PSI - Issue 28

I.J. Sánchez-Arce et al. / Procedia Structural Integrity 28 (2020) 1084–1093 Sánchez-Arce et al. / Structural Integrity Procedia 00 (2019) 000–000

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3. Numerical simulations 3.1. Meshless method The NNRPIM method considers that the geometry to be analyzed is composed of a set of nodes � � � , � , … , � � � � which can be either randomly or uniformly distributed in the surface or volume, and ‘n’ corresponds to the number of nodes inside the domain . Then, a Voronoï diagram is created from resulting in the ‘backing’ mesh for the method and it provides the base for selecting the ‘influence cell’ � [13]. Taking a node from as the point of interest, all the nodes inside influence . Therefore, the function relating can be described as the sum of two shape functions, one radial based and one polynomial, as shown in Equation (1) [11,13,21]. � � � � � � � � � � � ��� �� � � � � � ��� � � �1� where � � � is the ‘radial basis function’ (RBF), and � � � is a non-constant coefficient of � � � [12]. Similarly, � � � is a polynomial basis function and � � � are their non-constant coefficients [21]. Various RBFs exist, here one related to the ‘Euclidean norm’ (distance between two points in space) � �� � was chosen, the multi quadratics radial basis functions (Equation (21)). � �� � � � �� � � � � � �2� �� � �� � � � � � � � � � � � � � � � � � � � �3� Although and are parameters that have to be optimized, their values were already optimized as � �����1 and � ��9999 [11,12]. The non-constant coefficients � � � and � � � can be obtained by applying the interpolation function (1) to all nodes (n) in [11,21]. The interpolation function (1) can be represented as a matrix [13], as follows: � �� � � �4� Where: � � � � � � � � � � � ⋯ � � � � � � � � � � � � ⋯ � � � � ⋮ ⋮ ⋱ ⋮ � � � � � � � � ⋯ � � � � � �5� � � � ��� � � ��� � � ⋯ ��� � � ��� � � ��� � � ⋯ ��� � � ⋮ ⋮ ⋱ ⋮ ��� � � ��� � � ⋯ ��� � � � �6� Then, Equation (1) can be expressed as: � � � � � � � ; � � � � � �� � � � � �Φ� � ; Ψ� � � � � � � �7� where, � is the total moment matrix. Subsequently, Equation (7) can be expressed as: � � � �Φ � � � � � ��� �8� Where � � is the shape function value at [13]. Subsequently, the partial derivatives with respect to and of � � . Further details of the method in [11–13,21]. Subsequently, the weak-Galerkin form is used to apply the material matrix and the natural and essential boundary conditions [12]. First, the equilibrium condition is established considering the stress tensor and the body forces , as shown in (9). � � � � �9� Then, the natural and essential boundary conditions are established: