Large strain solid dynamics in OpenFOAM

Abstract

An industry-driven computational framework for the numerical simulation of extremely large strain solid dynamics is presented. This work focuses on the tailor-made implementation, from scratch, of the TOtal Lagrangian Upwind Cell Centred Finite Volume Method for Hyperbolic conservation laws (TOUCH) into the CFD-based open source platform OpenFOAM. Crucially, the proposed framework bridges the gap between CFD and large strain solid dynamics. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies. The TOUCH scheme shows excellent behaviour in highly nonlinear (bending dominated) nearly incompressible scenarios and overcomes the current drawbacks of existing industry computer codes.

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Student Submission for the 4th OpenFOAM User Conference 2016, Cologne - Germany
Large strain solid dynamics in OpenFOAM
Jibran Haider a, Chun Hean Lee a, Antonio J. Gil a, Javier Bonet band Antonio Huerta c
m.j.haider@swansea.ac.uk
(a) Zienkiewicz Centre for Computational Engineering, College of Engineering,
Swansea University, Bay Campus, SA1 8EN, United Kingdom
(b) University of Greenwich, London, SE10 9LS, United Kingdom
(c) Laboratory of Computational Methods and Numerical Analysis (LaCàN),
Universitat Politèchnica de Catalunya, UPC BarcelonaTech, 08034, Barcelona, Spain
Abstract:
An industry-driven computational framework for the numerical simulation of extremely large strain solid dynamics
is presented. This work focuses on the tailor-made implementation, from scratch, of the TOtal Lagrangian Upwind
Cell Centred Finite Volume Method for Hyperbolic conservation laws (TOUCH [1]) into the CFD-based open
source platform OpenFOAM. Crucially, the proposed framework bridges the gap between CFD and large strain
solid dynamics. A series of challenging numerical examples are examined in order to assess the robustness and
accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies
[2,3,4,5,6,7,8]. The TOUCH scheme shows excellent behaviour in highly nonlinear (bending dominated)
nearly incompressible scenarios and overcomes the current drawbacks of existing industry computer codes.
1 Introduction
Current computer codes (e.g. PAM-CRASH, ANSYS AUTODYN, LS-DYNA, ABAQUS, Altair HyperCrash)
used in industry for the simulation of large-scale solid mechanics problems are typically based on the use of
traditional second order displacement based Finite Element formulations. However, it is well-known that these
formulations present a number of shortcomings, namely (1) reduced accuracy for strains and stresses in compari-
son with displacements; (2) high frequency noise in the vicinity of shocks and (3) numerical instabilities associated
with shear (or bending) locking, volumetric locking and pressure checker-boarding.
Over the past few decades, various attempts have been reported at aiming to solve solid mechanics problems using
the displacement-based Finite Volume Method [9,10,11]. However, most of the proposed methodologies have
been restricted to the case of small strain linear elasticity, with very limited effort directed towards dealing with
large strain nearly incompressible materials.
To address the shortcomings identified above, a novel mixed-based methodology tailor-made for emerging (indus-
trial) solid mechanics problems has been recently proposed [1]. The mixed-based approach is written in the form
of a system of first order hyperbolic conservation laws, widely known in CFD community. The primary variables
of interest are linear momentum and deformation gradient (also known as fibre map). Essentially, the formulation
has been proven to be very efficient in simulating sophisticated dynamical behaviour of a solid [1].
