particle finite element method (pfem) · 2017-06-22 · finite element method for the analysis of...
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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
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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
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1 of 4 – What is the PFEM?
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MOTIVATION:
• Fluid • Free surface • Deformable solid • Fluid-structure-interaction • Fluid separation
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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
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‘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.
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The Particle Finite Element Method (PFEM)
Numerical model based on:
- lagrangian approach
- fast re-meshing algorithm
- boundary recognition method
- the classic FEM
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solid mechanics fluid mechanics
The remeshing step allows large deformations
- the classic FEM
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The PFEM
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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.
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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
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• 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…
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(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
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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
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The PFEM
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The PFEM
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Water depth h=9.3 cm
Period T= 1.91s
Pressure sensor are inserted at the free surface level
COMPARISON
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Pressure in time: comparison
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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
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2 of 4 - PFEM vs SPH
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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…!
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The free surface is very spiky, did
you try to use ‘sub-station’?
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one month later…
‘sub-station’ = surface tension
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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)
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• 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!)
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3 of 4 – PFEM application fields
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Stability of objects knocked by waves COUPLINGS
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Punta Langosteira Harbour – A Coruña (Spain) 150 tons each concrete block
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.
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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
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LANDSLIDES The PFEM
55
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LANDSLIDES The PFEM
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COUPLINGS LIQUEFACTION
57
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COUPLINGS LIQUEFACTION Barcelona Harbour
OK ????
58
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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
.)
65
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COUPLINGS RESCUE AND OBSERVATION ROBOTS
67
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UUV (Unmanned Underwater Vehicle) in water tank COUPLINGS
68
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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
70
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Falling box
0v
nt
LAGRANGIAN CONTROL VOLUME
NEW APPROACH TO THE UUV CALCULATION COUPLINGS
71
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0v 0v0v
new nodes
nodes removed
nodes kept
nt 1nt
LAGRANGIAN CONTROL VOLUME
NEW APPROACH TO THE UUV CALCULATION COUPLINGS
72
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COUPLINGS UUV (Unmanned Underwater Vehicle)
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OIL-GAS DRILLING PROCESSES
74
COUPLINGS
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25 December 2003 _ Devore, California 76
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COUPLINGS
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COUPLINGS
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Earthquake and Tsunami at Miyako Japan , March 2011
79
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TSUNAMI CONSEQUENCES… COUPLINGS
80
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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
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PFEM IN SOLIDS
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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
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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
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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
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• 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
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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
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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.
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‘Melting chair and flow around it’
PFEM embedded in Eulerian CFD
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4 of 4 – PFEM future
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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)
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• 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…
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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.
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THANK YOU FOR YOUR ATENTION!
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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
99
www.cimne.com/pfem www.cimne.com/kratos www.cimne.com/dempack