Real
GapHeatTransfer::computeQpOffDiagJacobian( unsigned jvar )
{
computeGapValues();
if (!_has_info)
return 0;
unsigned coupled_component(0);
bool active(false);
if ( _xdisp_coupled && jvar == _xdisp_var )
{
coupled_component = 0;
active = true;
}
else if ( _ydisp_coupled && jvar == _ydisp_var )
{
coupled_component = 1;
active = true;
}
else if ( _zdisp_coupled && jvar == _zdisp_var )
{
coupled_component = 2;
active = true;
}
Real dRdx(0);
if ( active )
{
// Compute dR/du_[xyz]
// Residual is based on
// h_gap = h_conduction() + h_contact() + h_radiation();
// grad_t = (_u[_qp] - _gap_temp) * h_gap;
// So we need
// (_u[_qp] - _gap_temp) * (dh_gap/du_[xyz]);
// Assuming dh_contact/du_[xyz] = dh_radiation/du_[xyz] = 0,
// we need dh_conduction/du_[xyz]
// Given
// h_conduction = gapK / gapLength, then
// dh_conduction/du_[xyz] = -gapK/gapLength^2 * dgapLength/du_[xyz]
// Given
// gapLength = ((u_x-m_x)^2+(u_y-m_y)^2+(u_z-m_z)^2)^1/2
// where m_[xyz] is the master coordinate, then
// dGapLength/du_[xyz] = 1/2*((u_x-m_x)^2+(u_y-m_y)^2+(u_z-m_z)^2)^(-1/2)*2*(u_[xyz]-m_[xyz])
// = (u_[xyz]-m_[xyz])/gapLength
// This is the normal vector.
const Real gapL = gapLength();
// THIS IS NOT THE NORMAL WE NEED.
// WE NEED THE NORMAL FROM THE CONSTRAINT, THE NORMAL FROM THE
// MASTER SURFACE. HOWEVER, THIS IS TRICKY SINCE THE NORMAL
// FROM THE MASTER SURFACE WAS COMPUTED FOR A POINT ASSOCIATED
// WITH A SLAVE NODE. NOW WE ARE AT A SLAVE INTEGRATION POINT.
//
// HOW DO WE GET THE NORMAL WE NEED?
//
// Until we have the normal we need,
// we'll hope that the one we have is close to the negative of the one we need.
const Point & normal( _normals[_qp] );
const Real dgap = dgapLength( -normal(coupled_component) );
dRdx = -(_u[_qp]-_gap_temp)*_edge_multiplier*_gap_conductance[_qp]/gapL * dgap;
}
return _test[_i][_qp] * dRdx * _phi[_j][_qp];
}
//
// sets angle derivatives dR'/dth
//
void Node2d::setDr()
{
dRdx(0,0) = dRdx(1,1) = w2n(1,0); // -sin(th)
dRdx(0,1) = w2n(0,0); // cos(th)
dRdx(1,0) = -w2n(0,0); // -cos(th)
}
//
// sets angle derivatives
//
void Node::setDr(bool local)
{
// for dS'*R', with dS the incremental change
if (local)
{
#if 0
dRdx = w2n.block<3,3>(0,0) * dRidx;
dRdy = w2n.block<3,3>(0,0) * dRidy;
dRdz = w2n.block<3,3>(0,0) * dRidz;
#endif
dRdx = dRidx * w2n.block<3,3>(0,0);
dRdy = dRidy * w2n.block<3,3>(0,0);
dRdz = dRidz * w2n.block<3,3>(0,0);
}
else
{
double x2 = qrot.x() * 2.0;
double y2 = qrot.y() * 2.0;
double z2 = qrot.z() * 2.0;
double w2 = qrot.w() * 2.0;
double xw = qrot.x()/qrot.w(); // these are problematic for w near zero
double yw = qrot.y()/qrot.w();
double zw = qrot.z()/qrot.w();
// dR/dx
dRdx(0,0) = 0.0;
dRdx(0,1) = y2-zw*x2;
dRdx(0,2) = z2+yw*x2;
dRdx(1,0) = y2+zw*x2;
dRdx(1,1) = -2.0*x2;
dRdx(1,2) = w2-xw*x2;
dRdx(2,0) = z2-yw*x2;
dRdx(2,1) = -dRdx(1,2);
dRdx(2,2) = dRdx(1,1);
// dR/dy
dRdy(0,0) = -2.0*y2;
dRdy(0,1) = x2-zw*y2;
dRdy(0,2) = (-w2)+yw*y2;
dRdy(1,0) = x2+zw*y2;
dRdy(1,1) = 0.0;
dRdy(1,2) = dRdx(2,0);
dRdy(2,0) = -dRdy(0,2);
dRdy(2,1) = dRdx(0,2);
dRdy(2,2) = dRdy(0,0);
// dR/dz
dRdz(0,0) = -2.0*z2;
dRdz(0,1) = w2-zw*z2;
dRdz(0,2) = dRdy(1,0);
dRdz(1,0) = -dRdz(0,1);
dRdz(1,1) = dRdz(0,0);
dRdz(1,2) = dRdx(0,1);
dRdz(2,0) = dRdy(0,1);
dRdz(2,1) = dRdx(1,0);
dRdz(2,2) = 0.0;
}
}
