Analysis of spatial interpolation in the material-point method

This paper analyses different types of spatial interpolation for the material-point method. The interpolations include quadratic elements and cubic splines in addition to the standard linear shape functions usually applied. For the small-strain problem of a vibrating bar, the best results are obtained using quadratic elements. It is shown that for more complex problems, the use of partially negative shape functions is inconsistent with the material-point method in its current form, necessitating other types of interpolation such as cubic splines in order to obtain smoother representations of field quantities. The properties of different interpolation functions are analysed using numerical examples, including the classical cantilevered beam problem.