Particle Finite Element Method (PFEM) · 2017-06-22 · finite element method for the analysis of...

Miguel A. Celigueta, S. Latorre, S. Idelsohn , E. Oñate,

J.M. Carbonell, P. Ryzhakov, A. Franci , J. Marti

International Center for Numerical Methods in Engineering

Technical University of Catalonia (UPC)

Barcelona. Spain

Particle Finite Element Method (PFEM)

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OUTLINE 1 What is the ‘PFEM’? • Origins • Theory • Performance • Advantages and disadvantages 2 PFEM vs SPH • Conceptual differences and points in common. 3 PFEM Application fields • Waves, erosion, tsunamis, dams. • Landslides • Forging, machining • Fire, melting 4 Immediate future

1 of 4 – What is the PFEM?

MOTIVATION:

• Fluid • Free surface • Deformable solid • Fluid-structure-interaction • Fluid separation

MOTIVATION:

Even nowadays it is still hard to solve this type of problems with Eulerian CFD’s due to the double interface solid-fluid and air-fluid

‘MFEM’ Idelsohn, S. R., Onate, E., Calvo, N., & Del Pin, F. (2003). The meshless finite element method. International Journal for Numerical Methods in Engineering, 58(6), 893-912.

‘PFEM’ Oñate, E., Idelsohn, S. R., Del Pin, F., & Aubry, R. (2004). The particle finite element method. An overview. International Journal of Computational Methods, 1(02), 267-307.

The Particle Finite Element Method (PFEM)

Numerical model based on:

- lagrangian approach

- fast re-meshing algorithm

- boundary recognition method

- the classic FEM

7

solid mechanics fluid mechanics

The remeshing step allows large deformations

- the classic FEM

8

The PFEM

BROKEN DAM… The PFEM

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0VF

tVF

t+dtVF

tt

t +dt t

0V

tV

t +dt V

0u

du

Fluid

x1 , u1

x2 , u2

0t

0Гv

0Гt

tГv

t +dt Гv

tГt

t +dt Гt

UPDATED LAGRANGIAN FORMULATION

Initial configuration

Current configuration

Next (updated)

configuration

We seek for equilibrium at t + Dt

The PFEM

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New cloud of nodes n+1C

Solid node

Fixed boundary node

Fluid node Initial cloud of nodes nC

Finite element mesh nM

nx , nu , np

n+1u , n+1p n+1x

The PFEM

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Cloud of nodes

Delaunay triangulation

The PFEM

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Cloud of

nodes

ALPHA

SHAPE:

Boundary

element if

r(x)≥α h(x)

a1 a2

a2<a1

Delaunay

triangulation

Boundary surface recognition The PFEM

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After Delaunay Tesselation (re-connection of the nodes) Alpha-Shape method (free surface recognition) Optionally, mesh repairing operations are carried out: • Node shifting • Node deletion • Node addition

These operations exchange the ‘discretization error’ (distorted elements) by a ‘mapping error’, hopefully with a net gain.

0i

i

u

x

(continuity equation)

iij

ji

i fx

pxDt

Du

(momentum equation)

Governing equations

FLUID SOLVER: A predictor-corrector or a predictor-multicorrector fractional step method.

Navier-Stokes equations for incompressible flow (via FEM) :

The PFEM

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Predictor-corrector scheme

• Kamran, K., Rossi, R., Oñate, E., & Idelsohn, S. R. (2013). A compressible Lagrangian framework for the simulation of the underwater implosion of large air bubbles. Computer Methods in Applied Mechanics and Engineering, 255, 210-225.

• Ryzhakov, P. B., Rossi, R., Idelsohn, S. R., & Oñate, E. (2010). A monolithic Lagrangian approach for fluid–structure interaction problems. Computational mechanics, 46(6), 883-899.

• Oñate, E., Franci, A., & Carbonell, J. M. (2014). A particle finite element method (PFEM) for coupled thermal analysis of quasi and fully incompressible flows and fluid-structure interaction problems. In Numerical Simulations of Coupled Problems in Engineering (pp. 129-156). Springer International Publishing.

• Oñate, E., Idelsohn, S. R., Del Pin, F., & Aubry, R. (2004). The particle finite element method—an overview. International Journal of Computational Methods, 1(02), 267-307.

• Carbonell, J. M., Oñate, E., & Suárez, B. (2009). Modeling of ground excavation with the particle finite-element method. Journal of engineering mechanics, 136(4), 455-463.

• Onate, E., Idelsohn, S. R., Celigueta, M. A., & Rossi, R. (2008). Advances in the particle finite element method for the analysis of fluid–multibody interaction and bed erosion in free surface flows. Computer methods in applied mechanics and engineering, 197(19), 1777-1800.

