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

  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 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 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 BoussinesqBuoyancy::element_time_derivative
  ( bool compute_jacobian,
    AssemblyContext & context )
  {
    // The number of local degrees of freedom in each variable.
    const unsigned int n_u_dofs = context.get_dof_indices(_flow_vars.u()).size();
    const unsigned int n_T_dofs = context.get_dof_indices(_temp_vars.T()).size();

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

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

    // Get Jacobians
    libMesh::DenseSubMatrix<libMesh::Number> &KuT = context.get_elem_jacobian(_flow_vars.u(), _temp_vars.T()); // R_{u},{T}
    libMesh::DenseSubMatrix<libMesh::Number> &KvT = context.get_elem_jacobian(_flow_vars.v(), _temp_vars.T()); // R_{v},{T}
    libMesh::DenseSubMatrix<libMesh::Number>* KwT = NULL;



    if( this->_flow_vars.dim() == 3 )
      {
        Fw  = &context.get_elem_residual(_flow_vars.w()); // R_{w}
        KwT = &context.get_elem_jacobian(_flow_vars.w(), _temp_vars.T()); // R_{w},{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++)
      {
        // Compute the solution & its gradient at the old Newton iterate.
        libMesh::Number T;
        T = context.interior_value(_temp_vars.T(), 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++)
          {
            Fu(i) += -_rho*_beta_T*(T - _T_ref)*_g(0)*vel_phi[i][qp]*JxW[qp];
            Fv(i) += -_rho*_beta_T*(T - _T_ref)*_g(1)*vel_phi[i][qp]*JxW[qp];

            if (this->_flow_vars.dim() == 3)
              (*Fw)(i) += -_rho*_beta_T*(T - _T_ref)*_g(2)*vel_phi[i][qp]*JxW[qp];

            if (compute_jacobian)
              {
                for (unsigned int j=0; j != n_T_dofs; j++)
                  {
                    KuT(i,j) += context.get_elem_solution_derivative() *
                      -_rho*_beta_T*_g(0)*vel_phi[i][qp]*T_phi[j][qp]*JxW[qp];
                    KvT(i,j) += context.get_elem_solution_derivative() *
                      -_rho*_beta_T*_g(1)*vel_phi[i][qp]*T_phi[j][qp]*JxW[qp];

                    if (this->_flow_vars.dim() == 3)
                      (*KwT)(i,j) += context.get_elem_solution_derivative() *
                        -_rho*_beta_T*_g(2)*vel_phi[i][qp]*T_phi[j][qp]*JxW[qp];

                  } // End j dof loop
              } // End compute_jacobian check

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

//.........这里部分代码省略.........

            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));
            const libMesh::Real gamma_v = (grad_y(0)+grad_v(0));
            const libMesh::Real gamma_w = (grad_z(0)+grad_w(0));

            const libMesh::Real x_term = C1*gamma_u;
            const libMesh::Real y_term = C1*gamma_v;
            const libMesh::Real z_term = C1*gamma_w;

            const libMesh::Real dt_term = dgradu_dt(0)*gamma_u + dgradv_dt(0)*gamma_v + dgradw_dt(0)*gamma_w;

            Fu(i) += u_diag_factor + x_term*dt_term;
            Fv(i) += v_diag_factor + y_term*dt_term;
            Fw(i) += w_diag_factor + z_term*dt_term;
          }

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

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

                    libMesh::Real common_factor = _lambda_factor*this->_A*jac*u_gradphi_I(0);

                    const libMesh::Real diag_term_1 = common_factor*tau(0,0)*u_gradphi_J(0)*context.get_elem_solution_rate_derivative();

                    const libMesh::Real dgamma_du = ( u_gradphi_J(0)*(grad_x(0)+grad_u(0)) );

                    const libMesh::Real dgamma_dv = ( u_gradphi_J(0)*(grad_y(0)+grad_v(0)) );

                    const libMesh::Real dgamma_dw = ( u_gradphi_J(0)*(grad_z(0)+grad_w(0)) );

                    const libMesh::Real diag_term_2_factor = common_factor*C(0,0,0,0)*context.get_elem_solution_derivative();

                    Kuu(i,j) += diag_term_1 + dgradu_dt(0)*diag_term_2_factor*dgamma_du;
                    Kuv(i,j) += dgradu_dt(0)*diag_term_2_factor*dgamma_dv;
                    Kuw(i,j) += dgradu_dt(0)*diag_term_2_factor*dgamma_dw;

