Principle of the method
======================




Equations to be solved
--------------------

We consider solution :math:`(u,s)` of a linear elastic problem verifying:


* the equilibrium equations:

:math:`\mathrm{\{}\begin{array}{c}\text{Lu}\mathrm{=}q\text{dans}\Omega \\ {\sigma }_{\text{ij}}{n}_{j}\mathrm{=}{\stackrel{ˉ}{t}}_{i}\text{sur}{\Gamma }_{t}\end{array}`

with :math:`L={}^{t}\mathrm{BDB}` elasticity operator


* compatibility equations:

:math:`\mathrm{\{}\begin{array}{c}\varepsilon \mathrm{=}\text{Bu}\\ u\mathrm{=}\stackrel{ˉ}{u}\text{sur}{\Gamma }_{u}\end{array}`

with :math:`\Gamma ={\Gamma }_{u}\cup {\Gamma }_{t}`


* the law of behavior:

:math:`\sigma =D\varepsilon`

The problem discretized by finite elements consists in finding the :math:`({u}_{h},{\sigma }_{h})` solution of:


.. _RefEquation 2.1-1:

:math:`{u}_{h}=N{\stackrel{ˉ}{u}}_{h}` eq 2.1-1

checking :math:`K{\stackrel{ˉ}{u}}_{h}=\text{f}`

with :math:`K=\underset{\Omega }{\int }{}^{t}\text{}(\mathrm{BN})D(\mathrm{BN})d\Omega`

:math:`\text{f}=\underset{\Omega }{\int }{}^{t}\text{}\mathrm{Nq}d\Omega +\underset{{G}_{t}}{\int }{}^{t}\text{}N\stackrel{ˉ}{t}\mathrm{dG}`

where:

:math:`{\stackrel{ˉ}{\text{u}}}_{h}` represents the nodal unknowns of displacement

:math:`N` the associated form functions

The constraints are calculated from the displacements by the relationship:


.. _RefEquation 2.1-2:

:math:`{\sigma }_{h}={\mathrm{DBu}}_{h}` eq 2.1-2




Error estimator and effectiveness index
--------------------------------------------


.. csv-table::

    "We note", ":math:`e=u-{u}_{h}` ", "the error on the trips"
    "", ":math:`{e}_{s}=s-{s}_{h}` ", "the constraint error"


The energy norm for error :math:`e` is written as:

:math:`\parallel \text{e}\parallel ={(\underset{\Omega }{\int }{}^{t}\text{}\mathrm{eLe}d\Omega )}^{1/2}`

in the case of elasticity


.. _RefEquation 2.2-1:

:math:`={(\underset{\Omega }{\int }{}^{t}\text{}{e}_{\sigma }{D}^{\text{-}1}{e}_{\sigma }d\Omega )}^{1/2}` eq 2.2-1

The overall error above is broken down into the following sum of elementary errors:

:math:`{\parallel e\parallel }^{2}=\sum _{i=1}^{N}{\parallel e\parallel }_{i}^{2}`


.. csv-table::

    "where", ":math:`N` is the total number of items."
    "", ":math:`{\parallel e\parallel }_{i}` represents the local error indicator on the :math:`i` element."


The aim is to estimate the exact error using equation [:ref:`éq 2.2-1 <éq 2.2-1>`] formulated in constraints. The basic idea of the method is to build a new constraint field noted :math:`{\sigma }^{\text{*}}` from :math:`{\sigma }_{h}` and such as:

:math:`{e}_{\sigma }\approx {e}_{\sigma }^{\text{*}}={\sigma }^{\text{*}}-{\sigma }_{h}`

The error estimator will then be written as:

:math:`{}^{0}\text{}\parallel e\parallel ={(\underset{\Omega }{\int }{}^{t}\text{}{e}_{\sigma }^{\text{*}}{D}^{\text{-}1}{e}_{\sigma }^{\text{*}}d\Omega )}^{1/2}`

The quality of the estimator is measured by the quantity :math:`\mathrm{\theta }`, called the estimator effectiveness index:

:math:`\mathrm{\theta }=\frac{{}^{0}\text{}\parallel e\parallel }{\parallel e\parallel }`

An error estimator is said to be asymptotically accurate if :math:`\mathrm{\theta }\to 1` when :math:`\parallel e\parallel \to 0` (or when :math:`h\to 0`), which means that the estimated error will always converge to the exact error when the error decreases.

Obviously, the reliability of :math:`{}^{0}\text{}\parallel e\parallel` depends on the "quality" of :math:`{\sigma }^{\text{*}}`.

The two versions of the ZHU - ZIENKIEWICZ estimator differ at this level (see [:ref:`§3 <§3>`]).




Construction of an asymptotically accurate estimator
----------------------------------------

The characterization of such an estimator is provided by the following theorem (see [:ref:`bib 2 <bib 2>`]).

**Theorem**

If :math:`\parallel {e}^{\text{*}}\parallel =\parallel {u-u}^{\text{*}}\parallel` is the error norm of the reconstructed solution, then the error estimator :math:`{}^{0}\text{}\parallel e\parallel` defined earlier is asymptotically accurate

If :math:`\frac{\parallel {e}^{\text{*}}\parallel }{\parallel e\parallel }\to 0` when :math:`\parallel e\parallel \to 0`

This condition is met if the convergence rate with :math:`h` of :math:`\parallel {e}^{\text{*}}\parallel` is greater than that of :math:`\parallel e\parallel`. Typically, if we assume that the exact error of the finite element approximation converges to :math:`\parallel e\parallel =0({h}^{p})`

and the error of the reconstructed solution in

:math:`\parallel {e}^{\text{*}}\parallel =0({h}^{p\text{+}\alpha })` with :math:`\alpha >0`

then a simple calculation gives:

:math:`1-0({h}^{\alpha })\le \mathrm{\theta }\le 1+0({h}^{\alpha })`

and so :math:`\mathrm{\theta }\to 1` when :math:`h\to 0`