Many fluid, solid or mixed solvers used in PFEM…

(1) Solve for variables at the solid under prescribed

boundary tractions:

(2) Solve for variables at the fluid under

prescribed boundary velocities: Gt Gv

tVF

vSF

Fluid-Solid interface

Fluid Domain tVF

tVS

tFS

Solid Domain OPTION 1: STAGGERED SCHEME COUPLINGS

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FLUID-STRUCTURE INTERACTION OPTION 1: THROUGH A STAGGERED SCHEME (only for rigid-body structures)

COUPLINGS

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Collapse of a water column on a deformable membrane (2D)

COUPLINGS

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FLUID-STRUCTURE INTERACTION OPTION 2: Unified PFEM formulation for Lagrangian continua

Fluid domain

Fixed boundary

Solid

t M

Fti = - b Fvi Sign(Vti) Fni = K1(hc - h) – K2 abs(Vni)

Fti

Fni

e

i

Vni

Vti

h < hc

Frictional contact between solids

Contact elements are introduced

between the solid-solid interfaces

during mesh generation

Contact forces

Contact elements at the fixed boundary

t+Dt M

h < hc

Solid

Solid

Contact interface

EO , MA Celigueta S Idelsohn 2006 Oliver et al ( Contact domain method ) 2006-2012

COUPLINGS

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COUPLINGS

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COUPLINGS

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COUPLINGS

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FSI and contact

COUPLINGS

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Fluid

Solid

i j

l k

m

k

n k t k

k

h k

t k

t V

t t

Surface erosion due to fluid forces

0

t

t t g t dt = dt > H

c Then release node k

Fluid

Solid

i j

l k

m

“Worn” domain Wk

k

i j

l

n

0

t

m V t 4 h k

2

t t = m g t

g t =

1 2

V t n

= V

2h k

COUPLINGS

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COUPLINGS

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Computing times for PFEM

Particles

Time

The PFEM

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PFEM vs EULERIAN CFD

PFEM PROS • Free surface tracking • Large deformations • Natural interaction with structures • No need of estimation of the domain or bounding box PFEM CONS • Cost (10%-15% extra for re-meshing but no need of level set) • Delaunay Tesselation hard to parallelize • Conservation not totally fulfilled when re-meshing

The PFEM

The PFEM

Water depth h=9.3 cm

Period T= 1.91s

Pressure sensor are inserted at the free surface level

COMPARISON

Pressure in time: comparison

3D dam break

Data for the 3D experiment are taken from Spheric Workshop conference

May 2006 http://w3.uniroma1.it/cmar/SPHERIC/SPHERICWorkshop.htm

R.Issa and D. Violeau

Set up of the experiment

The PFEM

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EXAMPLE: 3D dambreak

3D MODEL

The vertical wall that supported the water column is drawn up with a velocity v=1.5m/s.

Experimental data refer to the

variation of PRESSURE on 8 points over the step that represents the obstacle.

Obstacle

The PFEM

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3D dam break

Pressure variation on the points on the vertical side of the step

P1 P2

P3 P4

The PFEM

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3D dam break

Pressure variation on the points on the horizontal side of the step

P5 P6

P7 P8

The PFEM

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3D dambreak

Pink line: Fine mesh 56 000 nodes Green line: Coarse mesh 14 000 nodes

P1

P6

The PFEM

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Collapse of a water column on a deformable membrane (3D)

NUMERICAL EXAMPLES (I/II)

Comparison 2D vs 3D

A. Franci, E. Oñate, J.M. Carbonell Unified Lagrangian formulation for solid and fluid mechanics and FSI problems. CMAME, 2015

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COUPLINGS

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COUPLINGS

2 of 4 - PFEM vs SPH

MFEM Idelsohn, S. R., Onate, E., Calvo, N., & Del Pin, F. (2003). The meshless finite element method. International Journal for Numerical Methods in Engineering, 58(6), 893-912.

PFEM Oñate, E., Idelsohn, S. R., Del Pin, F., & Aubry, R. (2004). The particle finite element method. An overview. International Journal of Computational Methods, 1(02), 267-307.

SPH (fluids) Cleary, P.W. and J.J. Monaghan (1993). Boundary interactions and transition to turbulence for standard CFD problems using SPH. Proc. 6 '~ Computational Techniques and Applications Conf., Canberra, 157-165,

SPH Gingold, R.A. and J.J. Monaghan (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not R. Astro. Soc, 181,375-389,

FEM Strang, Gilbert; Fix, George (1973). An Analysis of The Finite Element Method. Prentice Hall. ISBN 0-13-032946-0.

