large strain solid dynamics in openfoam

Introduction Governing equations Numerical methodology Results Conclusions

Large strain solid dynamics in OpenFOAM

Jibran Haider a, b, Chun Hean Lee a, Antonio J. Gil a, Javier Bonet c & Antonio Huerta b

a Zienkiewicz Centre for Computational Engineering (ZCCE),College of Engineering, Swansea University, UK

b Laboratory of Computational Methods and Numerical Analysis (LaCàN),Universitat Politèchnica de Catalunya (UPC BarcelonaTech), Spain

c University of Greenwich, London, UK

The 4th Annual OpenFOAM User Conference (11th - 13th October 2016)

12 th October 2016http://www.jibranhaider.weebly.com

Funded by the Erasmus Mundus SEED PhD Programme and ESI Group

October 16, 2016

Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 1

www.jibranhaider.weebly.com


Research group at Swansea University

Dr. Antonio J. Gil

Associate Professor

Dr. Chun Hean Lee

Research Fellow

Prof. Javier Bonet

University of Greenwich

Prof. Antonio Huerta

UPC BarcelonaTech

Dr. RogelioOrtigosa

Postdoc

Jibran Haider

Research Assistant

Osama I.Hassan

Research Assistant

Roman Poya

Research Assistant

Emilio G. Blanco

Research Assistant

AtaollahGhavamian

Research Assistant


http://www.swansea.ac.uk/staff/engineering/a.j.gil/

http://www.swansea.ac.uk/staff/engineering/c.h.lee/

http://www.lacan.upc.edu/huerta



http://www.jibranhaider.weebly.com


http://www.jibranhaider.weebly.com


Scheme of presentation

1. Introduction

2. Governing equations

3. Numerical methodology

4. Results

5. Conclusions




1. Introduction



4. Results

5. Conclusions



Fast transient dynamics

Objectives

• Simulate fast-transient solid dynamic problems.

• Develop an industry-driven library of low order numericalschemes.

Solid dynamics in OpenFOAM [Jasak & Weller, 2000]

× Standard displacement based implicit dynamics

× Linear elastic material with small strain deformation

× Locking in nearly incompressible scenarios

× First order convergence for stresses and strains

× Poor performance in shock dominated scenarios

OpenFOAM solid mechanics community [Ivankovic et al.]

• [Cardiff et al., 2012; 2014; 2016] −→ displacement based + pressure instabilities +moderate strains + ....



Proposed solid formulation

• First order conservation laws similar to the one used in CFD community.

• Entitled TOtal Lagrangian Upwind Cell-centred FVM for Hyperbolic conservation laws(TOUCH).

• Programmed in the open-source CFD software OpenFOAM.

TOUCH scheme[Haider et al., 2016; Lee et al., 2013]

X Mixed explicit dynamics

X Complex constitutive models

X Large strain deformation

X No bending and volumtric locking

X Second order convergence for stresses andstrains

v = 100 m/s(0.5, 0.5, 0.5)

(−0.5,−0.5,−0.5)

[Punch cube]

Aim is to bridge the gap between CFD and computational solid dynamics.




1. Introduction



4. Results

5. Conclusions



Total Lagrangian formulation

Conservation laws

• Linear momentum

∂p∂t

= ∇0 · P(F) + ρ0b; p = ρ0v

• Deformation gradient

∂F∂t

= ∇0 ·(

1ρ0

p⊗ I)

; CURL F = 0

Additional equations

• Total energy

∂E∂t

= ∇0 ·(

1ρ0

PT p− Q)

+ s

An appropriate constitutive model is required to close the system.



Hyperbolic system

First order conservation laws

∂U∂t

= ∇0 ·F(U) + S

U =

p

F

E

; F =

P(F)

1ρ0

p⊗ I1ρ0

(PT p)− Q

; S =

ρ0b

0

s

• Geometry update

∂x∂t

=1ρ0

p; x = X + u

Adapt CFD technology to the proposed formulation.

Develop an efficient low order numerical scheme for transient solid dynamics.




