A stabilised Total Lagrangian Element-Free Galerkin method for transient nonlinear solid dynamics

Abstract

This paper presents a new stabilised Element-Free Galerkin (EFG) method tailored for large strain transient solid dynamics. The method employs a mixed formulation that combines the Total Lagrangian conservation laws for linear momentum with an additional set of geometric strain measures. The main aim of this paper is to adapt the well-established Streamline Upwind Petrov–Galerkin (SUPG) stabilisation methodology to the context of EFG, presenting three key contributions. Firstly, a variational consistent EFG computational framework is introduced, emphasising behaviours associated with nearly incompressible materials. Secondly, the suppression of non-physical numerical artefacts, such as zero-energy modes and locking, through a well-established stabilisation procedure. Thirdly, the stability of the SUPG formulation is demonstrated using the time rate of Hamiltonian of the system, ensuring non-negative entropy production throughout the entire simulation. To assess the stability, robustness and performance of the proposed algorithm, several benchmark examples in the context of isothermal hyperelasticity and large strain plasticity are examined. Results show that the proposed algorithm effectively addresses spurious modes, including hour-glassing and spurious pressure fluctuations commonly observed in classical displacement-based EFG frameworks.

1 Introduction

Computational modelling of practical engineering problems under large deformation and dynamic conditions presents significant challenges. The calculation of material deformation in such scenarios typically relies on the use of Finite Element Method (FEM). The accuracy of FEM simulation is tied to the element used for discretising the computational domain. In cases of very large deformation, potentially accompanied by topological changes, meshfree methods become the preferred choice, especially in scenarios such as fracture and fragmentation simulations with complex crack propagation.

The Smooth Particle Hydrodynamic (SPH) method, introduced by Gingold and Monaghan [1], stands as the earliest meshfree approach. Initially applied for modelling astrophysical problems, Monaghan and co-workers have further refined and enhanced the robustness and applicability of the SPH method [2,3,4,5,6]. The fundamental concept involves approximating a function at a target point (particle) based on the function value at a set of particles surrounding the target. These values at neighbouring particles are weighted by a kernel function, also known as a window function or weight function. Liu and co-workers have contributed to the field by developing the Reproducing Kernel Particle Method (RKPM) with a correction function for the kernel to improve accuracy near domain boundaries [7,8,9]. Similar to SPH, RKPM estimates the function and its derivatives based on the kernel function. Applications of RKPM extend to simulating rubber-like materials under impact loads [7]. Chen et al. [10] have demonstrated that RKPM provides higher solution accuracy under large deformation compared to FEM due to its smoother shape functions. Jun et al. [11] compared RKPM against standard FEM for accuracy and computational efficiency in handling large deformation nonlinear elasticity. Chen et al. [12] have explored the effectiveness of RKPM in modelling problems with localised deformation and its ability to alleviate volumetric locking.

The construction of meshfree formulation takes an alternative approach based on the Moving Least Square (MLS) approximation. Nayroles et al. [13] pioneered the development of the Diffuse Element Method (DEM), employing the MLS approximation in Galerkin formulation. Subsequent refinement by Belytschko et al. [14] led to the introduction of Element-Free Galerkin (EFG) method, finding application in modelling solid mechanics problems under quasi-static condition [15,16,17]. In-depth investigations into the EFG method were carried out by Huerta et al. [18, 19], particularly focusing on the volumetric locking phenomena. Their study revealed that the number of non-physical locking modes remains independent of the domain support size. Whilst using a larger domain support decreases the eigenvalue of the locking mode, it does not entirely eliminate volumetric locking. Interestingly, their results demonstrated the EFG method with quadratic consistency exhibits more locking modes than with standard bi-quadratic finite elements.

In the context of explicit transient solid dynamic analysis, recent advancements have propelled the widespread development of the SPH method [20,21,22,23,24,25,26,27,28,29,30]. Conversely, the application of the EFG method for explicit solid dynamic problems remains limited. Belytschko et al. [31] developed a three-dimensional explicit dynamic framework based on the classical displacement-based formulation. Their study focussed on the well-known Taylor bar benchmark problem, offering valuable insights into the inelastic behaviour of solids. Horton et al. [32] further extended the EFG method’s application by developing a Total Lagrangian explicit dynamic EFG framework for surgical application. Smith et al. [33] proposed a Total Lagrangian explicit EFG formulation featuring a nodal integration method, with a focus on evaluating the accuracy and computational efficiency through modelling metal forming and stretch blow moulding processes. Bourantas et al. [34] proposed an explicit Total Lagrangian EFG formulation, combining it with a regularised weight function that closely approximate the Kronecker delta condition. This allows for the enforcement of essential boundary conditions without additional effort. A similar weight function was applied by Kahwash et al. [35] in the simulation of machining composite using the explicit EFG method.

All the EFG methods previously discussed were established based on the classical displacement-based formulation. Additionally, an ad-hoc strain smoothing procedure, typically employed in the context of Reproducing Kernel Particle Method, is required to avoid locking difficulties and instability issues. An alternative widely employed in the context of solid dynamics is the mixed-based methodology [36,37,38,39,40,41,42,43,44,45,46]. In this methodology, the motion of a deformable body is described using a system of first-order conservation laws. One of the objectives of this work is to introduce a Total Lagrangian EFG framework using the mixed-based methodology. Additionally, a variational consistent Petrov–Galerkin stabilisation procedure is implemented. The proposed framework is advanced in time using a second-order two-step Runge–Kutta explicit time integrator. One critical aspect that requires careful consideration is the overall stability of the algorithm. In this work, we employ a Streamline-Upwind Petrov–Galerkin (SUPG) approach, introducing a residual-based numerical stabilisation [47,48,49] to ensure the production of total numerical entropy throughout the entire simulation. This is demonstrated by monitoring of the time rate of the Hamiltonian [50, 51] of the system via the standard Coleman–Noll procedure.

The paper is organised as follows. In Sect. 2, we provide a summary of the first-order system of Total Lagrangian conservation laws for isothermal hyperelasticity. Section 3 introduces fundamental concepts such as Hamiltonian, conjugate fields and Hessian operators, necessary for the remainder of the paper. Section 4 presents the variational formulation of the problem and a stability estimate expressed in terms of the Hamiltonian free energy. Section 5 details the Element-Free Galerkin spatial discretisation, with a particular focus on the residual-based numerical dissipation. Explicit type of time integrator is discussed. In Sect. 6, we present several challenging numerical examples to assess the stability and robustness of the proposed EFG algorithm, with comparisons against an alternative mixed-based Total Lagrangian SPH implementation [26]. Finally, Sect. 7 presents some concluding remarks. For completeness, Appendix A includes an algorithmic flowchart illustrating the resulting numerical scheme.

2 Total Lagrangian conservation laws description

Consider the three dimensional deformation of an isothermal body with material density $\rho _R$. The body undergoes motion from its initial undeformed configuration occupying a volume $\Omega _R$ with boundary $\partial \Omega _R$ to its current deformed configuration at time t occupying a volume $\Omega$ with boundary $\partial \Omega$ (see Fig. 1). The time dependent motion is defined by the deformation mapping $\varvec{x} = \varvec{\phi } (\varvec{X}, t)$, which adheres to a set of Total Lagrangian first-order conservation laws described by [20,21,22, 26, 36, 37, 39, 42, 43, 52, 53]

(1a)

(1b)

(1c)

(1d)

Fig. 1

Three dimensional body motion

Here, $\varvec{p} =\rho _R \varvec{v}$ is the linear momentum per unit of undeformed volume, $\varvec{v}$ is the velocity field, $\varvec{F}$ is the deformation gradient (or fibre map), $\varvec{H}$ is the co-factor of the deformation (or area map), J is the Jacobian of the deformation (or volume map), $\varvec{P}$ is the first Piola–Kirchhoff stress tensor and $\varvec{f}_R$ 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 [42, 52, 54, 55], DIV and CURL represent the material divergence and material curl operators, respectively.

