A variationally consistent Streamline Upwind Petrov–Galerkin Smooth Particle Hydrodynamics algorithm for large strain solid dynamics

Abstract

This paper presents a new Smooth Particle Hydrodynamics (SPH) computational framework for explicit fast solid dynamics. The proposed methodology explores the use of the Streamline Upwind Petrov Galerkin (SUPG) stabilisation methodology as an alternative to the Jameson-Schmidt-Turkel (JST) stabilisation recently presented by the authors in [1] in the context of a conservation law formulation of fast solid dynamics. The work introduced in this paper puts forward three advantageous features over the recent JST-SPH framework. First, the variationally consistent nature of the SUPG stabilisation allows for the introduction of a locally preserving angular momentum procedure which can be solved in a monolithic manner in conjunction with the rest of the system equations. This differs from the JST-SPH framework, where an a posteriori projection procedure was required to ensure global angular momentum preservation. Second, evaluation of expensive harmonic and bi-harmonic operators, necessary for the JST stabilisation, is circumvented in the new SUPG-SPH framework. Third, the SUPG-SPH framework is more accurate (for the same number of degrees of freedom) than its JST-SPH counterpart and its accuracy is comparable to that of the robust (but computationally more demanding) Petrov Galerkin Finite Element Method (PG-FEM) technique explored by the authors in [2][3][4][5], as shown in the numerical examples included. A series of numerical examples are analysed in order to benchmark and assess the robustness and effectiveness of the proposed algorithm. The resulting SUPG-SPH framework is therefore accurate, robust and computationally efficient, three key desired features that will allow the authors in forthcoming publications to explore its applicability in large scale simulations.

A Total Lagrangian upwind Smooth Particle Hydrodynamics algorithm for large strain explicit solid dynamics

“…Consider the three dimensional deformation of an isothermal body of material density ρ 0 moving from its initial undeformed configuration occupying a volume V , of boundary ∂V , to Figure 1). The time dependent motion is defined through a deformation mapping x = φ(X, t) which satisfies the following set of Total Lagrangian first order conservation laws [1,2,27,[34][35][36][37][38] Here, p := ρ 0 v is the linear momentum per unit of undeformed volume, v is the velocity field, F is the deformation gradient (or fibre map), H is the co-factor of the deformation (or area map), J is the Jacobian of the deformation (or volume map), P is the first Piola-Kirchhoff stress tensor and f 0 is a body force term per unit of undeformed volume. The symbol represents the tensor cross product between vectors and/or second order tensors in the sense of [25,30,39,40], DIV and CURL represent the material divergence and material curl operators as defined in expressions (5) and (7) of Reference [25], respectively, and ∇ 0 represents the material gradient operator defined as ∇ 0 := ∂ ∂X .…”

“…The classical (displacement-based) Smooth Particle Hydrodynamics (SPH) Lagrangian formalism [4][5][6][7][8][9][10] is well-known to suffer from a number of severe drawbacks, namely: (1) tensile instability, spurious pressure and hour-glassing [11,12]; (2) numerical issues associated with conservation, consistency, long term stability and convergence [13,14] and (3) the reduced order of convergence for derived variables (e.g. stresses and strains) [15][16][17][18].…”

“…In very recent works [1,2], some of the authors of the present manuscript have successfully introduced a mixed-based SPH computational framework for explicit fast solid dynamics, where the conservation of linear momentum p is solved along with conservation equations for the deformation gradient F , its co-factor H and its Jacobian J. Specifically, the SPH discretisation of the new system of conservation laws {p, F , H, J} was introduced through a family of wellestablished stabilisation strategies, namely a Jameson-Schmidt-Turkel (JST) algorithm [25,26] and a variationally consistent Streamline Upwind Petrov Galerkin (SUPG) algorithm [2,27]. Both computational methodologies were capable of eliminating spurious hourglass-like modes and spurious pressure oscillations in nearly incompressible scenarios.…”

Structure‐preserving space‐time discretization of a mixed formulation for quasi‐incompressible large strain elasticity in principal stretches

“…In contrast to (9), the closed-form representation (12) does not need to compute the eigenvectors. For more background on the spectral decomposition, we refer to the works of Morman 39 and Ting.…”

“…For more background on the spectral decomposition, we refer to the works of Morman 39 and Ting. 40 The numerical treatment of the closed-form expression (12) is discussed in the works of Simo and Taylor 19 and Miehe. 36,37 See the work of Miehe 41 for a detailed comparison of alternative approaches based either on (9) or the closed-form expression (12).…”