Radial function, the interpolation function can be defined at

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 ,

( 1)

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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,

( 2)

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,

( 3)

 

( 4)

 

( 5)

 

( 6)

 

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,

,
,

( 7)

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,

( 8)

The inclusion of the following polynomial term is an extra-requirement that guarantees unique approximation 10, 32,

( 9)

The computation of the shape functions is written in a matrix form as

( 10)

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,

( 11)

and polynomial moment matrix  is defined as,

( 12)

Since the distance is directionless,, i.e. , matrix   is symmetric. A single solution is obtained if the inverse of the radial moment matrix   exists,

( 13)

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

( 14)

where the interpolation function vector  is defined by

( 15)

and the residual vector , with no relevant physical meaning, is expressed as follows,

( 16)

Since

 ,

( 17)

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,

( 18)

From equation (14) it is possible to write

( 19)

Since the moment matrix   does not depend on the variable , equation (19) can be rewritten as,

( 20)

The partial derivatives of the MQ-RBF vector , with respect to a generic variable , can be obtained for each component  with the expression,

( 21)

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.