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2 Governing equations
Consider the three dimensional deformation of an elastic body moving from its initial configuration occupying a
volume Ω0, of boundary ∂Ω0, to a current configuration at time toccupying a volume Ω, of boundary ∂Ω. The
motion is defined through a deformation mapping x=φ(X, t)which satisfies the following set of mixed-based
Total Lagrangian conservation laws [2,3,4,5,6,7,8]:
∂p
∂t =DIVP+f0;(1a)
∂F
∂t =DIV 1
ρ0
p⊗I.(1b)
Here, prepresents the linear momentum per unit of undeformed volume, ρ0is the material density, Fis the
deformation gradient (or fibre map), Pis the first Piola-Kirchhoff stress tensor, f0is a material body force term,
Iis the second-order identity tensor and DIV represents the material divergence operator [6]. The above system
(1a-1b) can alternatively be written in a concise manner as:
∂U
∂t =∂FI
∂XI
+S;∀I= 1,2,3,(2)
where Uis the vector of conserved variables and FIis the flux vector in the I-th material direction and Sis the
material source term. Their respective components, with the consideration of surface fluxes via a material unit
outward normal N, are:
U=p
F,FN=FINI=P N
1
ρ0p⊗N,S=f0
0.(3)
For closure of system (2), it is necessary to introduce an appropriate constitutive model to relate Pwith F, obeying
the principle of objectivity and thermodynamic consistency. Finally, for the complete definition of the Initial
Boundary Value Problem (IBVP), initial and boundary (essential and natural) conditions must also be specified.
3 Numerical methodology
From the spatial discretisation viewpoint, the above system (2) is discretised using the standard cell centred finite
volume algorithm as shown in Figure 1. The application of the Gauss divergence theorem on the integral form of
(2) leads to its spatial approximation for an arbitrary cell e:
dUe
dt =1
Ωe
0ZΩe
0
∂FI
∂XI
dΩ0=1
Ωe
0Z∂Ωe
0
FNdA ≈1
Ωe
0X
f∈Λf
e
FC
Nef (U−
f,U+
f)kCef k.(4)
In above equations, Ωe
0denotes the control volume of cell e,Λf
erepresents the set of surfaces fof cell e,Nef :=
Cef /kCef kand kCef kdenote the material unit outward surface normal and the surface area at face fof cell e,
and FC
Nef (U−
f,U+
f)represents the numerical flux computed using the left and right states of variable Uat face f,
namely U−
fand U+
f. Specifically, acoustic Riemann solver [1,2] and appropriate monotonicity-preserving linear
reconstruction procedure [2] are used for the evaluation of the numerical flux.
From the time discretisation viewpoint, an explicit one-step two-stage Total Variation Diminishing Runge-Kutta
(TVD-RK) time integrator [2] has been employed in order to update the resulting semi-discrete system (4).
4 Numerical results
4.1 Swinging cube
As presented in References [1,4,5,6,7,8], this example shows a cube of unit side length with symmetric boundary
conditions at faces X= 0,Y= 0 and Z= 0 and skew-symmetric boundary conditions at faces X= 1m, Y= 1m
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eFC
Ne f
kCe f kΩe
0
Figure 1: Cell centred Finite Volume Method
10−2 10−1 100
10−7
10−6
10−5
10−4
Grid Size (m)
L2 Norm Error
vx
vy
vZ
Slope = 2
(a) Velocities
10−2 10−1 100
10−7
10−6
10−5
10−4
Grid Size (m)
L2 Norm Error
Pxx
Pyy
Pzz
Slope = 2
(b) Stresses
Figure 2: Low dispersion cube: L2norm convergence of components of (a) Velocities; and (b) Stresses at a
particular time t= 0.004 s.
and Z= 1m. The main aim of this example is to assess the convergence behaviour of the proposed TOUCH
scheme. For small deformations, the problem has a closed-form solution for the displacement field described as
u(X, t) = U0cos √3
2cdπt!


Asin πX1
2cos πX2
2cos πX3
2
Bcos πX1
2sin πX2
2cos πX3
2
Ccos πX1
2cos πX2
2sin πX3
2

, cd=sλ+ 2µ
ρ0
.(5)
A linear elastic material is chosen with a Poisson’s ratio of ν= 0.3, Young’s modulus E= 1.7×107Pa and
density ρ0= 1.1×103kg/m3. The solution parameters are set as A=B=C=1and U0= 5 ×10−4m.
Fig. 2shows the expected second order convergence pattern (L2norm errors) of the linear momentum pand the
components of the first Piola-Kirchhoff stress tensor P, as compared to the analytical solution given in (5).
