Simulation of impacts on elastic–viscoplastic solids with the flux-difference splitting finite volume method applied to non-uniform quadrilateral meshes
- Thomas Heuzé^{1}Email author
https://doi.org/10.1186/s40323-018-0101-z
© The Author(s) 2018
Received: 9 November 2017
Accepted: 3 April 2018
Published: 2 May 2018
Abstract
The flux-difference splitting finite volume method (Leveque in J Comput Phys 131:327–353, 1997; Leveque in Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press, 2002) is here employed to perform numerical simulation of impacts on elastic–viscoplastic solids on bidimensional non-uniform quadrilateral meshes. The formulation is second order accurate in space through flux limiters, embeds the corner transport upwind method, and uses a fractional-step method to compute the relaxation operator. Elastic–viscoplastic constitutive models falling within the framework of generalized standard materials (Halphen and Nguyen in J Mech 14:667–688, 1975) in small strains are considered. Many test cases are proposed and two particular viscoplastic constitutive models are studied, on which comparisons with finite element solutions show a very good accuracy of the finite volume solutions, both on stresses and viscoplastic strains.
Keywords
Introduction
The numerical simulation of hyperbolic initial boundary value problems including extreme loading conditions such as impacts requires the ability to accurately capture and track the fronts of shock waves induced in the medium. Indeed, this permits to correctly follow the path of waves and hence understand the mechanical phenomena occuring within that medium. For solid-type media, it also allows for an accurate assessment of the propagation of irreversible strains and hence of residual stresses and distortions within the structure. High speed forming processes like electromagnetic material forming [1–3] are some application examples of severe loading conditions in which the track of wave fronts is important both for understanding the development of irreversible strains in the workpiece and optimizing its final shape. Hence, these problems require numerical schemes capable to rewrite the film of history of loading undergone by any material point with sufficient precision to permit the understanding of particular physical phenomena of interest, while freeing oneself from any numerical disturbance that might impair that understanding. In particular, numerical schemes able to represent regular as well as discontinuous solutions are of interest; more precisely, they should meet both high orders of accuracy in regions where the solution is smooth and a high resolution of discontinuities when they occur without any numerical spurious oscillations appearing in their vicinity.
The numerical simulation of impacts on dissipative solids has been and is again mainly performed with the classical finite element method coupled with centered differences or Newmark finite difference schemes in time [4, 5], which is implemented in many industrial codes. Indeed, the finite element method is still popular in the solid mechanics community for, among others, its easy implementation of nonlinearities of partial differential equations, that is for solid-type media it enables to account easily for history-dependent constitutive equations through appropriate integration algorithms [6] and storage of internal variables at integration points in each element. However, on the one hand the amount of artificial viscosity added to numerical time integrators required to reduce the high frequency noise in the vicinity of shocks is hard to assess properly in order to remove the sole spurious oscillations, without destroying the accuracy of the numerical solution. On the other hand, finite elements do not use any feature of the characteristic structure of the set of hyperbolic equations, and is hence not the best suited method to accurately capture discontinuous solutions.
The finite volume method, initially developed for the simulation of gas dynamics [7, 8], has gained recently more and more interest for problems involving impacts on solid media (see e.g. [9–17]). This family of methods show some advantages to achieve an accurate tracking of wavefronts; among others (i) the continuity of fields is not enforced on the mesh in its cell-centered version, that allows for capturing discontinuous solutions, (ii) the characteristic structure of hyperbolic equations can be introduced within the numerical solution, either through the explicit solution of a Riemann problem at cell interfaces, or in an implicit way through the construction of the numerical scheme, (iii) the same order of convergence is achieved for both the velocity and stress fields [12], and (iv) the amount of numerical viscosity introduced can be controlled locally as a function of the local regularity of the solution, so that to permit the elimination of spurious numerical oscillations while preserving a high order of accuracy in more regular zones.
