A corrected transport-velocity formulation for fluid and structural mechanics with SPH

Particle shifting techniques (PST) have been used to improve the accuracy of the smoothed particle hydrodynamics (SPH) method. Shifting ensures that the particles are distributed homogeneously in space. This may be performed by moving the particles using a transport velocity. In this paper, we propose an extension to the class of transport velocity formulation (TVF) methods. We derive the equations in a consistent manner and show that there are additional terms that significantly improve the accuracy of the method. In particular, we apply this to the entropically damped artificial compressibility SPH method. We identify the free-surface particles and their normals using a simple approach and thereby adapt the method for free-surface problems. We show how the new method can be applied to the problem of elastic dynamics. We consider a suite of benchmark problems involving both fluid and solid mechanics to demonstrate the accuracy and applicability of the method. The implementation is open source, and the manuscript is fully reproducible.