C++ AssemblyContext::get_elem_jacobian方法代码示例

  void AveragedTurbine<Mu>::nonlocal_time_derivative(bool compute_jacobian,
				                 AssemblyContext& context,
				                 CachedValues& /* cache */ )
  {
    libMesh::DenseSubMatrix<libMesh::Number> &Kss =
            context.get_elem_jacobian(this->fan_speed_var(), this->fan_speed_var()); // R_{s},{s}

    libMesh::DenseSubVector<libMesh::Number> &Fs =
            context.get_elem_residual(this->fan_speed_var()); // R_{s}

    const std::vector<libMesh::dof_id_type>& dof_indices =
      context.get_dof_indices(this->fan_speed_var());

    const libMesh::Number fan_speed =
      context.get_system().current_solution(dof_indices[0]);

    const libMesh::Number output_torque =
      this->torque_function(libMesh::Point(0), fan_speed);

    Fs(0) += output_torque;

    if (compute_jacobian)
      {
        // FIXME: we should replace this FEM with a hook to the AD fparser stuff
        const libMesh::Number epsilon = 1e-6;
        const libMesh::Number output_torque_deriv =
          (this->torque_function(libMesh::Point(0), fan_speed+epsilon) -
           this->torque_function(libMesh::Point(0), fan_speed-epsilon)) / (2*epsilon);

        Kss(0,0) += output_torque_deriv * context.get_elem_solution_derivative();
      }

    return;
  }

  void HeatConduction<K>::mass_residual( bool compute_jacobian,
				      AssemblyContext& context,
				      CachedValues& /*cache*/ )
  {
    // First we get some references to cell-specific data that
    // will be used to assemble the linear system.

    // Element Jacobian * quadrature weights for interior integration
    const std::vector<libMesh::Real> &JxW = 
      context.get_element_fe(_temp_vars.T_var())->get_JxW();

    // The shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& phi = 
      context.get_element_fe(_temp_vars.T_var())->get_phi();

    // The number of local degrees of freedom in each variable
    const unsigned int n_T_dofs = context.get_dof_indices(_temp_vars.T_var()).size();

    // The subvectors and submatrices we need to fill:
    libMesh::DenseSubVector<libMesh::Real> &F =
      context.get_elem_residual(_temp_vars.T_var());

    libMesh::DenseSubMatrix<libMesh::Real> &M =
      context.get_elem_jacobian(_temp_vars.T_var(), _temp_vars.T_var());

    unsigned int n_qpoints = context.get_element_qrule().n_points();
    
    for (unsigned int qp = 0; qp != n_qpoints; ++qp)
      {
	// For the mass residual, we need to be a little careful.
	// The time integrator is handling the time-discretization
	// for us so we need to supply M(u_fixed)*u' for the residual.
	// u_fixed will be given by the fixed_interior_value function
	// while u' will be given by the interior_rate function.
        libMesh::Real T_dot;
        context.interior_rate(_temp_vars.T_var(), qp, T_dot);

	for (unsigned int i = 0; i != n_T_dofs; ++i)
	  {
	    F(i) -= JxW[qp]*(_rho*_Cp*T_dot*phi[i][qp] );

	    if( compute_jacobian )
              {
                for (unsigned int j=0; j != n_T_dofs; j++)
                  {
		    // We're assuming rho, cp are constant w.r.t. T here.
                    M(i,j) -=
                      context.get_elem_solution_rate_derivative()
                        * JxW[qp]*_rho*_Cp*phi[j][qp]*phi[i][qp] ;
                  }
              }// End of check on Jacobian

	  } // End of element dof loop

      } // End of the quadrature point loop

    return;
  }

  void BoundaryConditions::pin_value( AssemblyContext& context,
                                      const CachedValues& /*cache*/,
                                      const bool request_jacobian,
                                      const VariableIndex var, 
                                      const double pin_value,
                                      const libMesh::Point& pin_location, 
                                      const double penalty )
  {
    if (context.get_elem().contains_point(pin_location))
      {
        libMesh::FEGenericBase<libMesh::Real>* elem_fe = NULL; 
        context.get_element_fe( var, elem_fe );

	libMesh::DenseSubVector<libMesh::Number> &F_var = context.get_elem_residual(var); // residual
	libMesh::DenseSubMatrix<libMesh::Number> &K_var = context.get_elem_jacobian(var,var); // jacobian

	// The number of local degrees of freedom in p variable.
	const unsigned int n_var_dofs = context.get_dof_indices(var).size();

	libMesh::Number var_value = context.point_value(var, pin_location);

	libMesh::FEType fe_type = elem_fe->get_fe_type();
      
	libMesh::Point point_loc_in_masterelem = 
	  libMesh::FEInterface::inverse_map(context.get_dim(), fe_type, &context.get_elem(), pin_location);

	std::vector<libMesh::Real> phi(n_var_dofs);

	for (unsigned int i=0; i != n_var_dofs; i++)
          {
            phi[i] = libMesh::FEInterface::shape( context.get_dim(), fe_type, &context.get_elem(), i, 
                                                  point_loc_in_masterelem );
          }
      
	for (unsigned int i=0; i != n_var_dofs; i++)
	  {
	    F_var(i) += penalty*(var_value - pin_value)*phi[i];
	  
	    /** \todo What the hell is the context.get_elem_solution_derivative() all about? */
	    if (request_jacobian && context.get_elem_solution_derivative())
	      {
		libmesh_assert (context.get_elem_solution_derivative() == 1.0);
	      
		for (unsigned int j=0; j != n_var_dofs; j++)
		  K_var(i,j) += penalty*phi[i]*phi[j];

	      } // End if request_jacobian
	  } // End i loop
      } // End if pin_location

    return;
  }

  void ScalarODE::nonlocal_constraint(bool compute_jacobian,
				      AssemblyContext& context,
				      CachedValues& /* cache */ )
  {
    libMesh::DenseSubMatrix<libMesh::Number> &Kss =
            context.get_elem_jacobian(_scalar_ode_var, _scalar_ode_var); // R_{s},{s}

    libMesh::DenseSubVector<libMesh::Number> &Fs =
            context.get_elem_residual(_scalar_ode_var); // R_{s}

    const libMesh::Number constraint =
      (*constraint_function)(context, libMesh::Point(0),
                             context.get_time());

