# A Solution Procedure to Improve 3D Solid Finite Element Analysis with an Enrichment Scheme
## 1 Introduction  
During the last decades, the finite element method (FEM) has been widely used for analyzing various engineering problems and still many researchers have been devoted to improving the accuracy of solutions [1; 2; 3; 4]. Despite its notable success, the standard finite element method (ST-FEM) has some drawbacks, especially in solution accuracy which is affected by the finite element mesh and the use of linear elements such as 2D 3-node triangular and 3D 4-node tetrahedral elements that have considerably low accuracy in stress and strain prediction [5; 6; 7]. There are several ways to improve the accuracy, such as increasing mesh density and adding nodes to use higher-order finite elements. In addition, numerous studies have been carried out to overcome these limitations of the standard FEM (ST-FEM). As a result, several numerical methods have been developed, including extended FEM (EX-FEM), enriched FEM (EN-FEM), and smoothed FEM (S-FEM) [6; 7; 8; 9; 10]. EX- and EN-FEM improve the solution accuracy by supplementing their solution space [11; 12; 13]. S-FEM improves the accuracy by using information from neighboring elements [14; 15; 16; 17]. Among numerical methods, an enrichment scheme expands the solution space by introducing cover functions to additional degrees of freedom (DOFs) compared to the standard DOFs of ST-FEM [18; 19]. It can enhance finite element analysis solution like the \(p\)-version of the ST-FEM but does not require additional nodes. Also, different cover functions can be applied to each node. Therefore, the solution accuracy of the finite element analysis result could be effectively improved through adaptively applying thecover function to the local region where the predictive capability needs to be improved. [19]. However, the presence of linear dependence (LD) problems has limited the use of enriched finite elements. When the LD problem occurs, it is difficult to obtain reliable solutions because the stiffness matrix becomes a singular matrix. Various approaches have been proposed to address the LD problem in EN-FEM [20; 21; 22], and effective ways to avoid the LD problem have been suggested in recent studies [23; 24; 25]. One such solution proposed by [PERSON] involves suppressing all degrees of freedom corresponding to the cover function at the node where essential boundary conditions are imposed for 3-node triangular and 4-node tetrahedral elements [20]. [PERSON] and [PERSON] proposed the use of piecewise linear shape functions as the partition of unity functions for geometry and displacement interpolations to effectively resolve the LD problem for 4-node quadrilateral, 8-node hexahedral, 6-node prismatic, and 5-node pyramidal elements [23; 24]. The feasibility of improving solution accuracy through the adaptive use of cover functions was also demonstrated in these studies [24]. Several studies have proposed procedures for the use of enriched finite elements to automatically enhance the accuracy of finite element solutions [19; 25]. [PERSON] and [PERSON] proposed a method for automatically applying cover functions based on the estimated error for each node, but their study was restricted to 2D and 3D linear elements and involved artificial coefficients [19]. [PERSON] and [PERSON] presented a procedure for enhancing the analysis accuracy in a 2D problem with a 2D 4-node quadrilateral element free from the LD problem. This study reduced the number of artificial coefficients compared to prior studies and demonstrated the feasibility of automatic accuracy improvement [25]. The aim of this paper is to propose a procedure that automatically enhances the accuracy of 3D solid finite element solutions using enriched 3D solid finite elements, including 4-node tetrahedral, 5-node pyramidal, 6-node prism, and 8-node hexahedral elements.