  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.