    Fs(0) += constraint;

    if (compute_jacobian)
      {
        // FIXME: we should replace this hacky FDM with a hook to the
        // AD fparser stuff
        libMesh::DenseSubVector<libMesh::Number> &Us =
          const_cast<libMesh::DenseSubVector<libMesh::Number>&>
            (context.get_elem_solution(_scalar_ode_var)); // U_{s}

        const libMesh::Number s = Us(0);
        Us(0) = s + this->_epsilon;
        libMesh::Number constraint_jacobian =
          (*constraint_function)(context, libMesh::Point(0),
                                 context.get_time());

        Us(0) = s - this->_epsilon;
        constraint_jacobian -=
          (*constraint_function)(context, libMesh::Point(0),
                                 context.get_time());
           
        Us(0) = s;
        constraint_jacobian /= (2*this->_epsilon);

        Kss(0,0) += constraint_jacobian *
          context.get_elem_solution_derivative();
      }

    return;
  }

  void AveragedTurbine<Mu>::nonlocal_mass_residual( bool compute_jacobian,
				                AssemblyContext& context,
				                CachedValues& /* cache */ )
  {
    libMesh::DenseSubMatrix<libMesh::Number> &Kss =
            context.get_elem_jacobian(this->fan_speed_var(), this->fan_speed_var()); // R_{s},{s}

    libMesh::DenseSubVector<libMesh::Number> &Fs =
            context.get_elem_residual(this->fan_speed_var()); // R_{s}

    const libMesh::DenseSubVector<libMesh::Number> &Us =
      context.get_elem_solution_rate(this->fan_speed_var());

    const libMesh::Number& fan_speed = Us(0);

    Fs(0) -= this->moment_of_inertia * fan_speed;

    if (compute_jacobian)
      {
        Kss(0,0) -= this->moment_of_inertia * context.get_elem_solution_rate_derivative();
      }

    return;
  }

  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++)
//.........这里部分代码省略.........

  void AxisymmetricBoussinesqBuoyancy::element_time_derivative( bool compute_jacobian,
								AssemblyContext& context,
								CachedValues& /*cache*/ )
  {
#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->BeginTimer("AxisymmetricBoussinesqBuoyancy::element_time_derivative");
#endif

    // The number of local degrees of freedom in each variable.
    const unsigned int n_u_dofs = context.get_dof_indices(_flow_vars.u_var()).size();
    const unsigned int n_T_dofs = context.get_dof_indices(_temp_vars.T_var()).size();

    // Element Jacobian * quadrature weights for interior integration.
    const std::vector<libMesh::Real> &JxW =
      context.get_element_fe(_flow_vars.u_var())->get_JxW();

    // The velocity shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& vel_phi =
      context.get_element_fe(_flow_vars.u_var())->get_phi();

    // The temperature shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& T_phi =
      context.get_element_fe(_temp_vars.T_var())->get_phi();

    // Physical location of the quadrature points
    const std::vector<libMesh::Point>& u_qpoint =
      context.get_element_fe(_flow_vars.u_var())->get_xyz();

    // Get residuals
    libMesh::DenseSubVector<libMesh::Number> &Fr = context.get_elem_residual(_flow_vars.u_var()); // R_{r}
    libMesh::DenseSubVector<libMesh::Number> &Fz = context.get_elem_residual(_flow_vars.v_var()); // R_{z}

    // Get Jacobians
    libMesh::DenseSubMatrix<libMesh::Number> &KrT = context.get_elem_jacobian(_flow_vars.u_var(), _temp_vars.T_var()); // R_{r},{T}
    libMesh::DenseSubMatrix<libMesh::Number> &KzT = context.get_elem_jacobian(_flow_vars.v_var(), _temp_vars.T_var()); // R_{z},{T}

    // Now we will build the element Jacobian and residual.
    // Constructing the residual requires the solution and its
    // gradient from the previous timestep.  This must be
    // calculated at each quadrature point by summing the
    // solution degree-of-freedom values by the appropriate
    // weight functions.
    unsigned int n_qpoints = context.get_element_qrule().n_points();

    for (unsigned int qp=0; qp != n_qpoints; qp++)
      {
	const libMesh::Number r = u_qpoint[qp](0);

	// Compute the solution & its gradient at the old Newton iterate.
	libMesh::Number T;
	T = context.interior_value(_temp_vars.T_var(), qp);

	// First, an i-loop over the velocity degrees of freedom.
	// We know that n_u_dofs == n_v_dofs so we can compute contributions
	// for both at the same time.
	for (unsigned int i=0; i != n_u_dofs; i++)
	  {
	    Fr(i) += -_rho*_beta_T*(T - _T_ref)*_g(0)*vel_phi[i][qp]*r*JxW[qp];
	    Fz(i) += -_rho*_beta_T*(T - _T_ref)*_g(1)*vel_phi[i][qp]*r*JxW[qp];

	    if (compute_jacobian && context.get_elem_solution_derivative())
	      {
		for (unsigned int j=0; j != n_T_dofs; j++)
		  {
		    const libMesh::Number val =
                      -_rho*_beta_T*vel_phi[i][qp]*T_phi[j][qp]*r*JxW[qp]
                      * context.get_elem_solution_derivative();
		    KrT(i,j) += val*_g(0);
		    KzT(i,j) += val*_g(1);
		  } // End j dof loop
	      } // End compute_jacobian check

	  } // End i dof loop
      } // End quadrature loop

#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->EndTimer("AxisymmetricBoussinesqBuoyancy::element_time_derivative");
#endif

    return;
  }

  void VelocityPenalty<Mu>::element_time_derivative( bool compute_jacobian,
					         AssemblyContext& context,
					         CachedValues& /* cache */ )
  {
#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->BeginTimer("VelocityPenalty::element_time_derivative");
#endif

    // Element Jacobian * quadrature weights for interior integration
    const std::vector<libMesh::Real> &JxW = 
      context.get_element_fe(this->_flow_vars.u_var())->get_JxW();

