Moving-least-squares-particle hydrodynamics - I. Consistency and stability

The Smooth-Particle-Hydrodynamics (SPH) method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of continuum mechanics as in the Þnite-element method. This derivation is modiÞed to replace the SPH interpolant with the Moving-Least-Squares (MLS) interpolant of Lancaster and Saulkaskas, and deÞne a new particle volume which ensures thermodynamic compatibility. A variable-rank modiÞcation of the MLS interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent ßow reduces the number of neighbours of a particle to less than the minimum required. A surprise beneÞt of the Galerkin SPH derivation is a theoretical justiÞcation of a common ad hoc technique for variable-h SPH. The new MLSPH method is conservative if an anti-symmetric quadrature rule for the sti¤ness matrix elements can be supplied. In this paper, a simple one-point collocation rule is used to retain similarity with SPH, leading to a non-conservative method. Several examples document how MLSPH renders dramatic improvements due to the linear consistency of its gradients on three canonical di¦culties of the SPH method: spurious boundary e¤ects, erroneous rates of strain and rotation and tension instability. Two of these examples are non-linear Lagrangian patch tests with analytic solutions with which MLSPH agrees almost exactly. The examples also show that MLSPH is not absolutely stable if the problems are run to very long times. A linear stability analysis explains both why it is more stable than SPH and not yet absolutely stable and an argument is made that for realistic dynamic problems MLSPH is stable enough. The notion of coherent particles, for which the numerical stability is identical to the physical stability, is introduced. The new method is easily retroÞtted into a generic SPH code and some observations on performance are made. Copyright