cache_inputs cache_residual_cfbc compute_local_function compute_local_jacobian driving_stress explicit_driving_stress function_callback init_impl jacobian_callback m_alpha m_bc_mask m_bc_values m_beta_ice_free_bedrock m_boundary_integral m_callback_data m_coefficients m_dirichletScale m_driving_stress_x m_driving_stress_y m_element_index m_epsilon_ssa m_gc m_node_type m_q1_element m_rho_g m_snes monitor_function monitor_jacobian PointwiseNuHAndBeta quad_point_values solve solve_nocache solve_with_reason SSAFEM ~SSAFEM

driving_stress() void pism::stressbalance::SSAFEM::driving_stress ( const fem::Element &  E, const Coefficients *  x, Vector2d *  result  ) const protected Compute the the driving stress at quadrature points. The surface elevation \( h \) is defined by \begin{equation} h = \begin{cases} b + H, & \mathrm{grounded},\\ z_{sl} + \alpha H, & \mathrm{shelf}, \end{cases} \end{equation} where \( b \) is the bed elevation, \( H \) is the ice thickness, and \( z_{sl} \) is the sea level, and \( \alpha \) is the ratio of densities of ice and sea water. Note that if \( b \), \( H \), and \( z_{sl} \) are bilinear (or \( C^{1} \) in general), \( h \) is continuous but not continuously differentiable. In other words, the gradient of \( h \) is not continuous, so near the grounding line we cannot compute the gravitational driving stress \( \tau_{d} = - \rho g H \nabla h \) using the \( Q_1 \) or \( P_1 \) FE basis expansion of \( h \). We differentiate the equation above instead, getting \begin{equation} \nabla h = \begin{cases} \nabla b + \nabla H, & \mathrm{grounded},\\ \nabla z_{sl} + \alpha \nabla H, & \mathrm{shelf}. \end{cases} \end{equation} and use basis expansions of \( \nabla b \) and \( \nabla H \). Overall, we have \begin{align*} \tau_{d} &= \rho g H \nabla h\\ &= - \rho g H \begin{cases} \nabla b + \nabla H, & \mathrm{grounded},\\ \alpha \nabla H, & \mathrm{shelf}, \end{cases} \end{align*} because \( z = z_{sl} \) defines the geoid surface and so its gradient does not contribute to the driving stress. Definition at line 441 of file SSAFEM.cc. References pism::stressbalance::SSAFEM::Coefficients::bed, pism::fem::Element::chi(), pism::fem::Germ::dx, pism::fem::Germ::dy, pism::mask::grounded(), pism::k, m_alpha, m_gc, m_rho_g, pism::GeometryCalculator::mask(), n, pism::fem::Element::n_chi(), pism::fem::Element::n_pts(), pism::stressbalance::SSAFEM::Coefficients::sea_level, pism::stressbalance::SSAFEM::Coefficients::thickness, pism::Vector2d::u, pism::Vector2d::v, and pism::fem::Germ::val.