Since the early work of Wilkins [18] and Trangenstein et al. [19], several authors have proposed many ways to simulate impacts on dissipative solid media, such as elastic–plastic and elastic–viscoplastic solids, with this class of methods. These can be merely classified into Eulerian approaches, generally based on a fractional-step method to treat the irreversible processes [9, 10, 13, 17, 20] and used for extremely high strain, strain rate and pressure problems, and lagrangian approaches [14, 16, 21] that allow to follow the path of material particles and hence account for refined history-dependent constitutive equations though limited by mesh entanglement, both being coupled with an approximate Riemann or WENO solver. Eulerian approaches are written in a conservative form with a relaxation operator containing inelastic terms [22], and are often based on the so-called Maxwell-type relaxation approach [9–11, 13, 17, 23] which actually refers to an adapted version of Perzyna’s elastic–viscoplastic solids [24], in fact in its perfect viscoplasticity version since no hardening rules is generally accounted for, obtained by means of a relaxation process of stresses [11, 17]. Lagrangian approaches have been less investigated and have been so far more treated in elastoplasticity [14, 16] using classical integration of constitutive equations [6] coupled with acoustic Riemann solvers, or using simplified elastic–plastic Riemann solvers [21].
We are interested in this work in elastic–viscoplastic systems, whose study is here focused on the isothermal and linearized geometrical framework, leading to a nonhomogeneous system of partial differential equations, generating a system of weakly discontinuous waves beyond the viscoplastic yield, following a discontinuous (elastic) wave due to the transition between elastic and elastic–viscoplastic ranges. This work intends to apply the flux-difference splitting finite volume method, whose formalism has been made popular by Leveque [7, 25], for the simulation of impacts on elastic–viscoplastic solid media on bidimensional non-uniform quadrilateral meshes. Its derivation for these unstructured meshes follows classical ones [26, 27] for first order terms, but the process of limitation of waves required to achieve high resolution methods here accounts for different orientations between the current and upwind edges. Moreover, the approach is here derived using the class of generalized standard materials [28] (GSM) that describes a convenient framework to define thermodynamically consistent viscoplastic constitutive models, which can embed refined viscoplastic models with respect to these already used with such approach [9, 17], some particular creep and hardening rules being considered in this work. The viscoplastic relaxation system is solved by means of a fractional-step method, whose convection part is solved with the flux-difference splitting formalism.
The paper is organized as follows. First, the elastic–viscoplastic constitutive model, the governing balance laws and the characteristic analysis are presented in “Elastic–viscoplastic Initial Boundary Value Problem” section. Next, the flux-difference splitting finite volume method is presented for bidimensional non-uniform quadrilateral meshes in “The flux-difference splitting finite volume method” section. “Computation of the viscoplastic part” section discusses the asymptotic limit of the elastic–viscoplastic system, and the fractional-step method used to compute the viscoplastic part of the behaviour. At last, several test cases are presented in “Applications” section, mainly conducted within the two-dimensional plane strain assumption, on which comparisons with finite element solutions allow to show the good accuracy of finite volume solutions, both on stresses and viscoplastic strains. The viscoplastic flow computed with two viscoplastic constitutive models is compared on the last example. Especially, a very simple Chaboche-type [29] viscoplastic model coupled with Prager’s linear kinematic [30] hardening, and a more refined Chaboche–Nouailhas’ [31] one coupled with the Armstrong–Frederick’s [32, 33] kinematic nonlinear hardening law are considered.
Elastic–viscoplastic Initial Boundary Value Problem
Elastic–viscoplastic constitutive model
Balance laws and quasi-linear form
Characteristic analysis
The flux-difference splitting finite volume method
High order fluxes
Many limiting functions exist and permit to obtain different known finite volume schemes [36]. Some of them enable the numerical scheme to satisfy a non-increasing total variation, so that the appearance of spurious numerical oscillations can be avoided in the vicinity of discontinuities. The Superbee limiter defined by \(\phi (\theta )= \max (0,\min (1,2\theta ),\min (2,\theta ))\) falls in this family, and is used in the following of this work. More generally, the limitation of wave strength amounts to add locally some numerical viscosity, and to locally lower the order of accuracy to properly capture discontinuities. In zones where the solution field is more regular, the limitation is not active and an accuracy of order two can be reached.
Transverse fluxes
Computation of the viscoplastic part
Asymptotic limit of the elastic–viscoplastic relaxation system
Properties of numerical schemes
A system of hyperbolic conservation laws with relaxation is said to be stiff when the relaxation time \(\tau \) is small compared to the time scale determined by the characteristic speeds of the system and some appropriate length scales [42], or put in another way, the wave-propagation behavior of interest occurs on a much slower time scale than the fastest time scales of the ordinary differential equation (ODE) arising from the source term. This means that if the solution is perturbed away from its equilibrium condition, then it rapidly relaxes back towards the equilibrium.
