void ElasticMembranePressure<PressureType>::element_time_derivative
( bool compute_jacobian, AssemblyContext & context )
{
unsigned int u_var = this->_disp_vars.u();
unsigned int v_var = this->_disp_vars.v();
unsigned int w_var = this->_disp_vars.w();
const unsigned int n_u_dofs = context.get_dof_indices(u_var).size();
const std::vector<libMesh::Real> &JxW =
this->get_fe(context)->get_JxW();
const std::vector<std::vector<libMesh::Real> >& u_phi =
this->get_fe(context)->get_phi();
const MultiphysicsSystem & system = context.get_multiphysics_system();
unsigned int u_dot_var = system.get_second_order_dot_var(u_var);
unsigned int v_dot_var = system.get_second_order_dot_var(v_var);
unsigned int w_dot_var = system.get_second_order_dot_var(w_var);
libMesh::DenseSubVector<libMesh::Number> &Fu = context.get_elem_residual(u_dot_var);
libMesh::DenseSubVector<libMesh::Number> &Fv = context.get_elem_residual(v_dot_var);
libMesh::DenseSubVector<libMesh::Number> &Fw = context.get_elem_residual(w_dot_var);
libMesh::DenseSubMatrix<libMesh::Number>& Kuv = context.get_elem_jacobian(u_dot_var,v_var);
libMesh::DenseSubMatrix<libMesh::Number>& Kuw = context.get_elem_jacobian(u_dot_var,w_var);
libMesh::DenseSubMatrix<libMesh::Number>& Kvu = context.get_elem_jacobian(v_dot_var,u_var);
libMesh::DenseSubMatrix<libMesh::Number>& Kvw = context.get_elem_jacobian(v_dot_var,w_var);
libMesh::DenseSubMatrix<libMesh::Number>& Kwu = context.get_elem_jacobian(w_dot_var,u_var);
libMesh::DenseSubMatrix<libMesh::Number>& Kwv = context.get_elem_jacobian(w_dot_var,v_var);
unsigned int n_qpoints = context.get_element_qrule().n_points();
// All shape function gradients are w.r.t. master element coordinates
const std::vector<std::vector<libMesh::Real> >& dphi_dxi =
this->get_fe(context)->get_dphidxi();
const std::vector<std::vector<libMesh::Real> >& dphi_deta =
this->get_fe(context)->get_dphideta();
const libMesh::DenseSubVector<libMesh::Number>& u_coeffs = context.get_elem_solution( u_var );
const libMesh::DenseSubVector<libMesh::Number>& v_coeffs = context.get_elem_solution( v_var );
const libMesh::DenseSubVector<libMesh::Number>& w_coeffs = context.get_elem_solution( w_var );
const std::vector<libMesh::RealGradient>& dxdxi = this->get_fe(context)->get_dxyzdxi();
const std::vector<libMesh::RealGradient>& dxdeta = this->get_fe(context)->get_dxyzdeta();
for (unsigned int qp=0; qp != n_qpoints; qp++)
{
// sqrt(det(a_cov)), a_cov being the covariant metric tensor of undeformed body
libMesh::Real sqrt_a = sqrt( dxdxi[qp]*dxdxi[qp]*dxdeta[qp]*dxdeta[qp]
- dxdxi[qp]*dxdeta[qp]*dxdeta[qp]*dxdxi[qp] );
// Gradients are w.r.t. master element coordinates
libMesh::Gradient grad_u, grad_v, grad_w;
for( unsigned int d = 0; d < n_u_dofs; d++ )
{
libMesh::RealGradient u_gradphi( dphi_dxi[d][qp], dphi_deta[d][qp] );
grad_u += u_coeffs(d)*u_gradphi;
grad_v += v_coeffs(d)*u_gradphi;
grad_w += w_coeffs(d)*u_gradphi;
}
libMesh::RealGradient dudxi( grad_u(0), grad_v(0), grad_w(0) );
libMesh::RealGradient dudeta( grad_u(1), grad_v(1), grad_w(1) );
libMesh::RealGradient A_1 = dxdxi[qp] + dudxi;
libMesh::RealGradient A_2 = dxdeta[qp] + dudeta;
libMesh::RealGradient A_3 = A_1.cross(A_2);
// Compute pressure at this quadrature point
libMesh::Real press = (*_pressure)(context,qp);
// Small optimization
libMesh::Real p_over_sa = press/sqrt_a;
/* The formula here is actually
P*\sqrt{\frac{A}{a}}*A_3, where A_3 is a unit vector
But, |A_3| = \sqrt{A} so the normalizing part kills
the \sqrt{A} in the numerator, so we can leave it out
and *not* normalize A_3.
