The closed form reproducing polynomial particle shape functions for meshfree particle methods

It has been known that reproducing kernel particle (RKP) shape functions with Kronecker delta property are not available in simple forms. Thus, in this paper, we construct highly regular piecewise polynomial reproducing polynomial particle (RPP) shape functions that satisfy the Kronecker delta property. Like RKP shape functions, the RPP shape functions reproduce the complete polynomials of degree k   for an integer k⩾0k⩾0. Moreover, the particles associated with these RPP shape functions are either uniformly distributed in (-∞,∞)(-∞,∞) or non-uniformly distributed in [0,∞)[0,∞) (or in a compact set [a, b]).