# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 4 MLS-Procedure-Based Mesh-Free Error Recovery  
Due to its completeness and continuity [37], the moving least squares (MLS) approach can successfully interpolate data with reasonable accuracy. The MLS approach provides the displacement or stress estimate at a node via the interpolation of the displacement or stress in a weighted least-squares sense at the mesh-free local domains. The nodes can be expressed in 2-D as x1-xn, where x1 = (x1,y1). The displacement (or stress) estimate \(u^{h}(x)\)can be expressed as the product of the polynomial basis, \(\mathbf{P}(x)\), and a coefficients vector, \(a(x)\).  
\[u^{h}(x)=p^{T}(x)a(x)=\sum_{j=1}^{m}p_{j}(x)a_{j}(x) \tag{12}\]  
where m is polynomial basis number and the vector of coefficient, and \(a(x)\) is given by  
\[a(x)=\{a_{0}(x)a_{1}(x)\ldots\ \ldots\ \ldots a_{m-1}(x)a_{m}(x)\}^{T} \tag{13}\]  
In this study, for linear element, m is taken to be six, and for quadratic elements, m is taken to be nine in the basis function \(\mathbf{P}(x)\).  
It is possible to obtain the vector of coefficients \(\mathbf{a}(x)\) by minimizing a weighted residual in the following manner.  
\[J=\sum_{j=1}^{n}w(x-x_{I})\left[p^{T}(x_{I})a(x)-u_{I}\right]^{2} \tag{14}\]  
\[\frac{\partial J}{\partial a}=A(x)a(x)-B(x)u_{s}=0 \tag{15}\]  
When the weighted residual is minimized, the relation of the coefficient vector is developed.  
\[a(x)=\mathrm{A}^{-1}(x)B(x)u_{s} \tag{16}\]  
The MLS moment matrix, \(A(x)\), and B(x) are given as  
\[A(x)=\sum_{i=i}^{n}w_{i}(x-x_{i})p^{T}(x_{I})p(x_{I}) \tag{17}\]  
\[B(x)=\left[w_{I}(x-x_{I})p(x_{I})\right.,\ldots\ldots,\ldots w_{n}(x-x_{n})\left. p(x_{n})\right] \tag{18}\]  
The displacement (or stress) estimate may be expressed as:  
\[u^{h}(x)=\sum_{I=1}^{n}\sum_{j=1}^{m}p_{j}(x)\mathrm{A}^{-1}(x)B(x)_{jI}u_{I} \tag{19}\]  
where \(u_{s}=\) nodal parameter vector of the displacement or stress, \(A(x)=\) MLS moment matrix, \(n=\) number of nodes \(i\), and \(w(x-x_{i})=\) weighting function in 2D associated with each node (weight takes a value of one in the vicinity of the node where the function and its derivatives are to be computed and becomes zero outside a region \(\Omega\)i surrounding the node \(x_{i}\))  
The weights are created using the distance \(\overline{d}=\|\mathrm{x}-\mathrm{x}_{i}\|/\mathrm{d}_{\mathrm{m}}\). The domain of influence for the radial weight is in the circular form, i.e., the support boundary is circular (Figure 1). The \(\|\mathrm{x}-\mathrm{x}_{i}\|\) is the node x's distance from node \(\mathrm{x}_{i}\) and \(\mathrm{d}_{\mathrm{m}}\) is the influence domain size of the node xi. The support size of the \(\mathrm{I}\)th node, \(\mathrm{d}_{\mathrm{m}}\), is computed by \(\mathrm{d}_{\mathrm{m}}=\mathrm{d}_{\mathrm{m}ax}\)\(\mathrm{c}_{\mathrm{I}}\). The value of \(\mathrm{c}_{\mathrm{I}}\) is calculated from distances of the neighbour nodes. The dilation parameter (\(\mathrm{d}_{\mathrm{m}ax}\)) is used as 3.0 in the analysis. The cubic spline weight function for MLS interpolation given below is considered in the present study.  
\[w\left(\overline{d}\right)=\left\{\begin{array}{ll}\frac{2}{3}-4\overline{d }^{2}+4\overline{d}^{3}&for\overline{d}\leq\frac{1}{2}\\ \frac{4}{3}-4\overline{d}+4\overline{d}^{2}-\frac{1}{3}\overline{d}^{3}&for 1\leq\overline{d}\leq\frac{1}{2}\\ 0&for\overline{d}>1\end{array}\right\} \tag{20}\]  
The mesh-free MLS-based recovery estimation and mesh-improvement strategies were employed in two-dimensional FEM-based computer program. The mesh-dependent patch-based conventional error-recovery estimations and incompressible elastic formulation were also implemented. The program was run on i7-configuration computer with processor of 2.6 GHz and RAM of 16 GB to obtain the computational results. Flow chart for MLS-based recovery technique coupled with adaptive analysis of incompressible elastic problems is shown in Figure 2.