A treatment of zero-energy modes in the smoothed particle hydrodynamics method

This paper describes the development and testing of an approach for treatment of zero-energy modes in the Smoothed Particle Hydrodynamics (SPH) method. The zero-energy modes are a consequence of the fact that field variables and their derivatives are calculated at the same points, so that an alternating field variable has a zero gradient at the particles. An alternative discretisation method that uses two types of particles, velocity particles where the velocity is known and stress particles where the stress is known, is proposed as a solution to this problem. This approach prevents the zero-energy modes from occurring, and also is a probable solution to the tensile instability problem. 1D and 2D algorithms are presented and test results shown, demonstrating that it does solve the zero-energy mode problem. This approach also offered a method to satisfy the stress-free boundary condition with out the need to explicitly enforce it, as is required in conventional normalised SPH. This was achieved by placing only the velocity points (points where the velocity is known) at material boundaries. The proposed type of discretisation and boundary treatment significantly improves performance and simplifies penalty based contact algorithm for SPH. (C) 2000 Elsevier Science S.A. All rights reserved.