Solving stiff hyperbolic equations with relaxation can be even more challenging than solving stiff ODEs. Indeed in a stiff hyperbolic equation, the fastest reactions are often not in equilibrium everywhere. The stiffness of the relaxation operator can cause many difficulties to numerical schemes, particularly on coarse, underresolved (\(\Delta t \gg \tau \)) grids. Stiff source terms and underresolved numerical methods, though stable, may yield spurious nonphysical or poor numerical solutions [46]. Accordingly, numerical schemes should be designed so that to satisfy some particular properties to ensure asymptotic convergence, accuracy, and stability.
In particular, numerical schemes should (i) use coarse grids [42] that do not resolve the small relaxation time \(\tau \), and still remain bounded by the Courant–Friedrichs–Lewy (CFL) stability constraint, governed by the sole convection part of the system, (ii) be asymptotic preserving [42, 47] meaning that the numerical scheme should be consistent with the asymptotic limit \(\tau \rightarrow 0\) for fixed \(\Delta x\), \(\Delta t\), that is the limiting scheme is a good discretization of the equilibrium system (47) even if the source term is underresolved. The numerical schemes should also (iii) be asymptotic accurate [47], that is preserve the order of accuracy in the stiff limit, (iv) strong stability preserving [47], strong stability is maintained at discrete level, and (v) well-balanced [43], preserving steady state numerically.
Fractional-step or splitting methods
In this work, the Godunov splitting (49) coupled with an implicit backward Euler ODE solver will be used for stiff problems (e.g. see [9, 17]), while the Strang splitting and a backward differentiation formula at order two (BDF2) will be used for less stiff problems. However, more complex and higher order time integrators for stiff relaxation terms exist, but are not considered in this work. These are in general not based on splitting methods, and are still the purpose of current researches. Among others, the family of implicit–explicit IMEX Runge–Kutta schemes [47] allows to define high order schemes using an explicit time discretization for numerical flux and an implicit (DIRK [49]) one for the relaxation operator. Other approaches like ADER-WENO schemes [43] are also available.
Applications
Plane waves in a one-dimensional finite medium with Riemann-type initial conditions
Chaboche’s viscoplastic constitutive model
Numerical elastic–plastic asymptotic limit
As \(\tau \) tends to zero, the computed elastic–viscoplastic solution should tend to the elastic–plastic one. On this test case, numerical solutions computed with the Strang and Godunov splitting are compared to that computed with classical P1-finite elements. The ODE system (50) computed for the finite volume numerical solution is solved by means of an implicit backward Euler scheme for the Godunov splitting and a backward differentiation formula at order two (BDF2) for the Strang splitting. Then the viscoplastic strain is updated explicitly (with a forward Euler scheme) at the end of the time step with the viscoplastic flow rule [(combining (13), (52) and (53)]. The finite element solution is coupled with a central difference explicit time integrator, uses a lumped mass matrix [4], and the constitutive equations are integrated with a radial return algorithm [6].
Material parameters
\(E = 200\) GPa | \(\sigma _{y} = 400\) MPa |
\(\nu = 0.3\) | \(D = 10\) GPa |
\(\rho =7800\) kg/m\(^{3}\) | \(K = 24.3\) MPa s |
\(N = 4.37\) |
In Fig. 5a, a moderately low relaxation time is considered, viscous effects are observable since the numerical elastic–viscoplastic solutions are smoother than the elastic–plastic one. Moreover, an increased apparent tensile yield stress is observable. FEM and Godunov splitting solutions are superposed while the Strang one appears slightly in advance. In Fig. 5b, a lower relaxation time \(\tau \) is set; viscous effects are now less apparent, but the same global behaviour than previously is observed. For a very low relaxation time (see Fig. 5c), the numerical solutions now conform as well as their respective possibility to the elastic–plastic analytical solution. A local overshoot occurs in the viscoplastic strains computed by the Godunov splitting scheme, and small oscillations appear: they do not subsist if the viscoplastic flow rule is solved implicitly together with the source term. The Strang solution shows an error in computing the sound speed of plastic waves, actually it shows a stiff behaviour. Indeed, this very low relaxation time leads to compute a numerical solution on an underresolved grid, since it becomes far smaller than the time step dicted by the convection part of the system. It is well-known that though Strang splitting can yield second-order accuracy for smooth solutions and non-stiff problems, it may fail to correctly compute wave speeds on underresolved grids, even with L-stable ODE solvers (see e.g. [46]).
Energy balance
As the relaxation parameter \(\tau \) decreases, the energy numerically dissipated by the Godunov splitting slightly increases, while the Strang time evolution of the total energy decreases markedly: it is associated to the appearance of its stiff behaviour (Fig. 5c).
