An adaptive approach to remove tensile instability in SPH for weakly compressible fluids

Smoothed Particle Hydrodynamics (SPH) is plagued by the phenomenon of tensile instability, which is the occurrence of short wavelength zero energy modes resulting in unphysical clustering of particles. The root cause of the instability is the shape of derivative of the compactly supported kernel function which may yield negative stiffness in the particle interaction under certain circumstances. In this work, an adaptive algorithm is developed to remove tensile instability in SPH for weakly compressible fluids. Herein, a B-spline function is used as the SPH kernel and the knots of the B-spline are adapted to change the shape of the kernel, thereby satisfying the condition associated with stability. The knot-shifting criterion is based on the particle movement within the influence domain. This enables the prevention of instability in fluid problems where excessive rearrangement of particle positions occurs. A 1D dispersion analysis of an Oldroyd B fluid material model is performed to show how the algorithm prevents instabilities for short wavelengths but ensures accuracy at large wavelengths. The efficacy of the approach is demonstrated through a few benchmark fluid dynamics simulations where a visco-elastic Oldroyd B material model and a non-viscous Eulerian fluid material model are considered.