Large strain solid dynamics in OpenFOAM

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

a Zienkiewicz Centre for Computational Engineering, Swansea University, UKb Laboratori de Calcul Numeric (LaCaN), UPC BarcelonaTech, Spain

c University of Greenwich, London, UK

Research outline

Objectives:

• Simulate fast-transient solid dynamic problems.

• Develop a fast and efficient low order numerical

scheme.

Key features:

X An upwind cell-centred FVM Total Lagrangian scheme (TOUCH).

X Utilises an explicit Runge-Kutta time integrator.

X Programmed in the open-source CFD software OpenFOAM.

X Overcomes the shortcomings of linear tetrahedral

elements in standard displacement based

FEM/FVM formulations:

• Equal order of convergence for velocities and stresses.

• No volumetric locking for nearly incompressible materials.

• Excellent performance in bending and shock dominatedscenarios.

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 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

Q1-P0 FEM Proposed FVM

First order conservation laws

1. Linear momentum:

2. Deformation gradient:

3. Total energy:

d

dt

∫Ω0

p dΩ0 =

∫∂Ω0

t dA +

∫Ω0

ρ0b dΩ0

d

dt

∫Ω0

F dΩ0 =

∫∂Ω0

p

ρ0⊗N dA

d

dt

∫Ω0

E dΩ0 =

∫∂Ω0

p

ρ0· t dA−

∫∂Ω0

Q ·N dA +

∫Ω0

s dΩ0

• Hyperbolic laws in differential form:∂U∂t

=∂F I

∂XI+ S, ∀ I = 1, 2, 3

Cell centred FVM discretisation

Standard face-based CC-FVM

e FCNe f

‖Ce f‖ Ωe0

dU e

dt=

1

Ωe0

∑f∈Λf

e

FCN ef

(U−f ,U+f ) ‖Cef‖

Node-based CC-FVM

FCNea

‖Cea‖

Ωe0

e

dU e

dt=

1

Ωe0

∑a∈Λa

e

FCN ea

(U−a ,U+a ) ‖Cea‖

• Gradient calculation through least squares minimisation −→ Ge

• Satisfaction of monotonicity through Barth and Jespersen limiter −→ φe

• Linear reconstruction procedure for second order spatial accuracy −→ U+,− (φe, Ge)

Lagrangian contact dynamics

Contact flux:

FCN = F INI =

tC

1ρ0pC ⊗N

1ρ0pC · tC −Q ·N

Acoustic Riemann solver:

FCN = FC

NAve+ FC

NStab

=1

2

[FN(U−f ) + FN(U+

f )]− 1

2

∫ U+f

U−f

|AN | dU︸ ︷︷ ︸Upwinding stabilisation

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

Explicit time integrationTotal Variation Diminishing Runge-Kutta scheme:

1st RK stage −→ U?e = Un

e + ∆t Un

e (Une , t

n)

2nd RK stage −→ U??e = U?

e + ∆t U?

e(U?e, t

n+1)

Un+1e =

1

2(Un

e + U??e )

with stability criterion:

∆t = αCFLhmin

cmaxp

Numerical results

Shock scenario

6 7 8 9 10

x 10−3

−7.5

−5

−2.5

0

2.5

5x 10

7

Time (sec)

Str

ess

(Pa)

AnalyticalTOUCH (1st order)TOUCH (2nd order w/o limiter)TOUCH (2nd order with limiter)JST VCFVM

Mesh convergence

10−2 10−1 100

10−8

10−7

10−6

10−5

10−4

10−3

Grid Size (m)

Str

ess

Err

or

slope = 1L1 norm (1st order)

L2 norm (1st order)

slope = 2L1 norm (2nd order)

L2 norm (2nd order)

Structured vs Unstructured

Pressure (Pa)

Complex twisting

Pressure (Pa)

Flapping structure

Pressure (Pa)

Von Mises plasticity

Constrained-TOUCH Penalised-TOUCH Hyperelastic-GLACE

Plastic strain

Bar rebound

Pressure (Pa)

Torus impact

Pressure (Pa)

On-going work

1. An advanced Roe’s Riemann solver.

2. Robust shock capturing algorithm.

3. Ability to handle tetrahedral elements.

Future work

1. Extension to Fluid-Structure Interaction(FSI) problems.

2. Implementation of ArbitraryLagrangian-Eulerian (ALE) formulation.

References[1] 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, International Journal for Numerical Methods

in Engineering, 109(3) : 407–456, 2017.

[2] C. H. Lee, A. J. Gil and J. Bonet. Development of a cell centred upwind finite volume algorithm for a new conservation law formulation in structural dynamics. Computers and Structures, 118 : 13–38, 2013.

Website: http://www.jibranhaider.weebly.com Email:m.j.haider,c.h.lee,a.j.gil@swansea.ac.uk