                    Kvu(i,j) += dgradv_dt(0)*diag_term_2_factor*dgamma_du;
                    Kvv(i,j) += diag_term_1 + dgradv_dt(0)*diag_term_2_factor*dgamma_dv;
                    Kvw(i,j) += dgradv_dt(0)*diag_term_2_factor*dgamma_dw;

                    Kwu(i,j) += dgradw_dt(0)*diag_term_2_factor*dgamma_du;
                    Kwv(i,j) += dgradw_dt(0)*diag_term_2_factor*dgamma_dv;
                    Kww(i,j) += diag_term_1 + dgradw_dt(0)*diag_term_2_factor*dgamma_dw;

                    const libMesh::Real C1 = common_factor*C(0,0,0,0);

                    const libMesh::Real gamma_u = (grad_x(0)+grad_u(0));
                    const libMesh::Real gamma_v = (grad_y(0)+grad_v(0));
                    const libMesh::Real gamma_w = (grad_z(0)+grad_w(0));

                    const libMesh::Real x_term = C1*gamma_u;
                    const libMesh::Real y_term = C1*gamma_v;
                    const libMesh::Real z_term = C1*gamma_w;

                    const libMesh::Real ddtterm_du = u_gradphi_J(0)*(gamma_u*context.get_elem_solution_rate_derivative()
                                                                     + dgradu_dt(0)*context.get_elem_solution_derivative());

                    const libMesh::Real ddtterm_dv = u_gradphi_J(0)*(gamma_v*context.get_elem_solution_rate_derivative()
                                                                     + dgradv_dt(0)*context.get_elem_solution_derivative());

                    const libMesh::Real ddtterm_dw = u_gradphi_J(0)*(gamma_w*context.get_elem_solution_rate_derivative()
                                                                     + dgradw_dt(0)*context.get_elem_solution_derivative());

                    Kuu(i,j) += x_term*ddtterm_du;
                    Kuv(i,j) += x_term*ddtterm_dv;
                    Kuw(i,j) += x_term*ddtterm_dw;

                    Kvu(i,j) += y_term*ddtterm_du;
                    Kvv(i,j) += y_term*ddtterm_dv;
                    Kvw(i,j) += y_term*ddtterm_dw;

                    Kwu(i,j) += z_term*ddtterm_du;
                    Kwv(i,j) += z_term*ddtterm_dv;
                    Kww(i,j) += z_term*ddtterm_dw;

                    const libMesh::Real dt_term = dgradu_dt(0)*gamma_u + dgradv_dt(0)*gamma_v + dgradw_dt(0)*gamma_w;

                    // Here, we're missing derivatives of C(0,0,0,0) w.r.t. strain
                    // Nonzero for hyperelasticity models
                    const libMesh::Real dxterm_du = C1*u_gradphi_J(0)*context.get_elem_solution_derivative();
                    const libMesh::Real dyterm_dv = dxterm_du;
                    const libMesh::Real dzterm_dw = dxterm_du;

                    Kuu(i,j) += dxterm_du*dt_term;
                    Kvv(i,j) += dyterm_dv*dt_term;
                    Kww(i,j) += dzterm_dw*dt_term;

                  } // end j-loop
              } // end i-loop
          } // end if(compute_jacobian)
      } // end qp loop
  }

  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 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.........

//.........这里部分代码省略.........
        {
            U(2) = context.interior_value( this->_flow_vars.w_var(), 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 );

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

        libMesh::RealGradient d_residual_dT = _rho_ref*_beta_T*_g;
        // d_residual_dU = 0
        libMesh::RealGradient residual = (T-_T_ref)*d_residual_dT;

        for (unsigned int i=0; i != n_u_dofs; i++)
        {
            libMesh::Real test_func = this->_rho*U*u_gradphi[i][qp] +
                                      mu_qp*( u_hessphi[i][qp](0,0) + u_hessphi[i][qp](1,1) + u_hessphi[i][qp](2,2) );
            Fu(i) += -tau_M*residual(0)*test_func*JxW[qp];

            Fv(i) += -tau_M*residual(1)*test_func*JxW[qp];

            if (_dim == 3)
            {
                (*Fw)(i) += -tau_M*residual(2)*test_func*JxW[qp];
            }

            if (compute_jacobian)
            {
                libMesh::Gradient d_test_func_dU = this->_rho*u_gradphi[i][qp];
                // d_test_func_dT = 0

