Large strain solid dynamics in OpenFOAM - ESI Group Submission for the 4th OpenFOAM User Conference...

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 b and Antonio Huerta c

[email protected]

(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 dynamicsis presented. This work focuses on the tailor-made implementation, from scratch, of the TOtal Lagrangian UpwindCell Centred Finite Volume Method for Hyperbolic conservation laws (TOUCH [1]) into the CFD-based opensource platform OpenFOAM. Crucially, the proposed framework bridges the gap between CFD and large strainsolid dynamics. A series of challenging numerical examples are examined in order to assess the robustness andaccuracy 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 IntroductionCurrent 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 oftraditional second order displacement based Finite Element formulations. However, it is well-known that theseformulations 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 associatedwith 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 usingthe displacement-based Finite Volume Method [9, 10, 11]. However, most of the proposed methodologies havebeen restricted to the case of small strain linear elasticity, with very limited effort directed towards dealing withlarge 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 formof a system of first order hyperbolic conservation laws, widely known in CFD community. The primary variablesof interest are linear momentum and deformation gradient (also known as fibre map). Essentially, the formulationhas been proven to be very efficient in simulating sophisticated dynamical behaviour of a solid [1].

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2 Governing equationsConsider the three dimensional deformation of an elastic body moving from its initial configuration occupying avolume Ω0, of boundary ∂Ω0, to a current configuration at time t occupying a volume Ω, of boundary ∂Ω. Themotion is defined through a deformation mapping x = φ(X, t) which satisfies the following set of mixed-basedTotal Lagrangian conservation laws [2, 3, 4, 5, 6, 7, 8]:

∂p

∂t= DIVP + f0; (1a)

∂F

∂t= DIV

(1

ρ0p⊗ I

). (1b)

Here, p represents the linear momentum per unit of undeformed volume, ρ0 is the material density, F is thedeformation gradient (or fibre map), P is the first Piola-Kirchhoff stress tensor, f0 is a material body force term,I is 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 U is the vector of conserved variables and FI is the flux vector in the I-th material direction and S is thematerial source term. Their respective components, with the consideration of surface fluxes via a material unitoutward normalN , are:

U =

[pF

], FN = FINI =

[PN

1ρ0p⊗N

], S =

[f0

0

]. (3)

For closure of system (2), it is necessary to introduce an appropriate constitutive model to relateP with F , obeyingthe principle of objectivity and thermodynamic consistency. Finally, for the complete definition of the InitialBoundary Value Problem (IBVP), initial and boundary (essential and natural) conditions must also be specified.

3 Numerical methodologyFrom the spatial discretisation viewpoint, the above system (2) is discretised using the standard cell centred finitevolume 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

Ωe0

∫Ωe

0

∂FI

∂XIdΩ0 =

1

Ωe0

∫∂Ωe

0

FN dA ≈ 1

Ωe0

∑f∈Λf

e

FCNef

(U−f ,U+f ) ‖Cef‖. (4)

In above equations, Ωe0 denotes the control volume of cell e, Λfe represents the set of surfaces f of cell e, Nef :=Cef/‖Cef‖ and ‖Cef‖ denote the material unit outward surface normal and the surface area at face f of cell e,and FC

Nef(U−f ,U+

f ) represents the numerical flux computed using the left and right states of variable U at face f ,namely U−f and U+

f . Specifically, acoustic Riemann solver [1, 2] and appropriate monotonicity-preserving linearreconstruction 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 cubeAs presented in References [1, 4, 5, 6, 7, 8], this example shows a cube of unit side length with symmetric boundaryconditions at facesX = 0, Y = 0 and Z = 0 and skew-symmetric boundary conditions at facesX = 1m, Y = 1m

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e FCNe f

‖Ce f‖ Ωe0

Figure 1: Cell centred Finite Volume Method

10−2

10−1

100

10−7

10−6

10−5

10−4

Grid Size (m)

L2

No

rm E

rro

r

vx

vy

vZ

Slope = 2

(a) Velocities

10−2

10−1

100

10−7

10−6

10−5

10−4

Grid Size (m)

L2

No

rm E

rro

r

Pxx

Pyy

Pzz

Slope = 2

(b) Stresses

Figure 2: Low dispersion cube: L2 norm convergence of components of (a) Velocities; and (b) Stresses at aparticular time t = 0.004 s.

and Z = 1m. The main aim of this example is to assess the convergence behaviour of the proposed TOUCHscheme. For small deformations, the problem has a closed-form solution for the displacement field described as

u(X, t) = U0 cos

(√3

2cdπt

)A sin(πX1

2

)cos(πX2

2

)cos(πX3

2

)B cos

(πX1

2

)sin(πX2

2

)cos(πX3

2

)C cos

(πX1

2

)cos(πX2

2

)sin(πX3

2

) , cd =

√λ+ 2µ

ρ0. (5)

A linear elastic material is chosen with a Poisson’s ratio of ν = 0.3, Young’s modulus E = 1.7 × 107 Pa anddensity ρ0 = 1.1 × 103 kg/m3. The solution parameters are set as A = B = C = 1 and U0 = 5 × 10−4 m.Fig. 2 shows the expected second order convergence pattern (L2 norm errors) of the linear momentum p and thecomponents of the first Piola-Kirchhoff stress tensor P , as compared to the analytical solution given in (5).

