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The correct evaluation of gradients is at the cornerstone of the smoothedparticle hydrodynamics (SPH) technique. Using an integral approach to estimategradients has proven to enhance accuracy substantially. Such approach retainsthe Lagrangian structure of SPH equations and is fully conservative. But, inpractice, it is difficult to make the Lagrangian formulation totally consistentto an exact partition of the unity.  In this paper we study, among other things, the connection between the choiceof the volume elements (VEs), which enters in the SPH summations, and theaccuracy in the gradient estimation within the integral approach scheme (ISPH).A new variant of VEs are proposed which improve the partition of the unity andis fully compatible with the Lagrangian formulation of SPH, including thegrad-h corrections. Using analytic considerations, simple static toy models in1D, and a few full 3D test cases, we show that any improvement in the partitionof the unity also leads to a better calculation of gradients when the integralapproach is used jointly. Additionally, we propose an easy-to-implementmodification of the ISPH scheme, which makes it more flexible and better suitedto handle sharp density contrasts.  The ISPH code built with the proposed scheme has been validated with a goodnumber of standard tests, some of them involving contact discontinuities. Theperformance of the code was excellent in all of them, showing that animprovement in the partition of the unity is not detrimental of the goodconservation of energy, momentum, and entropy typical of Lagrangian schemes.

Conservative, density-based smoothed particle hydrodynamics with improved partition of the unity and better estimation of gradients