Expression (1a) represents the standard linear momentum conservation equation, whilst the rest of the Eqs. (1b)–(1d) represent a supplementary set of geometric conservation laws for $\{\varvec{F},\varvec{H},J\}$. Additionally, appropriate involutions [20, 39] must be satisfied by some of the strain variables $\{ \varvec{F}, \varvec{H} \}$ [56] of the system as

$$\begin{aligned} \text {CURL} \varvec{F} = \varvec{0}; \quad \text {DIV} \varvec{H} = \varvec{0}. \end{aligned}$$

(2)

Furthermore, the above system (1a)–(1d) can be expressed in a concise manner as

$$\begin{aligned} \frac{\partial \varvec{\mathcal {U}}}{\partial t} + \sum _{I = 1}^3 \frac{\partial \varvec{\mathcal {F}}_I}{\partial X_I} = \varvec{\mathcal {S}}, \end{aligned}$$

(3)

where $\varvec{\mathcal {U}}$ denotes the set of conservation variables, $\varvec{\mathcal {S}}$ the source term and $\varvec{\mathcal {F}}_I$ the flux vector in the material Cartesian direction I, given as below

(4)

Here, $\varvec{E}_I$ is the I-th unit vector of the Cartesian basis defined as

$$\begin{aligned} \varvec{E}_1 = \left[ \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right] ; \quad \varvec{E}_2 = \left[ \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right] ; \quad \varvec{E}_3 = \left[ \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right] . \end{aligned}$$

(5)

In addition to initial and boundary (essential and natural) conditions required for the complete definition of the initial boundary value problem, closure of system (1a)–(1d) requires the introduction of a suitable constitutive law. This law must fulfill a series of physical requirements, including thermodynamic consistency (via the Coleman–Noll procedure [57]) and the principle of objectivity. In the case of reversible elasticity, this can be achieved via the introduction of a suitable polyconvex strain energy potential W [55] such that stress $\varvec{P}$ and the triplet set of strains $\{\varvec{F}, \varvec{H}, J \}$ are related via

(6)

with the conjugate stresses defined by

$$\begin{aligned} \varvec{\Sigma }_{\varvec{F}}= & {} \frac{\partial W (\varvec{F}, \varvec{H}, J)}{\partial \varvec{F}}; \quad \varvec{\Sigma }_{\varvec{H}} = \frac{\partial W (\varvec{F}, \varvec{H}, J)}{\partial \varvec{H}};\nonumber \\ \Sigma _J= & {} \frac{\partial W (\varvec{F}, \varvec{H}, J)}{\partial J}. \end{aligned}$$

(7)

The inherent characteristic of the polyconvex model [58, 59] facilitates the transformation of the system of conservation laws into its dual set of hyperbolic equations, expressed in terms of the entropy conjugates of the conservation variables [42, 52]. This transformation will be summarised in the following section. Crucially, the polyconvex model also establishes a one-to-one mapping between stresses and strains, and vice versa.

3 Entropy-based conservation laws for solid dynamics

To establish an appropriate entropy-based system, we explore the convex Hamiltonian free energy $\mathcal {H}$ expressed in relation to both the linear momentum $\varvec{p}$ and a triplet set of deformation measures, namely $\{ \varvec{F}, \varvec{H}, J \}$. In the context of an isothermal process, the Hamiltonian represents the summation of kinetic and elastic energy contributions per unit of undeformed volume. Mathematically, this is described by

$$\begin{aligned} \mathcal {H} (\varvec{X},t) = \hat{\mathcal {H}} (\varvec{\mathcal {U}}) = \underbrace{\frac{1}{2 \rho _R} \varvec{p} \cdot \varvec{p}}_{\text {kinetic energy}} + \underbrace{W (\varvec{F}, \varvec{H}, J)}_{\text {convex strain energy}}, \end{aligned}$$

where $\mathcal {H}(\varvec{X},t)$ and $\hat{\mathcal {H}} (\varvec{\mathcal {U}})$ represent alternative functional representations of the same magnitude. With this in mind, a set of conjugate entropy variables $\varvec{\mathcal {V}}$ can be subsequently evaluated by taking derivatives of $\hat{\mathcal {H}} (\varvec{\mathcal {U}})$ (with respect to $\varvec{\mathcal {U}}$) to give

$$\begin{aligned} \varvec{\mathcal {V}}= \frac{\partial \hat{\mathcal {H}} (\varvec{\mathcal {U}})}{\partial \varvec{\mathcal {U}}}= \left[ \begin{array}{c} \varvec{v} \\ \frac{\partial W}{\partial \varvec{F}} \\ \frac{\partial W}{\partial \varvec{H}} \\ \frac{\partial W}{\partial J}\\ \end{array} \right] = \left[ \begin{array}{c} \varvec{v} \\ \varvec{\Sigma _F} \\ \varvec{\Sigma _H} \\ {\Sigma }_J \\ \end{array} \right] . \end{aligned}$$

(8)

Note here that $\varvec{v}$ is the velocity field and $\{\varvec{\Sigma _F}, \varvec{\Sigma _H}, \Sigma _J \}$ are appropriate conjugate stresses to $\{ \varvec{F}, \varvec{H}, J\}$ respectively (as already defined in (7)).

Taking further derivatives of expression (8) enable the Hessian operator of the Hamiltonian $\left[ \mathbb {H}_{\mathcal {H}} \right]$ to be

$$\begin{aligned} \left[ \mathbb {H}_{\mathcal {H}} \right] = \frac{\partial \varvec{\mathcal {V}}}{\partial \varvec{\mathcal {U}}} = \frac{\partial ^2 \hat{\mathcal {H}} (\varvec{\mathcal {U}})}{\partial \varvec{\mathcal {U}} \partial \varvec{\mathcal {U}}} = \left[ \begin{array}{cc} \frac{1}{\rho _R} \varvec{I} &{} \varvec{0} \\ \varvec{0} &{} \left[ \mathbb {H}_{W} \right] \end{array} \right] . \end{aligned}$$

(9)

With $\varvec{I}$ representing the second order identity tensor, it is interesting to note that, with the use of a polyconvex strain energy functional $W (\varvec{F}, \varvec{H}, J)$, $\left[ \mathbb {H}_W \right]$ is inherently a positive definite matrix defined as

$$\begin{aligned} \left[ \mathbb {H}_W \right]= & {} \left[ \begin{array}{ccc} \frac{\partial ^2 W}{\partial \varvec{F} \partial \varvec{F}} &{} \frac{\partial ^2 W}{\partial \varvec{F} \partial \varvec{H}} &{} \frac{\partial ^2 W}{\partial \varvec{F} \partial J} \\ \frac{\partial ^2 W}{\partial \varvec{H} \partial \varvec{F}} &{} \frac{\partial ^2 W}{\partial \varvec{H} \partial \varvec{H}} &{} \frac{\partial ^2 W}{\partial \varvec{H} \partial J} \\ \frac{\partial ^2 W}{\partial J \partial \varvec{F}} &{} \frac{\partial ^2 W}{\partial J \partial \varvec{H}} &{} \frac{\partial ^2 W}{\partial J \partial J} \end{array} \right] \nonumber \\= & {} \left[ \begin{array}{ccc} \varvec{W_{FF}} &{} \varvec{W}_{\varvec{FH}} &{} \varvec{W}_{\varvec{F}J} \\ \varvec{W_{HF}} &{} \varvec{W}_{\varvec{HH}} &{} \varvec{W}_{\varvec{H}J} \\ \varvec{W}_{J\varvec{F}} &{} \varvec{W}_{J\varvec{H}} &{} W_{JJ} \end{array} \right] . \end{aligned}$$

(10)

Pre-multiplying system (3) with the Hessian operator (9) results in an alternative set of conjugate entropy-based conservation laws as

$$\begin{aligned} \frac{\partial \varvec{\mathcal {V}}}{\partial t} = - \sum _{I=1}^3 \left[ \mathbb {H}_{\mathcal {H}} \right] \frac{\partial \varvec{\mathcal {F}}_I}{\partial X_I} + \left[ \mathbb {H}_{\mathcal {H}} \right] \varvec{\mathcal {S}}. \end{aligned}$$

(11)

The above expression can then be particularised to the specific set of conjugate variables considered in this paper, namely $\varvec{\mathcal {V}} = \{ \varvec{v}, \varvec{\Sigma _F}, \varvec{\Sigma _H}, \Sigma _J \}$. For instance, the evolution equation for the conjugate entropy velocity field $\varvec{v}$ reads

$$\begin{aligned} \frac{\partial \varvec{v}}{\partial t} = \frac{1}{\rho _R} \text {DIV} \varvec{P} + \frac{1}{\rho _R} \varvec{f}_R. \end{aligned}$$

(12)

Furthermore, the evolution equations associated with the conjugate stresses $\{ \varvec{\Sigma _F}, \varvec{\Sigma _H}, \Sigma _J \}$ can also be formulated in a similar manner as

(13)

Since the stress–strain relationship is a one-to-one mapping (due to the use of a polyconvex model), the stress rate equations presented in (13) align consistently with the strain-based Eqs. (1b)–(1d), ensuring objectivity. Indeed, expressions (12) and (13) will be used in the next section when defining the stabilised conjugate fields in the Petrov–Galerkin formulation.

4 Petrov–Galerkin variational formulation

This section introduces the stabilised variational statement for the set of conservation laws (1) via the use of suitable work conjugates [60]. To achieve this, and following References [39, 42, 43], we employ appropriate stabilised conjugate variables $\delta \varvec{\mathcal {V}}^{st}$ = $\left[ \delta \varvec{v}^{st}, \delta \varvec{\Sigma }_{\varvec{F}}^{st}, \delta \varvec{\Sigma }_{\varvec{H}}^{st}, \delta \Sigma _J^{st} \right] ^T$ by utilising the entropy-based system ((12) and (13)) alongside the involution Eq. (2). These considerations lead to the expressions for the stabilised conjugate velocity and stresses as described by

$$\begin{aligned} \delta \varvec{v}^{st}&= \delta \varvec{v} - \frac{\tau _{\varvec{p}}}{\rho _R} ( \text {DIV} \delta \varvec{\Sigma _F} - \varvec{F} \times \text {CURL} \delta \varvec{\Sigma _H}\nonumber \\&\quad + \varvec{H} \varvec{\nabla }_0 \delta \Sigma _J ) \end{aligned}$$

(14a)

and

(15)

respectively. The units of the above four $\tau$ stabilising parameters (i.e. $\tau _{\varvec{p}}, \tau _{\varvec{F}}, \tau _{\varvec{H}}, \tau _J$) are those of time and are chosen as a fraction of the time step for explicit integration schemes [61,62,63]. The residuals $\varvec{\mathcal {R}}$ = $\left[ \varvec{\mathcal {R}}_{\varvec{p}}, \varvec{\mathcal {R}}_{\varvec{F}}, \varvec{\mathcal {R}}_{\varvec{H}}, \mathcal {R}_J \right] ^T$ of the conservation laws (1) are defined by

(16)

where the dot over a variable is used to denote differentiation in time.

