void FEMap::compute_face_map(int dim, const std::vector<Real>& qw,
const Elem* side)
{
libmesh_assert (side != NULL);
START_LOG("compute_face_map()", "FEMap");
// The number of quadrature points.
const unsigned int n_qp = qw.size();
switch (dim)
{
case 1:
{
// A 1D finite element, currently assumed to be in 1D space
// This means the boundary is a "0D finite element", a
// NODEELEM.
// Resize the vectors to hold data at the quadrature points
{
this->xyz.resize(n_qp);
normals.resize(n_qp);
this->JxW.resize(n_qp);
}
// If we have no quadrature points, there's nothing else to do
if (!n_qp)
break;
// We need to look back at the full edge to figure out the normal
// vector
const Elem *elem = side->parent();
libmesh_assert (elem);
if (side->node(0) == elem->node(0))
normals[0] = Point(-1.);
else
{
libmesh_assert (side->node(0) == elem->node(1));
normals[0] = Point(1.);
}
// Calculate x at the point
libmesh_assert (this->psi_map.size() == 1);
// In the unlikely event we have multiple quadrature
// points, they'll be in the same place
for (unsigned int p=0; p<n_qp; p++)
{
this->xyz[p].zero();
this->xyz[p].add_scaled (side->point(0), this->psi_map[0][p]);
normals[p] = normals[0];
this->JxW[p] = 1.0*qw[p];
}
// done computing the map
break;
}
case 2:
{
// A 2D finite element living in either 2D or 3D space.
// This means the boundary is a 1D finite element, i.e.
// and EDGE2 or EDGE3.
// Resize the vectors to hold data at the quadrature points
{
this->xyz.resize(n_qp);
this->dxyzdxi_map.resize(n_qp);
this->d2xyzdxi2_map.resize(n_qp);
this->tangents.resize(n_qp);
this->normals.resize(n_qp);
this->curvatures.resize(n_qp);
this->JxW.resize(n_qp);
}
// Clear the entities that will be summed
// Compute the tangent & normal at the quadrature point
for (unsigned int p=0; p<n_qp; p++)
{
this->tangents[p].resize(LIBMESH_DIM-1); // 1 Tangent in 2D, 2 in 3D
this->xyz[p].zero();
this->dxyzdxi_map[p].zero();
this->d2xyzdxi2_map[p].zero();
}
// compute x, dxdxi at the quadrature points
for (unsigned int i=0; i<this->psi_map.size(); i++) // sum over the nodes
{
const Point& side_point = side->point(i);
for (unsigned int p=0; p<n_qp; p++) // for each quadrature point...
{
this->xyz[p].add_scaled (side_point, this->psi_map[i][p]);
this->dxyzdxi_map[p].add_scaled (side_point, this->dpsidxi_map[i][p]);
this->d2xyzdxi2_map[p].add_scaled(side_point, this->d2psidxi2_map[i][p]);
}
}
// Compute the tangent & normal at the quadrature point
for (unsigned int p=0; p<n_qp; p++)
{
// The first tangent comes from just the edge's Jacobian
this->tangents[p][0] = this->dxyzdxi_map[p].unit();
#if LIBMESH_DIM == 2
// For a 2D element living in 2D, the normal is given directly
// from the entries in the edge Jacobian.
this->normals[p] = (Point(this->dxyzdxi_map[p](1), -this->dxyzdxi_map[p](0), 0.)).unit();
#elif LIBMESH_DIM == 3
// For a 2D element living in 3D, there is a second tangent.
// For the second tangent, we need to refer to the full
// element's (not just the edge's) Jacobian.
const Elem *elem = side->parent();
libmesh_assert (elem != NULL);
// Inverse map xyz[p] to a reference point on the parent...
Point reference_point = FE<2,LAGRANGE>::inverse_map(elem, this->xyz[p]);
// Get dxyz/dxi and dxyz/deta from the parent map.
