Conservation of Momentum and Tensile Instability in Particle Methods

This paper considers conservation of momentum and tensile instability in transient dynamic particle methods. Local particle methods are defined as representing a continuous body with particles which discretize the body. The particles carry the field variables and are used for interpolation purposes. They also are discrete chunks of matter which must have mass and volume which in sum represent the total mass and volume of the body. In particle methods, a function is approximated by summing data-weighted shape functions over the particles, Conservation of momentum is considered from the standpoint of Newtons third law, and the conditions which the shape functions must satisfy in order to conserve momentum are derived. The tensile instability is considered from the standpoint of the strength of interaction between particles, and the concept of area vectors which define the surfaces on which the stresses act is introduced. The local nature of the approximation is shown to produce area vectors and forces which decrease as the distance betw,een particles increases. This situation is shown to be similar to other particle systems, such as gravitational systems of patticles or molecular dynamics. The instability is shown to be the expected behavior of the physical system represented by the discretization scheme. Finally, these concepts are applied to a specific example, that of moving least squares.