Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations

Abstract

SUMMARYSmooth particle hydrodynamics (SPH) is a robust and conceptually simple method which su ers from unsatisfactory performance due to lack of consistency. The kernel function can be corrected to enforce the consistency conditions and improve the accuracy. For simplicity in this paper the SPH method with the corrected kernel is referred to as corrected smooth particle hydrodynamics (CSPH). The numerical solutions of CSPH can be further improved by introducing an integration correction which also enables the method to pass patch tests. It is also shown that the nodal integration of this corrected SPH method su ers from spurious singular modes. This spatial instability results from under integration of the weak form, and it is treated by a least-squares stabilization procedure which is discussed in detail in Section 4. The e ects of the stabilization and improvement in the accuracy are illustrated via examples. Further, the application of CSPH method to metal-forming simulations is discussed by formulating the governing equation associated with the process. Finally, the numerical examples showing the e ectiveness of the method in simulating metal-forming problems are presented.

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].…”

“…In particular, the second order (harmonic) and fourth order (biharmonic) operators employed in the JST stabilisation can be readily obtained by closed-form differentiation of the interpolating kernel functions. A quintic kernel approximation [6] is suitably used, due to the presence of the biharmonic JST operator. This paper is structured as follows: section 2 outlines the theoretical framework of the p-F mixed formulation system of equations; section 3 focuses on SPH spatial discretisation of the governing equations.…”

“…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.…”