Point dx_dxi = FE<2,LAGRANGE>::map_xi (elem, reference_point);
Point dx_deta = FE<2,LAGRANGE>::map_eta(elem, reference_point);
// The second tangent vector is formed by crossing these vectors.
tangents[p][1] = dx_dxi.cross(dx_deta).unit();
// Finally, the normal in this case is given by crossing these
// two tangents.
normals[p] = tangents[p][0].cross(tangents[p][1]).unit();
#endif
// The curvature is computed via the familiar Frenet formula:
// curvature = [d^2(x) / d (xi)^2] dot [normal]
// For a reference, see:
// F.S. Merritt, Mathematics Manual, 1962, McGraw-Hill, p. 310
//
// Note: The sign convention here is different from the
// 3D case. Concave-upward curves (smiles) have a positive
// curvature. Concave-downward curves (frowns) have a
// negative curvature. Be sure to take that into account!
const Real numerator = this->d2xyzdxi2_map[p] * this->normals[p];
const Real denominator = this->dxyzdxi_map[p].size_sq();
libmesh_assert (denominator != 0);
curvatures[p] = numerator / denominator;
}
// compute the jacobian at the quadrature points
for (unsigned int p=0; p<n_qp; p++)
{
const Real jac = this->dxyzdxi_map[p].size();
libmesh_assert (jac > 0.);
this->JxW[p] = jac*qw[p];
}
// done computing the map
break;
}
case 3:
{
// A 3D finite element living in 3D space.
// Resize the vectors to hold data at the quadrature points
{
this->xyz.resize(n_qp);
this->dxyzdxi_map.resize(n_qp);
this->dxyzdeta_map.resize(n_qp);
this->d2xyzdxi2_map.resize(n_qp);
this->d2xyzdxideta_map.resize(n_qp);
this->d2xyzdeta2_map.resize(n_qp);
this->tangents.resize(n_qp);
this->normals.resize(n_qp);
this->curvatures.resize(n_qp);
this->JxW.resize(n_qp);
}
// Clear the entities that will be summed
for (unsigned int p=0; p<n_qp; p++)
{
this->tangents[p].resize(LIBMESH_DIM-1); // 1 Tangent in 2D, 2 in 3D
this->xyz[p].zero();
this->dxyzdxi_map[p].zero();
this->dxyzdeta_map[p].zero();
this->d2xyzdxi2_map[p].zero();
this->d2xyzdxideta_map[p].zero();
this->d2xyzdeta2_map[p].zero();
}
// compute x, dxdxi at the quadrature points
for (unsigned int i=0; i<this->psi_map.size(); i++) // sum over the nodes
{
const Point& side_point = side->point(i);
for (unsigned int p=0; p<n_qp; p++) // for each quadrature point...
{
this->xyz[p].add_scaled (side_point, this->psi_map[i][p]);
this->dxyzdxi_map[p].add_scaled (side_point, this->dpsidxi_map[i][p]);
this->dxyzdeta_map[p].add_scaled(side_point, this->dpsideta_map[i][p]);
this->d2xyzdxi2_map[p].add_scaled (side_point, this->d2psidxi2_map[i][p]);
this->d2xyzdxideta_map[p].add_scaled(side_point, this->d2psidxideta_map[i][p]);
this->d2xyzdeta2_map[p].add_scaled (side_point, this->d2psideta2_map[i][p]);
}
}
// Compute the tangents, normal, and curvature at the quadrature point
for (unsigned int p=0; p<n_qp; p++)
{
const Point n = this->dxyzdxi_map[p].cross(this->dxyzdeta_map[p]);
this->normals[p] = n.unit();
this->tangents[p][0] = this->dxyzdxi_map[p].unit();
this->tangents[p][1] = n.cross(this->dxyzdxi_map[p]).unit();
// Compute curvature using the typical nomenclature
// of the first and second fundamental forms.
