN;
    100:   const PetscInt  Ncomp = dim;
    101:   PetscInt        c, d;

    103:   for (c = 0; c < Ncomp; ++c) {
    104:     for (d = 0; d < dim; ++d) {
    105:       f0[c] += u[d] * u_x[c*dim+d];
    106:     }
    107:   }
    108:   f0[0] += u_t[0];
    109:   f0[1] += u_t[1];

    111:   f0[0] -= ( Re*((1.0L/2.0L)*PetscSinReal(2*t + 2*x[0]) + PetscSinReal(2*t + x[0] + x[1]) + PetscCosReal(t + x[0] - x[1])) + 2.0*PetscSinReal(t + x[0])*PetscSinReal(t + x[1]))/Re;
    112:   f0[1] -= (-Re*((1.0L/2.0L)*PetscSinReal(2*t + 2*x[1]) + PetscSinReal(2*t + x[0] + x[1]) + PetscCosReal(t + x[0] - x[1])) + 2.0*PetscCosReal(t + x[0])*PetscCosReal(t + x[1]))/Re;
    113: }

    115: static void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    116:                  const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    117:                  const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    118:                  PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    119: {
    120:   const PetscReal Re    = REYN;
    121:   const PetscInt  Ncomp = dim;
    122:   PetscInt        comp, d;

    124:   for (comp = 0; comp < Ncomp; ++comp) {
    125:     for (d = 0; d < dim; ++d) {
    126:       f1[comp*dim+d] = 1.0/Re * u_x[comp*dim+d];
    127:     }
    128:     f1[comp*dim+comp] -= u[Ncomp];
    129:   }
    130: }

    132: static void f0_p(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    133:                  const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    134:                  const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    135:                  PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    136: {
    137:   PetscInt d;
    138:   for (d = 0, f0[0] = 0.0; d < dim; ++d) f0[0] += u_x[d*dim+d];
    139: }

    141: static void f1_p(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    142:                  const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    143:                  const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    144:                  PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    145: {
    146:   PetscInt d;
    147:   for (d = 0; d < dim; ++d) f1[d] = 0.0;
    148: }

    150: /*
    151:   (psi_i, u_j grad_j u_i) ==> (\psi_i, \phi_j grad_j u_i)
    152: */
    153: static void g0_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    154:                   const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    155:                   const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    156:                   PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    157: {
    158:   PetscInt NcI = dim, NcJ = dim;
    159:   PetscInt fc, gc;
    160:   PetscInt d;

    162:   for (d = 0; d < dim; ++d) {
    163:     g0[d*dim+d] = u_tShift;
    164:   }

    166:   for (fc = 0; fc < NcI; ++fc) {
    167:     for (gc = 0; gc < NcJ; ++gc) {
    168:       g0[fc*NcJ+gc] += u_x[fc*NcJ+gc];
    169:     }
    170:   }
    171: }

    173: /*
    174:   (psi_i, u_j grad_j u_i) ==> (\psi_i, \u_j grad_j \phi_i)
    175: */
    176: static void g1_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    177:                   const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    178:                   const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    179:                   PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g1[])
    180: {
    181:   PetscInt NcI = dim;
    182:   PetscInt NcJ = dim;
    183:   PetscInt fc, gc, dg;
    184:   for (fc = 0; fc < NcI; ++fc) {
    185:     for (gc = 0; gc < NcJ; ++gc) {
    186:       for (dg = 0; dg < dim; ++dg) {
    187:         /* kronecker delta */
    188:         if (fc == gc) {
    189:           g1[(fc*NcJ+gc)*dim+dg] += u[dg];
    190:         }
    191:       }
    192:     }
    193:   }
    194: }

    196: /* < q, \nabla\cdot u >
    197:    NcompI = 1, NcompJ = dim */
    198: static void g1_pu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    199:                   const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    200:                   const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    201:                   PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g1[])
    202: {
    203:   PetscInt d;
    204:   for (d = 0; d < dim; ++d) g1[d*dim+d] = 1.0; /* \frac{\partial\phi^{u_d}}{\partial x_d} */
    205: }

    207: /* -< \nabla\cdot v, p >
    208:     NcompI = dim, NcompJ = 1 */
    209: static void g2_up(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    210:                   const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    211:                   const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    212:                   PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
    213: {
    214:   PetscInt d;
    215:   for (d = 0; d < dim; ++d) g2[d*dim+d] = -1.0; /* \frac{\partial\psi^{u_d}}{\partial x_d} */
    216: }

    218: /* < \nabla v, \nabla u + {\nabla u}^T >
    219:    This just gives \nabla u, give the perdiagonal for the transpose */
    220: static void g3_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    221:                   const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    222:                   const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    223:                   PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    224: {
    225:   const PetscReal Re    = REYN;
    226:   const PetscInt  Ncomp = dim;
    227:   PetscInt        compI, d;

    229:   for (compI = 0; compI < Ncomp; ++compI) {
    230:     for (d = 0; d < dim; ++d) {
    231:       g3[((compI*Ncomp+compI)*dim+d)*dim+d] = 1.0/Re;
    232:     }
    233:   }
    234: }

