# The integro-differential equations governing the physical phenomenon,

The meshless method applied in this work is the RPIM 10, 25, 26. The RPIM is an interpolator meshless method which enforces nodal connectivity using the influence-domain concept. To solve the integro-differential equations governing the physical phenomenon, the RPIM uses a background cloud of integration points, constructed using integration cells and the Gauss-Legendre quadrature rule.

2.1. Nodal Connectivity and Numerical Integration

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Several meshless methods use the concept of influence domain due to its simplicity. As FEM, meshless methods are discrete numerical methods. However, instead of discretizing the problem domain in elements and nodes, meshless methods discretize the problem domain using just nodes. The nodal connectivity in FEM is predefined by a finite element mesh defined in the preprocessing phase. Thus, the nodes of each element interact directly with each other and the nodes belonging to the element boundary interact with the nodes of neighbor finite elements.

In meshless methods, after the nodal discretization, the nodal connectivity is established with the “influence-domain” concept. The nodal connectivity in meshless methods is not a pre-established information (as in FEM) and it is assured by the overlap of the influence-domains 10. Since this technique is very simple to implement, it has been used to support the development of several meshless techniques 7, 10, 25, 27–29. Generally, the influence-domains are obtained by searching radially enough nodes inside a defined area (2D problems) or a defined volume (3D problems). Nevertheless, it has been observed that the size or shape variation of these influence-domains affects the performance of the meshless method along the problem domain 10. Thus, regardless the used meshless technique, the literature proposes that each 2D influence-domain should possess approximately n = 9, 16 nodes 7, 10, 25, 27–29. In figure 1(a) is presented an example of an influence domain of an interest point  (which could be a node or an integration point). In this work, for the 2D analysis each influence domain contains XX nodes. For all 3D analyses, each influence domain contains YY nodes.

In order to solve the integro-differential equations governing the discrete numerical methods – the Galerkin weak formulation – a background integration mesh is required. This numerical integration process represents a significant percentage of the total computational cost of the analysis.

In the FEM, the construction of the integration mesh is simplified by the existence of the element mesh, since each element is geometrically coincident with each integration cell. Additionally, the FEM shape functions are known polynomial functions. Therefore, accurate well-known relations 30, 31 can be used to predefine the number of integration points per each integration cell.

In the case of meshless methods, the accurate definition of the background integration mesh is a more challenging task. In meshless methods it is not possible to accurately define a priori the background integration mesh because the shape function degree is generally unknown. In this work it is used the numerical integration scheme suggested in previous RPIM works 25, 26. Generally, in the RPIM formulation, as figure 1(b) represents,  the entire domain is divided in a regular grid creating quadrilateral integration cells and then, respecting the Gauss-Legendre quadrature rule 10 (Figure 1), each cell is filled with integration points.

In this work, since every analysis is performed using the FEM and the RPIM, for the 2D analyses, the background integration cell lattice is assumed coincident with the FEM mesh. Then, inside each quadrilateral 3×3 Gauss points are inserted, as figure 1(b) shows. For the 3D analyses, since the 3D FEM meshes are built with tetrahedrons, inside each tetrahedron is considered one integration point, whose spatial position is coincident with the volume centre of the tetrahedron and whose integration weight is coincident with the tetrahedron volume. The literature shows that both these integration schemes allow to integrate accurately the Galerkin weak form 10.