// For reference, see:
// 1) http://mathworld.wolfram.com/MeanCurvature.html
// (note -- they are using inward normal)
// 2) F.S. Merritt, Mathematics Manual, 1962, McGraw-Hill
const Real L = -this->d2xyzdxi2_map[p] * this->normals[p];
const Real M = -this->d2xyzdxideta_map[p] * this->normals[p];
const Real N = -this->d2xyzdeta2_map[p] * this->normals[p];
const Real E = this->dxyzdxi_map[p].size_sq();
const Real F = this->dxyzdxi_map[p] * this->dxyzdeta_map[p];
const Real G = this->dxyzdeta_map[p].size_sq();
const Real numerator = E*N -2.*F*M + G*L;
const Real denominator = E*G - F*F;
libmesh_assert (denominator != 0.);
curvatures[p] = 0.5*numerator/denominator;
}
// compute the jacobian at the quadrature points, see
// http://sp81.msi.umn.edu:999/fluent/fidap/help/theory/thtoc.htm
for (unsigned int p=0; p<n_qp; p++)
{
const Real g11 = (dxdxi_map(p)*dxdxi_map(p) +
dydxi_map(p)*dydxi_map(p) +
dzdxi_map(p)*dzdxi_map(p));
const Real g12 = (dxdxi_map(p)*dxdeta_map(p) +
dydxi_map(p)*dydeta_map(p) +
dzdxi_map(p)*dzdeta_map(p));
const Real g21 = g12;
const Real g22 = (dxdeta_map(p)*dxdeta_map(p) +
dydeta_map(p)*dydeta_map(p) +
dzdeta_map(p)*dzdeta_map(p));
const Real jac = std::sqrt(g11*g22 - g12*g21);
libmesh_assert (jac > 0.);
this->JxW[p] = jac*qw[p];
}
// done computing the map
break;
}
default:
libmesh_error();
}
STOP_LOG("compute_face_map()", "FEMap");
}
void FEMap::compute_single_point_map(const unsigned int dim,
const std::vector<Real>& qw,
const Elem* elem,
unsigned int p)
{
libmesh_assert(elem);
switch (dim)
{
//--------------------------------------------------------------------
// 0D
case 0:
{
xyz[p] = elem->point(0);
jac[p] = 1.0;
JxW[p] = qw[p];
break;
}
//--------------------------------------------------------------------
// 1D
case 1:
{
// Clear the entities that will be summed
xyz[p].zero();
dxyzdxi_map[p].zero();
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
d2xyzdxi2_map[p].zero();
#endif
// compute x, dx, d2x at the quadrature point
for (unsigned int i=0; i<phi_map.size(); i++) // sum over the nodes
{
// Reference to the point, helps eliminate
// exessive temporaries in the inner loop
const Point& elem_point = elem->point(i);
xyz[p].add_scaled (elem_point, phi_map[i][p] );
dxyzdxi_map[p].add_scaled (elem_point, dphidxi_map[i][p]);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
d2xyzdxi2_map[p].add_scaled(elem_point, d2phidxi2_map[i][p]);
#endif
}
// Compute the jacobian
//
// 1D elements can live in 2D or 3D space.
// The transformation matrix from local->global
// coordinates is
//
// T = | dx/dxi |
// | dy/dxi |
// | dz/dxi |
//
// The generalized determinant of T (from the
// so-called "normal" eqns.) is
// jac = "det(T)" = sqrt(det(T'T))
//
// where T'= transpose of T, so
//
// jac = sqrt( (dx/dxi)^2 + (dy/dxi)^2 + (dz/dxi)^2 )
jac[p] = dxyzdxi_map[p].size();
if (jac[p] <= 0.)
{
libMesh::err << "ERROR: negative Jacobian: "
<< jac[p]
<< " in element "
<< elem->id()
<< std::endl;
libmesh_error();
}
// The inverse Jacobian entries also come from the
// generalized inverse of T (see also the 2D element
// living in 3D code).
const Real jacm2 = 1./jac[p]/jac[p];
dxidx_map[p] = jacm2*dxdxi_map(p);
#if LIBMESH_DIM > 1
dxidy_map[p] = jacm2*dydxi_map(p);
#endif
#if LIBMESH_DIM > 2
dxidz_map[p] = jacm2*dzdxi_map(p);
#endif
JxW[p] = jac[p]*qw[p];
// done computing the map
break;
}
//--------------------------------------------------------------------
// 2D
case 2:
{
//------------------------------------------------------------------
// Compute the (x,y) values at the quadrature points,
// the Jacobian at the quadrature points
xyz[p].zero();
dxyzdxi_map[p].zero();
dxyzdeta_map[p].zero();
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
d2xyzdxi2_map[p].zero();
d2xyzdxideta_map[p].zero();
d2xyzdeta2_map[p].zero();
#endif
// compute (x,y) at the quadrature points, derivatives once
for (unsigned int i=0; i<phi_map.size(); i++) // sum over the nodes
{
// Reference to the point, helps eliminate
// exessive temporaries in the inner loop
const Point& elem_point = elem->point(i);
xyz[p].add_scaled (elem_point, phi_map[i][p] );
dxyzdxi_map[p].add_scaled (elem_point, dphidxi_map[i][p] );
dxyzdeta_map[p].add_scaled (elem_point, dphideta_map[i][p]);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
d2xyzdxi2_map[p].add_scaled (elem_point, d2phidxi2_map[i][p]);
d2xyzdxideta_map[p].add_scaled (elem_point, d2phidxideta_map[i][p]);
d2xyzdeta2_map[p].add_scaled (elem_point, d2phideta2_map[i][p]);
#endif
}
// compute the jacobian once
const Real dx_dxi = dxdxi_map(p),
dx_deta = dxdeta_map(p),
dy_dxi = dydxi_map(p),
dy_deta = dydeta_map(p);
#if LIBMESH_DIM == 2
// Compute the Jacobian. This assumes the 2D face
// lives in 2D space
//
// Symbolically, the matrix determinant is
//
// | dx/dxi dx/deta |
// jac = | dy/dxi dy/deta |
//
// jac = dx/dxi*dy/deta - dx/deta*dy/dxi
jac[p] = (dx_dxi*dy_deta - dx_deta*dy_dxi);
if (jac[p] <= 0.)