    236: static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
    237: {

    241:   options->mms = 1;

    243:   PetscOptionsBegin(comm, "", "Navier-Stokes Equation Options", "DMPLEX");
    244:   PetscOptionsInt("-mms", "The manufactured solution to use", "ex46.c", options->mms, &options->mms, NULL);
    245:   PetscOptionsEnd();
    246:   return 0;
    247: }

    249: static PetscErrorCode CreateMesh(MPI_Comm comm, DM *dm, AppCtx *ctx)
    250: {
    252:   DMCreate(comm, dm);
    253:   DMSetType(*dm, DMPLEX);
    254:   DMSetFromOptions(*dm);
    255:   DMViewFromOptions(*dm, NULL, "-dm_view");
    256:   return 0;
    257: }

    259: static PetscErrorCode SetupProblem(DM dm, AppCtx *ctx)
    260: {
    261:   PetscDS        ds;
    262:   DMLabel        label;
    263:   const PetscInt id = 1;
    264:   PetscInt       dim;

    267:   DMGetDimension(dm, &dim);
    268:   DMGetDS(dm, &ds);
    269:   DMGetLabel(dm, "marker", &label);
    270:   switch (dim) {
    271:   case 2:
    272:     switch (ctx->mms) {
    273:     case 1:
    274:       PetscDSSetResidual(ds, 0, f0_mms1_u, f1_u);
    275:       PetscDSSetResidual(ds, 1, f0_p, f1_p);
    276:       PetscDSSetJacobian(ds, 0, 0, g0_uu, g1_uu, NULL,  g3_uu);
    277:       PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_up, NULL);
    278:       PetscDSSetJacobian(ds, 1, 0, NULL, g1_pu, NULL,  NULL);
    279:       PetscDSSetExactSolution(ds, 0, mms1_u_2d, ctx);
    280:       PetscDSSetExactSolution(ds, 1, mms1_p_2d, ctx);
    281:       DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) mms1_u_2d, NULL, ctx, NULL);
    282:       break;
    283:     case 2:
    284:       PetscDSSetResidual(ds, 0, f0_mms2_u, f1_u);
    285:       PetscDSSetResidual(ds, 1, f0_p, f1_p);
    286:       PetscDSSetJacobian(ds, 0, 0, g0_uu, g1_uu, NULL,  g3_uu);
    287:       PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_up, NULL);
    288:       PetscDSSetJacobian(ds, 1, 0, NULL, g1_pu, NULL,  NULL);
    289:       PetscDSSetExactSolution(ds, 0, mms2_u_2d, ctx);
    290:       PetscDSSetExactSolution(ds, 1, mms2_p_2d, ctx);
    291:       DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) mms2_u_2d, NULL, ctx, NULL);
    292:       break;
    293:     default: SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Invalid MMS %D", ctx->mms);
    294:     }
    295:     break;
    296:   default: SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Invalid dimension %D", dim);
    297:   }
    298:   return 0;
    299: }

    301: static PetscErrorCode SetupDiscretization(DM dm, AppCtx *ctx)
    302: {
    303:   MPI_Comm        comm;
    304:   DM              cdm = dm;
    305:   PetscFE         fe[2];
    306:   PetscInt        dim;
    307:   PetscBool       simplex;

    310:   PetscObjectGetComm((PetscObject) dm, &comm);
    311:   DMGetDimension(dm, &dim);
    312:   DMPlexIsSimplex(dm, &simplex);
    313:   PetscFECreateDefault(comm, dim, dim, simplex, "vel_", PETSC_DEFAULT, &fe[0]);
    314:   PetscObjectSetName((PetscObject) fe[0], "velocity");
    315:   PetscFECreateDefault(comm, dim, 1, simplex, "pres_", PETSC_DEFAULT, &fe[1]);
    316:   PetscFECopyQuadrature(fe[0], fe[1]);
    317:   PetscObjectSetName((PetscObject) fe[1], "pressure");
    318:   /* Set discretization and boundary conditions for each mesh */
    319:   DMSetField(dm, 0, NULL, (PetscObject) fe[0]);
    320:   DMSetField(dm, 1, NULL, (PetscObject) fe[1]);
    321:   DMCreateDS(dm);
    322:   SetupProblem(dm, ctx);
    323:   while (cdm) {
    324:     PetscObject  pressure;
    325:     MatNullSpace nsp;

    327:     DMGetField(cdm, 1, NULL, &pressure);
    328:     MatNullSpaceCreate(PetscObjectComm(pressure), PETSC_TRUE, 0, NULL, &nsp);
    329:     PetscObjectCompose(pressure, "nullspace", (PetscObject) nsp);
    330:     MatNullSpaceDestroy(&nsp);

    332:     DMCopyDisc(dm, cdm);
    333:     DMGetCoarseDM(cdm, &cdm);
    334:   }
    335:   PetscFEDestroy(&fe[0]);
    336:   PetscFEDestroy(&fe[1]);
    337:   return 0;
    338: }