I find SPH in every congress I go to…!

The free surface is very spiky, did

you try to use ‘sub-station’?

one month later…

‘sub-station’ = surface tension

2 of 4 - PFEM vs SPH

• The points carry the information

• The points move with the fluid velocity (ALE possible)

• The continuum is approximated by:

Elements (PFEM) Kernel function (SPH)

• Can interact with structures easily

• Free surface is detected with a geometrical criterion

• Well suited for transient fluids with free surface

• Sub-optimal for transient flows with no free surface

• …

• SPH community is big, PFEM community is small (though FEM community is big, too!)

3 of 4 – PFEM application fields

Stability of objects knocked by waves COUPLINGS

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Punta Langosteira Harbour – A Coruña (Spain) 150 tons each concrete block

.

COUPLINGS

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hmax

Thmax

VT

T(s)

h ola(m)

Ttotal

Parametrized overtopping mass of water COUPLINGS

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The PFEM LANDSLIDES

LANDSLIDES The PFEM

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LANDSLIDES The PFEM

COUPLINGS LIQUEFACTION

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COUPLINGS LIQUEFACTION Barcelona Harbour

OK ????

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LIQUEFACTION COUPLINGS Barcelona Harbour

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LIQUEFACTION COUPLINGS Barcelona Harbour

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LIQUEFACTION COUPLINGS Barcelona Harbour

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COUPLINGS LIQUEFACTION Barcelona Harbour

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COUPLINGS LIQUEFACTION Barcelona Harbour

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-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

CAJON

DE

SP

LA

ZA

MIE

NT

O[m

]

REAL

10 sec

20 sec

30 sec

40 sec

50 sec

Densidad:

M = 2000 kg/m3

R = 1800 kg/m3

Coeficiente de rozamiento:

= 0.6

(si Dx > 10m) = 0.35

COUPLINGS Barcelona Harbour LIQUEFACTION D

ISP

LAC

EMEN

T (m

.)

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COUPLINGS RESCUE AND OBSERVATION ROBOTS

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UUV (Unmanned Underwater Vehicle) in water tank COUPLINGS

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Underwater objects in large 3D domains with waves COUPLINGS

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Falling box

NEW APPROACH TO THE UUV CALCULATION COUPLINGS

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Falling box

0v

nt

LAGRANGIAN CONTROL VOLUME

NEW APPROACH TO THE UUV CALCULATION COUPLINGS

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0v 0v0v

new nodes

nodes removed

nodes kept

nt 1nt

LAGRANGIAN CONTROL VOLUME

NEW APPROACH TO THE UUV CALCULATION COUPLINGS

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COUPLINGS UUV (Unmanned Underwater Vehicle)

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OIL-GAS DRILLING PROCESSES

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COUPLINGS

25 December 2003 _ Devore, California 76

COUPLINGS

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COUPLINGS

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Earthquake and Tsunami at Miyako Japan , March 2011

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TSUNAMI CONSEQUENCES… COUPLINGS

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TSUNAMI CONSEQUENCES… COUPLINGS

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Cutting with PFEM and thermo-mechanical coupling

TEMPERATURE RIGID TOOL

MACHINING PFEM IN SOLIDS

PFEM IN SOLIDS

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Anaysis of damages in the nuclear reactor

pressure vessel caused by the dropping of corium. Objective:

NUMERICAL EXAMPLES (II/II)

Alessandro Franci PARTICLES 2015 Barcelona, 28th September

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Anaysis of damages in the nuclear reactor

pressure vessel caused by the dropping of corium. Objective:

Result of melting of the reactor’s

components;

Highly viscous and corrosive

material;

It can reach

T>2800° C.

NUMERICAL EXAMPLES (II/II)

Alessandro Franci PARTICLES 2015 Barcelona, 28th September

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Detail of the rod melting

𝑡 = 12.3𝑠 𝑡 = 13.1𝑠 𝑡 = 14.7𝑠

𝑡 = 15.1𝑠 𝑡 = 15.5𝑠 𝑡 = 15.9𝑠

𝑡 = 16.3𝑠 𝑡 = 16.7𝑠 𝑡 = 17.1𝑠

NUMERICAL EXAMPLES (II/II)

Alessandro Franci PARTICLES 2015 Barcelona, 28th September

• Velocity of the nodes on the top side was fixed to

zero.

• Imposing face heat flux during 70 sec.

• Viscosity is a function of the temperature

• In the symmetry faces

MODEL SET-UP

For the polymer:

For the air:

• Slip conditions in the vertical walls.

• Fix velocity, Temperature=298 K, Yf=0 and Yo=0.23

at the botton boundary.

