.. _results:

===============================
 Post-processing and results
===============================

Results are shown in terms of:

* energy density, i.e. the energy of the system is evaluated at each node of the beam during the whole simulation time as:

  .. math::
       \begin{aligned}
           H_{density} \left( x,t,u,\dot{u},u' \right) = \frac{1}{2} \left( \dot{u}^2 + c^2 u' ^2 \right)
       \end{aligned}

* energy charge, i.e. the total energy of the system evaluated in the discretised simulation time

  .. math::
       \begin{aligned}
           H_{charge} \left( t \right) = \int_{0}^l \left[ \frac{1}{2} \left( u' ^2 + c^2 \dot{u}^2 \right) dx \right]
       \end{aligned}

Both quantities are calculated by using the discrete Legendre transform. In the discrete settings, energy density and charge are respectively detemined as follow:

.. math::
     \begin{aligned}
         H_{density}^{i,j} = \frac{1}{2} \left( (p_t^{i,j})^2 + c^2 (p_x^{i,j})^2 \right) \; \; \text{for} \; i=0 \dots l \text{,} \; j=0 \dots \tau
     \end{aligned}

.. math::
     \begin{aligned}
         H_{charge}^j = \sum\limits_{i=0}^l \left[ \frac{1}{2} \left( (p_t^{i})^2 + c^2 (p_x^{i})^2 \right) \right] \; \text{for} \; j=0 \dots \tau
     \end{aligned}

In the `1D_wave_equation <https://github.com/THREAD-3-2/1D_wave_equation/tree/main/wave_equation>`_,
the energy density is evaluated as follow:

.. code-block::

       for j=1:n_t
            for i=1:n_s
                T_kin(i,j) = 1/2*rho*pp_t(i,j)^2;
                V_pot(i,j) = 1/2*k_pot*pp_s(i,j)^2;
                HH(i,j) = T_kin(i,j)+V_pot(i,j);
                HH_charge_node(i,j) = 1/2*(pp_t(i,j)^2+k_pot/rho*pp_s(i,j)^2);
                if not(i == 1)
                HH_charge_elem(i-1,j) = 1/2*delta_s*(HH_charge_node(i-1,j)+HH_charge_node(i,j));
                end
            end
        end

where n_t is the number of time steps, n_s is the number of nodes, T_kin is the kinetic energy, V_pot is the potential energy, rho is the density and
k_pot is the modulus of elasticity of the string. Moreover, instead of computing the energy charge itself, in the Matlab code its error during simulation is evaluated.

.. code-block::

        for j=1:n_t
            HH_charge(j) = sum(HH_charge_elem(:,j));
            HH_error(j) = HH_charge(1)-HH_charge(j);
        end


For the particular example problem, the above quantities are shown in Figure 1 and 2. It can be notice the good behaviour of the energy charge with an error in time of the order of :math:`10^{-15}` in Figure 2.

.. figure:: /figures/energy_density.png

   Figure 1. Energy density

.. figure:: /figures/energy_error.png

   Figure 2. Energy charge: error