    // The shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& u_phi = 
      context.get_element_fe(this->_flow_vars.u_var())->get_phi();

    const std::vector<libMesh::Point>& u_qpoint = 
      context.get_element_fe(this->_flow_vars.u_var())->get_xyz();

    // The number of local degrees of freedom in each variable
    const unsigned int n_u_dofs = context.get_dof_indices(this->_flow_vars.u_var()).size();

    // The subvectors and submatrices we need to fill:
    libMesh::DenseSubMatrix<libMesh::Number> &Kuu = context.get_elem_jacobian(this->_flow_vars.u_var(), this->_flow_vars.u_var()); // R_{u},{u}
    libMesh::DenseSubMatrix<libMesh::Number> &Kuv = context.get_elem_jacobian(this->_flow_vars.u_var(), this->_flow_vars.v_var()); // R_{u},{v}
    libMesh::DenseSubMatrix<libMesh::Number> &Kvu = context.get_elem_jacobian(this->_flow_vars.v_var(), this->_flow_vars.u_var()); // R_{v},{u}
    libMesh::DenseSubMatrix<libMesh::Number> &Kvv = context.get_elem_jacobian(this->_flow_vars.v_var(), this->_flow_vars.v_var()); // R_{v},{v}

    libMesh::DenseSubMatrix<libMesh::Number>* Kwu = NULL;
    libMesh::DenseSubMatrix<libMesh::Number>* Kwv = NULL;
    libMesh::DenseSubMatrix<libMesh::Number>* Kww = NULL;
    libMesh::DenseSubMatrix<libMesh::Number>* Kuw = NULL;
    libMesh::DenseSubMatrix<libMesh::Number>* Kvw = NULL;

    libMesh::DenseSubVector<libMesh::Number> &Fu = context.get_elem_residual(this->_flow_vars.u_var()); // R_{u}
    libMesh::DenseSubVector<libMesh::Number> &Fv = context.get_elem_residual(this->_flow_vars.v_var()); // R_{v}
    libMesh::DenseSubVector<libMesh::Number>* Fw = NULL;

    if( this->_dim == 3 )
      {
        Kuw = &context.get_elem_jacobian(this->_flow_vars.u_var(), this->_flow_vars.w_var()); // R_{u},{w}
        Kvw = &context.get_elem_jacobian(this->_flow_vars.v_var(), this->_flow_vars.w_var()); // R_{v},{w}

        Kwu = &context.get_elem_jacobian(this->_flow_vars.w_var(), this->_flow_vars.u_var()); // R_{w},{u}
        Kwv = &context.get_elem_jacobian(this->_flow_vars.w_var(), this->_flow_vars.v_var()); // R_{w},{v}
        Kww = &context.get_elem_jacobian(this->_flow_vars.w_var(), this->_flow_vars.w_var()); // R_{w},{w}
        Fw  = &context.get_elem_residual(this->_flow_vars.w_var()); // R_{w}
      }

    unsigned int n_qpoints = context.get_element_qrule().n_points();

    for (unsigned int qp=0; qp != n_qpoints; qp++)
      {
        // Compute the solution at the old Newton iterate.
        libMesh::Number u, v;
        u = context.interior_value(this->_flow_vars.u_var(), qp);
        v = context.interior_value(this->_flow_vars.v_var(), qp);

        libMesh::NumberVectorValue U(u,v);
        if (this->_dim == 3)
          U(2) = context.interior_value(this->_flow_vars.w_var(), qp); // w

        libMesh::NumberVectorValue F;
        libMesh::NumberTensorValue dFdU;
        libMesh::NumberTensorValue* dFdU_ptr =
          compute_jacobian ? &dFdU : NULL;
        if (!this->compute_force(u_qpoint[qp], context, U, F, dFdU_ptr))
          continue;

        const libMesh::Real jac = JxW[qp];

        for (unsigned int i=0; i != n_u_dofs; i++)
          {
            const libMesh::Number jac_i = jac * u_phi[i][qp];

            Fu(i) += F(0)*jac_i;

            Fv(i) += F(1)*jac_i;
            if( this->_dim == 3 )
              {
                (*Fw)(i) += F(2)*jac_i;
              }

	    if( compute_jacobian )
              {
                for (unsigned int j=0; j != n_u_dofs; j++)
                  {
                    const libMesh::Number jac_ij = context.get_elem_solution_derivative() * jac_i * u_phi[j][qp];
                    Kuu(i,j) += jac_ij * dFdU(0,0);
                    Kuv(i,j) += jac_ij * dFdU(0,1);
                    Kvu(i,j) += jac_ij * dFdU(1,0);
                    Kvv(i,j) += jac_ij * dFdU(1,1);

                    if( this->_dim == 3 )
                      {
                        (*Kuw)(i,j) += jac_ij * dFdU(0,2);
                        (*Kvw)(i,j) += jac_ij * dFdU(1,2);

                        (*Kwu)(i,j) += jac_ij * dFdU(2,0);
                        (*Kwv)(i,j) += jac_ij * dFdU(2,1);
//.........这里部分代码省略.........

  void ElasticCableRayleighDamping<StressStrainLaw>::damping_residual( bool compute_jacobian,
                                                                       AssemblyContext& context,
                                                                       CachedValues& /*cache*/)
  {
    // First, do the "mass" contribution
    this->mass_residual_impl(compute_jacobian,
                               context,
                               &libMesh::FEMContext::interior_rate,
                               &libMesh::DiffContext::get_elem_solution_rate_derivative,
                               _mu_factor);

    // Now do the stiffness contribution
    const unsigned int n_u_dofs = context.get_dof_indices(this->_disp_vars.u()).size();

    const std::vector<libMesh::Real> &JxW =
      this->get_fe(context)->get_JxW();

    // Residuals that we're populating
    libMesh::DenseSubVector<libMesh::Number> &Fu = context.get_elem_residual(this->_disp_vars.u());
    libMesh::DenseSubVector<libMesh::Number> &Fv = context.get_elem_residual(this->_disp_vars.v());
    libMesh::DenseSubVector<libMesh::Number> &Fw = context.get_elem_residual(this->_disp_vars.w());

