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.

This paper presents a new Smooth Particle Hydrodynamics (SPH) computational framework for large strain explicit solid dynamics. A mixed-based set of Total Lagrangian conservation laws [1,2] is presented in terms of the linear momentum and an extended set of geometric strain measures, comprised of the deformation gradient, its co-factor and the Jacobian. Taking advantage of this representation, the main aim of this paper is the adaptation of the very efficient Jameson-Schmidt-Turkel (JST) algorithm [3], extensively used in computational fluid dynamics, to a SPH based discretisation of the mixed-based set of conservation laws, with three key distinct novelties. First, a conservative JST-based SPH computational framework is presented with emphasis in nearly incompressible materials. Second, the suppression of numerical instabilities associated with the non-physical zero-energy modes is addressed through a well-established stabilisation procedure. Third, the use of a discrete angular momentum projection algorithm is presented in conjunction with a monolithic Total Variation Diminishing Runge-Kutta time integrator in order to guarantee the global conservation of angular momentum. For completeness, exact enforcement of essential boundary conditions is incorporated through the use of a Lagrange multiplier projection technique. A series of challenging numerical examples (e.g. in the near incompressibility regime) are examined in order to assess the robustness and accuracy of the proposed algorithm. The obtained results are benchmarked against a wide spectrum of alternative numerical strategies.

Smooth particle Hydrodynamics (SPH) is one of the most effective meshless techniques used in computational mechanics. SPH approximations are simple and allow greater flexibility in various engineering applications. However, modelling of particle-boundary interactions in SPH computations has always been considered an aspect that requires further research. A number of techniques have been developed to model particle-boundary interactions in SPH and allied methods. In this paper, an innovative approach is introduced to handle the contact between Lagrangian SPH particles and rigid solid boundaries. The formulation of boundary contact forces are derived based on a variational formulation, thus directly ensuring the conservativeness of the governing equations. In addition, the new elegant boundary contact force terms maintain the simplicity of the SPH governing equations.

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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.