The Petrov–Galerkin type of weak form for the system under consideration is established by multiplying the above residuals $\varvec{\mathcal {R}}$ with their appropriate stabilised work conjugate virtual fields $\delta \varvec{\mathcal {V}}^{st}$, and integrating over the material domain $\Omega _R$ of the body, to give

$$\begin{aligned} 0= & {} \int _{\Omega _R} \left( \delta \varvec{v}^{st} \cdot \varvec{\mathcal {R}_p} + \delta \varvec{\Sigma }_{\varvec{F}}^{st}: \varvec{\mathcal {R}}_{\varvec{F}} + \delta \varvec{\Sigma }_{\varvec{H}}^{st}: \varvec{\mathcal {R}}_{\varvec{H}}\right. \nonumber \\{} & {} \left. + \delta \Sigma _J^{st} \mathcal {R}_J\right) \, d{\Omega _R}. \end{aligned}$$

(17)

By re-grouping expression (17) according to each virtual conjugate variable, it is possible to extract first the terms containing the virtual velocity $\delta \varvec{v}$ as

$$\begin{aligned} 0 = \int _{\Omega _R} \left( \delta \varvec{v} \cdot \varvec{\mathcal {R}}_{\varvec{p}} + \mathcal {D}^{\text {SUPG}}_{\varvec{p}} \right) \, d \Omega _R. \end{aligned}$$

(18)

The dissipation term of the linear momentum equation, namely $\mathcal {D}_{\varvec{p}}^{\text {SUPG}}$, is described by

(19)

Integrating by parts the first term on the right-hand side of (18), and expanding the resulting equation, yields

$$\begin{aligned} \int _{\Omega _R} \delta \varvec{v} \cdot \frac{\partial \varvec{p}}{\partial t} \, d \Omega _R= & {} \int _{\Omega _R} \delta \varvec{v} \cdot \varvec{f}_R \, d \Omega _R + \int _{\partial \Omega _R} \delta \varvec{v} \cdot \varvec{t}_B \, dA \nonumber \\{} & {} - \int _{\Omega _R} \varvec{P}^{st}: \varvec{\nabla }_0 \delta \varvec{v} \, d \Omega _R. \end{aligned}$$

(20)

The stabilised first Piola–Kirchhoff stress tensor is expressed as

(21)

where the stabilised conjugate stresses are given by

$$\begin{aligned} \begin{aligned} \varvec{\Sigma }_{\varvec{F}}^{st}&= \varvec{\Sigma }_{\varvec{F}} + \varvec{W}_{\varvec{FF}}: \left( - \tau _{\varvec{F}} \varvec{\mathcal {R}}_{\varvec{F}} \right) \\&\quad + \varvec{W}_{\varvec{HF}}: \left( - \tau _{\varvec{H}} \varvec{\mathcal {R}}_{\varvec{H}} \right) + \varvec{W}_{J\varvec{F}} \left( - \tau _J \mathcal {R}_J \right) ; \\ \varvec{\Sigma }_{\varvec{H}}^{st}&= \varvec{\Sigma }_{\varvec{H}} + \varvec{W}_{\varvec{FH}}: \left( - \tau _{\varvec{F}} \varvec{\mathcal {R}}_{\varvec{F}} \right) \\&\quad + \varvec{W}_{\varvec{HH}}: \left( - \tau _{\varvec{H}} \varvec{\mathcal {R}}_{\varvec{H}} \right) + \varvec{W}_{J\varvec{H}} \left( - \tau _J \mathcal {R}_J \right) ; \\ \Sigma _J^{st}&= \Sigma _J + \varvec{W}_{\varvec{F}J}: \left( - \tau _{\varvec{F}} \varvec{\mathcal {R}}_{\varvec{F}} \right) \\&\quad + \varvec{W}_{\varvec{H}J}: \left( - \tau _{\varvec{H}} \varvec{\mathcal {R}}_{\varvec{H}} \right) + W_{JJ} \left( - \tau _J \mathcal {R}_J \right) . \end{aligned} \end{aligned}$$

(22)

Remark 1

Following a Variational Multi-Scale (VMS) stabilisation procedure [49, 64,65,66,67,68,69,70], these stresses (22) can be alternatively written in terms of the stabilised strain measures $\{\varvec{F}^{st}, \varvec{H}^{st}, J^{st} \}$ as

$$\begin{aligned}{} & {} \varvec{\Sigma }_{\varvec{F}}^{st} \approx \varvec{\Sigma }_{\varvec{F}} (\varvec{F}^{st}, \varvec{H}^{st}, J^{st}); \quad \varvec{\Sigma }_{\varvec{H}}^{st} \approx \varvec{\Sigma }_{\varvec{H}} (\varvec{F}^{st}, \varvec{H}^{st}, J^{st}); \quad \nonumber \\{} & {} \Sigma _{J}^{st} \approx \Sigma _J (\varvec{F}^{st}, \varvec{H}^{st}, J^{st}), \end{aligned}$$

(23)

where

$$\begin{aligned} \varvec{F}^{st}= & {} \varvec{F} - \tau _{\varvec{F}} \varvec{\mathcal {R}}_{\varvec{F}}; \quad \varvec{H}^{st} = \varvec{H} - \tau _{\varvec{H}} \varvec{\mathcal {R}}_{\varvec{H}}; \nonumber \\ J^{st}= & {} J-\tau _{J} \varvec{\mathcal {R}}_{J}. \end{aligned}$$

(24)

In the above expressions, the residual terms $\{ \varvec{\mathcal {R}}_{\varvec{F}}, \varvec{\mathcal {R}}_{\varvec{H}}, \mathcal {R}_J \}$ represent the difference between the time rate of the corresponding strain variable and its evaluation in terms of the material gradient of the velocities. To reduce the implicitness of the formulation, a simple procedure was proposed [37] whereby the above time residuals are replaced by their time integrated geometric equivalents $\{ \varvec{\mathcal {R}}_{\varvec{F}}^{\varvec{x}}, \varvec{\mathcal {R}}_{\varvec{H}}^{\varvec{x}}, \mathcal {R}_{J}^{\varvec{x}} \}$ to give

$$\begin{aligned} \varvec{F}^{st} = \varvec{F} + \varvec{F}^{\prime }; \quad \varvec{H}^{st} = \varvec{H} + \varvec{H}^{\prime }; \quad J^{st} = J + J^{\prime }, \end{aligned}$$

(25)

with

(26a)

(26b)

(26c)

Here, $\{ \xi _{\varvec{F}}, \xi _{\varvec{H}}, \xi _J \}$ are dimensionless stabilisation parameters. Given that the subgrid-scale terms $\{ \varvec{F}^{\prime }, \varvec{H}^{\prime }, J^{\prime } \}$ are residual-based, their contributions are considered small with respect to coarse-scale terms $\{ \varvec{F}, \varvec{H}, J \}$.

Finally, the stabilised conjugate stress measures defined in (15) can be introduced into the weighted residual Eq. (17) resulting in a stabilised set of strain update equations as

(27a)

(27b)

(27c)

with the SUPG dissipation term given by

$$\begin{aligned} \mathcal {D}_J^{\text {SUPG}} = \frac{1}{\rho _R} \left( - \tau _{\varvec{p}} \varvec{\mathcal {R}}_{\varvec{p}} \right) \cdot \left( \varvec{H} \varvec{\nabla }_0 \delta \Sigma _J \right) . \end{aligned}$$

(28)

To ensure the discrete satisfaction of the involutions (2), we introduce only the residual-based SUPG dissipation terms into the linear momentum conservation Eq. (20) and the volume update (27c). Further explanations can be found in References [37, 42]. By setting $\tau _J = 0$, and assigning $\tau _{\varvec{F}} = \tau _{\varvec{H}} = \tau$ and $\xi _{\varvec{F}} = \xi _{\varvec{H}} = \xi$, expressions (20) and (27) are fully decoupled and can be solved sequentially. Equations (27a) and (27b) are first solved simultaneously to obtain $\dot{\varvec{F}}$ and $\dot{\varvec{H}}$, which can then be substituted into (20) to deduce $\dot{\varvec{p}}$. Once $\dot{\varvec{p}}$ is determined, ${\dot{J}}$ can finally be obtained from (27c). Thus, the time discrete version of the underlying conservation formulation is reduced to considering four stabilising parameters $\{ \tau _{\varvec{p}}, \xi _J, \tau , \xi \}$.