    //Grab the Jacobian matrix as submatrices
    //libMesh::DenseMatrix<libMesh::Number> &K = context.get_elem_jacobian();
    libMesh::DenseSubMatrix<libMesh::Number> &Kuu = context.get_elem_jacobian(this->_disp_vars.u(),this->_disp_vars.u());
    libMesh::DenseSubMatrix<libMesh::Number> &Kuv = context.get_elem_jacobian(this->_disp_vars.u(),this->_disp_vars.v());
    libMesh::DenseSubMatrix<libMesh::Number> &Kuw = context.get_elem_jacobian(this->_disp_vars.u(),this->_disp_vars.w());
    libMesh::DenseSubMatrix<libMesh::Number> &Kvu = context.get_elem_jacobian(this->_disp_vars.v(),this->_disp_vars.u());
    libMesh::DenseSubMatrix<libMesh::Number> &Kvv = context.get_elem_jacobian(this->_disp_vars.v(),this->_disp_vars.v());
    libMesh::DenseSubMatrix<libMesh::Number> &Kvw = context.get_elem_jacobian(this->_disp_vars.v(),this->_disp_vars.w());
    libMesh::DenseSubMatrix<libMesh::Number> &Kwu = context.get_elem_jacobian(this->_disp_vars.w(),this->_disp_vars.u());
    libMesh::DenseSubMatrix<libMesh::Number> &Kwv = context.get_elem_jacobian(this->_disp_vars.w(),this->_disp_vars.v());
    libMesh::DenseSubMatrix<libMesh::Number> &Kww = context.get_elem_jacobian(this->_disp_vars.w(),this->_disp_vars.w());

    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 libMesh::DenseSubVector<libMesh::Number>& u_coeffs = context.get_elem_solution( this->_disp_vars.u() );
    const libMesh::DenseSubVector<libMesh::Number>& v_coeffs = context.get_elem_solution( this->_disp_vars.v() );
    const libMesh::DenseSubVector<libMesh::Number>& w_coeffs = context.get_elem_solution( this->_disp_vars.w() );

    const libMesh::DenseSubVector<libMesh::Number>& dudt_coeffs = context.get_elem_solution_rate( this->_disp_vars.u() );
    const libMesh::DenseSubVector<libMesh::Number>& dvdt_coeffs = context.get_elem_solution_rate( this->_disp_vars.v() );
    const libMesh::DenseSubVector<libMesh::Number>& dwdt_coeffs = context.get_elem_solution_rate( this->_disp_vars.w() );

    // Need these to build up the covariant and contravariant metric tensors
    const std::vector<libMesh::RealGradient>& dxdxi  = this->get_fe(context)->get_dxyzdxi();

    const unsigned int dim = 1; // The cable dimension is always 1 for this physics

    for (unsigned int qp=0; qp != n_qpoints; qp++)
      {
        // Gradients are w.r.t. master element coordinates
        libMesh::Gradient grad_u, grad_v, grad_w;
        libMesh::Gradient dgradu_dt, dgradv_dt, dgradw_dt;

        for( unsigned int d = 0; d < n_u_dofs; d++ )
          {
            libMesh::RealGradient u_gradphi( dphi_dxi[d][qp] );
            grad_u += u_coeffs(d)*u_gradphi;
            grad_v += v_coeffs(d)*u_gradphi;
            grad_w += w_coeffs(d)*u_gradphi;

            dgradu_dt += dudt_coeffs(d)*u_gradphi;
            dgradv_dt += dvdt_coeffs(d)*u_gradphi;
            dgradw_dt += dwdt_coeffs(d)*u_gradphi;
          }

        libMesh::RealGradient grad_x( dxdxi[qp](0) );
        libMesh::RealGradient grad_y( dxdxi[qp](1) );
        libMesh::RealGradient grad_z( dxdxi[qp](2) );

        libMesh::TensorValue<libMesh::Real> a_cov, a_contra, A_cov, A_contra;
        libMesh::Real lambda_sq = 0;

        this->compute_metric_tensors( qp, *(this->get_fe(context)), context,
                                      grad_u, grad_v, grad_w,
                                      a_cov, a_contra, A_cov, A_contra,
                                      lambda_sq );

        // Compute stress tensor
        libMesh::TensorValue<libMesh::Real> tau;
        ElasticityTensor C;
        this->_stress_strain_law.compute_stress_and_elasticity(dim,a_contra,a_cov,A_contra,A_cov,tau,C);

        libMesh::Real jac = JxW[qp];

        for (unsigned int i=0; i != n_u_dofs; i++)
          {
            libMesh::RealGradient u_gradphi( dphi_dxi[i][qp] );

            libMesh::Real u_diag_factor = _lambda_factor*this->_A*jac*tau(0,0)*dgradu_dt(0)*u_gradphi(0);
            libMesh::Real v_diag_factor = _lambda_factor*this->_A*jac*tau(0,0)*dgradv_dt(0)*u_gradphi(0);
            libMesh::Real w_diag_factor = _lambda_factor*this->_A*jac*tau(0,0)*dgradw_dt(0)*u_gradphi(0);

            const libMesh::Real C1 = _lambda_factor*this->_A*jac*C(0,0,0,0)*u_gradphi(0);

            const libMesh::Real gamma_u = (grad_x(0)+grad_u(0));
//.........这里部分代码省略.........

  void HeatTransfer::mass_residual( bool compute_jacobian,
				    AssemblyContext& context,
				    CachedValues& /*cache*/ )
  {
#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->BeginTimer("HeatTransfer::mass_residual");
#endif

    // First we get some references to cell-specific data that
    // will be used to assemble the linear system.