*/
libMesh::RealGradient traction = p_over_sa*A_3;
for (unsigned int i=0; i != n_u_dofs; i++)
{
// Small optimization
libMesh::Real phi_times_jac = u_phi[i][qp]*JxW[qp];
Fu(i) -= traction(0)*phi_times_jac;
Fv(i) -= traction(1)*phi_times_jac;
Fw(i) -= traction(2)*phi_times_jac;
if( compute_jacobian )
{
for (unsigned int j=0; j != n_u_dofs; j++)
{
libMesh::RealGradient u_gradphi( dphi_dxi[j][qp], dphi_deta[j][qp] );
const libMesh::Real dt0_dv = p_over_sa*(u_gradphi(0)*A_2(2) - A_1(2)*u_gradphi(1));
const libMesh::Real dt0_dw = p_over_sa*(A_1(1)*u_gradphi(1) - u_gradphi(0)*A_2(1));
const libMesh::Real dt1_du = p_over_sa*(A_1(2)*u_gradphi(1) - u_gradphi(0)*A_2(2));
const libMesh::Real dt1_dw = p_over_sa*(u_gradphi(0)*A_2(0) - A_1(0)*u_gradphi(1));
const libMesh::Real dt2_du = p_over_sa*(u_gradphi(0)*A_2(1) - A_1(1)*u_gradphi(1));
const libMesh::Real dt2_dv = p_over_sa*(A_1(0)*u_gradphi(1) - u_gradphi(0)*A_2(0));
Kuv(i,j) -= dt0_dv*phi_times_jac;
Kuw(i,j) -= dt0_dw*phi_times_jac;
Kvu(i,j) -= dt1_du*phi_times_jac;
Kvw(i,j) -= dt1_dw*phi_times_jac;
Kwu(i,j) -= dt2_du*phi_times_jac;
Kwv(i,j) -= dt2_dv*phi_times_jac;
}
}
}
}
}
void ElasticMembraneConstantPressure::element_time_derivative( bool compute_jacobian,
AssemblyContext& context,
CachedValues& /*cache*/ )
{
const unsigned int n_u_dofs = context.get_dof_indices(_disp_vars.u()).size();
const std::vector<libMesh::Real> &JxW =
this->get_fe(context)->get_JxW();
const std::vector<std::vector<libMesh::Real> >& u_phi =
this->get_fe(context)->get_phi();
libMesh::DenseSubVector<libMesh::Number> &Fu = context.get_elem_residual(_disp_vars.u());
libMesh::DenseSubVector<libMesh::Number> &Fv = context.get_elem_residual(_disp_vars.v());
libMesh::DenseSubVector<libMesh::Number> &Fw = context.get_elem_residual(_disp_vars.w());
libMesh::DenseSubMatrix<libMesh::Number>& Kuv = context.get_elem_jacobian(_disp_vars.u(),_disp_vars.v());
libMesh::DenseSubMatrix<libMesh::Number>& Kuw = context.get_elem_jacobian(_disp_vars.u(),_disp_vars.w());
libMesh::DenseSubMatrix<libMesh::Number>& Kvu = context.get_elem_jacobian(_disp_vars.v(),_disp_vars.u());
libMesh::DenseSubMatrix<libMesh::Number>& Kvw = context.get_elem_jacobian(_disp_vars.v(),_disp_vars.w());
libMesh::DenseSubMatrix<libMesh::Number>& Kwu = context.get_elem_jacobian(_disp_vars.w(),_disp_vars.u());
libMesh::DenseSubMatrix<libMesh::Number>& Kwv = context.get_elem_jacobian(_disp_vars.w(),_disp_vars.v());
unsigned int n_qpoints = context.get_element_qrule().n_points();
// All shape function gradients are w.r.t. master element coordinates
const std::vector<std::vector<libMesh::Real> >& dphi_dxi =
this->get_fe(context)->get_dphidxi();
const std::vector<std::vector<libMesh::Real> >& dphi_deta =
