A COMPUTATIONAL FRAMEWORK FOR

MULTI-MATERIAL FLUID-STRUCTURE

INTERACTION WITH DYNAMIC FRACTURE

Kevin G. Wang 1, Patrick Lea 2, Charbel Farhat 2,3,4

1 Department of Aerospace and Ocean Engineering,

Virginia Tech

2 Department of Aeronautics and Astronautics, 3 Department of Mechanical Engineering,

4 Institute for Computational and Mathematical Engineering,

Stanford University

experiment

FSI simulation

FSI problems with dynamic fracture

shock lithotripsy

pipeline explosion underwater explosion &

implosion

• shock waves at/near material interfaces

• large structural deformation

• fluid-induced fracture and/or fragmentation

• flow seepage through cracks

• multi-phase flow with large density/pressure

jump

• multiple time scales

computational challenges

MOTIVATION

before fracture after fracture

fluid 2

fluid 1

fluid 2

f-s interface

f-f interface

fluid 1

MATHEMATICAL MODELS

- Euler equations

where

- immiscible fluids / free surface

and

(level-set equation for interface tracking)

- equations of state (EOS)

Multi-material compressible fluids in Eulerian formulation

sss pep )1( (stiffened gas for liquids)

eeR

BeR

Ap

RR

0201

)2

1()1(001

left fluid right fluid

nvR

(JWL for explosion products)

MATHEMATICAL MODELS

- equation of motion

Elastic-plastic shells in Lagrangian formulation

- J2 plasticity, piecewise linear hardening

- EPS-based failure criterion before fracture after fracture

fluid 2

fluid 1

fluid 2

f-s interface

f-f interface

fluid 1

- 2nd-order Green’s strain tensor

Fluid-structure interaction

- Impermeable interface fluid-structure transmission conditions

(for inviscid fluid flows)

sss nunv

sss npn

“no interpenetration”

“equilibrium”

The standard finite volume semi-discretization

i Ci

- integrate over a control volume (Ci )

0)(

WF

t

W

0)()(

dnWFdt

W

ineij C

ijh

C

h

iji

- evaluate one numerical flux for each “facet” ( )ijC

j

ijC

ij

j

h

i

hijh EOSnWWRoednWF ),,,()(

ijC

- special treatment is required near material (i.e. fluid-fluid or fluid-structure)

interfaces

FIVER: FINITE VOLUME METHOD WITH MULTI-

MATERIAL RIEMANN SOLVERS

- Euler equations

1D fluid-structure Riemann problem

i j

fluid 1fluid 2

SF/S

* Wang, Rallu, Gerbeau, Farhat 2011

FIVER AT FLUID-STRUCTURE INTERFACE

0)(

WFW

ιjWW )0,(

ss nuV ),0(

*

ijW

0)(

WFW

jiWW )0,(

ss nuV ),0(

*

jiW

f-s transmission

condition

(embedded surface)

Fluid-fluid Riemann problem

i j

fluid 1fluid 2

SF/F

FIVER AT FLUID-STRUCTURE INTERFACE

(0 level-set)

0)(

WFW

)0,(W

**, jiij WWijW if

jiW

0

0if

- 2nd order accuracy at interfaces can be obtained by linear extrapolation

(Zeng, Main, Farhat, in press)

VALIDATION EXAMPLES

Underwater implosion

Flapping wings

experiment

simulation

experiment

simulation

-100-50050

100150200250300

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Sensor A

Time (ms)

pre

ss

ure

(p

si)

(Farhat, Wang, et. al. IJSS, 2013)

(Farhat, Lakshminarayan, JCP, 2014)

FSI WITH DYNAMIC FRACTURE

FEM-based computational methods for fracture

element deletion

XFEM

- element deletion

- extended finite element method (XFEM)

- cohesive elements, remeshing, etc.

• delete element if failure criterion (e.g. EPS)

is met

robust and easy to implement

no defined crack path

loss of mass, momentum, energy

explicitly tracks crack propagation

difficult to track multiple interacting cracks

- “generic” FSI methods are desirable

• enrich the FE basis locally with step functions

I

III

h qXfHtuXNtXu }))(()(){(),(

• enrichment (qI) is injected when a criterion for

crack nucleation or growth is met

INTERFACE REPRESENTATION

A generic representation of fractured interfaces

- phantom elements

• for element deletion: deleted elements

• for XFEM: duplicate elements near crack*

* originally proposed by Song and Belytschko (2006) to simplify theimplementation of XFEM in legacy FE codes

• can be easily extended to many other

FE-based fracture methods

- augmented fluid-structure interface (S*F/S)

• real elements + phantom elements

XFEM with phantom nodes/elements- level-set function (f) defined in S*F/S

• gives signed distance to the closest crack

implicit representation of crack: 0)(|*

/ S XX SF f

implicit representation of fractured interface: 0)(|*

/ S XX SF f

• in practice, the precise value of f is only needed near cracks

INTERFACE TRACKING

Tracking a fractured fluid-structure interface

S*F/S

D h

real

phantom

- an intersection based algorithm* is used to track the embedded

fluid-structure interface with respect to the non-body-fitted CFD grid

* Wang, Gretarsson, Main, Farhat, IJNMF, 2012

- an intersection point x* in S*F/S is accepted if and only if

0)( * xf

pre-flawed aluminum

pipe (specimen)

clamp (torque applied)

notch

457 mm

916 mm

detonation wave

Mylar diaphragm0.89 mm

41.28 mm

detonation tube

I beam

aluminum pipe

(specimen)

detonation tube

* Performed by J. Shepherd et al. at California Institute of Technology.

pressure gauge

PIPE FRACTURE DUE TO DETONATION

Fracture of aluminum pipe due to internal detonation

EXPERIMENTAL RESULT

Crack propagation patterns

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100

initial notch length (mm)

pe

ak b

last p

ressu

re (

kP

a)

EXPERIMENTAL RESULT

Blast pressure: the notch length series

- detonation modeled using the Chapman-Jouguet theory

- three fluid materials:

explosive gas (C2H2 + O2), detonation product, and water

- CFD grid: 1.4M nodes, 8.5M elements(on 100~200 proc. cores)

- initial notch: 25.4mm / 38.1mm / 50.8mm / 63.5mm / 76.2mm

HIGH-FIDELITY SIMULATION MODEL

Simulation setup

XFEM element deletion

notch is modeled as

an initial crack

notch is modeled by shells

with reduced thickness

HIGH-FIDELITY SIMULATION MODEL

Modeling the initial notch

SIMULATION RESULT

3D Visualization of simulation result

simulation with XFEM

simulation with XFEM

SIMULATION RESULT

Structural deformation, stress, and strain

simulation with XFEM

SIMULATION RESULT

2D cut-view of fluid pressure field

- XFEM: curving in the same direction or opposite directions

- element deletion: branching

- all these propagation patterns are observed in experiment

XFEM, L= 25.4 mm

XFEM, L= 38.1 mm

ED, L= 38.1 mm

VALIDATION

Fracture response

Blast pressure

VALIDATION

- summary

• simulations captured the crack

propagation modes

• linear trend in peak pressure vs. notch

length is captured

• discrepancy in peak blast pressure: 5~20%