2. Local method of minimization by least squares

In general, we want to approximate, in a least squares sense, the spatial distribution of constraints by \(\sigma (x)\) a polynomial function:

\(\stackrel{ˆ}{\sigma }(x)=\sum _{\begin{array}{}i=\mathrm{0,}\mathrm{...},p\end{array}}{a}_{i}{P}^{i}(x)\)

The problem comes down to finding the coefficients \({a}_{i}\) that minimize the functional:

\(\chi =\int \int {(\sigma -\stackrel{ˆ}{\sigma })}^{2}\mathrm{dx}\mathrm{dy}\)

The values of function \(\sigma\) are known here only at Gauss points: \({\sigma }^{k}=\sigma ({x}_{k})\)

The minimum will be reached if and only if:

\(\frac{\partial \chi }{\partial {a}_{i}}=0\begin{array}{}\forall i=\mathrm{0,}\mathrm{...},p\end{array}\)

In the context of the finite element in motion method, the following smoothing function is chosen:

\(\stackrel{ˆ}{\sigma }(x)=\sum _{i=1}^{n}{N}_{i}(x)\stackrel{ˆ}{{\sigma }_{i}}\)

where:

\({N}_{i}\) is the form function associated with node \(i\) on the finite element in question,

\(\stackrel{ˆ}{{\sigma }_{i}}\) is the value of the constraint at node \(i\) sought,

\(n\) the number of knots selected for smoothing.

So we have to solve the system:

\(\frac{\partial \chi }{\partial \stackrel{ˆ}{{\sigma }_{i}}}=0\forall i=\mathrm{1,}\mathrm{...},n\) eq 2-1

You can choose between two local smoothing methods: continuous smoothing or discreet smoothing.