As we all know, WCSPH has some issues of correct pressure prediction. In DSPH, what kind of pressure correction has been made to make it more accurate?

For WCSPH, a number of treatments have been proposed over the years to address the pressure fluctuations. We started off with the Shepard Filter which was crude, and more recently have moved to implement delta-SPH and particle shifting. Both techniques effectively help to mitigate the high-frequency noise which is generated by numerical problems in the SPH method.

The work on delta-SPH is considered to be "density diffusion" and hence the variation (i.e. fluctuations) in density are restricted. This feeds through to the pressure via the Equation of State. The work is actually very well described in the papers by Marrone et al. (2011) and Antuono et al. (2010). These are referenced in the wiki and guide where you will see that the density diffusion appears as an extra term in Equation (15): https://github.com/DualSPHysics/DualSPHysics/wiki/3.-SPH-formulation#35-deltasph

The inspiration of the class of "density diffusion" techniques for WCSPH lies in the fact that we use an Equation of State that is isothermal so that the energy evolution is decoupled from the conservation of mass and momentum. For a weakly compressible fluid, this is not completely physical and so density diffusion is required to compensate for this.

A strictly incompressible version of SPH, ISPH, has already been implemented in DualSPHysics (see papers by Chow et al. 2018, 2019). ISPH produces a far superior pressure field for incompressible flows. We plan to include this in future releases.

It's unlikely that the DBC gap will adversely affect pressure and hence the movement of the floating bodies since their motion (i.e. equations of motion) are due to integration of these quantities over the surface of the object. You can see that this makes little difference with the nice validation "Validation of a floating box interacting with waves" : The local pressures might be affected slightly, but the delta-SPH should improve this compared to other treatments.

## Comments

For WCSPH, a number of treatments have been proposed over the years to address the pressure fluctuations. We started off with the Shepard Filter which was crude, and more recently have moved to implement delta-SPH and particle shifting. Both techniques effectively help to mitigate the high-frequency noise which is generated by numerical problems in the SPH method.

The work on delta-SPH is considered to be "density diffusion" and hence the variation (i.e. fluctuations) in density are restricted. This feeds through to the pressure via the Equation of State. The work is actually very well described in the papers by Marrone et al. (2011) and Antuono et al. (2010). These are referenced in the wiki and guide where you will see that the density diffusion appears as an extra term in Equation (15): https://github.com/DualSPHysics/DualSPHysics/wiki/3.-SPH-formulation#35-deltasph

The inspiration of the class of "density diffusion" techniques for WCSPH lies in the fact that we use an Equation of State that is isothermal so that the energy evolution is decoupled from the conservation of mass and momentum. For a weakly compressible fluid, this is not completely physical and so density diffusion is required to compensate for this.

A strictly incompressible version of SPH, ISPH, has already been implemented in DualSPHysics (see papers by Chow et al. 2018, 2019). ISPH produces a far superior pressure field for incompressible flows. We plan to include this in future releases.

I hope this helps,

Ben

For the gap caused by DBC between floating bodies and fluid, will this affect pressure on the floating bodies?

It's unlikely that the DBC gap will adversely affect pressure and hence the movement of the floating bodies since their motion (i.e. equations of motion) are due to integration of these quantities over the surface of the object. You can see that this makes little difference with the nice validation "Validation of a floating box interacting with waves" :

The local pressures might be affected slightly, but the delta-SPH should improve this compared to other treatments.

Ben