4.2 Twisting column
In order to illustrate the applicability and robustness of the scheme, the 1 m squared cross section twisting column
already presented in References [1,4,6,7,8] is considered. The problem is initialised with a sinusoidal angular
velocity field relative to the origin given by ω0= [0,Ω sin(πY /2L),0]Trad/s, where Ωis the initial angular
velocity and L= 6 m is the length of the column. A nearly incompressible neo-Hookean material is used with
density ρ0= 1100 kg/m3, Youngs’s modulus E= 17 MPa and Poisson’s ratio ν= 0.45. A mesh refinement
analysis of the proposed TOUCH algorithm is shown in Figure 3, where smooth pressure distribution can be
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(a) 4×24 ×4cells (b) 8×48 ×8cells (c) 40 ×240 ×40 cells
(a)
(b)
(c)
Pressure (Pa)
Figure 3: Twisting column: Mesh refinement of deformed shapes with pressure distribution at t= 0.1s using
three different mesh sizes: (a) h= 1/4m; (b) h= 1/8m; and (c) h= 1/40 m.
observed. For benchmarking purposes, we simulate the exact same problem using alternative numerical strategies
with a higher Poisson’s ratio of value ν= 0.495, which can be considered as very nearly incompressible. Crucially,
all computational mixed methodologies presented produce very similar deformation patterns with smooth pressure
distribution and absence of locking (see Figure 4).
4.3 Taylor impact
Following References [3,7], a circular copper bar is dropped with an initial velocity v0= [0,−227,0]Tm/s
which impacts against a rigid frictionless wall. The initial radius and length of the bar are r0= 0.0032 m and
L= 0.0324 m. The main objective of this example is to show that the proposed TOUCH scheme can be used
without locking difficulties when experiencing plastic deformation. A von Mises hyperelastic-plastic material with
isotropic hardening is chosen and the material properties are such that density ρ0= 8.930 ×103kg/m3, Young’s
modulus E= 117 GPa, Poisson’s ratio ν= 0.35, yield stress, ¯τ0
y= 0.4GPa and hardening modulus H= 0.1
GPa. Clearly, very smooth pressure and plastic strain distributions are observed (see Figure 5).
4.4 Bar rebound
This example shows a circular bar, of radius r0= 0.0032 m and of length 0.0324 m, impacts against a rigid
frictionless wall with an initial velocity v0= [0,−150,0]Tm/s. A neo-Hookean material is used with density
ρ0= 8930 kg/m3, Young’s modulus E= 585 MPa and Poisson’s ratio ν= 0.45. An initial separation distance
between the bar and the wall is 0.004 m. Upon impact, the bar undergoes large compressive deformation until
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(a) (b) (c) (d) (e) (f)
Pressure (Pa)
Figure 4: Twisting column: Comparison of deformed shapes plotted with pressures at time t= 0.1s using
various numerical schemes: (a) TOUCH [1]; (b) B-bar method; (c) Petrov-Galerkin FEM [3]; (d) Hu-Washizu
type variational principle [12]; (e) Taylor-Hood (Q2-Q1) FEM; and (f) Jameson-Schmidt-Turkel SPH [13].
t= 12 ms when all kinetic energy of the bar is converted to potential energy. Soon afterwards, tensile (or
separating) forces start developing and the bounce-off motion begins. The sequence of the deformation process
plotted with the pressure distribution is shown in Figure 6. It is also interesting to point out that accurate solutions
can still be obtained using a very coarse mesh (see Figure 7), showing a clear advantage over industry codes.
4.5 Ring impact
In this example, we assess the contact collision of a rubber ring against a rigid wall. The geometry of the ring are
such that both inner and outer radius are 0.03 m and 0.04 m, respectively. The rubber ring is placed 0.004 m away
from the wall and is dropped with an initial velocity v0= [0,−0.59,0]T. A nearly incompressible neo-Hookean
model is used and the material properties are density ρ0= 1000 kg/m3, Youngs’s modulus E= 1 MPa and
Poisson’s ratio ν= 0.4. Its main objective is to assess the ability of the proposed algorithm for the simulation
of nonlinear contact behaviour. A mesh refinement study at time t= 0.18 s is shown in Figure 8. As expected,
similar deformation pattern is observed using two different mesh densities. More importantly, Figure 9shows the
global preservation of linear and angular momentum of the system.
