# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 1 Introduction  
The discretization of problem domains generates errors in the numerical technique. The finite element error and mesh adaptive analysis are implemented in finite element procedures to increase the reliability of finite element analyses. The recent research directions in finite element methods include the advancement of the finite element technique by overcoming its drawbacks and the enhancement of the method's reliability and efficiency. [PERSON] et al. [1] provide a survey of the finite element methods' vast range of applications. Several techniques have been proposed to recover displacements or their gradients and to improve the finite element solution's accuracy. A critical review of various error-estimation techniques to obtain the practical finite element results of linear and non-linear problems is summarized by [PERSON] and [PERSON] [2]. New finite element techniques are developed to decrease the reliance on mesh for the analysis of problems. [PERSON] et al. [3] summarized the surveys of mesh-free method developed to address the weaknesses of the finite element approach. Some the recent developments in alternative finite element methods (FEM) are the generalized finite element method [4], the extended finite element technique [5], the smoothed finite element method [6], the smoothed-particle-hydrodynamics method [7], thereproducing kernel particle method [8], and the moving-particle semi-implicit method [9]. A number of materials used in industry have very high bulk moduli in comparison with the shear modulus, i.e., nearly incompressible materials, and materials with large deformations may also be treated as incompressible. The displacement-based finite element approach has difficulties in the analysis of nearly incompressible materials, such as stiffness-matrix ill-conditioning, the locking phenomenon and spurious stresses. A summary of various methods used in the finite element analysis of incompressible and nearly incompressible elasticity is presented by [PERSON] and [PERSON] [10]. The developments over the past ten years in the application of various approaches in finite element methods to deal with volumetric locking are reviewed by [PERSON] and [PERSON] [11]. Mixed Kirchhof stress-displacement-pressure formulation [12], a displacement-pressure formulation, and a simple pressure-projection-stabilized method [13] can be used to deal with incompressibility constraints. [PERSON] et al. [14] investigated the effectiveness of selective reduced integration in overcoming the volumetric locking phenomenon in 2D and 3D solid elements. Numerous solutions to the problem of incompressible locking in displacement-based finite element approaches were been proposed by [PERSON] and [PERSON] [15]. [PERSON] et al. [16] applied a stabilized finite element procedure using a mixed formulation in which the momentum equation of the continuum is extended by a pressure equation that treats the incompressibility constraints, in order to solve transient linear and nonlinear solid dynamics in compressible and incompressible materials. The finite-strain-incompressible-elasticity problem was analyzed by [PERSON] et al. [17] using a variational technique based on a finite element method with a solution for volumetric constraints. The mixed meshless local Petrov-Galerkin approach was applied by [PERSON] et al. [18] to gradient elasticity, and they found high accuracy with a low order in the meshless approximation functions. The neural networks (ANN) technique was applied by [PERSON] et al. [19] to recuperate the stress errors in finite element solutions. The numerical errors due to discretization in the finite element method can be minimized using the error estimation and adaptive-mesh-optimization techniques. In incompressible and large deformations such as those for rubber materials and metal-forming processes, the accuracy is also lost due to volumetric locking, in addition to errors due to discretization.