4.1 Stability of the SUPG formulation

Before proceeding to the spatial discretisation of the weak form statements, it is crucial to demonstrate the stability of the SUPG formulation. This is achieved by examining the time variation of the Hamiltonian within the system and adopting specific choice of virtual fields, such as $\delta \varvec{\mathcal {V}} = \varvec{\mathcal {V}}$, to give

$$\begin{aligned}{} & {} \frac{d}{dt}\int _{\Omega _R} \hat{\mathcal {H}} (\varvec{\mathcal {U}}) \, d\Omega _R = \int _{\Omega _R} \varvec{{\mathcal {V}}}^T \frac{\partial \varvec{{\mathcal {U}}}}{\partial t}\, d{\Omega _R}\nonumber \\{} & {} \quad =\int _{\Omega _R} \left( \varvec{v} \cdot \frac{\partial \varvec{p}}{\partial t} \!+\! \varvec{\Sigma _F}: \frac{\partial \varvec{F}}{\partial t} \!+\! \varvec{\Sigma _H}: \frac{\partial \varvec{H}}{\partial t} \!+\! \Sigma _J \frac{\partial J}{\partial t} \right) \, d{\Omega _R}\nonumber \\{} & {} \quad = \int _{\Omega _R} \left( \varvec{v} \cdot \frac{\partial \varvec{p}}{\partial t} + \varvec{P}: \varvec{\nabla }_0 \varvec{v}\right) \, d{\Omega _R} \nonumber \\{} & {} \qquad - \int _{\Omega _R} \mathcal {D}_J^{\text {SUPG}} \, d \Omega _R. \end{aligned}$$

(29)

In the first line of (29), we use the conjugacy of the fields $\varvec{\mathcal {U}}$ and $\varvec{\mathcal {V}}$ as shown in expression (8). The three geometric Eqs. (27a)–(27c) have been substituted in the second line of (29). Subsequently, we substitute the linear momentum conservation Eq. (20) into the third line of (29) to give

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\int _{\Omega _R} \hat{\mathcal {H}} \, d\Omega _R - {\dot{\Pi }}_{\text {ext}} = - \mathcal {D}_{\text {total}}; \\&\mathcal {D}_{\text {total}} = \int _{\Omega _R} \mathcal {D}^{\text {SUPG}} \, d \Omega _R, \end{aligned} \end{aligned}$$

(30)

where the rate of external work being defined as

$$\begin{aligned}{} & {} {\dot{\Pi }}_{\text {ext}}= \int _{\Omega _R} \varvec{v} \cdot \varvec{f}_R \, d \Omega _R + \int _{\partial \Omega _R} \varvec{v} \cdot \varvec{t}_B \,dA; \nonumber \\{} & {} \mathcal {D}^{\text {SUPG}}=\mathcal {D}_{\varvec{p}}^{\text {SUPG}} + \mathcal {D}_J^{\text {SUPG}}. \end{aligned}$$

(31)

In the context of reversible elasticity and under time-independent external forces, the left-hand side of (30), representing the rate of total energy in the system, vanishes. This indicates that the term on the right-hand side denotes a potential dissipation introduced into the system through the addition of the SUPG stabilisation terms. It is important to demonstrate that this right-hand side is indeed dissipative (negative), leading to an overall decrease in the total energy of the system. To illustrate this, we adopt $\delta \varvec{v} = \frac{\tau _{\varvec{p}}}{\rho _R} \left( \varvec{\mathcal {R}}_{\varvec{p}} + 2 \varvec{H} \varvec{\nabla }_0 \Sigma _J \right)$ in expression (18) to render

$$\begin{aligned} 0= & {} \int _{\Omega _R} \frac{\tau _{\varvec{p}}}{\rho _R} \left( \varvec{\mathcal {R}}_{\varvec{p}} + 2 \varvec{H} \varvec{\nabla }_0 \Sigma _J \right) \cdot \varvec{\mathcal {R}}_{\varvec{p}} \, d \Omega _R \nonumber \\{} & {} +\int _{\Omega _R} \mathcal {D}_{\varvec{p}}^{\text {SUPG}} \, d \Omega _R. \end{aligned}$$

(32)

Similarly, adopting the expression

(33)

Equation (27) is transformed into

(34)

Combining Eqs. (32) and (34) with $\mathcal {D}^{\text {SUPG}}$ in (30), we obtain the following expression

$$\begin{aligned} \mathcal {D}^{\text {SUPG}}= & {} \left[ \tau _{\varvec{F}} \varvec{\mathcal {R}}_{\varvec{F}}: \quad \tau _{\varvec{H}} \varvec{\mathcal {R}}_{\varvec{H}}: \quad \tau _J \mathcal {R}_J \right] \left[ \mathbb {H}_W \right] \nonumber \\{} & {} \left[ \begin{array}{c} :\varvec{\mathcal {R}}_{\varvec{F}} \\ :\varvec{\mathcal {R}}_{\varvec{H}} \\ \mathcal {R}_J \end{array}\right] + \frac{\tau _{\varvec{p}}}{\rho _R} \varvec{\mathcal {R}}_{\varvec{p}} \cdot \varvec{\mathcal {R}}_{\varvec{p}} \ge 0, \end{aligned}$$

(35)

which, given that both $\rho _R$ and $\left[ \mathbb {H}_W \right]$ are positive, demonstrates the dissipative nature of the formulation.

5 Numerical scheme

5.1 Consistent Moving Least Square shape functions and their derivatives

This section is not dedicated to an exhaustive discussion of Moving Least Squares (MLS) interpolants. Excellent presentations on MLS approximation can be found in well-known references such as [71,72,73,74,75,76]. Here, we briefly revisit fundamental concepts to introduce the notation and the approach used in the subsequent sections.

Meshfree methods, also known as particle methods, are based on functional interpolation given by

$$\begin{aligned} \mathcal {U} (\varvec{X}, t) = \sum _{b=1}^N N_{b} (\varvec{X})\mathcal {U}_b (t). \end{aligned}$$

(36)

To ensure consistency, the shape functions, $N_b (\varvec{X})$, must be determined appropriately. In this work, we employ linear shape functions defined as

$$\begin{aligned} N_b (\varvec{X}) = w_b (\varvec{X}) \left( \varvec{\mathbb {P}} (\varvec{X}_b) \cdot \varvec{\alpha } (\varvec{X}) \right) , \end{aligned}$$

(37)

where the vector $\varvec{\alpha }$ is unknown and $\varvec{\mathbb {P}} (\varvec{X}) = \left[ 1, X_1, X_2, X_3 \right] ^T$ forms a basis of a linear polynomial space. The function $w (\varvec{X})$ is a weighting function with the properties of being positive, even and having compact support (see Remark 2). The unknown parameters $\varvec{\alpha } (\varvec{X})$ are determined by enforcing the so-called reproducibility or consistency condition. Specifically, in this case, the parameter $\varvec{\alpha }$ is computed by enforcing that any linear distribution is exactly interpolated, that is

$$\begin{aligned} \sum _{b = 1}^N N_b (\varvec{X}) \varvec{\mathbb {P}} (\varvec{X}_b) = \varvec{\mathbb {P}} (\varvec{X}). \end{aligned}$$

(38)

Substitution of expression (37) into (38) leads to a system of equations to be solved for $\varvec{\alpha }$

$$\begin{aligned} \varvec{A}_b (\varvec{X}) \varvec{\alpha }= & {} \varvec{\mathbb {P}} (\varvec{X}); \quad \varvec{A}_b (\varvec{X}) = \sum _{b=1}^N w_b (\varvec{X}) \varvec{\mathbb {P}} (\varvec{X}_b) \otimes \varvec{\mathbb {P}} (\varvec{X}_b).\nonumber \\ \end{aligned}$$

(39)

It is important to note that the matrix $\varvec{A}$ is positive definite and, therefore invertible, assuming a well-defined weighting function. The computational cost involved is primarily determined by the inversion of the matrix $\varvec{A}$, but explicit inversion is not necessary. Lower-Upper (LU) decomposition can be used instead.