{
libMesh::err << "ERROR: negative Jacobian: "
<< jac[p]
<< " in element "
<< elem->id()
<< std::endl;
libmesh_error();
}
JxW[p] = jac[p]*qw[p];
// Compute the shape function derivatives wrt x,y at the
// quadrature points
const Real inv_jac = 1./jac[p];
dxidx_map[p] = dy_deta*inv_jac; //dxi/dx = (1/J)*dy/deta
dxidy_map[p] = -dx_deta*inv_jac; //dxi/dy = -(1/J)*dx/deta
detadx_map[p] = -dy_dxi* inv_jac; //deta/dx = -(1/J)*dy/dxi
detady_map[p] = dx_dxi* inv_jac; //deta/dy = (1/J)*dx/dxi
dxidz_map[p] = detadz_map[p] = 0.;
#else
const Real dz_dxi = dzdxi_map(p),
dz_deta = dzdeta_map(p);
// Compute the Jacobian. This assumes a 2D face in
// 3D space.
//
// The transformation matrix T from local to global
// coordinates is
//
// | dx/dxi dx/deta |
// T = | dy/dxi dy/deta |
// | dz/dxi dz/deta |
// note det(T' T) = det(T')det(T) = det(T)det(T)
// so det(T) = std::sqrt(det(T' T))
//
//----------------------------------------------
// Notes:
//
// dX = R dXi -> R'dX = R'R dXi
// (R^-1)dX = dXi [(R'R)^-1 R']dX = dXi
//
// so R^-1 = (R'R)^-1 R'
//
// and R^-1 R = (R'R)^-1 R'R = I.
//
const Real g11 = (dx_dxi*dx_dxi +
dy_dxi*dy_dxi +
dz_dxi*dz_dxi);
const Real g12 = (dx_dxi*dx_deta +
dy_dxi*dy_deta +
dz_dxi*dz_deta);
const Real g21 = g12;
const Real g22 = (dx_deta*dx_deta +
dy_deta*dy_deta +
dz_deta*dz_deta);
const Real det = (g11*g22 - g12*g21);
if (det <= 0.)
{
libMesh::err << "ERROR: negative Jacobian! "
<< " in element "
<< elem->id()
<< std::endl;
libmesh_error();
}
const Real inv_det = 1./det;
jac[p] = std::sqrt(det);
JxW[p] = jac[p]*qw[p];
const Real g11inv = g22*inv_det;
const Real g12inv = -g12*inv_det;
const Real g21inv = -g21*inv_det;
const Real g22inv = g11*inv_det;
dxidx_map[p] = g11inv*dx_dxi + g12inv*dx_deta;
dxidy_map[p] = g11inv*dy_dxi + g12inv*dy_deta;
dxidz_map[p] = g11inv*dz_dxi + g12inv*dz_deta;
detadx_map[p] = g21inv*dx_dxi + g22inv*dx_deta;
detady_map[p] = g21inv*dy_dxi + g22inv*dy_deta;
detadz_map[p] = g21inv*dz_dxi + g22inv*dz_deta;
#endif
// done computing the map
break;
}
//--------------------------------------------------------------------
// 3D
case 3:
{
//------------------------------------------------------------------
// Compute the (x,y,z) values at the quadrature points,
// the Jacobian at the quadrature point
// Clear the entities that will be summed
xyz[p].zero ();
dxyzdxi_map[p].zero ();
dxyzdeta_map[p].zero ();
dxyzdzeta_map[p].zero ();
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
d2xyzdxi2_map[p].zero();
d2xyzdxideta_map[p].zero();
d2xyzdxidzeta_map[p].zero();
d2xyzdeta2_map[p].zero();
d2xyzdetadzeta_map[p].zero();
d2xyzdzeta2_map[p].zero();
#endif
