如何使用PETSc可扩展非线性方程求解器设置三维有限元求解器？

Question

在3.3中他们有grate news - 在GPU上仅使用PETC SNES进行FEM求解的示例。我是PETSc的新手并且有一个问题 - 我需要在3d空间中创建一个球体并对其施加力...所以我需要一个三维有限元模块（如果可能的话，在GPU上，我的情况下不需要MPI）。然而，当我see the simple example他们提供时，我有点伤痕累累：

static const char help[] = "Testbed for FEM operations on the GPU.\n\n";

#include<petscdmplex.h>
#include<petscsnes.h>

#define NUM_FIELDS 1
PetscInt spatialDim = 0;

typedef enum {LAPLACIAN = 0, ELASTICITY} OpType;

typedef struct {
  PetscFEM      fem;               /* REQUIRED to use DMPlexComputeResidualFEM() */
  DM            dm;                /* The solution DM */
  PetscInt      debug;             /* The debugging level */
  PetscMPIInt   rank;              /* The process rank */
  PetscMPIInt   numProcs;          /* The number of processes */
  PetscInt      dim;               /* The topological mesh dimension */
  PetscBool     interpolate;       /* Generate intermediate mesh elements */
  PetscReal     refinementLimit;   /* The largest allowable cell volume */
  PetscBool     refinementUniform; /* Uniformly refine the mesh */
  PetscInt      refinementRounds;  /* The number of uniform refinements */
  char          partitioner[2048]; /* The graph partitioner */
  PetscBool     computeFunction;   /* The flag for computing a residual */
  PetscBool     computeJacobian;   /* The flag for computing a Jacobian */
  PetscBool     gpu;               /* The flag for GPU integration */
  OpType        op;                /* The type of PDE operator (should use FFC/Ignition here) */
  PetscBool     showResidual, showJacobian;
  PetscLogEvent createMeshEvent, residualEvent, residualBatchEvent, jacobianEvent, jacobianBatchEvent, integrateBatchCPUEvent, integrateBatchGPUEvent, integrateGPUOnlyEvent;
  /* Element definition */
  PetscFE       fe[NUM_FIELDS];
  PetscFE       feAux[1];
  void (*f0Funcs[NUM_FIELDS])(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar f0[]);
  void (*f1Funcs[NUM_FIELDS])(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar f1[]);
  void (*g0Funcs[NUM_FIELDS*NUM_FIELDS])(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar g0[]);
  void (*g1Funcs[NUM_FIELDS*NUM_FIELDS])(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar g1[]);
  void (*g2Funcs[NUM_FIELDS*NUM_FIELDS])(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar g2[]);
  void (*g3Funcs[NUM_FIELDS*NUM_FIELDS])(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar g3[]);
  void (**exactFuncs)(const PetscReal x[], PetscScalar *u, void *ctx);
} AppCtx;

void quadratic_2d(const PetscReal x[], PetscScalar u[], void *ctx)
{
  u[0] = x[0]*x[0] + x[1]*x[1];
};

void quadratic_2d_elas(const PetscReal x[], PetscScalar u[], void *ctx)
{
  u[0] = x[0]*x[0] + x[1]*x[1];
  u[1] = x[0]*x[0] + x[1]*x[1];
};

void f0_lap(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar f0[])
{
  f0[0] = 4.0;
}

/* gradU[comp*dim+d] = {u_x, u_y} or {u_x, u_y, u_z} */
void f1_lap(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar f1[])
{
  PetscInt d;
  for (d = 0; d < spatialDim; ++d) {f1[d] = a[0]*gradU[d];}
}