To compute the material gradient of an arbitrary scalar function $\mathcal {U}$, it is necessary to evaluate the material gradient of the expression provided in Eq. (36), resulting in

$$\begin{aligned} \varvec{\nabla }_0 \mathcal {U} (\varvec{X}, t) = \sum _{b = 1}^N \varvec{\nabla }_0 N_b (\varvec{X}) \mathcal {U}_b (t). \end{aligned}$$

(40)

By employing the shape function described in (37), its derivatives can then be evaluated as

$$\begin{aligned}{} & {} \varvec{\nabla }_0 N_b (\varvec{X})\nonumber \\{} & {} \quad = \left[ \varvec{\nabla }_0 w_b (\varvec{X}) \otimes \varvec{\alpha } (\varvec{X}) + w_b (\varvec{X}) \left( \varvec{\nabla }_0 \varvec{\alpha } \right) ^T \right] \varvec{\mathbb {P}} (\varvec{X}_b), \end{aligned}$$

(41)

for which $\varvec{\nabla }_0 \varvec{\alpha }$ can be solved via the following system of equations

$$\begin{aligned} \varvec{A}_b \left( \varvec{\nabla }_0 \varvec{\alpha } \right) = \varvec{\nabla }_0 \varvec{\mathbb {P}} - \left( \varvec{\nabla }_0 \varvec{A}_b \right) \varvec{\alpha }. \end{aligned}$$

(42)

Exploiting LU factorisation of matrix $\varvec{A}$ facilitates the computation of shape function derivatives with minimal additional computational cost. Upon solving Eq. (39) for $\varvec{\alpha }$, the resulting values can be substituted into the right-hand side of expression (42). This substitution allows us to deduce $\varvec{\nabla }_0 \varvec{\alpha }$. Once $\varvec{\nabla }_0 \varvec{\alpha }$ is determined, the shape function derivatives can finally be obtained from Eq. (41). Note that the method for computing derivatives described in (41), which is based on the complete linear polynomial basis, differs from the approaches typically used in the corrected SPH method [20, 77, 78]. In the corrected SPH method, the gradient computation is explicitly modified to ensure first-order completeness.

Remark 2

In this study, we adopt the weighting function proposed by Most and Bucher [79], chosen for its advantageous property of satisfying the Kronecker delta. The weighting function is defined as

$$\begin{aligned} w_b (\varvec{X})= & {} w(\varvec{X}-\varvec{X}_b)\nonumber \\= & {} \frac{\left[ \left( \frac{\Vert \varvec{X}-\varvec{X}_b\Vert }{D_{\text {max}}}\right) ^2+\varepsilon \right] ^{-2} - \left( 1+\varepsilon \right) ^{-2}}{\varepsilon ^{-2}-\left( 1+ \varepsilon \right) ^{-2}}. \end{aligned}$$

(43)

Here, $D_{\text {max}}$ represents the radius of the support domain and $\varepsilon \ll 1$ is a small constant value. As recommended by [79], we set $\varepsilon = 10^{-5}$ in this work.

5.2 Element-Free Galerkin spatial discretisation

We employ Moving Least Squares (MLS) shape functions $N_b (\varvec{X})$ to interpolate all conservation variables and their time rates, yielding expressions such as

$$\begin{aligned} \varvec{p}_a= & {} \sum _{b \in \Lambda ^b_a} \varvec{p}_b (t) N_b (\varvec{X}_a); \quad \dot{\varvec{p}}_a = \sum _{b \in \Lambda ^b_a} \dot{\varvec{p}}_b (t) N_b (\varvec{X}); \quad \nonumber \\ \varvec{F}_a= & {} \sum _{b \in \Lambda ^b_a} \varvec{F}_b (t) N_b (\varvec{X}_a); \quad \ldots \end{aligned}$$

(44)

Similarly, the virtual work conjugates are expanded as

$$\begin{aligned} \delta \varvec{v}_a= & {} \sum _{b \in \Lambda ^b_a} \delta \varvec{v}_b N_b (\varvec{X}_a); \quad \delta \varvec{\Sigma }_{\varvec{F}_a} = \sum _{b \in \Lambda ^b_a} \delta \varvec{\Sigma }_{\varvec{F}_b} N_b (\varvec{X}_a); \quad \nonumber \\ \delta \varvec{\Sigma }_{\varvec{H}_a}= & {} \sum _{b \in \Lambda ^b_a} \delta \varvec{\Sigma }_{\varvec{H}_b} N_b (\varvec{X}_a); \quad \ldots \end{aligned}$$

(45)

Here, $\Lambda ^b_a$ represents the set of neighbouring particles b belonging to particle a.

Let us consider first the discretised weak statement for the linear momentum conservation equation. With the purpose of obtaining the equation that corresponds to particle a, we restrict $\delta \varvec{v}$ in Eq. (20) to a single particle $N_a \delta \varvec{v}_a$ before substituting into Eq. (20) to give, for each particle $a = 1, \ldots , N$,

$$\begin{aligned} \int _{\Omega _R} N_a \frac{\partial \varvec{p}}{\partial t} \, d \Omega _R= & {} \int _{\partial \Omega _R} N_a \varvec{t}_B \, dA + \int _{\Omega _R} N_a \varvec{f}_R \, d \Omega _R \nonumber \\{} & {} - \int _{\Omega _R} \varvec{P} (\varvec{F}^{st}) \varvec{\nabla }_0 N_a \, d \Omega _R. \end{aligned}$$

(46)

Once again, introducing the MLS interpolations for the time rate of linear momentum results in

$$\begin{aligned} \sum _{b \in \Lambda ^b_a} M_{ab} \dot{\varvec{p}}_b= & {} \int _{\partial \Omega _R} N_a \varvec{t}_B \, dA + \int _{\Omega _R} N_a \varvec{f}_R \, d \Omega _R \nonumber \\{} & {} - \int _{\Omega _R} \varvec{P} (\varvec{F}^{st}) \varvec{\nabla }_0 N_a \, d \Omega _R, \end{aligned}$$

(47)

where the consistent mass contribution is defined as $M_{ab} = \int _{\Omega _R} N_a N_b \, d \Omega _R$. Notice that $M_{ab}$ is called consistent mass-like component with units of volume. In expression (47), a VMS procedure employing a stabilised deformation gradient tensor is utilised. A simpler stabilisation term can be obtained using a specific hyperelastic model that considers only the diagonal components of the Hessian. This approach has been explored in our previous work [21] within the context of the SPH method.

Attention is now focussed on the discretised weak form for the triplet set of geometric conservation laws. In order to obtain the equations corresponding to particle a, we restrict $\{ \delta \varvec{\Sigma _F}, \delta \varvec{\Sigma _H}, \delta \Sigma _J \}$ to a single particle $\{ N_a \delta \varvec{\Sigma }_{\varvec{F}_a}, N_a \varvec{\Sigma }_{\varvec{H}_a}, N_a \Sigma _{J_a} \}$, leading to the following expressions

(48a)

(48b)

(48c)

5.3 Time integration

When considering our mixed-based system $\{\varvec{p}, \varvec{F}, \varvec{H}, J\}$ (1), an explicit time integrator becomes necessary due to the size of the system. For simplicity, we employ an explicit one-step two-stage Total Variation Diminishing Runge–Kutta (TVD-RK) scheme [20, 21, 36, 37, 39,40,41,42, 53]. This scheme is described by the following time update equations from time step $t^n$ to $t^{n+1}$

$$\begin{aligned} \begin{aligned} \varvec{\mathcal {U}}_a^{\star }&= \varvec{\mathcal {U}}_a^n + \Delta t\ \dot{\varvec{\mathcal {U}}}_a^n (\varvec{\mathcal {U}}_a^n, t^n); \\ \varvec{\mathcal {U}}_a^{\star \star }&= \varvec{\mathcal {U}}_a^{\star } + \Delta t\ \dot{\varvec{\mathcal {U}}}_a^{\star } (\varvec{\mathcal {U}}_a^{\star }, t^{n+1}); \\ \varvec{\mathcal {U}}_a^{n+1}&= \frac{1}{2}(\varvec{\mathcal {U}}_a^n + \varvec{\mathcal {U}}_a^{\star \star }). \end{aligned} \end{aligned}$$

(49)

In the current work, the same TVD-RK time integrator is also employed for updating the geometry [20, 21, 53]. This results in a monolithic time integration procedure where the unknowns $\varvec{\mathcal {U}} = \left[ \varvec{p}, \varvec{F}, \varvec{H}, J \right] ^T$ along with the geometry $\varvec{x}$ are all updated via (49). The maximum time step $\Delta t = t^{n+1}-t^n$ is determined by a standard Courant–Friedrichs–Lewy (CFL) condition [80] given as

$$\begin{aligned} \Delta t = \alpha _{CFL}\frac{h_{min}}{c_{p,max}}, \end{aligned}$$

(50)

where $c_{p,max}$ is the maximum p-wave speed, $h_{min}$ is the characteristic particle spacing within the computational domain and $\alpha _{CFL}$ is the CFL stability number. For the numerical computations presented in this paper, a value of $\alpha _{CFL} = 0.3$, unless otherwise stated, has been chosen to ensure both accuracy and stability [40] of the algorithm.

Finally, the solution procedure for the proposed Element-Free Galerkin method is detailed in Appendix A.

6 Numerical examples

The purpose of this section is to assess the accuracy and the robustness of the proposed numerical framework in modelling the dynamic behaviour of deformable bodies undergoing large deformation. To this end, a series of well-known benchmark examples is investigated. For all of the following examples, a CFL number $\alpha _{CFL} = 0.3$ is chosen and the SUPG stabilisation parameters used, unless otherwise stated, are set to be $\tau = \Delta t$, $\tau _{\varvec{p}} = 0.1 \Delta t$ and $\xi = \xi _J = 0.1$. In addition, the radius of the support domain $D_{max} = 2.1h$ is used in this study, where h is the characteristic particle spacing size. Eight-node hexahedral elements with eight Gauss quadrature points were used to discretise the problem domain.

