Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------


.. image:: images/Shape2.gif


.. _RefSchema_Shape2.gif:

:math:`K\mathrm{=}{10}^{6}`, :math:`\frac{\Phi }{{\Phi }_{0}}\mathrm{=}1.698`

:math:`F={\Phi }_{1}t` :math:`{\Phi }_{1}=7.2\times {10}^{21}` if :math:`t\in [\mathrm{0,}{t}_{p}=1728.98]={I}_{1}` :math:`\Rightarrow \Phi ={\Phi }_{1}`

:math:`F={\Phi }_{1}{t}_{p}` :math:`{\Phi }_{1}=7.2\times {10}^{21}` if :math:`t\in [:ref:`{t}_{p},{t}_{f}=2160.975 <{t}_{p},{t}_{f}=2160.975>`] = {I} _ {2} `:math:`\Rightarrow \Phi =0`

:math:`F={\Phi }_{1}{t}_{p}+2{\Phi }_{1}(t-{t}_{f})` :math:`{\Phi }_{1}=7.2\times {10}^{21}` if :math:`t\in [{t}_{f},{\mathrm{2t}}_{f}-{t}_{p}]={I}_{3}` :math:`\Rightarrow \Phi =2{\Phi }_{1}`

:math:`F={\Phi }_{1}t` :math:`{\Phi }_{1}=7.2\times {10}^{21}` if :math:`t>({\mathrm{2t}}_{f}-{t}_{p})={I}_{4}` :math:`\Rightarrow \Phi ={\Phi }_{1}`

:math:`p={\left[\frac{n+m}{m}{\sigma }^{n}{(\frac{1}{K}\frac{\Phi }{{\Phi }_{0}}+L)}^{\beta }t{e}^{-\frac{Q}{R(T+{T}_{0})}}\right]}^{\frac{m}{n+m}}` if :math:`t\in {I}_{1}`

:math:`p={\left[\frac{n+m}{m}{\sigma }^{n}{(\frac{1}{K}\frac{\Phi }{{\Phi }_{0}}+L)}^{\beta }{t}_{p}{e}^{-\frac{Q}{R(T+{T}_{0})}}\right]}^{\frac{m}{n+m}}={p}_{f}` if :math:`t\in {I}_{2}`

:math:`p={p}_{f}` to :math:`t={t}_{f}` :math:`L=0`

:math:`\dot{p}={\left[\frac{\sigma }{{p}^{\frac{1}{m}}}\right]}^{n}{(\frac{1}{K}\frac{2\Phi }{{\Phi }_{0}}+L)}^{\beta }{e}^{\frac{-Q}{R(T+{T}_{0})}}`

:math:`\dot{p}{p}^{\frac{n}{m}}={\sigma }^{n}{(\frac{1}{K}\frac{2\Phi }{{\Phi }_{0}}+L)}^{\beta }{e}^{\frac{-Q}{R(T+{T}_{0})}}`

:math:`{\dot{p}}^{\frac{m+n}{m}}=\frac{m+n}{m}{\sigma }^{n}{(\frac{1}{K}\frac{2\Phi }{{\Phi }_{0}}+L)}^{\beta }{e}^{\frac{-Q}{R(T+{T}_{0})}}`

:math:`p={\left[\frac{m+n}{m}{\sigma }^{n}{(\frac{1}{K}\frac{2\Phi }{{\Phi }_{0}}+L)}^{\beta }{e}^{\frac{-Q}{R(T+{T}_{0})}}((t-{t}_{f})2\beta +{t}_{p})\right]}^{\frac{m}{m+n}}` if :math:`t\in {I}_{3}`

:math:`p={\left[\frac{m+n}{m}{\sigma }^{n}{(\frac{1}{K}\frac{2\Phi }{{\Phi }_{0}}+L)}^{\beta }{e}^{\frac{-Q}{R(T+{T}_{0})}}(t+({t}_{f}-{t}_{p}(2\beta -2)))\right]}^{\frac{m}{m+n}}` if :math:`t\in {I}_{4}`

**Digital app**

:math:`\frac{1}{K}={10}^{-6}`; :math:`\frac{\Phi }{{\Phi }_{0}}=1.698`; :math:`\sigma =100`; :math:`\beta =1.2`

To :math:`t=3456.96`

:math:`p={(0.09067259953)}^{(\frac{m}{(n+m)})}=0.198332841`

:math:`\varepsilon =0.200569905`

To :math:`t=2592.97`

:math:`p={(0.06882302104)}^{(\frac{m}{(n+m)})}=0.164696317`

:math:`\varepsilon =0.166804179`


Reference quantities
----------------------


* Move :math:`\mathrm{DX}` to node :math:`\mathrm{N02}`


* :math:`\mathrm{SIXX}` constraint in :math:`\mathrm{MA1}` mesh


* Cumulative plastic deformation :math:`\mathrm{V1}` in mesh :math:`\mathrm{MA1}`


Benchmark result
---------------------


.. csv-table::

    "Size", "Knot or Knit", "Instant", "Reference"
    ":math:`\mathrm{V1}` "," :math:`\mathrm{MA1}` "," :math:`2.59297\times {10}^{3}` "," :math:`0.164696`"
    ":math:`\mathrm{DX}(m)` "," :math:`\mathrm{N02}` "," :math:`2.59297\times {10}^{3}` "," :math:`0.166804`"
    ":math:`\mathrm{V1}` "," :math:`\mathrm{MA1}` "," :math:`3.45696\times {10}^{3}` "," :math:`0.119833`"
    ":math:`\mathrm{DX}(m)` "," :math:`\mathrm{N02}` "," :math:`3.45696\times {10}^{3}` "," :math:`0.20057`"
    ":math:`\mathrm{SIYY}(\mathrm{Pa})` "," :math:`\mathrm{MA1}` "," :math:`3.45696\times {10}^{3}` "," :math:`100`"



Uncertainty about the solution
---------------------------

Analytical solution