  Tensile instability, often observed in smoothed particle hydrodynamics (SPH),
is a numerical artifact that manifests itself by unphysical clustering or
separation of particles. The instability originates in estimating the
derivatives of the smoothing functions which, when interact with material
constitution may result in negative stiffness in the discretized system. In the
present study, a stable formulation of SPH is developed where the kernel
function is continuously adapted at every material point depending on its state
of stress. Bspline basis function with a variable intermediate knot is used as
the kernel function. The shape of the kernel function is then modified by
changing the intermediate knot position such that the condition associated with
instability does not arise. While implementing the algorithm the simplicity and
computational efficiency of SPH are not compromised. One-dimensional dispersion
analysis is performed to understand the effect adaptive kernel on the
stability. Finally, the efficacy of the algorithm is demonstrated through some
benchmark elastic dynamics problems.