1. Introduction


3. Numerical methodologySpatial discretisationFlux computationInvolutionsEvolution

4. Results

5. Conclusions




1. Introduction



4. Results

5. Conclusions



Spatial discretisation

Conservation equations for an arbitrary element

dU e

dt=

1Ωe

0

∫Ωe

0

∂F I

∂XIdΩ0 −→ ∀ I = 1, 2, 3;

=1

Ωe0

∫∂Ωe

0

F INI︸︷︷︸FN

dA (Gauss Divergence theorem)

≈1

Ωe0

∑f∈Λf

e

FCNef‖Cef ‖

e FCNe f

‖Ce f‖ Ωe0

Traditional cell centred Finite Volume Method

dU e

dt=

1Ωe

0

∑f∈Λf

e

FCNef‖Cef ‖

; FCNef

=

tC

1ρ0

pC ⊗ N1ρ0

tC · pC




1. Introduction



4. Results

5. Conclusions



Lagrangian contact dynamics

Rankine-Hugoniot jump conditions

c JU K = JF K N

where JK = + −−wc J p K = J t K

c J F K =1ρ0

J p K⊗ N

c J E K =1ρ0

J PT p K · N

X, x

Y, y

Z, z

Ω+0

Ω−0

N+

N−

n−

n+

Ω+(t)

Ω−(t)

φ+

φ−

n−

n+

c−sc+s

c+pc−p

Time t = 0

Time t



Acoustic Riemann solver

Jump condition for linear momentum

cJpK = JtKNormal jump→ cpJpnK = JtnK

Tangential jump→ csJptK = JttK

Upwinding numerical stabilisation

pC=

[c−p p−n + c+p p+n

c−p + c+p

]+

[c−s p−t + c+s p+t

c−s + c+s

]︸︷︷︸

pCAve

+

[t+n − t−nc−p + c+p

]+

[t+t − t−tc−s + c+s

]︸︷︷︸

pCStab

tC =

[c+p t−n + c−p t+n

c−p + c+p

]+

[c+s t−t + c−s t+t

c−s + c+s

]︸︷︷︸

tCAve

+

[c−p c+p (p+n − p−n )

c−p + c+p

]+

[c−s c+s (p+t − p−t )

c−s + c+s

]︸︷︷︸

tCStab

How do we obtain U−,+?



Godunov’s method

• Piecewise constant representation in every cell.

• Methodology is first order accurate in space.

x

y

U

Ue

Uα1

Uα4

Uα2

Uα3

(a) Piecewise constant values×

x

y

U

Uα4

U α3

Uα2

Uα1

Ue

Uα3

Uα4

(b) Linear reconstructionX

× First order simulations suffer from excessive numerical dissipation.

X A linear reconstruction procedure is essential to increase spatial accuracy.



Linear reconstruction procedure

Gradient operator:

• Classical least squares minimisation procedure.

Ge =

∑α∈Λαe

deα ⊗ deα

−1 ∑α∈Λαe

(Uα − Ue

‖deα‖

)deα

Linear extrapolation to flux integration point:

Uf ,a = Ue + Ge ·[Xf ,a − Xe

]

de1α2

e1

α1

α2

α3

α4

α f1

α f2

α f3α f4α f5

e2

de2α4

Gradient correction procedure:

• Necessary for the satisfaction of monotonicity through Barth and Jespersen limiter (φe).

Uf ,a = Ue + φe Ge(Ue,Uα) ·[Xf ,a − Xe

]X Ensures that the spatial discretisation is second order accurate.




1. Introduction



4. Results

5. Conclusions



Godunov-type FVM

Standard FV update (CURL F 6= 0)

dFe

dt=

1Ωe

0

∑f∈Λ

fe

pCf

ρ0⊗ Cef X

Constrained FV update (CURL F = 0)[Dedner et al., 2002; Lee et al., 2013]

dFe

dt=

1Ωe

0

∑f∈Λ

fe

pCf

ρ0⊗ Cef X

• Algorithm is entitled ’C-TOUCH’.

pe

pCf −→

pe

Ge

ypC

f

←−

pa

Constrained transport schemes are widely used in Magnetohydrodynamics (MHD).




1. Introduction



4. Results

5. Conclusions



Time integration

Two stage Runge-Kutta time integration

1st RK stage −→ U∗e = Une + ∆t Un

e(Une , t

n)

2nd RK stage −→ U∗∗e = U∗e + ∆t U∗e (U∗e , tn+1)

Un+1e =

12

(Une + U∗∗e )

with stability constraint:

∆t = αCFLhmin

cp,max; cp,max = max

a(ca

p)

X An explicit Total Variation Diminishing Runge-Kutta time integration scheme.