4.6 Torus impact
A very challenging example is carried out in order to assess the robustness and effectiveness of the proposed
algorithm in highly nonlinear contact scenarios. A rubber torus, with an inner radius of 0.03 m and an outer radius
of 0.04 m, is initiated with an initial velocity field v0= [0,−3,0]Tagainst a rigid wall. A nearly incompressible
neo-Hookean model is used where the material properties are density ρ0= 1000 kg/m3, Youngs’s modulus E= 1
MPa and Poisson’s ratio ν= 0.4. Figure 10 shows a sequence of deformed states of the torus experiencing
extremely large deformation after impact. Smooth pressure profile is observed in the absence of locking. When
the torus comes into contact with the wall, its outer part suffers from compression and, on the other hand, the inner
part develops tension (see Figure 10).
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t= 10 µst= 30 µst= 50 µst= 80 µs
(a) Pressure (Pa)
t= 10 µst= 30 µst= 50 µst= 80 µs
(b) Plastic strain
Figure 5: Taylor impact: Time evolution of (a) Pressure; and (b) Plastic strain distribution along with the deforma-
tion in a quarter domain of a bar.
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t= 3 st= 6 ms t= 12 ms t= 18 ms t= 27 ms
Pressure (Pa)
Figure 6: Bar rebound: Time evolution of the deformation of a bar plotted with pressure distribution.
0 0.5 1 1.5 2 2.5 3
x 10−4
−20
−16
−12
−8
−4
0
4
8x 10−3
Time (sec)
y Dispacement (m)
Top (2880 cells)
Top (23040 cells)
Bottom (2880 cells)
Bottom (23040 cells)
Figure 7: Bar rebound: Time evolution of vertical displacement of the points on the top plane X= [0,0.0324,0]T
and the bottom plane X= [0,0,0]Tusing two different mesh sizes.
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(a) 420 cells (b) 6400 cells
Pressure (Pa)
Figure 8: Ring impact: Mesh refinement of deformed shapes with pressure distribution at t= 0.18 s using two
different mesh sizes: (a) 420; and (b) 6400 structured hexahedral cells.
0 0.01 0.02 0.03 0.04
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
Linear momentum (kg.m.s−1)
px (420 cells)
py (420 cells)
pz (420 cells)
px (3660 cells)
py (3660 cells)
pz (3660 cells)
px (25760 cells)
py (25760 cells)
pz (25760 cells)
(a) Linear momentum
0 0.01 0.02 0.03 0.04
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Angular momentum (N.m.s−1)
Ax (420 cells)
Ay (420 cells)
Az (420 cells)
Ax (3660 cells)
Ay (3660 cells)
Az (3660 cells)
Ax (25760 cells)
Ay (25760 cells)
Az (25760 cells)
(b) Angular momentum
Figure 9: Ring impact: Time evolution of components of (a) Linear momentum; and (b) Angular momentum using
various mesh sizes.
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t= 2 ms t= 4 ms t= 8 ms
t= 17 ms t= 28 ms t= 38 ms
Pressure (Pa)
Figure 10: Torus impact: Time evolution of the deformation plotted with pressure representation using the pro-
posed TOUCH scheme.
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5 Conclusions
This paper introduces an extremely fast industry-driven computational framework for the numerical simulation of
large strain solid dynamics. The formulation bridges the gap between CFD and large strain solid dynamics and it
has been implemented, from scratch, within the CFD-based open source code OpenFOAM. Following the works
of [2,7], a mixed formulation written in the form of a system of first order hyperbolic equations is employed. The
linear momentum pand the deformation gradient Fare regarded as primary conservation variables of this mixed-
based approach. Finally, a comprehensive list of challenging numerical examples has been presented. The overall
TOUCH scheme performs extremely well in large strain solid dynamics without any numerical instabilities.
6 Acknowledgements
The first author would like to acknowledge the financial support received through "The Erasmus Mundus Joint
Doctorate SEED" programme. The second, third and fourth authors gratefully acknowledge the financial support
provided by the Sêr Cymru National Research Network for Advanced Engineering and Materials, United Kingdom.
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