4.2 Twisting columnIn order to illustrate the applicability and robustness of the scheme, the 1 m squared cross section twisting columnalready presented in References [1, 4, 6, 7, 8] is considered. The problem is initialised with a sinusoidal angularvelocity field relative to the origin given by ω0 = [0,Ω sin(πY/2L), 0]T rad/s, where Ω is the initial angularvelocity and L = 6 m is the length of the column. A nearly incompressible neo-Hookean material is used withdensity ρ0 = 1100 kg/m3, Youngs’s modulus E = 17 MPa and Poisson’s ratio ν = 0.45. A mesh refinementanalysis of the proposed TOUCH algorithm is shown in Figure 3, where smooth pressure distribution can be

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(a) 4× 24× 4 cells (b) 8× 48× 8 cells (c) 40× 240× 40 cells

(a)

(b)

(c)

Pressure (Pa)

Figure 3: Twisting column: Mesh refinement of deformed shapes with pressure distribution at t = 0.1 s usingthree different mesh sizes: (a) h = 1/4 m; (b) h = 1/8 m; and (c) h = 1/40 m.

observed. For benchmarking purposes, we simulate the exact same problem using alternative numerical strategieswith 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 pressuredistribution and absence of locking (see Figure 4).

4.3 Taylor impactFollowing References [3, 7], a circular copper bar is dropped with an initial velocity v0 = [0,−227, 0]T m/swhich impacts against a rigid frictionless wall. The initial radius and length of the bar are r0 = 0.0032 m andL = 0.0324 m. The main objective of this example is to show that the proposed TOUCH scheme can be usedwithout locking difficulties when experiencing plastic deformation. A von Mises hyperelastic-plastic material withisotropic hardening is chosen and the material properties are such that density ρ0 = 8.930 × 103 kg/m3, Young’smodulus E = 117 GPa, Poisson’s ratio ν = 0.35, yield stress, τ0

y = 0.4 GPa and hardening modulus H = 0.1GPa. Clearly, very smooth pressure and plastic strain distributions are observed (see Figure 5).

4.4 Bar reboundThis example shows a circular bar, of radius r0 = 0.0032 m and of length 0.0324 m, impacts against a rigidfrictionless wall with an initial velocity v0 = [0,−150, 0]T m/s. A neo-Hookean material is used with densityρ0 = 8930 kg/m3, Young’s modulus E = 585 MPa and Poisson’s ratio ν = 0.45. An initial separation distancebetween 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 = 0.1 s usingvarious numerical schemes: (a) TOUCH [1]; (b) B-bar method; (c) Petrov-Galerkin FEM [3]; (d) Hu-Washizutype 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 (orseparating) forces start developing and the bounce-off motion begins. The sequence of the deformation processplotted with the pressure distribution is shown in Figure 6. It is also interesting to point out that accurate solutionscan still be obtained using a very coarse mesh (see Figure 7), showing a clear advantage over industry codes.

4.5 Ring impactIn this example, we assess the contact collision of a rubber ring against a rigid wall. The geometry of the ring aresuch that both inner and outer radius are 0.03 m and 0.04 m, respectively. The rubber ring is placed 0.004 m awayfrom the wall and is dropped with an initial velocity v0 = [0,−0.59, 0]T . A nearly incompressible neo-Hookeanmodel is used and the material properties are density ρ0 = 1000 kg/m3, Youngs’s modulus E = 1 MPa andPoisson’s ratio ν = 0.4. Its main objective is to assess the ability of the proposed algorithm for the simulationof 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 9 shows theglobal preservation of linear and angular momentum of the system.

