.. index:: fix continuum/weighted

fix continuum/weighted command
==============================

Syntax
""""""

fix ID group-ID continuum/weighted keyword value

* ID, group-ID are documented in :doc:`fix <fix>` command
* continuum/weighted = style name of this fix command
* zero or one keyword/value pairs may be appended
.. parsed-literal::

   keyword = {kernel_radius, kernel_type, compute}
     *kernel_radius* value = radius
       radius = Radius of the smoothing kernel
     *kernel_type* value = type
       type = Type of kernel {Top_Hat, Gaussian, Wendland}
     *compute* value = compute-type
       compute-type = Which tensor(s) to compute {stress, strain, stress_strain}



Examples
""""""""

.. parsed-literal::

   fix 1 all continuum/weighted kernel_radius 0.01 compute stress
   fix 1 all continuum/weighted kernel_radius 0.2 kernel_type Wendland compute stress_strain

Description
"""""""""""

If the *compute* keyword is set to either stress or stress_strain this fix calculates the complete stress tensor at
each particle according to :ref:`Goldhirsch <Goldhirsch>`. The formula is given by:

.. parsed-literal::

     sigma_{i,ab} = -1/2 sum_{j,k} f_{jk,a} r_{jk,b} integral_0^1 phi(**r**_i - **r**_j + s **r**_{jk}) ds
                  - sum_j m_j v'_{ij,a} v'_{ij,b} phi(r_i - r_j)
                  + sigma_wall_{i,ab}

where

.. parsed-literal::

     **v**'_{ij} = **v**_j - **V**_i
     **V**_i = **p**_i / rho_i
     **p**_i = sum_j m_j **v**_j phi(r_i - r_j)
     rho_i = sum_j m_j phi(r_i - r_j)
     phi(**r**) = H(radius - |r|) / Omega(radius)
     Omega(radius) = 4/3 pi radius^3

and

.. parsed-literal::

     **v** is the velocity vector
     **f**_{ij} force acting from j onto i
     **r**_{ij} vector from center of j to center of i
     m_i mass of particle i
     H is the Heavyside function

In case solid boundaries are present the last term is given according to :ref:`Weinhart et al. <Weinhart>` by

.. parsed-literal::

       sigma_wall_{i,ab} = - sum_{j,k} f_{jk,a} a_{jk,b} integral_0^1 phi(**r**_i - **r**_j + s **a**_{jk}) ds

where

.. parsed-literal::

       **a**_{jk} = **r**_j - c_{jk}

and **c**_{jk} is the contact point of particle *j* with wall *k* and the sum runs over all particles *j* and walls *k*.

If the *compute* keyword is set to either strain or stress_strain this fix calculates the incremental strain tensor at
each particle according to :ref:`Zhang et al. <Zhang>` The formula is given by

.. parsed-literal::

     epsilon_*i,ab* = 1/(2 rho_i) sum_*j,k* m_j m_k phi(**r**_{ij}) dt (**v**_{jk,a} grad_phi(**r**_{ik},b) + **v**_{jk,b} grad_phi(**r**_{ik},a))

where most of the variables are given as above and additionally

.. parsed-literal::

       **v**_{ij,a} is the a-th component of the velocity difference between i and j
       grad_phi(**r**_{ij},a) is the a-th component of the gradient of phi with respect to **r**_i
       dt is the time-step size

The following three kernel types are implemented at the moment:

* *Top_Hat* - Top hat kernel
.. parsed-literal::

           w(r) = a_t * 1 if q < 1 (q = r / kernel_radius)
                = 0 otherwise

* *Gaussian* - Gaussian kernel
.. parsed-literal::

           w(r) = a_g * exp(-q^2) (q = 3 r / kernel_radius)

* *Wendland* - Quinitic radial polynomial
.. parsed-literal::

           w(r) = a_w * (1-q/2)^4 (1+2q) (q = 2 r / kernel_radius)



Note that all kernels are equal to zero if *r > kernel_radius* (this implies a cut-off for the Gaussian). The constants *a* (different for each kernel) are chosen such that the integral of *w* over the ball of radius *kernel_radius* is equal to one. In case of the top hat kernel *a_t* is equal to the volume of this sphere.


----------


Restart, fix_modify, output, run start/stop, minimize info
""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

No information about this fix is written to :doc:`binary restart files <restart>`.  None of the :doc:`fix_modify <fix_modify>` options
are relevant to this fix.

The values can be dumped by using the *f_stressTensor_[i]* and/or *f_strainTensor_[i]* (0 <= i <= 8) values in :doc:`dump <dump>` commands

No parameter of this fix can be used with the *start/stop* keywords of
the :doc:`run <run>` command.  This fix is not invoked during :doc:`energy minimization <minimize>`.

Restrictions
""""""""""""


Strain computation does not work with the default *TOP_HAT* kernel as its derivative is zero.

In order to ensure that all particles in the kernel radius are considered make use of the :doc:`neigh_modify <neigh_modify>`
command. In particular the *contact_distance_factor* which should be set such that

.. parsed-literal::

     2*min(radius)*contact_distance_factor >= kernel_radius

Related commands
""""""""""""""""

:doc:`compute <compute>`, :doc:`compute stress/atom <compute_stress_atom>`,
:doc:`fix ave/atom <fix_ave_atom>`, :doc:`fix ave/histo <fix_ave_histo>`,
:doc:`fix ave/time <fix_ave_time>`, :doc:`fix ave/spatial <fix_ave_spatial>`,
:doc:`fix ave/euler <fix_ave_euler>`

**Default:** none

**References:**

.. _Goldhirsch:



**(Goldhirsch)** Goldhirsch; Stress, stress asymmetry and couple stress: from discrete particles to continuous fields, Granular Matter (2010)

.. _Weinhart:



**(Weinhart)** Weinhart, Thornton, Luding, Bokhove; From discrete particles to continuum fields near a boundary (2012)

.. _Zhang:



**(Zhang)** Zhang, Behringer, Goldhirsch; Coarse-Graining of a Physical Granular System, Progress of Theoretical Physics Supplement (2010)


.. _liws: http://www.cfdem.com
.. _ld: Manual.html
.. _lc: Section_commands.html#comm