Real
AuxChemElastic::computeDifferential(const Real & coupled_conserved, const Real & coupled_nonconserved)
{
// partial derivative of f_chem with respect to conserved variable
Real dfchem_dcons(0.0);
// partial derivative of f_chem with respect to nonconserved variable
Real dfchem_dnoncons(0.0);
// partial derivatives of self-elastic energies
Real dself_dcons(0.0);
Real dself_dnoncons(0.0);
// partial derivative of interaction elastic energy
Real dint_dcons(0.0);
Real dint_dnoncons(0.0);
Real first_term(0.0);
Real second_term(0.0);
dfchem_dcons = computeDfchemDcons(coupled_conserved, coupled_nonconserved);
dself_dcons = computeDselfDcons();
dint_dcons = computeDintDcons();
dfchem_dnoncons = computeDfchemDnoncons(coupled_conserved, coupled_nonconserved);
dself_dnoncons = computeDselfDnoncons();
dint_dnoncons = computeDintDnoncons();
first_term = (dfchem_dcons + dself_dcons + dint_dcons)*(_precip_conserved - coupled_conserved);
second_term = (dfchem_dnoncons + dself_dnoncons + dint_dnoncons)*(_precip_nonconserved - coupled_nonconserved);
return first_term + second_term;
}
Real
AuxChem::computeDifferential(const Real & coupled_conserved, const Real & coupled_nonconserved)
{
// partial derivative of f_chem with respect to conserved variable
Real dfchem_dcons(0.0);
// partial derivative of f_chem with respect to nonconserved variable
Real dfchem_dnoncons(0.0);
Real first_term(0.0);
Real second_term(0.0);
dfchem_dcons = computeDfchemDcons(coupled_conserved, coupled_nonconserved);
dfchem_dnoncons = computeDfchemDnoncons(coupled_conserved, coupled_nonconserved);
first_term = (dfchem_dcons)*(_precip_conserved - coupled_conserved);
second_term = (dfchem_dnoncons)*(_precip_nonconserved - coupled_nonconserved);
return first_term + second_term;
}
Real
StressDivergenceRZ::calculateJacobian( unsigned int ivar, unsigned int jvar )
{
// B^T_i * C * B_j
SymmTensor test, phi;
if (ivar == 0)
{
test.xx() = _grad_test[_i][_qp](0);
test.xy() = 0.5*_grad_test[_i][_qp](1);
test.zz() = _test[_i][_qp] / _q_point[_qp](0);
}
else
{
test.xy() = 0.5*_grad_test[_i][_qp](0);
test.yy() = _grad_test[_i][_qp](1);
}
if (jvar == 0)
{
phi.xx() = _grad_phi[_j][_qp](0);
phi.xy() = 0.5*_grad_phi[_j][_qp](1);
phi.zz() = _phi[_j][_qp] / _q_point[_qp](0);
}
else
{
phi.xy() = 0.5*_grad_phi[_j][_qp](0);
phi.yy() = _grad_phi[_j][_qp](1);
}
SymmTensor tmp( _Jacobian_mult[_qp] * phi );
Real first_term( test.doubleContraction( tmp ) );
Real val = 0.0;
// volumetric locking correction
// K = Bbar^T_i * C * Bbar^T_j where Bbar = B + Bvol
// K = B^T_i * C * B_j + Bvol^T_i * C * Bvol_j + B^T_i * C * Bvol_j + Bvol^T_i * C * B_j
if (_volumetric_locking_correction)
{
RealGradient new_test(2, 0.0);
RealGradient new_phi(2, 0.0);
new_test(0) = _grad_test[_i][_qp](0) + _test[_i][_qp] / _q_point[_qp](0);
new_test(1) = _grad_test[_i][_qp](1);
new_phi(0) = _grad_phi[_j][_qp](0) + _phi[_j][_qp] / _q_point[_qp](0);
new_phi(1) = _grad_phi[_j][_qp](1);
// Bvol^T_i * C * Bvol_j
val += _Jacobian_mult[_qp].sum_3x3() * (_avg_grad_test[_i][ivar] - new_test(ivar)) * (_avg_grad_phi[_j][jvar] - new_phi(jvar)) / 3.0;
// B^T_i * C * Bvol_j
RealGradient sum_3x1 = _Jacobian_mult[_qp].sum_3x1();
if (ivar == 0 && jvar == 0)
val += (sum_3x1(0) * test.xx() + sum_3x1(2) * test.zz()) * (_avg_grad_phi[_j][0] - new_phi(0));
else if (ivar == 0 && jvar == 1)
val += (sum_3x1(0) * test.xx() + sum_3x1(2) * test.zz()) * (_avg_grad_phi[_j][1] - new_phi(1));
else if (ivar == 1 && jvar == 0)
val += sum_3x1(1) * test.yy() * (_avg_grad_phi[_j][0] - new_phi(0));
else
val += sum_3x1(1) * test.yy() * (_avg_grad_phi[_j][1] - new_phi(1));
// Bvol^T_i * C * B_j
// tmp = C * B_j from above
if (ivar == 0)
val += (tmp.xx() + tmp.yy() + tmp.zz()) * (_avg_grad_test[_i][0] - new_test(0));
else
val += (tmp.xx() + tmp.yy() + tmp.zz()) * (_avg_grad_test[_i][1] - new_test(1));
}
return val / 3.0 + first_term;
}