                for (unsigned int j=0; j != n_u_dofs; ++j)
                {
                    Kuu(i,j) += -tau_M*residual(0)*d_test_func_dU(0)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                    Kuu(i,j) += -d_tau_M_dU(0)*u_phi[j][qp]*residual(0)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                    Kuv(i,j) += -tau_M*residual(0)*d_test_func_dU(1)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                    Kuv(i,j) += -d_tau_M_dU(1)*u_phi[j][qp]*residual(0)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                    Kvu(i,j) += -tau_M*residual(1)*d_test_func_dU(0)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                    Kvu(i,j) += -d_tau_M_dU(0)*u_phi[j][qp]*residual(1)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                    Kvv(i,j) += -tau_M*residual(1)*d_test_func_dU(1)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                    Kvv(i,j) += -d_tau_M_dU(1)*u_phi[j][qp]*residual(1)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                }

                for (unsigned int j=0; j != n_T_dofs; ++j)
                {
                    // KuT(i,j) += -tau_M*residual(0)*dtest_func_dT[j]*JxW[qp] * context.get_elem_solution_derivative();
                    KuT(i,j) += -tau_M*d_residual_dT(0)*T_phi[j][qp]*test_func*JxW[qp] * context.get_elem_solution_derivative();
                    // KvT(i,j) += -tau_M*residual(1)*dtest_func_dT[j]*JxW[qp] * context.get_elem_solution_derivative();
                    KvT(i,j) += -tau_M*d_residual_dT(1)*T_phi[j][qp]*test_func*JxW[qp] * context.get_elem_solution_derivative();
                }
                if (_dim == 3)
                {
                    for (unsigned int j=0; j != n_T_dofs; ++j)
                    {
                        // KwT(i,j) += -tau_M*residual(2)*dtest_func_dT[j]*JxW[qp] * context.get_elem_solution_derivative();
                        (*KwT)(i,j) += -tau_M*d_residual_dT(2)*T_phi[j][qp]*test_func*JxW[qp] * context.get_elem_solution_derivative();
                    }

                    for (unsigned int j=0; j != n_u_dofs; ++j)
                    {
                        (*Kuw)(i,j) += -tau_M*residual(0)*d_test_func_dU(2)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kuw)(i,j) += -d_tau_M_dU(2)*u_phi[j][qp]*residual(0)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kvw)(i,j) += -tau_M*residual(1)*d_test_func_dU(2)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kvw)(i,j) += -d_tau_M_dU(2)*u_phi[j][qp]*residual(1)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kwu)(i,j) += -tau_M*residual(2)*d_test_func_dU(0)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kwu)(i,j) += -d_tau_M_dU(0)*u_phi[j][qp]*residual(2)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kwv)(i,j) += -tau_M*residual(2)*d_test_func_dU(1)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kwv)(i,j) += -d_tau_M_dU(1)*u_phi[j][qp]*residual(2)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kww)(i,j) += -tau_M*residual(2)*d_test_func_dU(2)*u_phi[j][qp]*JxW[qp] * context.get_elem_solution_derivative();
                        (*Kww)(i,j) += -d_tau_M_dU(2)*u_phi[j][qp]*residual(2)*test_func*JxW[qp] * context.get_elem_solution_derivative();
                    }
                }

            } // End compute_jacobian check

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

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

    return;
}

  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 HeatConduction<K>::element_time_derivative( bool compute_jacobian,
						AssemblyContext& context,
						CachedValues& /*cache*/ )
  {
    // 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();

    // 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 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();

    // The subvectors and submatrices we need to fill:
    //
    // K_{\alpha \beta} = R_{\alpha},{\beta} = \partial{ R_{\alpha} } / \partial{ {\beta} } (where R denotes residual)
    // e.g., for \alpha = T and \beta = v we get: K_{Tu} = R_{T},{u}
    //

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

    libMesh::DenseSubVector<libMesh::Number> &FT = context.get_elem_residual(_temp_vars.T_var()); // R_{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++)
      {
	// Compute the solution & its gradient at the old Newton iterate.
	libMesh::Gradient grad_T;
	grad_T = context.interior_gradient(_temp_vars.T_var(), qp);

	// Compute the conductivity at this qp
	libMesh::Real _k_qp = this->_k(context, qp);
	
	// First, an i-loop over the  degrees of freedom.
	for (unsigned int i=0; i != n_T_dofs; i++)
	  {
	    FT(i) += JxW[qp]*(-_k_qp*(T_gradphi[i][qp]*grad_T));

	    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) += JxW[qp] * context.get_elem_solution_derivative() *
		      ( -_k_qp*(T_gradphi[i][qp]*T_gradphi[j][qp]) ); // diffusion term
		  } // end of the inner dof (j) loop

	      } // end - if (compute_jacobian && context.get_elem_solution_derivative())

	  } // end of the outer dof (i) loop
      } // end of the quadrature point (qp) loop

    return;
  }