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 Performance
 ===========
 
 Parallelism
 -----------
 
 Tamaas implements shared-memory parallelism using `thrust
 <https://github.com/thrust/thrust>`_ and `OpenMP <https://www.openmp.org/>`_
 [1]_. The number of threads can be controlled with the ``OMP_NUM_THREADS``
 environment variable, or alternatively to call the :func:`initialize
 <tamaas._tamaas.initialize>` function with the desired number of threads. The
 default behavior is to use as many threads as available cores on the system.
 
 
 Computational methods
 ---------------------
 
 Tamaas uses specialized numerical methods to efficiently solve elastic and
 elastoplastic periodic contact problems. Using a boundary integral formulation
 and a half-space geometry for the former allow (a) the focus of computational
 power to the contact interface since the bulk response can be represented
 exactly, (b) the use of the fast-Fourier transform for the computation of
 convolution integrals. In conjunction with a boundary integral formulation of
 the bulk state equations, a conjugate gradient approach is used to solve the
 contact problem.
 
 .. note::
 
    The above methods are state-of-the-art in the domain of rough surface
    contact. Below are selected publications detailing the methods used in
    elastic contact with and without adhesion:
 
    - Boundary integral formulation:
 
      - Stanley and Kato (`J. of Tribology, 1997
        <https://doi.org/10.1115/1.2833523>`_)
-
    - Conjugate Gradient:
 
      - Polonsky and Keer (`Wear, 1999
        <https://doi.org/10.1016/S0043-1648(99)00113-1>`_)
      - Rey, Anciaux and Molinari (`Computational Mechanics, 2017
        <https://doi.org/10.1007/s00466-017-1392-5>`_)
-
    - Frictional contact:
 
      - Condat (`J. of Optimization Theory and Applications, 2012
        <https://doi.org/10.1007/s10957-012-0245-9>`_)
 
 For elastic-plastic contact, Tamaas uses a similar approach by implementing a
 *volume* integral formulation of the bulk equilibrium equations. Thanks to
 kernel expressions that are directly formulated in the Fourier domain, the
 method reduces the algorithmic complexity, memory requirements and sampling
 errors compared to traditional volume integral methods (Frérot, Bonnet, Anciaux
 and Molinari, `Computer Methods in Applied Mechanics and Engineering, 2019
 <https://doi.org/10.1016/j.cma.2019.04.006>`_, `arXiv:1811.11558
 <https://arxiv.org/pdf/1811.11558.pdf>`_). The figure below shows a comparison
 of run times for an elasticity problem (only a single solve step) between Tamaas
 and `Akantu <https://gitlab.com/akantu/akantu>`_, a high-performance FEM code
 using the direct solver `MUMPS <http://mumps.enseeiht.fr/>`_.
 
 .. figure:: figures/complexity.svg
             :align: center
             :width: 75%
 
             Comparison of run times between the volume integral implementation
             of Tamaas and an FEM solve step with a Cholesky factorization
             performed by Akantu+MUMPS. :math:`N` is the total number of points.
 
 Further discussion about the elastic-plastic solver implemented in Tamaas can be
 found in Frérot, Bonnet, Anciaux and Molinari, (`Computer Methods in Applied
 Mechanics and Engineering, 2019 <https://doi.org/10.1016/j.cma.2019.04.006>`_,
 `arXiv:1811.11558 <https://arxiv.org/pdf/1811.11558.pdf>`_).
 
 
 .. [1] Thrust uses a backend system for thread-level parallelism, which can be
        changed to use `TBB
        <https://software.intel.com/content/www/us/en/develop/tools/threading-building-blocks.html>`_
        in the build options (``build-setup.conf``). You can experiment with it,
        but we do not formally support this backend. Note that this does not
        affect FFTW, which always uses OpenMP.