this->get_fe(context)->get_dphideta();
const libMesh::DenseSubVector<libMesh::Number>& u_coeffs = context.get_elem_solution( _disp_vars.u() );
const libMesh::DenseSubVector<libMesh::Number>& v_coeffs = context.get_elem_solution( _disp_vars.v() );
const libMesh::DenseSubVector<libMesh::Number>& w_coeffs = context.get_elem_solution( _disp_vars.w() );
const std::vector<libMesh::RealGradient>& dxdxi = this->get_fe(context)->get_dxyzdxi();
const std::vector<libMesh::RealGradient>& dxdeta = this->get_fe(context)->get_dxyzdeta();
for (unsigned int qp=0; qp != n_qpoints; qp++)
{
// sqrt(det(a_cov)), a_cov being the covariant metric tensor of undeformed body
libMesh::Real sqrt_a = sqrt( dxdxi[qp]*dxdxi[qp]*dxdeta[qp]*dxdeta[qp]
- dxdxi[qp]*dxdeta[qp]*dxdeta[qp]*dxdxi[qp] );
// Gradients are w.r.t. master element coordinates
libMesh::Gradient grad_u, grad_v, grad_w;
for( unsigned int d = 0; d < n_u_dofs; d++ )
{
libMesh::RealGradient u_gradphi( dphi_dxi[d][qp], dphi_deta[d][qp] );
grad_u += u_coeffs(d)*u_gradphi;
grad_v += v_coeffs(d)*u_gradphi;
grad_w += w_coeffs(d)*u_gradphi;
}
libMesh::RealGradient dudxi( grad_u(0), grad_v(0), grad_w(0) );
libMesh::RealGradient dudeta( grad_u(1), grad_v(1), grad_w(1) );
libMesh::RealGradient A_1 = dxdxi[qp] + dudxi;
libMesh::RealGradient A_2 = dxdeta[qp] + dudeta;
libMesh::RealGradient A_3 = A_1.cross(A_2);
/* The formula here is actually
P*\sqrt{\frac{A}{a}}*A_3, where A_3 is a unit vector
But, |A_3| = \sqrt{A} so the normalizing part kills
the \sqrt{A} in the numerator, so we can leave it out
and *not* normalize A_3.
*/
libMesh::RealGradient traction = _pressure/sqrt_a*A_3;
libMesh::Real jac = JxW[qp];
for (unsigned int i=0; i != n_u_dofs; i++)
{
Fu(i) -= traction(0)*u_phi[i][qp]*jac;
Fv(i) -= traction(1)*u_phi[i][qp]*jac;
Fw(i) -= traction(2)*u_phi[i][qp]*jac;
if( compute_jacobian )
{
for (unsigned int j=0; j != n_u_dofs; j++)
{
libMesh::RealGradient u_gradphi( dphi_dxi[j][qp], dphi_deta[j][qp] );
const libMesh::Real dt0_dv = _pressure/sqrt_a*(u_gradphi(0)*A_2(2) - A_1(2)*u_gradphi(1));
const libMesh::Real dt0_dw = _pressure/sqrt_a*(A_1(1)*u_gradphi(1) - u_gradphi(0)*A_2(1));
const libMesh::Real dt1_du = _pressure/sqrt_a*(A_1(2)*u_gradphi(1) - u_gradphi(0)*A_2(2));
const libMesh::Real dt1_dw = _pressure/sqrt_a*(u_gradphi(0)*A_2(0) - A_1(0)*u_gradphi(1));
const libMesh::Real dt2_du = _pressure/sqrt_a*(u_gradphi(0)*A_2(1) - A_1(1)*u_gradphi(1));
const libMesh::Real dt2_dv = _pressure/sqrt_a*(A_1(0)*u_gradphi(1) - u_gradphi(0)*A_2(0));
Kuv(i,j) -= dt0_dv*u_phi[i][qp]*jac;
Kuw(i,j) -= dt0_dw*u_phi[i][qp]*jac;
Kvu(i,j) -= dt1_du*u_phi[i][qp]*jac;
Kvw(i,j) -= dt1_dw*u_phi[i][qp]*jac;
Kwu(i,j) -= dt2_du*u_phi[i][qp]*jac;
Kwv(i,j) -= dt2_dv*u_phi[i][qp]*jac;
}
}
}
}
return;
}