Large strain computational solid dynamics: An upwind cell centred Finite Volume Method

Introduction Governing equations Numerical methodology Results Conclusions

Large strain computational solid dynamics:An upwind cell centred Finite Volume Method

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

World Congress in Computational Mechanics (24th - 29th July 2016)MS 703: Advances in Finite Element Methods for Tetrahedral Mesh Computations

http://www.jibranhaider.weebly.com

Funded by the Erasmus Mundus Programme and International Association for Computational Mechanics

August 2, 2016

Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 1

www.jibranhaider.weebly.com


Scheme of presentation

1. Introduction

2. Governing equations

3. Numerical methodology

4. Results

5. Conclusions



Fast transient solid dynamics

Displacement based FEM/FVM formulations

• Linear tetrahedral elements suffer from:

× Locking in nearly incompressible materials.

× First order for stresses and strains.

× Poor performance in shock scenarios.

Proposed mixed formulation [Haider et al., 2016]

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

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

X Programmed in the open-source CFD softwareOpenFOAM.

0 0.5 1

0

0.5

1

1.5

X-Coordinate

Y-C

oord

inate

t=0.03s

-1

-0.5

0

0.5

1x 10

7

-0.5 0 0.5 1 1.5

0

0.5

1

1.5

X-Coordinate

Y-C

oord

inate

t=0.0006s

-5

0

5x 10

9

Q1-P0 FEM

0 0.5 1

0

0.5

1

1.5

X-Coordinate

Y-C

oord

inate

t=0.03s

-1

-0.5

0

0.5

1x 10

7

-0.5 0 0.5 1 1.5

0

0.5

1

1.5

X-Coordinate

Y-C

oord

inate

t=0.0006s

-5

0

5x 10

9

Upwind FVM

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 = JtK

Normal jump→ cpJpnK = JtnKTangential jump→ csJptK = JttK

p+n , t+np−

n , t−n

c+pc−ppC

n , tCn

x

t

Normal jump

p+t , t+tp−

t , t−t

c+sc−s pCt , tC

t

x

t

Tangential jump

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

Linear reconstruction procedure + limiter (monotonicity) for U−,+.




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 convergenceHighly non-linear problemVon-Mises plasticityContact problems

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 = 0.3and αCFL = 0.3.


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 = 0.3and αCFL = 0.3.



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]

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,ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.


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,E = 17 MPa, ν = 0.495, αCFL = 0.3 and Ω = 105 rad/s.



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]

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,αCFL = 0.3, τ 0

y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.



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]


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



1. Introduction



4. Results

5. Conclusions



Conclusions and further research

Conclusions

• Upwind CC-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.



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.

• 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, G. Greto, S. Kulasegaram and J. Bonet. "A new Jameson-Schmidt-Turkel Smooth Particle

Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME .

• C. H. Lee, A. J. Gil, J. Bonet and S. Kulasegaram. "An efficient Streamline Upwind Petrov-Galerkin Smooth ParticleHydrodynamics algorithm for large strain explicit fast dynamics, CMAME .

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

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

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