    // Element Jacobian * quadrature weights for interior integration
    const std::vector<libMesh::Real> &JxW = 
      context.get_element_fe(_temp_vars.T_var())->get_JxW();

    // The shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& phi = 
      context.get_element_fe(_temp_vars.T_var())->get_phi();

    const std::vector<libMesh::Point>& u_qpoint = 
      context.get_element_fe(this->_flow_vars.u_var())->get_xyz();

    // The number of local degrees of freedom in each variable
    const unsigned int n_T_dofs = context.get_dof_indices(_temp_vars.T_var()).size();

    // The subvectors and submatrices we need to fill:
    libMesh::DenseSubVector<libMesh::Real> &F = context.get_elem_residual(_temp_vars.T_var());

    libMesh::DenseSubMatrix<libMesh::Real> &M = context.get_elem_jacobian(_temp_vars.T_var(), _temp_vars.T_var());

    unsigned int n_qpoints = context.get_element_qrule().n_points();

    for (unsigned int qp = 0; qp != n_qpoints; ++qp)
      {
	// For the mass residual, we need to be a little careful.
	// The time integrator is handling the time-discretization
	// for us so we need to supply M(u_fixed)*u for the residual.
	// u_fixed will be given by the fixed_interior_* functions
	// while u will be given by the interior_* functions.
	libMesh::Real T_dot = context.interior_value(_temp_vars.T_var(), qp);

        const libMesh::Number r = u_qpoint[qp](0);

        libMesh::Real jac = JxW[qp];

        if( _is_axisymmetric )
          {
            jac *= r;
          }

	for (unsigned int i = 0; i != n_T_dofs; ++i)
	  {
	    F(i) += _rho*_Cp*T_dot*phi[i][qp]*jac;

	    if( compute_jacobian )
              {
                for (unsigned int j=0; j != n_T_dofs; j++)
                  {
		    // We're assuming rho, cp are constant w.r.t. T here.
                    M(i,j) += _rho*_Cp*phi[j][qp]*phi[i][qp]*jac;
                  }
              }// End of check on Jacobian
          
	  } // End of element dof loop
      
      } // End of the quadrature point loop

#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->EndTimer("HeatTransfer::mass_residual");
#endif

    return;
  }

	void PracticeCDRinv::element_time_derivative( bool compute_jacobian,
						AssemblyContext& context,
						CachedValues& /*cache*/ ){
	
		// The number of local degrees of freedom in each variable.
    const unsigned int n_c_dofs = context.get_dof_indices(_c_var).size();

    // We get some references to cell-specific data that
    // will be used to assemble the linear system.

    // Element Jacobian * quadrature weights for interior integration.
    const std::vector<libMesh::Real> &JxW =
      context.get_element_fe(_c_var)->get_JxW();

    // The temperature shape function gradients (in global coords.)
    // at interior quadrature points.
    const std::vector<std::vector<libMesh::RealGradient> >& dphi =
      context.get_element_fe(_c_var)->get_dphi();
    const std::vector<std::vector<libMesh::Real> >& phi = context.get_element_fe(_c_var)->get_phi();

    const std::vector<libMesh::Point>& q_points = 
      context.get_element_fe(_c_var)->get_xyz();
    
  	libMesh::DenseSubMatrix<libMesh::Number> &J_c_zc = context.get_elem_jacobian(_c_var, _zc_var);
		libMesh::DenseSubMatrix<libMesh::Number> &J_c_c = context.get_elem_jacobian(_c_var, _c_var);
	
		libMesh::DenseSubMatrix<libMesh::Number> &J_zc_c = context.get_elem_jacobian(_zc_var, _c_var);
		libMesh::DenseSubMatrix<libMesh::Number> &J_zc_fc = context.get_elem_jacobian(_zc_var, _fc_var);
	
		libMesh::DenseSubMatrix<libMesh::Number> &J_fc_zc = context.get_elem_jacobian(_fc_var, _zc_var);
		libMesh::DenseSubMatrix<libMesh::Number> &J_fc_fc = context.get_elem_jacobian(_fc_var, _fc_var);
		
		libMesh::DenseSubVector<libMesh::Number> &Rc = context.get_elem_residual( _c_var );;
		libMesh::DenseSubVector<libMesh::Number> &Rzc = context.get_elem_residual( _zc_var );
		libMesh::DenseSubVector<libMesh::Number> &Rfc = context.get_elem_residual( _fc_var );

    // Now we will build the element Jacobian and residual.
    // Constructing the residual requires the solution and its
    // gradient from the previous timestep.  This must be
    // calculated at each quadrature point by summing the
    // solution degree-of-freedom values by the appropriate
    // weight functions.
    unsigned int n_qpoints = context.get_element_qrule().n_points();

    for (unsigned int qp=0; qp != n_qpoints; qp++){

			libMesh::Number 
	      c = context.interior_value(_c_var, qp),
	      zc = context.interior_value(_zc_var, qp),
	      fc = context.interior_value(_fc_var, qp);
	    libMesh::Gradient 
	      grad_c = context.interior_gradient(_c_var, qp),
	      grad_zc = context.interior_gradient(_zc_var, qp),
	      grad_fc = context.interior_gradient(_fc_var, qp);
			
	  	//location of quadrature point
	  	const libMesh::Real ptx = q_points[qp](0);
	  	const libMesh::Real pty = q_points[qp](1);
			
   		int xind, yind;
   		libMesh::Real xdist = 1.e10; libMesh::Real ydist = 1.e10;
   		for(int ii=0; ii<x_pts.size(); ii++){
   			libMesh::Real tmp = std::abs(ptx - x_pts[ii]);
   			if(xdist > tmp){
   				xdist = tmp;
   				xind = ii;
   			}
   			else
   				break;
   		} 
   		for(int jj=0; jj<y_pts[xind].size(); jj++){
   			libMesh::Real tmp = std::abs(pty - y_pts[xind][jj]);
   			if(ydist > tmp){
   				ydist = tmp;
   				yind = jj;
   			}
   			else
   				break;
   		}
   		libMesh::Real u = vel_field[xind][yind](0);
   		libMesh::Real v = vel_field[xind][yind](1);

	    libMesh::NumberVectorValue U     (u,     v);

	
			// First, an i-loop over the  degrees of freedom.
			for (unsigned int i=0; i != n_c_dofs; i++){
				