Convergence analysis
Figure 8a–c show the convergence curves associated to discontinuous solutions for the three values of the relaxation parameter \(\tau \). A very close convergence rate of about 0.5 for the stress and 0.8 for the velocity is observed for the three schemes in the case of initial discontinuous profile of the velocity. Indeed, because of the discontinuous solution, the source term is expected to be active only over thin regions where there are fast transients that cannot be resolved with a high accuracy [7]. Convergence curves are almost superposed for the highest value of the relaxation parameter. However, the constant in the Strang splitting convergence curves (Fig. 9b, c) becomes larger than these of the FEM and the Godunov splitting for lower values of \(\tau \), yielding a bigger error. Indeed, as \(\tau \) decreases, the Strang splitting shows a too stiff behaviour (Fig. 5c) on an underresolved grid (for example \(\Delta x =3 \times 10^{-2}\) m, 200 grid cells). As expected, this behaviour is improved for a lower grid size associated to a lower time step of the order of the time scale of the source term (for example \(\Delta x = 3 \times 10^{-3}\,\text{ m } \rightarrow \Delta t \approx 5 \times 10^{-7} \text {s} \sim \tau \)). But in general we do not want to use such a fine grid, because larger time steps are preferred to save computational cost.
Figure 9a–c show the convergence curves associated to smooth initial profile of the velocity for the three values of the relaxation parameter \(\tau \). Now, convergence rates of about 1.1 and 1.5 for stress and the velocity are observed, and are higher than these provided for discontinuous solutions. The FEM error appears globally to be the smallest, though that of the Godunov splitting solution is close to it in each case. However, the Strang splitting solution appears less efficient than the two others, although its error has decreased with respect to discontinuous solutions. But, as \(\tau \) decreases, it shows again a too stiff behaviour.
As a first draw of conclusion, for very stiff problems, the Godunov splitting should be preferred to the Strang one. For non-stiff ones, both can be used. However, it should be noticed that the present case of a plane wave is a hard test because of its one-dimensional strain state. Two-dimensional cases below yield less stiff solutions due to their multi-dimensional strain state.
Partial impact on a plane
Parameters of the partial impact test case
Geometry (m) | Loading |
---|---|
\(a= 0.5 \) | \(p = 1.5Y_H\) \(=1.05\) GPa |
\(L= 2\) |
But a clear improvement provided by finite volumes over the finite element solution can be observed by using a refined \(100\times 100\) quadrangle cartesian mesh, which provided the CFL number set at one, enables to properly capture the elastic discontinuity. Figure 14 shows the tensile wave reflected from the bottom side at time \(9.25 \times 10^{-4}\) s, captured in one cell with the finite volume solution, while finite elements exhibit spurious numerical oscillations around this wavefont. Finite volumes benefit here from flux limiters designed to achieve a nonincreasing total variation, applied to the convection part of the system (15).
Double-notched specimen with tensile initial velocity
Parameters of the double-notched specimen test case
Geometry | Loading |
---|---|
\(a= 1.5 \times 10^{-2}\) m | \(\bar{v} = 40\) m/s |
\(L= 3.73 \times 10^{-2}\) m | |
\(h = 1 \times 10^{-2}\) m | |
\(\alpha = \pi /6\) | |
\(l = 3 \times 10^{-3}\) m |
The nonzero initial velocity generates a tensile wave, which is first reflected on the right free boundary, as shown in Fig. 16. The finite element solution shows spurious oscillations upstream of the left wavefront, essentially due to the implicit time stepping, while the viscoplastic strains are close for both numerical solutions and show a conical spread pattern due to the plane of symmetry. Then, the left front of the tensile wave is reflected both on the notch and left plane of symmetry, which leads the normal stress \(\sigma _{11}\) to double on the symmetry line and to concentrate at the notch corner. After few waves interactions, Fig. 17 shows viscoplastic strains which have much increased at the notch corners, so does for the normal stress which shows shearing pattern due to multiple wave reflexions. Essentially, the two numerical solutions fit well, though the finite volume one shows less numerical spurious oscillations.
