PSI - Issue 77

D.C. Gonçalves et al. / Procedia Structural Integrity 77 (2026) 79–86 Gonçalves et al. / Structural Integrity Procedia 00 (2026) 000 – 000

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Fig. 1. Numerical integration scheme.

Fig. 2. The overlapping influence domains of two distinct integration points. To construct the shape functions at a given integration point, , the interpolation function is built by combining a Radial Basis Function (RBF) with a polynomial basis function (PBF), multiplied by the coefficients and , respectively, ( )=∑ ( − ) ( ) = +∑ ( ) ( ) = = { ( ) ( ) } { ( ( ) ) } (1) A constant, ( )={1} , and the Multiquadric RBF, defined as, , ( )=( 2 + 2 ) (2) being the Euclidean distance, and = 0.0001 and = 0.9999 shape parameters [11], were used in this work. The coefficients and must then be determined and are obtained by enforcing exact interpolation at the field nodes. Thus, the interpolation function is written for each influence node, and assuming the nodal values are known, we arrive at a system of equations that allows isolating the coefficients and . =[ 1 ( 1 ) 2 ( 1 ) ⋯ ( 1 ) 1 ( 2 ) 2 ( 2 ) ⋯ ( 2 ) ⋮ ⋮ ⋱ ⋮ 1 ( ) 2 ( ) ⋯ ( )] ( )+[ 1 1 1 1 2 2 ⋮ ⋮ ⋮ 1 ] ( ) (3) An extra set of equation must be added to the system to obtain a set of + equations with + unknowns, allowing to define, { } = [ ⏟ ] { ( ( ) ) } (4) being referred to as the moment matrix . Once the coefficient matrix is isolated, { ( ( ) ) } = − 1 { } (5) the coefficients are then substituted back into the original interpolation function in equation (1). ( ) = { ( ) ( )} − 1 { 0 } = { ( ) ( )}{ 0 } = ( ) (6) By analyzing the resulting equation, the shape function vector ( ) can be identified. The challenge lies in optimizing the parameters of the radial basis function (MQ-RBF) and the polynomial basis [11]. The procedure to obtain the final discrete system of equations and enforce boundary conditions is analogous to classic FEM.