				Rc(i) += JxW[qp]*(-_k*grad_zc*dphi[i][qp] + U*grad_zc*phi[i][qp] + 2*_R*zc*c*phi[i][qp]);
	      Rzc(i) += JxW[qp]*(-_k*grad_c*dphi[i][qp] - U*grad_c*phi[i][qp] + _R*c*c*phi[i][qp] + fc*phi[i][qp]);
     		Rfc(i) += JxW[qp]*(_beta*grad_fc*dphi[i][qp] + zc*phi[i][qp]);

				if (compute_jacobian){
					for (unsigned int j=0; j != n_c_dofs; j++){
						J_c_zc(i,j) += JxW[qp]*(-_k*dphi[j][qp]*dphi[i][qp] + U*dphi[j][qp]*phi[i][qp] 
															+ 2*_R*phi[j][qp]*c*phi[i][qp]);
						J_c_c(i,j) += JxW[qp]*(2*_R*zc*phi[j][qp]*phi[i][qp]);

						J_zc_c(i,j) += JxW[qp]*(-_k*dphi[j][qp]*dphi[i][qp] - U*dphi[j][qp]*phi[i][qp] 
																+ 2*_R*c*phi[j][qp]*phi[i][qp]);
//.........这里部分代码省略.........

  void VelocityPenaltyAdjointStabilization<Mu>::element_constraint( bool compute_jacobian,
                                                                AssemblyContext& context,
                                                                CachedValues& /*cache*/ )
  {
#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->BeginTimer("VelocityPenaltyAdjointStabilization::element_constraint");
#endif

    // The number of local degrees of freedom in each variable.
    const unsigned int n_p_dofs = context.get_dof_indices(this->_press_var.p()).size();
    const unsigned int n_u_dofs = context.get_dof_indices(this->_flow_vars.u()).size();

    // Element Jacobian * quadrature weights for interior integration.
    const std::vector<libMesh::Real> &JxW =
      context.get_element_fe(this->_flow_vars.u())->get_JxW();

    const std::vector<libMesh::Point>& u_qpoint = 
      context.get_element_fe(this->_flow_vars.u())->get_xyz();

    const std::vector<std::vector<libMesh::Real> >& u_phi =
      context.get_element_fe(this->_flow_vars.u())->get_phi();

    const std::vector<std::vector<libMesh::RealGradient> >& p_dphi =
      context.get_element_fe(this->_press_var.p())->get_dphi();

    libMesh::DenseSubVector<libMesh::Number> &Fp = context.get_elem_residual(this->_press_var.p()); // R_{p}

    libMesh::DenseSubMatrix<libMesh::Number> &Kpu = 
      context.get_elem_jacobian(this->_press_var.p(), this->_flow_vars.u()); // J_{pu}
    libMesh::DenseSubMatrix<libMesh::Number> &Kpv = 
      context.get_elem_jacobian(this->_press_var.p(), this->_flow_vars.v()); // J_{pv}
    libMesh::DenseSubMatrix<libMesh::Number> *Kpw = NULL;
 
    if(this->mesh_dim(context) == 3)
      {
        Kpw = &context.get_elem_jacobian
          (this->_press_var.p(), this->_flow_vars.w()); // J_{pw}
      }

    // Now we will build the element Jacobian and residual.
    // Constructing the residual requires the solution and its
    // gradient from the previous timestep.  This must be
    // calculated at each quadrature point by summing the
    // solution degree-of-freedom values by the appropriate
    // weight functions.
    unsigned int n_qpoints = context.get_element_qrule().n_points();

    libMesh::FEBase* fe = context.get_element_fe(this->_flow_vars.u());

    for (unsigned int qp=0; qp != n_qpoints; qp++)
      {
        libMesh::RealGradient g = this->_stab_helper.compute_g( fe, context, qp );
        libMesh::RealTensor G = this->_stab_helper.compute_G( fe, context, qp );

        libMesh::RealGradient U( context.interior_value( this->_flow_vars.u(), qp ),
                                 context.interior_value( this->_flow_vars.v(), qp ) );
        if( this->mesh_dim(context) == 3 )
          {
            U(2) = context.interior_value( this->_flow_vars.w(), qp );
          }

        // Compute the viscosity at this qp
        libMesh::Real mu_qp = this->_mu(context, qp);

        libMesh::Real tau_M;
        libMesh::Real d_tau_M_d_rho;
        libMesh::Gradient d_tau_M_dU;

        if (compute_jacobian)
          this->_stab_helper.compute_tau_momentum_and_derivs
            ( context, qp, g, G, this->_rho, U, mu_qp,
              tau_M, d_tau_M_d_rho, d_tau_M_dU,
              this->_is_steady );
        else
          tau_M = this->_stab_helper.compute_tau_momentum
                    ( context, qp, g, G, this->_rho, U, mu_qp,
                      this->_is_steady );

        libMesh::NumberVectorValue F;
        libMesh::NumberTensorValue dFdU;
        libMesh::NumberTensorValue* dFdU_ptr =
          compute_jacobian ? &dFdU : NULL;
        if (!this->compute_force(u_qpoint[qp], context, U, F, dFdU_ptr))
          continue;

        // First, an i-loop over the velocity degrees of freedom.
        // We know that n_u_dofs == n_v_dofs so we can compute contributions
        // for both at the same time.
        for (unsigned int i=0; i != n_p_dofs; i++)
          {
            Fp(i) += -tau_M*F*p_dphi[i][qp]*JxW[qp];

            if (compute_jacobian)
              {
                for (unsigned int j=0; j != n_u_dofs; ++j)
                  {
                    Kpu(i,j) += -d_tau_M_dU(0)*u_phi[j][qp]*F*p_dphi[i][qp]*JxW[qp]*context.get_elem_solution_derivative();
                    Kpv(i,j) += -d_tau_M_dU(1)*u_phi[j][qp]*F*p_dphi[i][qp]*JxW[qp]*context.get_elem_solution_derivative();
                    for (unsigned int d=0; d != 3; ++d)
                      {
//.........这里部分代码省略.........

  void BoundaryConditions::apply_neumann_axisymmetric( AssemblyContext& context,
                                                       const CachedValues& cache,
                                                       const bool request_jacobian,
                                                       const VariableIndex var,
                                                       const libMesh::Real sign,
                                                       SharedPtr<NeumannFuncObj> neumann_func ) const
  {
    libMesh::FEGenericBase<libMesh::Real>* side_fe = NULL; 
    context.get_side_fe( var, side_fe );

    // The number of local degrees of freedom
    const unsigned int n_var_dofs = context.get_dof_indices(var).size();
  
    // Element Jacobian * quadrature weight for side integration.
    const std::vector<libMesh::Real> &JxW_side = side_fe->get_JxW();

    // The var shape functions at side quadrature points.
    const std::vector<std::vector<libMesh::Real> >& var_phi_side =
      side_fe->get_phi();

    // Physical location of the quadrature points
    const std::vector<libMesh::Point>& var_qpoint =
      side_fe->get_xyz();

    const std::vector<libMesh::Point> &normals = side_fe->get_normals();

    libMesh::DenseSubVector<libMesh::Number> &F_var = context.get_elem_residual(var); // residual
    libMesh::DenseSubMatrix<libMesh::Number> &K_var = context.get_elem_jacobian(var,var); // jacobian

    unsigned int n_qpoints = context.get_side_qrule().n_points();

    for (unsigned int qp=0; qp != n_qpoints; qp++)
      {
	const libMesh::Point bc_value = neumann_func->value( context, cache, qp );
        libMesh::Point jac_value;
        if (request_jacobian)
          {
            jac_value = neumann_func->derivative( context, cache, qp );
          }

	const libMesh::Number r = var_qpoint[qp](0);

	for (unsigned int i=0; i != n_var_dofs; i++)
	  {
	    F_var(i) += sign*r*JxW_side[qp]*bc_value*normals[qp]*var_phi_side[i][qp];

	    if (request_jacobian)
	      {
		for (unsigned int j=0; j != n_var_dofs; j++)
		  {
		    K_var(i,j) += sign*r*JxW_side[qp]*jac_value*normals[qp]*
		      var_phi_side[i][qp]*var_phi_side[j][qp];
		  }
	      }
	  }
      } // End quadrature loop

    // Now must take care of the case that the boundary condition depends on variables
    // other than var.
    std::vector<VariableIndex> other_jac_vars = neumann_func->get_other_jac_vars();

    if( (other_jac_vars.size() > 0) && request_jacobian )
      {
	for( std::vector<VariableIndex>::const_iterator var2 = other_jac_vars.begin();
	     var2 != other_jac_vars.end();
	     var2++ )
	  {
            libMesh::FEGenericBase<libMesh::Real>* side_fe2 = NULL; 
            context.get_side_fe( *var2, side_fe2 );

            libMesh::DenseSubMatrix<libMesh::Number> &K_var2 = context.get_elem_jacobian(var,*var2); // jacobian

	    const unsigned int n_var2_dofs = context.get_dof_indices(*var2).size();
	    const std::vector<std::vector<libMesh::Real> >& var2_phi_side =
              side_fe2->get_phi();

	    for (unsigned int qp=0; qp != n_qpoints; qp++)
	      {
		const libMesh::Number r = var_qpoint[qp](0);

		const libMesh::Point jac_value = neumann_func->derivative( context, cache, qp, *var2 );

		for (unsigned int i=0; i != n_var_dofs; i++)
		  {
		    for (unsigned int j=0; j != n_var2_dofs; j++)
		      {
			K_var2(i,j) += sign*r*JxW_side[qp]*jac_value*normals[qp]*
			  var_phi_side[i][qp]*var2_phi_side[j][qp];
		      }
		  }
	      }
	  } // End loop over auxillary Jacobian variables
      }
    return;
  }

  void AxisymmetricHeatTransfer<Conductivity>::element_time_derivative( bool compute_jacobian,
									AssemblyContext& context,
									CachedValues& /*cache*/ )
  {
#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->BeginTimer("AxisymmetricHeatTransfer::element_time_derivative");
#endif

    // The number of local degrees of freedom in each variable.
    const unsigned int n_T_dofs = context.get_dof_indices(_T_var).size();
    const unsigned int n_u_dofs = context.get_dof_indices(_u_r_var).size();

    //TODO: check n_T_dofs is same as n_u_dofs, n_v_dofs, n_w_dofs

    // We get some references to cell-specific data that
    // will be used to assemble the linear system.

    // Element Jacobian * quadrature weights for interior integration.
    const std::vector<libMesh::Real> &JxW =
      context.get_element_fe(_T_var)->get_JxW();

    // The temperature shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& T_phi =
      context.get_element_fe(_T_var)->get_phi();

    // The velocity shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& vel_phi =
      context.get_element_fe(_u_r_var)->get_phi();

    // The temperature shape function gradients (in global coords.)
    // at interior quadrature points.
    const std::vector<std::vector<libMesh::RealGradient> >& T_gradphi =
      context.get_element_fe(_T_var)->get_dphi();

    // Physical location of the quadrature points
    const std::vector<libMesh::Point>& u_qpoint =
      context.get_element_fe(_u_r_var)->get_xyz();

    // The subvectors and submatrices we need to fill:
    libMesh::DenseSubVector<libMesh::Number> &FT = context.get_elem_residual(_T_var); // R_{T}

    libMesh::DenseSubMatrix<libMesh::Number> &KTT = context.get_elem_jacobian(_T_var, _T_var); // R_{T},{T}

    libMesh::DenseSubMatrix<libMesh::Number> &KTr = context.get_elem_jacobian(_T_var, _u_r_var); // R_{T},{r}
    libMesh::DenseSubMatrix<libMesh::Number> &KTz = context.get_elem_jacobian(_T_var, _u_z_var); // R_{T},{z}


    // Now we will build the element Jacobian and residual.
    // Constructing the residual requires the solution and its
    // gradient from the previous timestep.  This must be
    // calculated at each quadrature point by summing the
    // solution degree-of-freedom values by the appropriate
    // weight functions.
    unsigned int n_qpoints = context.get_element_qrule().n_points();

    for (unsigned int qp=0; qp != n_qpoints; qp++)
      {
	const libMesh::Number r = u_qpoint[qp](0);
      
	// Compute the solution & its gradient at the old Newton iterate.
	libMesh::Number u_r, u_z;
	u_r = context.interior_value(_u_r_var, qp);
	u_z = context.interior_value(_u_z_var, qp);

	libMesh::Gradient grad_T;
	grad_T = context.interior_gradient(_T_var, qp);

	libMesh::NumberVectorValue U (u_r,u_z);

	libMesh::Number k = this->_k( context, qp );