/* < \nabla v, \nabla u + {\nabla u}^T >
   This just gives \nabla u, give the perdiagonal for the transpose */
void g3_lap(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar g3[])
{
  PetscInt d;
  for (d = 0; d < spatialDim; ++d) {g3[d*spatialDim+d] = 1.0;}
}

void f0_elas(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar f0[])
{
  const PetscInt Ncomp = spatialDim;
  PetscInt       comp;

  for (comp = 0; comp < Ncomp; ++comp) f0[comp] = 3.0;
}

/* gradU[comp*dim+d] = {u_x, u_y, v_x, v_y} or {u_x, u_y, u_z, v_x, v_y, v_z, w_x, w_y, w_z}
   u[Ncomp]          = {p} */
void f1_elas(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar f1[])
{
  const PetscInt dim   = spatialDim;
  const PetscInt Ncomp = spatialDim;
  PetscInt       comp, d;

  for (comp = 0; comp < Ncomp; ++comp) {
    for (d = 0; d < dim; ++d) {
      f1[comp*dim+d] = 0.5*(gradU[comp*dim+d] + gradU[d*dim+comp]);
    }
    f1[comp*dim+comp] -= u[Ncomp];
  }
}

/* < \nabla v, \nabla u + {\nabla u}^T >
   This just gives \nabla u, give the perdiagonal for the transpose */
void g3_elas(const PetscScalar u[], const PetscScalar gradU[], const PetscScalar a[], const PetscScalar gradA[], const PetscReal x[], PetscScalar g3[])
{
  const PetscInt dim   = spatialDim;
  const PetscInt Ncomp = spatialDim;
  PetscInt       compI, d;

  for (compI = 0; compI < Ncomp; ++compI) {
    for (d = 0; d < dim; ++d) {
      g3[((compI*Ncomp+compI)*dim+d)*dim+d] = 1.0;
    }
  }
}

#undef __FUNCT__
#define __FUNCT__ "ProcessOptions"
PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
{
  const char     *opTypes[2] = {"laplacian", "elasticity"};
  PetscInt       op;
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  options->debug             = 0;
  options->dim               = 2;
  options->interpolate       = PETSC_FALSE;
  options->refinementLimit   = 0.0;
  options->refinementUniform = PETSC_FALSE;
  options->refinementRounds  = 1;
  options->computeFunction   = PETSC_FALSE;
  options->computeJacobian   = PETSC_FALSE;
  options->gpu               = PETSC_FALSE;
  options->op                = LAPLACIAN;
  options->showResidual      = PETSC_TRUE;
  options->showJacobian      = PETSC_TRUE;

  ierr = MPI_Comm_size(comm, &options->numProcs);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm, &options->rank);CHKERRQ(ierr);
  ierr = PetscOptionsBegin(comm, "", "Bratu Problem Options", "DMPLEX");CHKERRQ(ierr);
  ierr = PetscOptionsInt("-debug", "The debugging level", "ex52.c", options->debug, &options->debug, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsInt("-dim", "The topological mesh dimension", "ex52.c", options->dim, &options->dim, NULL);CHKERRQ(ierr);
  spatialDim = options->dim;
  ierr = PetscOptionsBool("-interpolate", "Generate intermediate mesh elements", "ex52.c", options->interpolate, &options->interpolate, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-refinement_limit", "The largest allowable cell volume", "ex52.c", options->refinementLimit, &options->refinementLimit, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-refinement_uniform", "Uniformly refine the mesh", "ex52.c", options->refinementUniform, &options->refinementUniform, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsInt("-refinement_rounds", "The number of uniform refinements", "ex52.c", options->refinementRounds, &options->refinementRounds, NULL);CHKERRQ(ierr);
  ierr = PetscStrcpy(options->partitioner, "chaco");CHKERRQ(ierr);
  ierr = PetscOptionsString("-partitioner", "The graph partitioner", "ex52.c", options->partitioner, options->partitioner, 2048, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-compute_function", "Compute the residual", "ex52.c", options->computeFunction, &options->computeFunction, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-compute_jacobian", "Compute the Jacobian", "ex52.c", options->computeJacobian, &options->computeJacobian, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-gpu", "Use the GPU for integration method", "ex52.c", options->gpu, &options->gpu, NULL);CHKERRQ(ierr);

  op = options->op;
  ierr = PetscOptionsEList("-op_type","Type of PDE operator","ex52.c",opTypes,2,opTypes[options->op],&op,NULL);CHKERRQ(ierr);
  options->op = (OpType) op;

