Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods

Abstract

SUMMARYThe paper discusses the problem of tension instability of particle-based methods such as smooth particle hydrodynamics (SPH) or corrected SPH (CSPH). It is shown that tension instability is a property of a continuum where the stress tensor is isotropic and the value of the pressure is a function of the density or volume ratio. The paper will show that, for this material model, the non-linear continuum equations fail to satisfy the stability condition in the presence of tension. Consequently, any discretization of this continuum will result in negative eigenvalues in the tangent sti ness matrix that will lead to instabilities in the time integration process. An important exception is the 1-D case where the continuum becomes stable but SPH or CSPH can still exhibit negative eigenvalues. The paper will show that these negative eigenvalues can be eliminated if a Lagrangian formulation is used whereby all derivatives are referred to a ÿxed reference conÿguration. The resulting formulation maintains the momentum preservation properties of its Eulerian equivalent. Finally a simple 1-D wave propagation example will be used to demonstrate that a stable solution can be obtained using Lagrangian CSPH without the need for any artiÿcial viscosity.

A Stabilised Total Lagrangian Corrected Smooth Particle Hydrodynamics Technique in Large Strain Explicit Fast Solid Dynamics

“…During the spatial discretisation, the border kernel supports are rendered incomplete by boundary edges, leading to kernel interpolations that are not partitions of unity. To improve the accuracy of the SPH interpolation near boundaries, and to exactly preserve momentum, corrections must be introduced on both the kernel and the kernel gradient [6,7,10]. The resulting corrected SPH (CSPH) formulation greatly enhances the accuracy and the consistency of the discretisation.…”

“…To achieve this, and following the work of Bonet and co-authors [6,7,10,11], the weak statement for the linear momentum evolution must be obtained through the use of work-conjugate principles [1] and integration by parts: Upon application of the particle integration on the above expression (10), and using the kernel approximation…”

“…Following References [6,7,[9][10][11], the first Piola-Kirchhoff stress tensors {P a , P b } are further approximated by using values of {F a , F b } estimated directly at the particles. In (11), corrections are introduced on the kernel gradient to yield∇ 0 W b (X a ), ensuring the correct evaluation of a gradient of a general linear function [6,10].…”

“…A series of benchmark examples are tested in order to assess the applicability and effectiveness of the p-F JST-CSPH mixed algorithm over the Total Lagrangian CSPH displacement based formulation [6,7,10].…”

“…An explicit Total Lagrangian mixed momentum/strains formulation [1][2][3][4][5], in the form of a system of first order conservation laws, has been recently proposed to overcome the shortcomings posed by the traditional second order displacement-based formulation, namely: (1) bending and volumetric locking difficulties; (2) hydrostatic pressure fluctuations; and (3) reduced order of convergence for derived variables. Following the work of Bonet and Kulasegaram [6,7], the main objective of this paper is the adaptation of Corrected Smooth Particle Hydrodynamics (CSPH) in the context of Total Lagrangian mixed formulation. Appropriate nodally conservative Jameson-Schmidt-Turkel (JST) stabilisation is introduced by taking advantage of the conservation laws.…”