From the summary: This paper describes the development and testing of an approach for treatment of zero-energy modes in the smoothed particle hydrodynamics 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 discretization 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. One- and two-dimensional algorithms are presented and test results shown, demonstrating that the approach does solve the zero-energy mode problem.