// compute (x,y,z) at the quadrature points,
// dxdxi, dydxi, dzdxi,
// dxdeta, dydeta, dzdeta,
// dxdzeta, dydzeta, dzdzeta all once
for (unsigned int i=0; i<phi_map.size(); i++) // sum over the nodes
{
// Reference to the point, helps eliminate
// exessive temporaries in the inner loop
const Point& elem_point = elem->point(i);
xyz[p].add_scaled (elem_point, phi_map[i][p] );
dxyzdxi_map[p].add_scaled (elem_point, dphidxi_map[i][p] );
dxyzdeta_map[p].add_scaled (elem_point, dphideta_map[i][p] );
dxyzdzeta_map[p].add_scaled (elem_point, dphidzeta_map[i][p]);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
d2xyzdxi2_map[p].add_scaled (elem_point,
d2phidxi2_map[i][p]);
d2xyzdxideta_map[p].add_scaled (elem_point,
d2phidxideta_map[i][p]);
d2xyzdxidzeta_map[p].add_scaled (elem_point,
d2phidxidzeta_map[i][p]);
d2xyzdeta2_map[p].add_scaled (elem_point,
d2phideta2_map[i][p]);
d2xyzdetadzeta_map[p].add_scaled (elem_point,
d2phidetadzeta_map[i][p]);
d2xyzdzeta2_map[p].add_scaled (elem_point,
d2phidzeta2_map[i][p]);
#endif
}
// compute the jacobian
const Real
dx_dxi = dxdxi_map(p), dy_dxi = dydxi_map(p), dz_dxi = dzdxi_map(p),
dx_deta = dxdeta_map(p), dy_deta = dydeta_map(p), dz_deta = dzdeta_map(p),
dx_dzeta = dxdzeta_map(p), dy_dzeta = dydzeta_map(p), dz_dzeta = dzdzeta_map(p);
// Symbolically, the matrix determinant is
//
// | dx/dxi dy/dxi dz/dxi |
// jac = | dx/deta dy/deta dz/deta |
// | dx/dzeta dy/dzeta dz/dzeta |
//
// jac = dx/dxi*(dy/deta*dz/dzeta - dz/deta*dy/dzeta) +
// dy/dxi*(dz/deta*dx/dzeta - dx/deta*dz/dzeta) +
// dz/dxi*(dx/deta*dy/dzeta - dy/deta*dx/dzeta)
jac[p] = (dx_dxi*(dy_deta*dz_dzeta - dz_deta*dy_dzeta) +
dy_dxi*(dz_deta*dx_dzeta - dx_deta*dz_dzeta) +
dz_dxi*(dx_deta*dy_dzeta - dy_deta*dx_dzeta));
if (jac[p] <= 0.)
{
libMesh::err << "ERROR: negative Jacobian: "
<< jac[p]
<< " in element "
<< elem->id()
<< std::endl;
libmesh_error();
}
JxW[p] = jac[p]*qw[p];
// Compute the shape function derivatives wrt x,y at the
// quadrature points
const Real inv_jac = 1./jac[p];
dxidx_map[p] = (dy_deta*dz_dzeta - dz_deta*dy_dzeta)*inv_jac;
dxidy_map[p] = (dz_deta*dx_dzeta - dx_deta*dz_dzeta)*inv_jac;
dxidz_map[p] = (dx_deta*dy_dzeta - dy_deta*dx_dzeta)*inv_jac;
detadx_map[p] = (dz_dxi*dy_dzeta - dy_dxi*dz_dzeta )*inv_jac;
detady_map[p] = (dx_dxi*dz_dzeta - dz_dxi*dx_dzeta )*inv_jac;
detadz_map[p] = (dy_dxi*dx_dzeta - dx_dxi*dy_dzeta )*inv_jac;
dzetadx_map[p] = (dy_dxi*dz_deta - dz_dxi*dy_deta )*inv_jac;
dzetady_map[p] = (dz_dxi*dx_deta - dx_dxi*dz_deta )*inv_jac;
dzetadz_map[p] = (dx_dxi*dy_deta - dy_dxi*dx_deta )*inv_jac;
// done computing the map
break;
}
default:
libmesh_error();
}
}