To account for large and reversible deformations, a neo-Hookean hyperelastic model is used. The associated strain energy functional $\Psi (\varvec{F})$ is decomposed into the summation of a deviatoric component ${\hat{\Psi }} (J^{-1/3} \varvec{F})$ and a volumetric component U(J) given by

$$\begin{aligned} \Psi (\varvec{F}) = {\hat{\Psi }} (J^{-1/3} \varvec{F}) + U(J), \end{aligned}$$

(51)

with

$$\begin{aligned}{} & {} {\hat{\Psi }} (J^{-1/3} \varvec{F}) = \frac{1}{2} \mu \left[ J^{-2/3} \left( \varvec{F}: \varvec{F} \right) - 3 \right] ;\nonumber \\{} & {} U (J) = \frac{1}{2} \kappa (J - 1)^2, \end{aligned}$$

(52)

where $\mu$ and $\kappa$ are the shear modulus and bulk modulus, respectively. In previous publications [55], some of the authors of this paper have shown that this model is polyconvex and, thus, rank-one convex.

Remark 3

In the present manuscript, consideration of irreversible processes is restricted to the case of an isothermal elasto-plastic model typically used in metal forming applications [81]. Particularly, thermal effects will be neglected. In order to model irrecoverable plastic behaviour, the standard rate-independent von Mises plasticity model with isotropic hardening is used and the basic structure is summarised for completeness in Appendix B.

Fig. 2

Bending column. Geometry, boundary and initial conditions

Fig. 3

Bending column. Comparison of deformed shapes at time $t=0.5~\text {s}$. The first three columns (left to right) show the particle refinement of a structure simulated using the proposed stabilised EFG algorithm, whereas the last column (on the right) shows a deformed structure via alternative in-house Total Lagrangian Godunov-type SPH algorithm [22]. Colour indicates pressure distribution

Fig. 4

Bending column. Time evolution of a the horizontal velocity and b the horizontal displacement at the tip end $\varvec{X} = [0.5, 6, 0.5]^T$ m

Fig. 5

Bending column. Time evolution of a different energy measures, such as kinetic energy, internal energy and total energy, and b global numerical dissipation

6.1 Applicability in bending-dominated scenario

This three-dimensional example [82, 83] involves a large strain vibration of a thick column, as illustrated in Fig. 2. The column has a fixed bottom, and its initial velocity varies linearly along its length L of 6 m and is given by

$$\begin{aligned} \varvec{v}^0 = \varvec{v} (\varvec{X}, t = 0) = V_0 \left[ \begin{array}{c} Y/L\\ 0 \\ 0 \end{array} \right] \, \, [\text {m}/\text {s}], \end{aligned}$$

(53)

with the value of $V_0 = 10$ m/s. The main objective of this problem is to assess the performance of the proposed algorithm in a scenario dominated by bending. A nearly incompressible neo-Hookean material model is employed with Young’s modulus $E=17~\text {MPa}$, Poisson’s ratio $\nu = 0.45$ and material density $\rho _R = 1100~\text {kg}/\text {m}^3$. The solution is obtained using the stabilised EFG formulation with three different uniform particle distributions, namely (Model 1) 1116, (Model 2) 3969, and (Model 3) 12,337 particles. For comparison, the same problem is also analysed using Godunov-type SPH approach proposed by Lee et al. [22].

Figure 3 displays the deformed column at the first bend at time $t=0.5~\text {s}$. Remarkably, the deformation pattern and pressure profile converge even with a small number of particles. Bending locking is effectively resolved by the mixed-based EFG algorithm proposed in this paper, without resorting to any ad-hoc regularisation procedure. This is a common issue in standard displacement-based EFG algorithm when attempting to model bending dominated behaviour. The time history plots for horizontal velocity and displacement are monitored at the position $\varvec{X} = [ 0.5, 6, 0.5 ]^T$ m. As shown in Fig. 4, the EFG results agree extremely well with those obtained using the Total Lagrangian Godunov-type SPH method [22].

To ensure adherence to the second law of thermodynamics, we monitor the global total energy (see Fig. 5). For all three models, the total numerical dissipation of the system decrease over time, with irreversibility caused by numerical stabilisation introduced into the SUPG algorithm. Figure 6 illustrates the locking-free deformation over time. Its pressure plot is smooth, demonstrating the applicability of the proposed EFG formulation to bending-dominated problems.

Additionally, we simulate the same problem using a higher value of Poisson’s ratio $\nu = 0.499$. We then compare the EFG results with those obtained using alternative Finite Element Method (FEM) implementations in Abaqus software, such as the standard linear tetrahedral FEM and a mixed displacement-pressure formulation. See Fig. 7 for the comparison. No spurious pressure modes are observed when using the proposed EFG approach.

Fig. 6

Bending column. Sequence of deformed structures with pressure distribution at time $t = \{0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00\}$ s (from left to right and top to bottom), obtained using the proposed stabilised EFG formulation

Fig. 7

Bending column. Comparison of deformed shapes at time $t=0.5~\text {s}$. The left-most figure shows the deformed structure using the proposed EFG algorithm. The other two figures display the deformed structure using alternative finite element methodologies, such as (middle) the standard linear tetrahedral FEM and (right) the mixed displacement-pressure FEM formulation

Fig. 8

Twisting column. Geometry, boundary and initial conditions

6.2 Robustness in highly nonlinear deformation

Fig. 9

Twisting column. Comparison of deformed shapes at time $t=0.1~\text {s}$. The first three columns (left to right) show the particle refinement of a structure simulated using the proposed stabilised EFG algorithm, whereas the last column (on the right) shows a deformed structure via an alternative in-house Total Lagrangian Godunov-type SPH algorithm [22]. Colour indicates pressure distribution

Fig. 10

Twisting column. Time history of twisting angle at four different locations, namely a $\varvec{X} = [-0.5, 6, 0.5]^T$ m, b $\varvec{X} = [0.5, 6, 0.5]^T$ m, c $\varvec{X} = [0.5, 6, -0.5]^T$ m and d $\varvec{X} = [-0.5, 6, -0.5]^T$ m

Fig. 11

Twisting column. Time evolution of a different energy measures, such as kinetic energy, strain energy and total energy, and b global numerical dissipation

Fig. 12

Twisting column. Comparison of a sequence of deformed shapes at time $t = \{0.02, 0.04, 0.06, 0.1\}$ s (from left to right) using (top row) the classical displacement-based EFG formulation and (bottom row) the proposed stabilised EFG formulation. Colour indicates pressure contour

Fig. 13

Twisting column. Components of the first Piola–Kirchhoff stress tensor at time $t = 0.054$ s for the nine figures on the left (first three columns), and at time $t=0.1$ s for the remaining figures (three columns on the right side)

The primary aim of this example [25, 41, 44, 45, 66, 84] is to rigorously examine the robustness of the proposed EFG algorithm in the case of extreme large deformations. The geometry of the column is exactly the same as the previous bending example. A nearly incompressible neo-Hookean constitutive model is used with Young’s modulus $E=17~\text {MPa}$, Poisson’s ratio $\nu = 0.45$ and material density $\rho _R = 1100~\text {kg}/\text {m}^3$. As shown in Fig. 8, the prescribed initial velocity condition for inducing a twist in the column is given as follows

$$\begin{aligned} \varvec{v}^0&= \, \varvec{v}\left( \varvec{X}, t = 0\right) = \varvec{\omega } \times \varvec{X}\, [\text {m}/\text {s}]; \\ \varvec{\omega }&= \left[ 0, \Omega \sin \left( \frac{\pi Y}{2 L}\right) , 0\right] ^T; \quad \Omega = 105~\text {rad s}^{-1}. \end{aligned}$$

To examine the convergence behaviour of the algorithm, we consider three different particle refinements: (Model 1) 1116, (Model 2) 3969 and (Model 3) 12,337 particles. A particle refinement study for the column is carried out in Fig. 9 at time t = 0.1 s. The EFG solutions converge progressively with increasing particle refinement. For benchmarking purposes, results are compared against an alternative in-house Godunov-type SPH algorithm [22]. Figure 10 monitors the evolution of the accumulated angle at four different corner positions, namely $\varvec{X} = [-0.5, 6, 0.5]^T$ m, $\varvec{X} = [0.5, 6, 0.5]^T$ m, $\varvec{X} = [0.5, 6, -0.5]^T$ m and $\varvec{X} = [-0.5, 6, -0.5]^T$ m. The comparison of the proposed EFG and SPH schemes, shown in Figs. 9 and 10, reveals nearly identical results. Figure 11 illustrates the time histories of different forms of energy, including kinetic energy, elastic strain energy and the total energy, where the latter is defined as the summation of kinetic energy and elastic strain energy. As anticipated, the global total energy of the system decreases over time throughout the entire simulation duration, ensuring the satisfaction of the second law of thermodynamics.

In order to demonstrate the performance of the proposed EFG algorithm in predominantly nearly incompressible behaviour, we have increased the value of Poisson’s ratio to $\nu = 0.4995$. Spurious pressure modes are observed when the classical displacement-based EFG algorithm is used. This deficiency can be overcome by employing the Petrov–Galerkin type of EFG algorithm. This is shown in Fig. 12. Additionally, a series of deformed shapes of the twisting column is included in Fig. 13, with contours indicating components of the first Piola.

