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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)
m
1
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
9
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
12
Cloud of
nodes
ALPHA
SHAPE:
Boundary
element if
r(x)≥α h(x)
a1 a2
a2<a1
Delaunay
triangulation
Boundary surface recognition The PFEM
13
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
15
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
18
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
20
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
21
COUPLINGS
22
COUPLINGS
23
COUPLINGS
24
FSI and contact
COUPLINGS
25
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
26
COUPLINGS
27
Computing times for PFEM
Particles
Time
The PFEM
28
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
35
3D dam break
Pressure variation on the points on the vertical side of the step
P1 P2
P3 P4
The PFEM
36
3D dam break
Pressure variation on the points on the horizontal side of the step
P5 P6
P7 P8
The PFEM
37
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
40
COUPLINGS
41
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
49
Punta Langosteira Harbour – A Coruña (Spain) 150 tons each concrete block
.
COUPLINGS
52
hmax
Thmax
VT
T(s)
h ola(m)
Ttotal
Parametrized overtopping mass of water COUPLINGS
53
54
The PFEM LANDSLIDES
LANDSLIDES The PFEM
55
LANDSLIDES The PFEM
COUPLINGS LIQUEFACTION
57
COUPLINGS LIQUEFACTION Barcelona Harbour
OK ????
58
LIQUEFACTION COUPLINGS Barcelona Harbour
59
LIQUEFACTION COUPLINGS Barcelona Harbour
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LIQUEFACTION COUPLINGS Barcelona Harbour
61
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
.)
65
COUPLINGS RESCUE AND OBSERVATION ROBOTS
67
UUV (Unmanned Underwater Vehicle) in water tank COUPLINGS
68
Underwater objects in large 3D domains with waves COUPLINGS
69
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
71
0v 0v0v
new nodes
nodes removed
nodes kept
nt 1nt
LAGRANGIAN CONTROL VOLUME
NEW APPROACH TO THE UUV CALCULATION COUPLINGS
72
COUPLINGS UUV (Unmanned Underwater Vehicle)
73
OIL-GAS DRILLING PROCESSES
74
COUPLINGS
25 December 2003 _ Devore, California 76
COUPLINGS
77
COUPLINGS
78
Earthquake and Tsunami at Miyako Japan , March 2011
79
TSUNAMI CONSEQUENCES… COUPLINGS
80
TSUNAMI CONSEQUENCES… COUPLINGS
81
Cutting with PFEM and thermo-mechanical coupling
TEMPERATURE RIGID TOOL
MACHINING PFEM IN SOLIDS
PFEM IN SOLIDS
87
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
88
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
89
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
International Center for Numerical Methods in Engineering
Technical University of Catalonia (UPC)
Barcelona. Spain
Particle Finite Element Method (PFEM)
m
99
www.cimne.com/pfem www.cimne.com/kratos www.cimne.com/dempack