  ierr = PetscOptionsBool("-show_residual", "Output the residual for verification", "ex52.c", options->showResidual, &options->showResidual, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-show_jacobian", "Output the Jacobian for verification", "ex52.c", options->showJacobian, &options->showJacobian, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();

  ierr = PetscLogEventRegister("CreateMesh",    DM_CLASSID,   &options->createMeshEvent);CHKERRQ(ierr);
  ierr = PetscLogEventRegister("Residual",      SNES_CLASSID, &options->residualEvent);CHKERRQ(ierr);
  ierr = PetscLogEventRegister("ResidualBatch", SNES_CLASSID, &options->residualBatchEvent);CHKERRQ(ierr);
  ierr = PetscLogEventRegister("Jacobian",      SNES_CLASSID, &options->jacobianEvent);CHKERRQ(ierr);
  ierr = PetscLogEventRegister("JacobianBatch", SNES_CLASSID, &options->jacobianBatchEvent);CHKERRQ(ierr);
  ierr = PetscLogEventRegister("IntegBatchCPU", SNES_CLASSID, &options->integrateBatchCPUEvent);CHKERRQ(ierr);
  ierr = PetscLogEventRegister("IntegBatchGPU", SNES_CLASSID, &options->integrateBatchGPUEvent);CHKERRQ(ierr);
  ierr = PetscLogEventRegister("IntegGPUOnly",  SNES_CLASSID, &options->integrateGPUOnlyEvent);CHKERRQ(ierr);
  PetscFunctionReturn(0);
};

#undef __FUNCT__
#define __FUNCT__ "CreateMesh"
PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
{
  PetscInt       dim               = user->dim;
  PetscBool      interpolate       = user->interpolate;
  PetscReal      refinementLimit   = user->refinementLimit;
  PetscBool      refinementUniform = user->refinementUniform;
  PetscInt       refinementRounds  = user->refinementRounds;
  const char     *partitioner      = user->partitioner;
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscLogEventBegin(user->createMeshEvent,0,0,0,0);CHKERRQ(ierr);
  ierr = DMPlexCreateBoxMesh(comm, dim, interpolate, dm);CHKERRQ(ierr);
  {
    DM refinedMesh     = NULL;
    DM distributedMesh = NULL;

    /* Refine mesh using a volume constraint */
    ierr = DMPlexSetRefinementLimit(*dm, refinementLimit);CHKERRQ(ierr);
    ierr = DMRefine(*dm, comm, &refinedMesh);CHKERRQ(ierr);
    if (refinedMesh) {
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = refinedMesh;
    }
    /* Distribute mesh over processes */
    ierr = DMPlexDistribute(*dm, partitioner, 0, NULL, &distributedMesh);CHKERRQ(ierr);
    if (distributedMesh) {
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = distributedMesh;
    }
    /* Use regular refinement in parallel */
    if (refinementUniform) {
      PetscInt r;

      ierr = DMPlexSetRefinementUniform(*dm, refinementUniform);CHKERRQ(ierr);
      for (r = 0; r < refinementRounds; ++r) {
        ierr = DMRefine(*dm, comm, &refinedMesh);CHKERRQ(ierr);
        if (refinedMesh) {
          ierr = DMDestroy(dm);CHKERRQ(ierr);
          *dm  = refinedMesh;
        }
      }
    }
  }
  ierr = PetscObjectSetName((PetscObject) *dm, "Mesh");CHKERRQ(ierr);
  ierr = DMSetFromOptions(*dm);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(user->createMeshEvent,0,0,0,0);CHKERRQ(ierr);

