Abstract
The Material Point Method (MPM) uses two kinds of space discretization. One is physical space, in which physical quantities are evaluated, and the other is calculation space, in which discretized domain movements are solved. In the course of the MPM process, physical quantities are mapped from material points to a numerical grid using interpolation functions. On the contrary, material points recover physical quantities from their motions, which are mapped from the numerical grid to material points using interpolation functions. In its original form, MPM treats each material point as a point. Later, an enhanced method called the Generalized Interpolation Material Point (GIMP) method was proposed in which the effect of the size of the controlling domain is considered. More recently, the Convected Particle Domain Interpolation (CPDI) method has been proposed, in which a material point is considered to be an arbitrarily shaped controlling domain. Thus MPM and its derivatives are gradually gaining improved and enhanced numerical characteristics.
The CPDI method, which is able to consider arbitrarily shaped domain integration, has been formulated as an interpolation function for a two-dimensional controlling domain using a direct integration technique. However, it is difficult to extend direct integration to three-dimensional problems because of a drastic increase in the number of terms, leading to computational complexity. In this paper, the authors employ a numerical integration technique for domain integration over the material points in the CPDI method. Called the Arbitrary Particle Domain Interpolation (APDI) method, it has been applied to three-dimensional analysis. The APDI method allows use of the same numerical procedure even if material points have controlling domains of different shapes. Through a demonstrative analysis of two- and three-dimensional numerical problems related to geomaterials, the applicability and effectiveness of the APDI method are clarified.
