A NURBS-based Error Reproducing Kernel Method with Applications in Solid Mechanics

A novel method for derivation of mesh-free shape functions is proposed. The first step in the method is to approximate a function and its derivatives through non-uniform-rational-B-spline (NURBS) basis functions. However since NURBS functions neither reproduce polynomials of degree higher than one nor interpolate the control points (also referred to as grid or nodal points), the approximated function leads to uncontrolled errors over the domain including the nodal points. Accordingly the error function in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Since any desired order of continuity in the approximation can be achieved through NURBS, the proposed error reproducing kernel method (ERKM) can even approximate functions with discontinuous derivatives. Moreover, thanks to the variation diminishing property of NURBS, it has advantages in representing sharp layers without the so-called Gibbs‘ or Runge’s phenomena. Since derivatives are reproduced within polynomial spaces of appropriately reduced dimensions, differentiability requirements of the kernel functions are avoided. Any compactly supported continuous function, monotonically decreasing on either side of its maximum, may be used as the weight function (unlike other mesh free approximations). As it turns out, a target function is mainly approximated via NURBS and error functions are just supposed to add corrections, whose magnitudes are typically an order less than those of the NURBS components. The proposed method is observed to be nearly insensitive to the support size of the weight function. The proposed method is next applied to some linear and nonlinear boundary value problems of typical interest in solid mechanics. Some of these results are compared with those obtained via the standard form of RKPM. In the process, the relative numerical advantages and accuracy of the new method are brought out to an extent.