Radial Point Interpolators (RPI) are used to obtain the RPIM shape functions, combining radial and polynomial basis functions. Thus, consider a function space defined in the domain . The finite dimensional space discretises the domain and it is defined by: , where is at least a function and is defined in the space of polynomials of degree less than . In this section only simplified two-dimensional domains are shown. Therefore, it is consider an interpolation function defined in an influence-domain of an interest point and discretized by a set of nodes: , being the number of nodes inside the influence-domain of. Notice that the domain is discretised by a nodal set defined by . The density of is identified by ,
Being the Euclidean norm.
The RPI constructs an interpolation function capable to pass through all nodes within the influence-domain, meaning that since the nodal function value is assumed to be at the node , , consequently, . Using a radial basis function and a polynomial basis function, the interpolation function can be defined at the interest point (not necessarily coincident with any ) by,
where and are the non-constant coefficient of and respectively. The integer is the number of nodes inside the influence-domain of the interest point . The vectors are defined as,
Being . This work uses the Multiquadrics Radial Basis Function (MQ-RBF) 10, 25, 26, which can be defined as , where is the distance between the interest point and the node , being . The and variables are the MQ-RBF shape parameters, which are fixed values determined in previous works 25, 26. The variation of these parameters can affect the performance of the MQ-RBFs. In the work of Wang and Liu 25 26 it was shown that the optimal values are and , which are the values used in this work. The original RPI formulation requires a complete polynomial basis function. For the two-dimensional space the following constant, linear and quadratic polynomial basis can be defined, respectively, as,
Nevertheless, it was shown in previous RPI research works 10, 12, 32 that using a constant basis increases the RPI formulation efficiency.
The coefficients and in equation ( 1) are determined by enforcing the interpolation to pass through all nodes within the influence-domain 10. The interpolation at the node is defined by,
The inclusion of the following polynomial term is an extra-requirement that guarantees unique approximation 10, 32,
The computation of the shape functions is written in a matrix form as
where is the complete moment matrix, is a null matrix defined by and the null vector can be represented by . The vector for function values is defined as . The radial moment matrix is represented as,
and polynomial moment matrix is defined as,
Since the distance is directionless,, i.e. , matrix is symmetric. A single solution is obtained if the inverse of the radial moment matrix exists,
The solvability of this system is usually guaranteed by the requirements 33. In this work, the influence-domain will always possess enough nodes to largely satisfy the previously mentioned condition. It is possible to obtain the interpolation with
where the interpolation function vector is defined by
and the residual vector , with no relevant physical meaning, is expressed as follows,
it is possible to obtain the partial derivatives of the interpolated field variable, with respect to a generic variable , which can be or , with the following expression,
From equation (14) it is possible to write
Since the moment matrix does not depend on the variable , equation (19) can be rewritten as,
The partial derivatives of the MQ-RBF vector , with respect to a generic variable , can be obtained for each component with the expression,
The RPI test functions depend exclusively on the distribution of scattered nodes 10. Previous works 10, 12, 25 show that RPI test functions possess the Kronecker delta property, facilitating the imposition of essential and natural boundary conditions. Since the obtained RPI test functions have a local compact support, it is possible to construct and assemble well-conditioned and banded stiffness matrix. If a polynomial basis is included, the RPI test functions have reproducing properties and possess the partition of unity property 10.