# Fracture Toughness Determination on an SCB Specim by Meshless Methods
## 2 Overview on Meshless Methods  
In this study, the 2D deformation theory in plane stress state was taken into account for all computational analyses. The RPIM and NNRPIM meshless method formulations were taken into account [25] extended to the LEFM theory. Generally, meshless methods discretize the problem domain and respective boundaries using a nodal distribution. This nodal set cannot be categorized as a mesh, because no previous information regarding the relationship between each node is required to build the interpolation functions for the unknown variational fields [26]. The RPIM is an interpolator meshless method assuring the nodal connectivity within the influence-domain conception, which is chiefly adopted by many meshless techniques because of its simplicity. Similar to the FEM, RPIM meshless method stands as a discrete computational approach. The main difference could be related to the discretization procedure. The elements and nodes discretize the Finite Element (FE) problem domain whereas meshless methods discretize the problem domain only by nodal points. Likewise, in the FEM, the nodal connectivity must be enforced in the pre-processing phase through a predefined FE mesh establishment. Hence, any element owns the intrinsic and adequate information to impose nodal connectivity. Despite the FEM, meshless methods do not possess predefined connectivity mesh, it is essential to generate one [27]. Therefore, generally, the nodal interdependency must be enforced by the influence-domain geometrical construction in meshless methods. It can be acquired once the nodal discretization is created [25]. This is an overlap associated with the influence-domains allowing setting up nodal connectivity. Considering an interest point in the problem domain, it is possible to establish its influence-domain by a radial search centered in that interest point. It led to collecting a definite number of nodes inside a fixed area or volume, the 2D and 3D problems, respectively. The literature proved that this approach is simple to figure out and to implement contributing to support the expansion of several meshless techniques. Regarding RPIM meshless method, the literature [25] recommends that any 2D influence-domain must include roughly \(n=[9;16]\) nodes. Besides, a background integration mesh is necessary to mathematically integrate the integro-differential equations governing the physical phenomenon [26; 28] due to the adaptation of the Galerkin weak-formulation. In the case of the NNRPIM, the natural neighbor's concept has been taken into account to impose the nodal connectivity as recommended by [PERSON] [29]. This mathematical notion could be described by the Voronoi diagram associated with the discretized domain. In the NNRPIM, the problem domain \(\Omega\ \in\mathbb{R}^{d}\), bounded by a physical boundary \(\Gamma\subset\Omega\), is discretized in numerous arbitrarily distributed points \(\mathbf{N}=\{n_{0},n_{1},\dots,n_{N}\}\) scattered in the space domain: \(\mathbf{X}=\{x_{1},x_{2},\dots,x_{N}\}\in\Omega\). The Voronoi diagram of \(\mathbf{N}\) is the domain partition determined by \(\Omega\) in sub-regions \(\mathbf{V}_{i}\), closed and convex where any sub-region \(\mathbf{V}_{i}\) is attendant with the node \(n_{i}\), in a way that any point in the interior of the \(\mathbf{V}_{i}\) is closer to \(n_{i}\) than any other node \(n_{j}\), where \(n_{j}\in\mathbf{N}\wedge j\
eq i,\mathbf{x}_{1}\in\Omega\). \[\mathbf{V}_{i}:=\Big{\{}\mathbf{x}_{1}\in\Omega\ \subset\ \mathbb{R}^{d}:\|\mathbf{x}_{1}-\mathbf{x}_ {i}\|<\|\mathbf{x}_{1}-\mathbf{x}_{j}\|,\forall i\
eq j\Big{\}} \tag{1}\]  
However, a set of Voronoi cells \(\mathbf{V}_{i}\) defines the Voronoi diagram, \(\mathbf{V}=\{V_{1},V_{2},\ldots,V_{N}\}\). In the literature [25; 30; 31], it is possible to find relevant works properly addressing the Voronoi construction procedure.