4.6 Torus impactA very challenging example is carried out in order to assess the robustness and effectiveness of the proposedalgorithm in highly nonlinear contact scenarios. A rubber torus, with an inner radius of 0.03 m and an outer radiusof 0.04 m, is initiated with an initial velocity field v0 = [0,−3, 0]T against a rigid wall. A nearly incompressibleneo-Hookean model is used where the material properties are density ρ0 = 1000 kg/m3, Youngs’s modulus E = 1MPa and Poisson’s ratio ν = 0.4. Figure 10 shows a sequence of deformed states of the torus experiencingextremely large deformation after impact. Smooth pressure profile is observed in the absence of locking. Whenthe torus comes into contact with the wall, its outer part suffers from compression and, on the other hand, the innerpart develops tension (see Figure 10).

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t = 10µs t = 30µs t = 50µs t = 80µs

(a) Pressure (Pa)

t = 10µs t = 30µs t = 50µs t = 80µs

(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 s t = 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 D

isp

acem

ent

(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 planeX = [0, 0.0324, 0]T

and the bottom planeX = [0, 0, 0]T using 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 twodifferent 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)

Line

ar m

omen

tum

(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)

Ang

ular

mom

entu

m (

N.m

.s−1

)

A

x (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 usingvarious 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 ConclusionsThis paper introduces an extremely fast industry-driven computational framework for the numerical simulation oflarge strain solid dynamics. The formulation bridges the gap between CFD and large strain solid dynamics and ithas been implemented, from scratch, within the CFD-based open source code OpenFOAM. Following the worksof [2, 7], a mixed formulation written in the form of a system of first order hyperbolic equations is employed. Thelinear momentum p and the deformation gradient F are regarded as primary conservation variables of this mixed-based approach. Finally, a comprehensive list of challenging numerical examples has been presented. The overallTOUCH scheme performs extremely well in large strain solid dynamics without any numerical instabilities.

6 AcknowledgementsThe first author would like to acknowledge the financial support received through "The Erasmus Mundus JointDoctorate SEED" programme. The second, third and fourth authors gratefully acknowledge the financial supportprovided by the Sêr Cymru National Research Network for Advanced Engineering and Materials, United Kingdom.

References[1] Haider J, Lee CH, Gil AJ, Bonet J. A first order hyperbolic framework for large strain computational solid

dynamics: An upwind cell centred Total Lagrangian scheme. IJNME 2016; DOI: 10.1002/nme.5293.

[2] Lee CH, Gil AJ, Bonet J. Development of a cell centred upwind finite volume algorithm for a new conserva-tion law formulation in structural dynamics. Computers and Structures 2013; 118:13–38.

[3] Lee CH, Gil AJ, Bonet J. Development of a stabilised Petrov–Galerkin formulation for conservation laws inLagrangian fast solid dynamics. CMAME 2014; 268:40–64.

[4] Gil AJ, Lee CH, Bonet J, Aguirre M. A stabilised Petrov–Galerkin formulation for linear tetrahedral elementsin compressible, nearly incompressible and truly incompressible fast dynamics. CMAME 2014; 276:659–690.

[5] Gil AJ, Lee CH, Bonet J, Ortigosa R. A first order hyperbolic framework for large strain computational soliddynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity.CMAME 2016; 300:146–181.

[6] Bonet J, Gil AJ, Lee CH, Aguirre M, Ortigosa R. A first order hyperbolic framework for large strain compu-tational solid dynamics. Part I: Total Lagrangian isothermal elasticity. CMAME 2015; 283:689–732.

[7] Aguirre M, Gil AJ, Bonet J, Carreño AA. A vertex centred finite volume Jameson–Schmidt–Turkel (JST)algorithm for a mixed conservation formulation in solid dynamics. JCP 2014; 259:672–699.

[8] Aguirre M, Gil AJ, Bonet J, Lee CH. An upwind vertex centred Finite Volume solver for Lagrangian soliddynamics. JCP 2015; 300:387–422.

[9] Jasak H, Weller H. Application of the finite volume method and unstructured meshes to linear elasticity.IJNME 2000; 48(2):267–287.

[10] Cardiff P, Karac A, Ivankovic A. Development of a finite volume contact solver based on the penalty method.Computational Materials Science 2012; 64:283–284.

[11] Cardiff P, Karac A, Ivankovic A. A large strain finite volume method for orthotropic bodies with generalmaterial orientations. CMAME 2014; 268:318–335.

[12] Bonet J, Gil AJ, Ortigosa R. A computational framework for polyconvex large strain elasticity. CMAME2015; 283:1061–1094.

[13] Lee CH, Gil AJ, Greto G, Kulasegaram S, Bonet J. A new Jameson-Schmidt-Turkel Smooth Particle Hydro-dynamics algorithm for large strain explicit fast dynamics. CMAME 2016; Under review.

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http://dx.doi.org/10.1002/nme.5293

Large strain solid dynamics in OpenFOAM - ESI Group Submission for the 4th OpenFOAM User Conference...

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