X Monolithic time update for geometry.




1. Introduction



4. ResultsMesh convergenceEnhanced reconstructionHighly non-linear problemVon-Mises plasticityContact problemsUnstructured meshesComplex geometries

5. Conclusions




1. Introduction




5. Conclusions



Low dispersion cube

X, x

Y, y

Z, z

(0, 0, 0)

(1, 1, 1)

Displacements scaled 300 times

t = 0 s t = 2 ms t = 4 ms t = 6 ms

Pressure (Pa)

Boundary conditions

1. Symmetric at:

X = 0, Y = 0, Z = 0

2. Skew-symmetric at:

X = 1, Y = 1, Z = 1

Analytical solution

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

)


Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.3[Haider et al., 2016] and αCFL = 0.3.[Aguirre et al., 2014]


Low dispersion cube: Mesh convergence

Velocity at t = 0.004 s

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

Stress at t = 0.004 s

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

X Demonstrates second order convergence for velocities and stresses.


Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.3[Haider et al., 2016] and αCFL = 0.3.[Aguirre et al., 2014]



1. Introduction




5. Conclusions



Bending dominated scenario

X, x

Y, y

(−0.5, 0, 0.5)

(0.5, 6,−0.5)

Z, z

L = 6m

v0 = [V Y/L, 0, 0]T

[Bending column]

Mesh convergence at t = 1.5 s

Pressure (Pa)

X Eliminates bending difficulty.


Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3,[Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.


Bending dominated scenario

Time = 0.5 s

(a) OpenFOAM least square gradient (b) Enhanced least square gradient

Pressure (Pa)


Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3,[Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.



1. Introduction




5. Conclusions



Twisting column

X, x

Y, y

(−0.5, 0, 0.5)

(0.5, 6,−0.5)

Z, z

ω0 = [0, Ω sin(πY/2L), 0]T

L

[Twisting column - Refinement]

[Twisting column - Comparison]

Mesh refinement at t = 0.1 s

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

(a) 4 × 24 × 4

(b) 8 × 48 × 8

(c) 40 × 240 × 40

Pressure (Pa)

X Demonstrates the robustness of the numerical scheme


Problem description: Nearly incompressible neo-Hookean material, ρ0 = 1100 kg/m3, E = 17 MPa,[Haider et al., 2016] ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.

[Gil et al., 2014]


Comparison of various alternative numerical schemes

t = 0.1 s

C-TOUCH P-TOUCH B-bar Taylor Hood PG-FEM Hu-Washizu JST-SPH SUPG-SPH

Pressure (Pa)


Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3,[Haider et al., 2016] E = 17 MPa, ν = 0.495, αCFL = 0.3 and Ω = 105 rad/s.

[Lee et al., 2016]



1. Introduction




5. Conclusions



Taylor impact

X, x

Y, y

v0

(−0.0032, 0, 0)

(0.0032, 0.0324, 0)

Z, z

r0

[Taylor impact]

[Taylor impact - Radius]

Evolution of pressure wave

t = 0.1µs t = 0.2µs t = 0.3µs t = 0.4µs t = 0.5µs t = 0.6µs

Pressure (Pa)

X Demonstrates the ability of the algorithm to simulate plastic behaviour.


Problem description: Hyperelastic-plastic material, ρ0 = 8930 kg/m3, E = 117 GPa, ν = 0.35,[Aguirre et al., 2014] αCFL = 0.3, τ 0

y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.[Lee et al., 2014]



1. Introduction




5. Conclusions



Bar rebound

X, x

Y, y

v0

(−0.0032, 0, 0)

(0.0032, 0.0324, 0)

Z, z

r0

0.004

[Bar rebound]

t = 3 ms t = 6 ms t = 12 ms t = 18 ms t = 27 ms

Pressure (Pa)

X Demonstrates the ability of the algorithm to simulate contact problems.


Problem description: Nearly incompressible neo-Hookean material, ρ0 = 8930 kg/m3, E = 585 MPa,[Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.


Bar rebound

X, x

Y, y

v0

(−0.0032, 0, 0)

(0.0032, 0.0324, 0)

Z, z

r0

0.004

y Displacement of the points X = [0, 0.0324, 0]T and X = [0, 0, 0]T

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)

X Demonstrates the ability of the algorithm to simulate contact problems.