• Fix pressure to zero in the top boundary.

• In the symmetry faces

𝜅𝜕𝑇/𝜕𝑛 = 0

𝜅𝜕𝑌𝑘/𝜕𝑛 = 0, 𝜅𝜕𝑇/𝜕𝑛 = 0

Melting and flow of a chair

Governing equations for the polymer

*Momentum equation

𝜌𝐷𝑉

𝐷𝑡= 𝛻. 2𝜇

𝛻𝑉 + 𝛻𝑇𝑉

2− 𝛻𝑝 + 𝜌𝑔

*Mass equation 𝐷𝜌

𝐷𝑡+ 𝜌𝛻. 𝑉 = 0

where 𝐷 𝐷𝑡 represents the material derivative.

If the Newtonian flow is nearly-incompressible, we have 𝑑𝑝 = 𝜅

𝜌 𝑑𝜌

where 𝜅 is the elastic bulk modulus of the fluid.

*Temperature equation 𝐷𝜌𝐶𝑇

𝐷𝑡= 𝛻. 𝜅𝛻𝑇 + 𝑄

Polymer

Governing equations for the air

• Mass equation

• Momentum equation

• Temperature equation

Air

𝜕𝜌

𝜕𝑡+ 𝛻. (𝜌𝑉) = 0

𝜕𝜌𝑉

𝜕𝑡+ 𝛻. 𝜌𝑉𝑉 = 𝛻. 2𝜇

𝛻𝑉 + 𝛻𝑇𝑉

2

−𝛻𝑝 + 𝜌f

𝜕𝜌𝐶𝑇

𝜕𝑡+ 𝛻. 𝜌𝐶𝑉𝑇 = 𝛻. 𝜅𝛻𝑇 + 𝑄

where 𝑄 represents radiation, pressure work, energy release.

‘Melting chair and flow around it’

PFEM embedded in Eulerian CFD

4 of 4 – PFEM future

PFEM vs EULERIAN CFD

PFEM PROS • Free Surface Tracking • Large deformations • Interaction with structures • No need of estimation of the domain or bounding box PFEM CONS • Cost (10%-15% extra for re-meshing but no need of level set) • Conservation not totally fulfilled when re-meshing • Delaunay Tesselation hard to parallelize

Higher order finite elements?

Several papers claim to have this issue solved (to be implemented)

• Kamran, K., Rossi, R., Oñate, E., & Idelsohn, S. R. (2013). A compressible Lagrangian framework for the simulation of the underwater implosion of large air bubbles. Computer Methods in Applied Mechanics and Engineering, 255, 210-225.

• Ryzhakov, P. B., Rossi, R., Idelsohn, S. R., & Oñate, E. (2010). A monolithic Lagrangian approach for fluid–structure interaction problems. Computational mechanics, 46(6), 883-899.

• Oñate, E., Franci, A., & Carbonell, J. M. (2014). A particle finite element method (PFEM) for coupled thermal analysis of quasi and fully incompressible flows and fluid-structure interaction problems. In Numerical Simulations of Coupled Problems in Engineering (pp. 129-156). Springer International Publishing.

• Oñate, E., Idelsohn, S. R., Del Pin, F., & Aubry, R. (2004). The particle finite element method—an overview. International Journal of Computational Methods, 1(02), 267-307.

• Carbonell, J. M., Oñate, E., & Suárez, B. (2009). Modeling of ground excavation with the particle finite-element method. Journal of engineering mechanics, 136(4), 455-463.

• Onate, E., Idelsohn, S. R., Celigueta, M. A., & Rossi, R. (2008). Advances in the particle finite element method for the analysis of fluid–multibody interaction and bed erosion in free surface flows. Computer methods in applied mechanics and engineering, 197(19), 1777-1800.

Many fluid, solid or mixed solvers used in PFEM…

After PFEM was implemented in the Kratos framework (several months): • Coupling with thermal solver required 1 hour of work (one-way coupling)

• Coupling with DEM required 2 days of work (two-way coupling)

• Parallelism (OpenMP) was ‘for free’. Assemblers and solvers were coded

previously by other developers in other fields.

THANK YOU FOR YOUR ATENTION!

Miguel A. Celigueta, S. Latorre, S. Idelsohn , E. Oñate,

J.M. Carbonell, P. Ryzhakov, A. Franci , J. Marti

maceli@cimne.upc.edu

International Center for Numerical Methods in Engineering

Technical University of Catalonia (UPC)

Barcelona. Spain

Particle Finite Element Method (PFEM)

m

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www.cimne.com/pfem www.cimne.com/kratos www.cimne.com/dempack