  user->dm = *dm;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SetupElement"
PetscErrorCode SetupElement(DM dm, AppCtx *user)
{
  const PetscInt  dim = user->dim;
  PetscFE         fem;
  PetscQuadrature q;
  DM              K;
  PetscSpace      P;
  PetscDualSpace  Q;
  PetscInt        order;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  /* Create space */
  ierr = PetscSpaceCreate(PetscObjectComm((PetscObject) dm), &P);CHKERRQ(ierr);
  ierr = PetscSpaceSetFromOptions(P);CHKERRQ(ierr);
  ierr = PetscSpacePolynomialSetNumVariables(P, dim);CHKERRQ(ierr);
  ierr = PetscSpaceSetUp(P);CHKERRQ(ierr);
  ierr = PetscSpaceGetOrder(P, &order);CHKERRQ(ierr);
  /* Create dual space */
  ierr = PetscDualSpaceCreate(PetscObjectComm((PetscObject) dm), &Q);CHKERRQ(ierr);
  ierr = PetscDualSpaceCreateReferenceCell(Q, dim, PETSC_TRUE, &K);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetDM(Q, K);CHKERRQ(ierr);
  ierr = DMDestroy(&K);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetOrder(Q, order);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetFromOptions(Q);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetUp(Q);CHKERRQ(ierr);
  /* Create element */
  ierr = PetscFECreate(PetscObjectComm((PetscObject) dm), &fem);CHKERRQ(ierr);
  ierr = PetscFESetFromOptions(fem);CHKERRQ(ierr);
  ierr = PetscFESetBasisSpace(fem, P);CHKERRQ(ierr);
  ierr = PetscFESetDualSpace(fem, Q);CHKERRQ(ierr);
  ierr = PetscFESetNumComponents(fem, 1);CHKERRQ(ierr);
  ierr = PetscFESetUp(fem);CHKERRQ(ierr);
  ierr = PetscSpaceDestroy(&P);CHKERRQ(ierr);
  ierr = PetscDualSpaceDestroy(&Q);CHKERRQ(ierr);
  /* Create quadrature */
  ierr = PetscDTGaussJacobiQuadrature(dim, order, -1.0, 1.0, &q);CHKERRQ(ierr);
  ierr = PetscFESetQuadrature(fem, q);CHKERRQ(ierr);
  ierr = PetscQuadratureDestroy(&q);CHKERRQ(ierr);
  user->fe[0] = fem;
  user->fem.fe = user->fe;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SetupMaterialElement"
PetscErrorCode SetupMaterialElement(DM dm, AppCtx *user)
{
  const PetscInt  dim = user->dim;
  const char     *prefix = "mat_";
  PetscFE         fem;
  PetscQuadrature q;
  DM              K;
  PetscSpace      P;
  PetscDualSpace  Q;
  PetscInt        order;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  /* Create space */
  ierr = PetscSpaceCreate(PetscObjectComm((PetscObject) dm), &P);CHKERRQ(ierr);
  ierr = PetscObjectSetOptionsPrefix((PetscObject) P, prefix);CHKERRQ(ierr);
  ierr = PetscSpaceSetFromOptions(P);CHKERRQ(ierr);
  ierr = PetscSpacePolynomialSetNumVariables(P, dim);CHKERRQ(ierr);
  ierr = PetscSpaceSetUp(P);CHKERRQ(ierr);
  ierr = PetscSpaceGetOrder(P, &order);CHKERRQ(ierr);
  /* Create dual space */
  ierr = PetscDualSpaceCreate(PetscObjectComm((PetscObject) dm), &Q);CHKERRQ(ierr);
  ierr = PetscObjectSetOptionsPrefix((PetscObject) Q, prefix);CHKERRQ(ierr);
  ierr = PetscDualSpaceCreateReferenceCell(Q, dim, PETSC_TRUE, &K);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetDM(Q, K);CHKERRQ(ierr);
  ierr = DMDestroy(&K);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetOrder(Q, order);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetFromOptions(Q);CHKERRQ(ierr);
  ierr = PetscDualSpaceSetUp(Q);CHKERRQ(ierr);
  /* Create element */
  ierr = PetscFECreate(PetscObjectComm((PetscObject) dm), &fem);CHKERRQ(ierr);
  ierr = PetscObjectSetOptionsPrefix((PetscObject) fem, prefix);CHKERRQ(ierr);
  ierr = PetscFESetFromOptions(fem);CHKERRQ(ierr);
  ierr = PetscFESetBasisSpace(fem, P);CHKERRQ(ierr);
  ierr = PetscFESetDualSpace(fem, Q);CHKERRQ(ierr);
  ierr = PetscFESetNumComponents(fem, 1);CHKERRQ(ierr);
  ierr = PetscSpaceDestroy(&P);CHKERRQ(ierr);
  ierr = PetscDualSpaceDestroy(&Q);CHKERRQ(ierr);
  /* Create quadrature */
  ierr = PetscDTGaussJacobiQuadrature(dim, PetscMax(order, 1), -1.0, 1.0, &q);CHKERRQ(ierr);
  ierr = PetscFESetQuadrature(fem, q);CHKERRQ(ierr);
  user->feAux[0]  = fem;
  user->fem.feAux = user->feAux;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DestroyElement"
PetscErrorCode DestroyElement(AppCtx *user)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscFEDestroy(&user->fe[0]);CHKERRQ(ierr);
  ierr = PetscFEDestroy(&user->feAux[0]);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SetupSection"
PetscErrorCode SetupSection(DM dm, AppCtx *user)
{
  PetscSection    section;
  PetscInt        dim   = user->dim;
  PetscInt        numBC = 0;
  PetscInt        numComp[1];
  const PetscInt *numDof;
  PetscErrorCode  ierr;