Fig. 14

Punch block. Geometry, boundary and velocity conditions

Fig. 15

Punch block. Comparison of deformed shapes at time $t=0.013~\text {s}$. The first three columns (left to right) show the particle refinement of a structure simulated using the proposed stabilised EFG algorithm, whereas the last column (on the right) shows a deformed structure via an alternative in-house Total Lagrangian Godunov-type SPH algorithm [22]. Colour indicates pressure distribution

Fig. 16

Punch block. Time evolution of a the vertical velocity and b the vertical displacement at the corner end $\varvec{X} = [1, 0.5, 0.1]^T$ m

Fig. 17

Punch block. Comparison of a sequence of deformed shapes at time $t= \{0.0035, 0.009, 0.013 \}$ s (from left to right) using (first row) the displacement-based EFG approach, and the proposed stabilised EFG algorithm with (second row) $\tau _{\varvec{p}} = 0$ and (third row) $\tau _{\varvec{p}} = 0.1 \Delta t$. Colour indicates pressure contour

Fig. 18

Punch block. A sequence of deformed structures at time $t=\{2.86, 4.55, 5.32, 6.77, 8.33, 9.47, 10.71, 11.97, 13.17\}$ ms (from left to right and top to bottom). Colour indicates pressure profile

6.3 Effectiveness in addressing zero-energy modes

This example, initially introduced in Reference [85], serves as a benchmark test case [21, 82, 86,87,88] for evaluating the performance of the proposed algorithm in suppressing zero-energy modes within a highly constrained scenario. The problem involves a block of $1\text { m} \times 0.5\text { m} \times 0.1\text { m}$ subjected to a uniform velocity field $\varvec{v}^0 = V\left[ 0, -1, 0 \right] ^T$, with $V = 10$ m/s, applied on a region of the top face, as illustrated in Fig. 14. The top surface remains free, whilst all other boundary surfaces are constrained using a roller support. A nearly incompressible neo-Hookean material with Young’s modulus $E=1~\text {MPa}$, Poisson’s ratio $\nu = 0.4995$ and material density $\rho _R = 1000~\text {kg}/\text {m}^3$ is employed.

To demonstrate particle convergence, three different particle refinements are employed, namely (Model 1) 2448, (Model 2) 4305 and (Model 3) 11,774 particles. Figure 15 reveals that despite increasing the number of particles from 2448 to 11,774, the predicted deformation patterns remain practically identical, with improved resolution in pressure. Figure 16 depicts the time evolution of vertical velocity and displacement at one of the corners, specifically at the position $\varvec{X} = [1, 0.5, 0.1]^T$ (m). The proposed EFG algorithm accurately captures the deformed shapes near the top surface, exhibiting good agreement with results obtained from an alternative Total Lagrangian Godunov-type SPH approach [22]. To emphasise the importance of the SUPG stabilisation term in the volume map $\mathcal {D}_J^{\text {SUPG}}$, the simulation is repeated with $\tau _{\varvec{p}} = 0$. As shown in Fig. 17, introducing an appropriate value for $\mathcal {D}_J^{\text {SUPG}}$ alleviates spurious modes even under highly constrained conditions. For comparative purposes, we also simulated this test case using the displacement-based EFG approach, where an incorrect deformation path and pressure instabilities were observed. Finally, Fig. 18 depicts a sequence of deformed states, with the contour plot indicating the pressure field.

6.4 Capability of the algorithm in capturing large plastic flows

In this section, we explore the simulation of the Taylor impact test (see Fig. 19), a recognised benchmark [22, 41, 83, 89,90,91] for transient solid dynamic. The primary objective of this example is to showcase the capability of the proposed stabilised EFG algorithm in reliably capturing large plastic flows under high speed impact. The bar, with an initial length and radius of 32.4 mm and 3.2 mm, impacts against the rigid wall with an initial velocity of 227 m/s. This impact induces significant plastic deformation, presenting a challenging model for simulation.

Fig. 19

Taylor impact bar. Geometry, boundary and initial conditions

Table 1 Taylor bar

Fig. 20

Taylor bar. A sequence of deformed structures (left to right) using different model refinements (from top to bottom). In terms of contour plot, von Mises stresses (right side) and equivalent plastic strain (left side)

Exploiting symmetry, we consider one quarter of the bar by enforcing appropriate boundary conditions. The contact between the rigid wall and the bar is frictionless and non-sticky, so that only the axial component of displacement is fixed at the bottom of the bar. A Hencky-based elasto-plastic model with linear isotropic hardening is employed. The simulation parameters are detailed in Table 1. For comparison purposes, three different models are considered, namely (Model 1) 1870, (Model 2) 3102 and (Model 3) 3942 particles.

Fig. 21

Taylor bar. A sequence of deformed structures together with the pressure contour plot at different simulation times (left to right) using different model refinements (from top to bottom)

Fig. 22

Taylor bar. Time history of a different forms of energy such as kinetic energy, internal energy (the sum of elastic strain energy and plastic dissipation) and total energy, and b numerical dissipated energy as function of model refinement

Fig. 23

Taylor bar. Evolution of a the radius of the bar measured at the impact interface and b the length of the bar, comparing with the Total Lagrangian Godunov-type SPH algorithm [22]

Table 2 Taylor bar

Fig. 24

Taylor bar. A sequence of deformed structures with equivalent plastic strain profile at time $t = \{16, 24, 32, 40, 48, 56, 64, 72, 80 \}~\upmu \hbox {s}$ (from left to right and top to bottom)

Fig. 25

Necking bar. a Problem set-up and b applied velocity profile

Table 3 Necking bar

A sequence of deformed shapes, along with the distribution of von Mises effective stress and equivalent plastic strain at different stages of the simulation, are shown in Fig. 20. Figure 21 illustrates the pressure evolution during the impact process, with a smooth pressure contour observed. Additionally, the time evolution of different forms of energy contributions is depicted in Fig. 22. Throughout the simulation process, the majority of the kinetic energy transforms into a combination of strain energy and plastic dissipation, with only a minimal amount converted into numerical dissipation. Figure 22 demonstrates that the total numerical dissipation introduced by the residual-based SUPG dissipation terms decreases with particle refinement. Crucially, the time rate of the accumulated numerical dissipation remains non-positive throughout the entire simulation, ensuring the satisfaction of the second law of thermodynamics for every time integration step. Finally, Fig. 23 illustrates the time evolution of the bar radius at the impact surface. The reduction in length of the bar over time is also monitored. The final radius at time $t = 80~\upmu \hbox {s}$ is compared with reference solutions [22, 25, 49, 90] and summarised in Table 2. The solution obtained using the proposed EFG method aligns remarkably well with other published results. For completeness, Fig. 24 presents a sequence of snapshots depicting how the bar deforms upon impact.

6.5 Accuracy in capturing necking region

The classical benchmark problem of the necking of a circular bar is studied [49, 87, 91,92,93]. As illustrated in Fig. 25, the geometry of the problem consists of a circular bar of radius 6.413 mm and length 53.34 mm. Due to symmetry in the sample, only one quarter of the geometry is simulated. In order to trigger necking in the central region, 1% reduction in the radius of the bar is considered at the center. The geometric imperfection linearly varies from the center to the top half of the sample (see Fig. 25a). Regarding the problem setup, a smooth time-dependent velocity profile is applied at the top end of the sample. The velocity profile is given by

$$\begin{aligned} \varvec{v}= & {} \left[ \begin{array}{c} 0\\ 0\\ V\end{array} \right] ;\\ V= & {} \left\{ \begin{array}{ll} V_0 + \left( V_1 - V_0\right) {\bar{t}}_1^3\left( 10-15{\bar{t}}_1 + 6{\bar{t}}_1^2\right) &{} \quad \text {if} \quad 0 \le t \le t_1\\ V_1 + \left( V_2 - V_1\right) {\bar{t}}_2^3\left( 10-15{\bar{t}}_2 + 6{\bar{t}}_2^2\right) &{} \quad \text {if} \quad t_1 \le t \le t_2\end{array} \right. \end{aligned}$$

where

$$\begin{aligned} {\bar{t}}_1 = \frac{t-t_0}{t_1 - t_0}; \quad {\bar{t}}_2 = \frac{t-t_1}{t_2-t1}, \end{aligned}$$

with the values of $t_0=V_0=V_2=0$, $t_1=0.0007~\text {s}$, $t_2=0.0014~\text {s}$ and $V_1=10~\text {m}/\text {s}$. The velocity curve is depicted in Fig. 25b as a function of time. In terms of the constitutive model, a von Mises elasto-plastic model with nonlinear isotropic hardening rule as reported in [91] is utilised. The nonlinear hardening curve is described by

Fig. 26

Necking bar. Deformed structure at the end of the simulation (total elongation of the bar 14 mm), using the three different model refinements Model 1, 2 and 3 (left to right). In terms of contour plot, and for each model, von Mises stress profile (right side) and equivalent plastic strain (left side)

Fig. 27

Necking bar. Pressure distribution in the deformed structure at the end of the simulation (total elongation of the bar 14 mm), using the three different model refinements Model 1, 2 and 3 (left to right)

$$\begin{aligned} \tau _y\left( {\bar{\varepsilon }}_p \right) = \tau _y^0 + H{\bar{\varepsilon }}_p + \left( \tau _y^\infty - \tau _y^0 \right) \left[ 1 - e^{-\delta {\bar{\varepsilon }}_p} \right] , \end{aligned}$$

where H is the hardening modulus, $\tau _y^\infty$ represents the saturated yield stress and $\delta$ is the saturation exponent. The material parameters used in the simulation are summarised in Table 3.