Problem description: Nearly incompressible neo-Hookean material, ρ0 = 8930 kg/m3, E = 585 MPa,[Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.


Torus impact

[Torus impact]

t = 2 ms t = 4 ms t = 8 ms

t = 17 ms t = 28 ms t = 28 ms

Pressure (Pa)


Problem description: Neo-Hookean material, ρ0 = 1000 kg/m3, E = 1 MPa, ν = 0.4, αCFL = 0.3 andv0 = −3 m/s.



1. Introduction




5. Conclusions



Spinning plate: Structured vs unstructured elements

X, x

Y, y(0.5, 0.5, 0)

ω0 = [0, 0, Ω]T

(−0.5,−0.5, 0)

Time = 0.15 s

(a) Structured 20× 20 cells (b) Unstructured 484 cells

Pressure (Pa)

X Demonstrates the ability of the framework to handle unstructured grids.


Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material,[Haider et al., 2016] ρ0 = 1000 kg/m3, E = 17 MPa, ν = 0.45 and αCFL = 0.3 and Ω = 105 rad/s.


Spinning plate: Structured vs unstructured elements

X, x

Y, y(0.5, 0.5, 0)

ω0 = [0, 0, Ω]T

(−0.5,−0.5, 0)

Displacement of point X = [0.5, 0.5, 0]T

0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

Time (sec)

Dis

pla

cem

ent

(m)

u

x structured

uy structured

ux unstructured

uy unstructured

X Demonstrates the ability of the framework to handle unstructured grids.


Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material,[Haider et al., 2016] ρ0 = 1000 kg/m3, E = 17 MPa, ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.


Flapping device

1. Introduction




5. Conclusions



Flapping device

t = 0 ms t = 25 ms t = 50 ms t = 75 ms

t = 100 ms t = 125 ms t = 175 ms t = 200 ms

Pressure (Pa)

[Flapping device]


Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3,E = 17 MPa, ν = 0.45, αCFL = 0.3.


Complex twisting

[Complex twisting]

t = 5 ms t = 10 ms t = 15 ms

t = 20 ms t = 25 ms t = 30 ms

Pressure (Pa)


Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3,E = 17 MPa, ν = 0.45, αCFL = 0.3.



1. Introduction



4. Results

5. Conclusions



Conclusions and on-going work

Conclusions

• Upwind cell centred FVM is presented for fast solid dynamic simulations within the OpenFOAMenvironment.

• Linear elements can be used without usual locking.

• Velocities and stresses display the same rate of convergence.

On-going work

• Investigation into an advanced Roe’s Riemann solver with robust shock capturing algorithm.

• Extension to multiple body and self contact.

• Ability to handle tetrahedral elements.

• Extension to fluid-structure interaction problems.



References

Published / accepted• J. Haider, C. H. Lee, A. J. Gil and J. Bonet. "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.

• C. H. Lee, A. J. Gil, G. Greto, S. Kulasegaram and J. Bonet. "A new Jameson-Schmidt-Turkel Smooth ParticleHydrodynamics algorithm for large strain explicit fast dynamics, CMAME (2016); 311: 71-111.

• A. J. Gil, C. H. Lee, J. Bonet and R. Ortigosa. "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.

• J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. "A first order hyperbolic framework for large straincomputational solid dynamics. Part I: Total Lagrangian isothermal elasticity", CMAME (2015); 283: 689-732.

• M. Aguirre, A. J. Gil, J. Bonet and C. H. Lee. "An upwind vertex centred Finite Volume solver for Lagrangian soliddynamics", JCP (2015); 300: 387-422.

• C. H. Lee, A. J. Gil and J. Bonet. "Development of a cell centred upwind finite volume algorithm for a newconservation law formulation in structural dynamics", Computers and Structures (2013); 118: 13-38.

Under review• C. H. Lee, A. J. Gil, O. I. Hassan, J. Bonet and S. Kulasegaram. "An efficient Streamline Upwind Petrov-Galerkin

Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME (2016).

In preparation• J. Haider, C. H. Lee, A. J. Gil, A. Huerta and J. Bonet. "Contact dynamics in OpenFOAM, JCP.

• A. J. Gil, J. Bonet, C. H. Lee, J. Haider and A. Huerta. "Adapted Roe’s Riemann solver in explicit fast soliddynamics, JCP.

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