Sudden velocity loading and unloading of a heterogeneous volume
Chaboche’s viscoplastic constitutive model
Parameters values for the heterogeneous volume test case
Geometry (m) | Loading |
---|---|
\(a= 10^{-3}\) | \(\bar{v}=40\) m/s |
\(R= 5 \times 10^{-4}\) | \(t_u = 7\times 10^{-8}\) s |
Matrix | Inclusion |
---|---|
\(E_M = 200\) GPa | \(E_I = 70\) GPa |
\(\nu _M = 0.3\) | \(\nu _I=0.34\) |
\(\rho _{M}=7800\) kg/m\(^{3}\) | \(\rho _{I}=2700\) kg/m\(^{3}\) |
\(\sigma _{y_I} = 350\) MPa | |
\(D_I = 10\) GPa | |
\(K_I=24.3\) MPa s | |
\(N_I=4.37\) |
A slot of compressive normal stress is first formed by the horizontal component of the velocity prescribed on the left side, and travels rightward in the matrix. Then, its first front interacts with the front interface between the matrix and the inclusion, and generates an intermediate state of stress and velocity due to the mismatch of elastic impedances of the matrix and the inclusion. In Fig. 19, the second front of the former stress slot interacts with the generated intermediate state and yields a tensile stress wave, while the first compressive loading keeps on travelling within the inclusion, and propagates viscoplastic strains. One can observe that the finite element stress field shows spurious numerical oscillations in the vicinity of discontinuities, especially close to lateral boundaries where tensile spurious stress states appear. Cumulated viscoplastic strains are almost identical for both numerical solutions, except close to the matrix/inclusion interface. Once the tensile wave has reflected on the left side of the volume, it propagates rightward, following the initial compression slot, as shown in Fig. 20. The former compression slot has interacted with the back side of the inclusion interface, generating an important growth of cumulated viscoplastic strains close to this area. Note also that the front of the tensile slot has been curved after reflexion first on the circular matrix/inclusion interface, second on the boundary of the volume. Generally speaking, the finite volume solution allows to obtain the same viscoplastic strains than these of the finite element solution without the spurious numerical oscillations on stresses obtained with the finite element solution.
Chaboche–Nouailhas’ viscoplastic constitutive model
Figure 23 shows the cumulated viscoplastic strain field in the inclusion after the compression slot is passed, computed by means of finite volume method with Chaboche and Chaboche–Nouailhas’ viscoplastic constitutive models. The latter leads to much more important cumulated viscoplastic strain than the former. Indeed, for an overstress close to \(10^2\) MPa, the effective cumulated strain rate has largely increased for that model (see Fig. 21), which yields a more important viscoplastic flow. As the compression slot passes within the inclusion, viscoplastic flow concentrates in the inclusion close to the rear part of the interface with the matrix (see Fig. 23). This is provided by the circular geometry of the inclusion that defines a concentrating profile as the compression wave travels rightward. In Figs. 24 and 25, isovalues of the effective viscoplastic strain rate are shown at two successive instants, as the compression slot passes. The observed profiles appear quite different in terms of the chosen viscoplastic constitutive model. In particular, the saturation of the overstress predicted by expression (60) combined with the particular rear geometry of the interface inclusion/matrix yield an effective strain rate that reaches very high numerical values on a narrow band. This small example illustrates the importance of the chosen viscoplastic constitutive model on the propagated viscoplastic strains in a dynamic process.
Conclusion
In this work, the flux-difference splitting finite volume method [7, 25] has been employed to perform numerical simulation of impacts on elastic–viscoplastic solids on bidimensional non-uniform quadrilateral meshes. The elastic–viscoplastic system of equations identifies itself with a relaxation system with threshold, whose asymptotic limit yields an elastic–plastic system. The linearized geometrical framework considered here leads to a linear hyperbolic system with a nonlinear source term, driven by the viscoplastic part of the behaviour. This relaxation system is solved by means of a fractional-step method (Strang or Godunov splitting), whose convection part is solved with the flux-difference splitting formalism applied here to bidimensional non-uniform quadrilateral meshes. Several test cases have been proposed, and show the good accuracy of the computed finite volume solutions in terms of both stresses and viscoplastic strains with respect to finite element ones. The flux limiters used to compute the convection part enable to remove spurious numerical oscillations shown in the finite element solution close to the elastic discontinuity. In addition, two viscoplastic constitutive models have been tested to illustrate the genericity of the approach, and their influence on the viscoplastic flow has been shown on the heterogeneous volume test case. Notice that this work is straightforward extendable to three-dimensional meshes following [53] for example.
Declarations
Author's contributions
TH developed the idea, conducted numerical experiments and wrote the paper. The author read and approved the final manuscript.
Acknowledgements
None.
Competing interests
The author declares that he has no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
None.
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