  PetscFunctionBeginUser;
  ierr = PetscFEGetNumComponents(user->fe[0], &numComp[0]);CHKERRQ(ierr);
  ierr = PetscFEGetNumDof(user->fe[0], &numDof);CHKERRQ(ierr);
  ierr = DMPlexCreateSection(dm, dim, 1, numComp, numDof, numBC, NULL, NULL, NULL, &section);CHKERRQ(ierr);
  ierr = DMSetDefaultSection(dm, section);CHKERRQ(ierr);
  ierr = PetscSectionDestroy(&section);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SetupMaterial"
PetscErrorCode SetupMaterial(DM dm, DM dmAux, AppCtx *user)
{
  Vec            epsilon;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMCreateLocalVector(dmAux, &epsilon);CHKERRQ(ierr);
  ierr = VecSet(epsilon, 1.0);CHKERRQ(ierr);
  ierr = PetscObjectCompose((PetscObject) dm, "A", (PetscObject) epsilon);CHKERRQ(ierr);
  ierr = VecDestroy(&epsilon);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "main"
int main(int argc, char **argv)
{
  DM             dm, dmAux;
  SNES           snes;
  AppCtx         user;
  PetscInt       numComp;
  PetscErrorCode ierr;

  ierr = PetscInitialize(&argc, &argv, NULL, help);CHKERRQ(ierr);
#if !defined(PETSC_HAVE_CUDA) && !defined(PETSC_HAVE_OPENCL)
  SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_SUP, "This example requires CUDA or OpenCL support.");
#endif
  ierr = ProcessOptions(PETSC_COMM_WORLD, &user);CHKERRQ(ierr);
  ierr = SNESCreate(PETSC_COMM_WORLD, &snes);CHKERRQ(ierr);
  ierr = CreateMesh(PETSC_COMM_WORLD, &user, &dm);CHKERRQ(ierr);
  ierr = SNESSetDM(snes, dm);CHKERRQ(ierr);