In order to investigate the effect of particle refinement, the simulation is performed with (Model 1) 4050, (Model 2) 7488 and (Model 3) 15,004 particles. Non-uniform particle distributions are used to have more particles at the critical (necking) area. An adaptive value for the radius of the support domain $D_{\text {max}}$ is chosen, so that enough neighbouring particles are available for each integration point.

Fig. 28

Necking bar. Normalised radius reduction at the necking region versus normalised elongation compared against published results [25, 49, 91]

Fig. 29

Necking bar. A sequence of deformed structures with von Mises profile at time $t = \{0.30, 0.45, 0.60, 0.75, 0.90, 0.105, 0.120, 0.135, 0.140 \}$ ms (from left to right and top to bottom)

Distributions of the effective stress and equivalent plastic strain at the end of the simulation are depicted in Fig. 26 for the three models. Notably, the deformation profile at the critical necking area, as well as the effective stress and equivalent plastic strain distributions, converge as the model is refined. Moreover, by using the proposed stabilised EFG, Fig. 27 displays a stable and smooth pressure contour observed at the end of the simulation, qualitatively consistent with the results reported in [25]. This is, however, not the case for the classical displacement-based EFG where spurious pressure modes are observed. For a quantitative assessment, the normalised bar radius in the necking zone is plotted as a function of the normalised axial elongation (see Fig. 28). The obtained results are in good agreement with the experimental results from [92] and the numerical results reported in [25, 91]. Figure 29 shows the distribution of von Mises effective stress at various deformation stages, with symmetric mirroring applied for ease of visualisation.

7 Conclusions

This paper introduced a novel Element-Free Galerkin (EFG) computational framework tailored for modelling solid dynamics, with a particular emphasis on hyperelasticity and plasticity. The methodology stemmed from a system of Total Lagrangian first-order conservation equations, solving the linear momentum conservation equation $\varvec{p}$ together with the supplementary set of geometric conservation equations $\{ \varvec{F}, \varvec{H}, J \}$. From a spatial discretisation perspective, and taking advantage of the hyperbolic system, we introduced a robust EFG method that incorporates the well-established Streamline Upwind Petrov–Galerkin stabilisation technique. This ensured non-negative numerical entropy production, a critical consideration in the context of hyperelastic and plastic material behaviour. The discrete variation of the Hamiltonian was monitored numerically to validate the stability of the approach. The semi-discrete equations were then advanced in time using a Total Variation Diminishing (TVD) two-stage Runge–Kutta explicit time integrator. Through a series of numerical examples presented in this paper, we demonstrated the capability of the algorithm to address pressure instabilities, commonly encountered in the

Table 4 A summary of the semi-discrete version of Petrov–Galerkin Element-Free Galerkin method

displacement-based EFG approach. We observed global production of both physical (such as plastic dissipation) and numerical entropy (due to the involvement of the residual-based Petrov–Galerkin stabilisation) throughout the entire simulation. Additionally, we illustrated the benefits of using an EFG approach in plasticity scenarios, such as preventing particle clumping near impact zones in high-speed Taylor impact situations and achieving good resolution accuracy during the process of plastic necking. Future work will focus on optimising the balance between accuracy and computational time, and on comparing processing times between EFG and other mesh-free techniques, such as Reproducing Kernel Particle Method and Smooth Particle Hydrodynamics Method.

Appendices

Appendix A: Solution procedure

For the sake of completeness, the semi-discrete equations described in (47) and (48a)–(48c) are summarised in Table 4. Following the usual assembly procedure [81], these expressions can then be collected into a single system of equations written in vector format as

$$\begin{aligned} \varvec{\textrm{M}} \dot{\varvec{\textrm{U}}}^{\varvec{p}}= & {} \varvec{\textrm{E}}^{\varvec{p}} - \varvec{\textrm{T}}^{\varvec{p}}; \quad \varvec{\textrm{M}} \dot{\varvec{\textrm{U}}}^{\varvec{F}} = \varvec{\textrm{T}}^{\varvec{F}}; \quad \varvec{\textrm{M}} \dot{\varvec{\textrm{U}}}^{\varvec{H}} = \varvec{\textrm{T}}^{\varvec{H}}; \quad \varvec{\textrm{M}} \nonumber \\ \dot{\varvec{\textrm{U}}}^{J}= & {} \varvec{\textrm{T}}^{J}. \end{aligned}$$

(A.1)

Here, $\varvec{\textrm{M}}$ represents a consistent mass-like matrix, $\varvec{\textrm{U}}$ is the vector of nodal unknowns

$$\begin{aligned} \varvec{\textrm{M}}= & {} \left[ \begin{array}{cccc} M_{11} \varvec{I}_{3 \times 3} &{} M_{12} \varvec{I}_{3 \times 3} &{} \dots &{} M_{1N} \varvec{I}_{3 \times 3} \\ M_{21} \varvec{I}_{3 \times 3} &{} M_{22} \varvec{I}_{3 \times 3} &{} \dots &{} M_{2N} \varvec{I}_{3 \times 3} \\ \vdots &{} \vdots &{} &{} \vdots \\ M_{N1} \varvec{I}_{3 \times 3} &{} M_{N2} \varvec{I}_{3 \times 3} &{} \dots &{} M_{NN} \varvec{I}_{3 \times 3} \end{array}\right] ; \nonumber \\ \varvec{\textrm{U}}^{[\bullet ]}= & {} \left[ \begin{array}{c} \left[ \bullet \right] _1 \\ \left[ \bullet \right] _2 \\ \vdots \\ \left[ \bullet \right] _N \end{array} \right] \end{aligned}$$

(A.2)

with $\varvec{I}_{3 \times 3}$ represents the second order identity tensor. Moreover, $\varvec{\textrm{E}}^{[ \bullet ]}$ and $\varvec{\textrm{T}}^{[ \bullet ]}$ are the equivalent external and internal vectors defined as

$$\begin{aligned} \varvec{\textrm{E}}^{[\bullet ]} = \left[ \begin{array}{c} \varvec{E}^{\left[ \bullet \right] }_1 \\ \varvec{E}^{\left[ \bullet \right] }_2 \\ \vdots \\ \varvec{E}^{\left[ \bullet \right] }_N \end{array} \right] ; \quad \varvec{\textrm{T}}^{[\bullet ]} = \left[ \begin{array}{c} \varvec{T}^{\left[ \bullet \right] }_1 \\ \varvec{T}^{\left[ \bullet \right] }_2 \\ \vdots \\ \varvec{T}^{\left[ \bullet \right] }_N \end{array} \right] . \end{aligned}$$

(A.3)

To further improve computational efficiency, the previously described consistent mass-like matrix contributions can be replaced by its lumped counterpart $\varvec{\textrm{M}}^L$ whilst still maintaining the order of convergence [42]. This diagonal mass matrix is shown as below

$$\begin{aligned} \varvec{\textrm{M}}^L= & {} \left[ \begin{array}{cccc} M_{11}^L \varvec{I}_{3 \times 3} &{} \varvec{0}_{3 \times 3} &{} \dots &{} \varvec{0}_{3 \times 3} \\ \varvec{0}_{3 \times 3} &{} M_{22}^L \varvec{I}_{3 \times 3} &{} \dots &{} \varvec{0}_{3 \times 3} \\ \vdots &{} \vdots &{} &{} \vdots \\ \varvec{0}_{3 \times 3} &{} \varvec{0}_{3 \times 3} &{} \dots &{} M_{NN}^L \varvec{I}_{3 \times 3} \end{array} \right] ; \nonumber \\ M_{aa}^L= & {} \sum _{b=1}^N M_{ab}. \end{aligned}$$

(A.4)

Algorithm 1 details the algorithmic description of the proposed Element-Free Galerkin method presented above. It incorporates all necessary numerical ingredients and procedures, providing a clear guide for implementing and applying the method in practical computational simulations.

Algorithm 1

Element-free Galerkin method for solid dynamics

Appendix B: von Mises plasticity model

The standard algorithm to ensure that the Kirchhoff stress satisfies a von Mises type of plastic constraint is summarised here for completeness in Algorithm 2. In the case of nonlinear hardening rule (such as the one presented in Sect. 6.5), the plastic multiplier has to be obtained via the enforcement of the yield condition described in (54). This in general leads to the solution of nonlinear equations which would require a Newton–Raphson iterative procedure. Details can be found in Reference [26].

Algorithm 2

Time update of first Piola Kirchoff stress tensor