    340: static PetscErrorCode MonitorError(TS ts, PetscInt step, PetscReal crtime, Vec u, void *ctx)
    341: {
    342:   PetscSimplePointFunc funcs[2];
    343:   void                *ctxs[2];
    344:   DM                   dm;
    345:   PetscDS              ds;
    346:   PetscReal            ferrors[2];

    349:   TSGetDM(ts, &dm);
    350:   DMGetDS(dm, &ds);
    351:   PetscDSGetExactSolution(ds, 0, &funcs[0], &ctxs[0]);
    352:   PetscDSGetExactSolution(ds, 1, &funcs[1], &ctxs[1]);
    353:   DMComputeL2FieldDiff(dm, crtime, funcs, ctxs, u, ferrors);
    354:   PetscPrintf(PETSC_COMM_WORLD, "Timestep: %04d time = %-8.4g \t L_2 Error: [%2.3g, %2.3g]\n", (int) step, (double) crtime, (double) ferrors[0], (double) ferrors[1]);
    355:   return 0;
    356: }

    358: int main(int argc, char **argv)
    359: {
    360:   AppCtx         ctx;
    361:   DM             dm;
    362:   TS             ts;
    363:   Vec            u, r;

    365:   PetscInitialize(&argc, &argv, NULL, help);
    366:   ProcessOptions(PETSC_COMM_WORLD, &ctx);
    367:   CreateMesh(PETSC_COMM_WORLD, &dm, &ctx);
    368:   DMSetApplicationContext(dm, &ctx);
    369:   SetupDiscretization(dm, &ctx);
    370:   DMPlexCreateClosureIndex(dm, NULL);

    372:   DMCreateGlobalVector(dm, &u);
    373:   VecDuplicate(u, &r);

    375:   TSCreate(PETSC_COMM_WORLD, &ts);
    376:   TSMonitorSet(ts, MonitorError, &ctx, NULL);
    377:   TSSetDM(ts, dm);
    378:   DMTSSetBoundaryLocal(dm, DMPlexTSComputeBoundary, &ctx);
    379:   DMTSSetIFunctionLocal(dm, DMPlexTSComputeIFunctionFEM, &ctx);
    380:   DMTSSetIJacobianLocal(dm, DMPlexTSComputeIJacobianFEM, &ctx);
    381:   TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER);
    382:   TSSetFromOptions(ts);
    383:   DMTSCheckFromOptions(ts, u);

    385:   {
    386:     PetscSimplePointFunc funcs[2];
    387:     void                *ctxs[2];
    388:     PetscDS              ds;

    390:     DMGetDS(dm, &ds);
    391:     PetscDSGetExactSolution(ds, 0, &funcs[0], &ctxs[0]);
    392:     PetscDSGetExactSolution(ds, 1, &funcs[1], &ctxs[1]);
    393:     DMProjectFunction(dm, 0.0, funcs, ctxs, INSERT_ALL_VALUES, u);
    394:   }
    395:   TSSolve(ts, u);
    396:   VecViewFromOptions(u, NULL, "-sol_vec_view");

    398:   VecDestroy(&u);
    399:   VecDestroy(&r);
    400:   TSDestroy(&ts);
    401:   DMDestroy(&dm);
    402:   PetscFinalize();
    403:   return 0;
    404: }

    406: /*TEST

    408:   # Full solves
    409:   test:
    410:     suffix: 2d_p2p1_r1
    411:     requires: !single triangle
    412:     filter: sed -e "s~ATOL~RTOL~g" -e "s~ABS~RELATIVE~g"
    413:     args: -dm_refine 1 -vel_petscspace_degree 2 -pres_petscspace_degree 1 \
    414:           -ts_type beuler -ts_max_steps 10 -ts_dt 0.1 -ts_monitor -dmts_check \
    415:           -snes_monitor_short -snes_converged_reason \
    416:           -ksp_monitor_short -ksp_converged_reason \
    417:           -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type full \
    418:             -fieldsplit_velocity_pc_type lu \
    419:             -fieldsplit_pressure_ksp_rtol 1.0e-10 -fieldsplit_pressure_pc_type jacobi

    421:   test:
    422:     suffix: 2d_q2q1_r1
    423:     requires: !single
    424:     filter: sed -e "s~ATOL~RTOL~g" -e "s~ABS~RELATIVE~g" -e "s~ 0\]~ 0.0\]~g"
    425:     args: -dm_plex_simplex 0 -dm_refine 1 -vel_petscspace_degree 2 -pres_petscspace_degree 1 \
    426:           -ts_type beuler -ts_max_steps 10 -ts_dt 0.1 -ts_monitor -dmts_check \
    427:           -snes_monitor_short -snes_converged_reason \
    428:           -ksp_monitor_short -ksp_converged_reason \
    429:           -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type full \
    430:             -fieldsplit_velocity_pc_type lu \
    431:             -fieldsplit_pressure_ksp_rtol 1.0e-10 -fieldsplit_pressure_pc_type jacobi

    433: TEST*/