  ierr = SetupElement(user.dm, &user);CHKERRQ(ierr);
  ierr = DMClone(user.dm, &dmAux);CHKERRQ(ierr);
  ierr = PetscObjectCompose((PetscObject) dm, "dmAux", (PetscObject) dmAux);CHKERRQ(ierr);
  ierr = SetupMaterialElement(dmAux, &user);CHKERRQ(ierr);
  ierr = PetscFEGetNumComponents(user.fe[0], &numComp);CHKERRQ(ierr);
  ierr = PetscMalloc(numComp * sizeof(void (*)(const PetscReal[], PetscScalar *, void *)), &user.exactFuncs);CHKERRQ(ierr);
  switch (user.op) {
  case LAPLACIAN:
    user.f0Funcs[0]    = f0_lap;
    user.f1Funcs[0]    = f1_lap;
    user.g0Funcs[0]    = NULL;
    user.g1Funcs[0]    = NULL;
    user.g2Funcs[0]    = NULL;
    user.g3Funcs[0]    = g3_lap;
    user.exactFuncs[0] = quadratic_2d;
    break;
  case ELASTICITY:
    user.f0Funcs[0]    = f0_elas;
    user.f1Funcs[0]    = f1_elas;
    user.g0Funcs[0]    = NULL;
    user.g1Funcs[0]    = NULL;
    user.g2Funcs[0]    = NULL;
    user.g3Funcs[0]    = g3_elas;
    user.exactFuncs[0] = quadratic_2d_elas;
    break;
  default:
    SETERRQ1(PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Invalid PDE operator %d", user.op);
  }
  user.fem.f0Funcs = user.f0Funcs;
  user.fem.f1Funcs = user.f1Funcs;
  user.fem.g0Funcs = user.g0Funcs;
  user.fem.g1Funcs = user.g1Funcs;
  user.fem.g2Funcs = user.g2Funcs;
  user.fem.g3Funcs = user.g3Funcs;
  user.fem.bcFuncs = user.exactFuncs;
  user.fem.bcCtxs  = NULL;
  ierr = SetupSection(dm, &user);CHKERRQ(ierr);
  ierr = SetupSection(dmAux, &user);CHKERRQ(ierr);
  ierr = SetupMaterial(dm, dmAux, &user);CHKERRQ(ierr);

  ierr = DMSNESSetFunctionLocal(dm,  (PetscErrorCode (*)(DM,Vec,Vec,void*))DMPlexComputeResidualFEM,&user);CHKERRQ(ierr);
  ierr = DMSNESSetJacobianLocal(dm,  (PetscErrorCode (*)(DM,Vec,Mat,Mat,void*))DMPlexComputeJacobianFEM,&user);CHKERRQ(ierr);
  if (user.computeFunction) {
    Vec X, F;

    ierr = DMGetGlobalVector(dm, &X);CHKERRQ(ierr);
    ierr = DMGetGlobalVector(dm, &F);CHKERRQ(ierr);
    ierr = DMPlexProjectFunction(dm, user.fe, user.exactFuncs, NULL, INSERT_VALUES, X);CHKERRQ(ierr);
    ierr = SNESComputeFunction(snes, X, F);CHKERRQ(ierr);
    ierr = DMRestoreGlobalVector(dm, &X);CHKERRQ(ierr);
    ierr = DMRestoreGlobalVector(dm, &F);CHKERRQ(ierr);
  }
  if (user.computeJacobian) {
    Vec          X;
    Mat          J;

    ierr = DMGetGlobalVector(dm, &X);CHKERRQ(ierr);
    ierr = DMSetMatType(dm,MATAIJ);CHKERRQ(ierr);
    ierr = DMCreateMatrix(dm, &J);CHKERRQ(ierr);
    ierr = SNESComputeJacobian(snes, X, J, J);CHKERRQ(ierr);
    ierr = MatDestroy(&J);CHKERRQ(ierr);
    ierr = DMRestoreGlobalVector(dm, &X);CHKERRQ(ierr);
  }
  ierr = PetscFree(user.exactFuncs);CHKERRQ(ierr);
  ierr = DestroyElement(&user);CHKERRQ(ierr);
  ierr = DMDestroy(&dmAux);CHKERRQ(ierr);
  ierr = DMDestroy(&dm);CHKERRQ(ierr);
  ierr = SNESDestroy(&snes);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return 0;
}

干净易读，像代码一样......