        // FIXME - once we have T-dependent k, we'll need its
        // derivatives in Jacobians
	// libMesh::Number dk_dT = this->_k.deriv( T );

	// First, an i-loop over the  degrees of freedom.
	for (unsigned int i=0; i != n_T_dofs; i++)
	  {
	    FT(i) += JxW[qp]*r*
	      (-_rho*_Cp*T_phi[i][qp]*(U*grad_T)    // convection term
	       -k*(T_gradphi[i][qp]*grad_T) );  // diffusion term

	    if (compute_jacobian)
	      {
		libmesh_assert (context.get_elem_solution_derivative() == 1.0);

		for (unsigned int j=0; j != n_T_dofs; j++)
		  {
		    // TODO: precompute some terms like:
		    //   _rho*_Cp*T_phi[i][qp]*(vel_phi[j][qp]*T_grad_phi[j][qp])

		    KTT(i,j) += JxW[qp] * context.get_elem_solution_derivative() *r*
		      (-_rho*_Cp*T_phi[i][qp]*(U*T_gradphi[j][qp])  // convection term
		       -k*(T_gradphi[i][qp]*T_gradphi[j][qp])); // diffusion term
		  } // end of the inner dof (j) loop

#if 0
		if( dk_dT != 0.0 )
		{
		  for (unsigned int j=0; j != n_T_dofs; j++)
//.........这里部分代码省略.........

  void HeatTransfer::element_time_derivative( bool compute_jacobian,
					      AssemblyContext& context,
					      CachedValues& /*cache*/ )
  {
#ifdef GRINS_USE_GRVY_TIMERS
    this->_timer->BeginTimer("HeatTransfer::element_time_derivative");
#endif

    // The number of local degrees of freedom in each variable.
    const unsigned int n_T_dofs = context.get_dof_indices(_temp_vars.T_var()).size();
    const unsigned int n_u_dofs = context.get_dof_indices(_flow_vars.u_var()).size();

    //TODO: check n_T_dofs is same as n_u_dofs, n_v_dofs, n_w_dofs

    // We get some references to cell-specific data that
    // will be used to assemble the linear system.

    // Element Jacobian * quadrature weights for interior integration.
    const std::vector<libMesh::Real> &JxW =
      context.get_element_fe(_temp_vars.T_var())->get_JxW();

    // The temperature shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& T_phi =
      context.get_element_fe(_temp_vars.T_var())->get_phi();

    // The velocity shape functions at interior quadrature points.
    const std::vector<std::vector<libMesh::Real> >& vel_phi =
      context.get_element_fe(_flow_vars.u_var())->get_phi();

    // The temperature shape function gradients (in global coords.)
    // at interior quadrature points.
    const std::vector<std::vector<libMesh::RealGradient> >& T_gradphi =
      context.get_element_fe(_temp_vars.T_var())->get_dphi();

    const std::vector<libMesh::Point>& u_qpoint = 
      context.get_element_fe(this->_flow_vars.u_var())->get_xyz();

    libMesh::DenseSubMatrix<libMesh::Number> &KTT = context.get_elem_jacobian(_temp_vars.T_var(), _temp_vars.T_var()); // R_{T},{T}

    libMesh::DenseSubMatrix<libMesh::Number> &KTu = context.get_elem_jacobian(_temp_vars.T_var(), _flow_vars.u_var()); // R_{T},{u}
    libMesh::DenseSubMatrix<libMesh::Number> &KTv = context.get_elem_jacobian(_temp_vars.T_var(), _flow_vars.v_var()); // R_{T},{v}
    libMesh::DenseSubMatrix<libMesh::Number>* KTw = NULL;

    libMesh::DenseSubVector<libMesh::Number> &FT = context.get_elem_residual(_temp_vars.T_var()); // R_{T}

    if( this->_dim == 3 )
      {
        KTw = &context.get_elem_jacobian(_temp_vars.T_var(), _flow_vars.w_var()); // R_{T},{w}
      }

    // Now we will build the element Jacobian and residual.
    // Constructing the residual requires the solution and its
    // gradient from the previous timestep.  This must be
    // calculated at each quadrature point by summing the
    // solution degree-of-freedom values by the appropriate
    // weight functions.
    unsigned int n_qpoints = context.get_element_qrule().n_points();

    for (unsigned int qp=0; qp != n_qpoints; qp++)
      {
	// Compute the solution & its gradient at the old Newton iterate.
	libMesh::Number u, v;
	u = context.interior_value(_flow_vars.u_var(), qp);
	v = context.interior_value(_flow_vars.v_var(), qp);

	libMesh::Gradient grad_T;
	grad_T = context.interior_gradient(_temp_vars.T_var(), qp);

	libMesh::NumberVectorValue U (u,v);
	if (_dim == 3)
	  U(2) = context.interior_value(_flow_vars.w_var(), qp);

        const libMesh::Number r = u_qpoint[qp](0);

        libMesh::Real jac = JxW[qp];

        if( _is_axisymmetric )
          {
            jac *= r;
          }

	// First, an i-loop over the  degrees of freedom.
	for (unsigned int i=0; i != n_T_dofs; i++)
	  {
	    FT(i) += jac *
	      (-_rho*_Cp*T_phi[i][qp]*(U*grad_T)    // convection term
	       -_k*(T_gradphi[i][qp]*grad_T) );  // diffusion term

	    if (compute_jacobian)
	      {
		for (unsigned int j=0; j != n_T_dofs; j++)
		  {
		    // TODO: precompute some terms like:
		    //   _rho*_Cp*T_phi[i][qp]*(vel_phi[j][qp]*T_grad_phi[j][qp])

		    KTT(i,j) += jac *
		      (-_rho*_Cp*T_phi[i][qp]*(U*T_gradphi[j][qp])  // convection term
		       -_k*(T_gradphi[i][qp]*T_gradphi[j][qp])); // diffusion term
		  } // end of the inner dof (j) loop

//.........这里部分代码省略.........