然而阅读它让我头疼，因为来自子弹phisix \ gamedev backgrownd我没有看到3个主要的东西：在哪里设置尺寸，创建网格，并且施加力？

任何人都可以解释如何使用PETSc SNES设置三维有限元求解器（高亮显示如何设置尺寸，进给网格，施加力并插入结果）？

Answer 1

我没有使用这些库的经验，但这至少是一个开始（还不完整的答案）。我看到的一件事非常令人费解的是主程序中似乎没有循环。不要感觉不好，缺乏评论，编码风格和在线文档使得这很难理解。

至少在函数中创建了网格，从这里调用（在main中）：

ierr = CreateMesh(PETSC_COMM_WORLD, &user, &dm);CHKERRQ(ierr);

该功能在此处的代码中进一步定义：

PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
{
  PetscInt       dim               = user->dim;
  PetscBool      interpolate       = user->interpolate;
  PetscReal      refinementLimit   = user->refinementLimit;
  PetscBool      refinementUniform = user->refinementUniform;
  PetscInt       refinementRounds  = user->refinementRounds;
  const char     *partitioner      = user->partitioner;
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = PetscLogEventBegin(user->createMeshEvent,0,0,0,0);CHKERRQ(ierr);
  ierr = DMPlexCreateBoxMesh(comm, dim, interpolate, dm);CHKERRQ(ierr);
  {
    DM refinedMesh     = NULL;
    DM distributedMesh = NULL;

    /* Refine mesh using a volume constraint */
    ierr = DMPlexSetRefinementLimit(*dm, refinementLimit);CHKERRQ(ierr);
    ierr = DMRefine(*dm, comm, &refinedMesh);CHKERRQ(ierr);
   if (refinedMesh) {
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = refinedMesh;
    }
    /* Distribute mesh over processes */
    ierr = DMPlexDistribute(*dm, partitioner, 0, NULL, &distributedMesh);CHKERRQ(ierr);
    if (distributedMesh) {
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = distributedMesh;
    }
    /* Use regular refinement in parallel */
    if (refinementUniform) {
      PetscInt r;

      ierr = DMPlexSetRefinementUniform(*dm, refinementUniform);CHKERRQ(ierr);
      for (r = 0; r < refinementRounds; ++r) {
        ierr = DMRefine(*dm, comm, &refinedMesh);CHKERRQ(ierr);
        if (refinedMesh) {
          ierr = DMDestroy(dm);CHKERRQ(ierr);
          *dm  = refinedMesh;
        }
      }
    }
  }
  ierr = PetscObjectSetName((PetscObject) *dm, "Mesh");CHKERRQ(ierr);
  ierr = DMSetFromOptions(*dm);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(user->createMeshEvent,0,0,0,0);CHKERRQ(ierr);

  user->dm = *dm;
  PetscFunctionReturn(0);
}

Answer 2

1）PETSc选项通常从命令行设置。请参阅PetscOptionsInt()，这是PetscOptionsGetInt（）的并行版本。相关的代码行是：

ierr = PetscOptionsInt("-dim", "The topological mesh dimension", "ex52.c", options->dim, &options->dim, NULL);CHKERRQ(ierr);

2）Jonathan已经提到了网格创建功能：

PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm) { ... }

3）在SNESSetFunction帮助中，您将能够看到正在尝试解决f'(x) x = -f(x)，其中为f'(x)和f(x)形成矩阵（见http://acts.nersc.gov/events/Workshop2003/slides/Gropp.pdf第71页。因此，力量进入f'(x)和f(x)的矩阵的装配。解决等式f'(x) x = -f(x)的代码的相关部分是：

ierr = SNESComputeFunction(snes, X, F);CHKERRQ(ierr);

4）要查看SNESComputFunction(ones, X, F);函数调用的结果，您可能希望像src/snes/examples/tutorials/ex12.c一样使用像VecChop（）/ VecView（）这样的函数。

最后，如果您不希望手上有很多时间，我强烈建议您考虑是否可以在GPU上使用以下替代方案 - 使用更高级别的软件包，例如MOOSE或FEniCS直接与PETSc整合。使用更高级别的包将为您节省大量时间。例如，在FEniCS中，指定方程的弱形式而不是手工组装矩阵。 FEniCS的另一个有用之处在于它可以指定球形网格。在FEniCS文档中的以下page中，相关行仅为mesh = UnitSphere(10)。