Titolo presentazione Massimiliano Cremonesi sottotitolo · A.Franci, E. Onate, J.M. Carbonell, "Unified Lagrangian formulation for solid and mechanics and FSI problems". Computer

Titolo presentazione

sottotitolo

Milano, XX mese 20XX

Numerical simulation of propagating landslides

Massimiliano Cremonesi

Umberto Perego, Attilio Frangi, Francesco Ferri, Simone Meduri

Department of Civil and Environmental Engineering

Politecnico di Milano


Landslides in Europe

Spatial distribution of fatal (death, injury, and missing) landslides

The territory of Europe is highly exposed to slope processes due to its geological, geomorphological,

and climate variations. Such slope processes include slope mass movements that can be further

categorized into landslides, soil slips, debris flow, and rock falls. Recently, triggered by

increasingly frequent extreme weather events, mass movements in many European countries have

become common natural phenomena and have caused considerable damage and economic

losses.

• 1.3 to 3.6 million Europeans live in landslide prone areas

• 8000 to 20,000 km of roads and railways are highly exposed to landslides

Haque, U., Blum, P., da Silva, P.F. et al. Landslides (2016) 13: 1545. doi:10.1007/s10346-016-0689-3

“The highest losses per percentage of gross domestic product (GDP) with 0.19 % worldwide

however occur in Italy, which along with the USA, Japan, and India suffer the highest economic

impact from landslides” (Haque et al., 2016)


Landslides in Italy

Italian National Institute for Environmental Protection and Research presented in 2016 a database of

528903 active landslides on the Italian territory covering 22.176 km2, which represents 7,3% of the

national territory.

Trigila A., Iadanza C., Bussettini M., Lastoria B., Barbano A. (2015) Dissesto idrogeologico in Italia: pericolosità e indicatori di rischio. Rapporto 2015. ISPRA, Rapporti 233/2015 (ISBN 978-88-448-0751-1)


Landslides

Catastrophic landslides are exceptional natural hazard conditioned by several topographic

and environmental aspects related to soil properties, geological structure, lithology and

weathering conditions, slope morphology, land cover, and water flow.

Triggering mechanisms:

• the rapid changes in groundwater level and/or flow

• precipitation (both average and peak)

• natural erosion

• snowmelt

• earthquakes and volcanic processes

• human activities (excavations, mining, deforestation, irrigation …)

Often, it can be a combination of natural and human activities that induces landslides

The numerical simulation can be is useful for preventing and mitigating the consequences ofthese events.


Objectives

Two different objectives:

- Given a stable slope in a geostatic state, predict

- the insurgence of instability under evolving external conditions (heavy rainfalls, earthquakes, excavations, constructions of civil infrastructures or buildings,….)

- Given an unstable slope, simulate the landslide runout and predict

- the landslide path,

- the runout distance,

- the landslide action onto existing buildings or infrastructures,

- the generated waves in the case of a landslide impinging into a water reservoir,….


Numerical simulation of propagating landslides: what we need?

Equations of motion

Constitutive behavior

Numerical tool

accounting for extremely large displacements and deformations

track free-surfaces and interfaces

accounting for mixing of different constituents

Landslide-structure interaction

Boundary conditions

Equations of motion


Landslides – fluid or solid?


Balance equations

Navier-Stokes equations are written in ALE form in a moving reference domain:

c = convective velocity defined as:u fluid velocity

v mesh velocity

momentum conservation:

mass conservation:

v = 0 (i.e. c = u) => mesh is fixed => standard Eulerian description is recovered

v = u (i.e. c = 0) => mesh moves at fluid velocity => Lagrangian description is obtained

Lagrangian description is adopted almost everywhere


Why Lagrangian?

Typically in fluid-dynamics Eulerian or ALE formulations are preferred.

Advantages of Lagrangian approach

• Natural treatment free-surfaces

• No convective terms (eqs. are still non-linear)

• Interfaces are automatically defined by the particles (nodes) position

• Fluid-structure interaction

Disadvantages of Lagrangian approach

• Excessive mesh distortion (if mesh-based solver are used)

• Complex definition of outflow and inflow

• Boundary condition can be difficult to imposed



Equations of motion


Numerical tool





Boundary conditions


Modelling of soil behavior

A simplified model as been chosen for the landslide constitutive behavior.

The granular material has been modeled as an visco-plastic non-Newtonian

Bingham-like constitutive model.

The shear stress is related to the shear strain 𝛾:

𝑝 = pressurec = cohesion𝜑 = friction angle𝜇 = viscosity

Cohesive-frictional flowing granular material behavior modeled by Mohr-Coulomb

criterion

The Cauchy stress tensor is decomposed into its hydrostatic and deviatoriccomponents:

Papanastasiou regularization:

(Papanastasiou TC. Flows of materials with yield. Journal of Rheology 1987; 31:385–404)



Equations of motion


Numerical tool





Boundary conditions

• Finite element method

• Fast remeshing


Perform the Delaunay

triangulation

Identify the external and

internal boundaries

Solve the linearized

Navier-Stokes equation

Move the mesh nodes to

the new position

Check

convergence

Check mesh

distortion

t = tn+1t = tn+1

k=k+1

yes

yes

no

no

The steps of method

2D

3D


alpha shape method

Triangulation and boundary identification

alpha-shape method: remove the unnecessary triangles to find the real shape using a

criterion based on the mesh distortion ( Edelsbrunner & Mucke(1994); Oñate et al(2004) )

the minimal distance between two

nodes in the elementeh

eR

criterion h

Re1

the radius of the circumcircle

mean value of heh

Delaunay triangulation

2D

3D


Alpha-shape

Separation of a particle

Inclusion of a particle

the motion of the separated particles is governed by the body force and the initial

velocity which they are subjected to


Space and time discretization

Delaunay Tessellation to regenerate frequently the connectivity:

• only triangles (in 2D) and tetrahedra (in 3D) can be used;

• to avoid interpolation from mesh to mesh only linear shape functions can be used (P1)

space and time

discretization

mixed formulation (velocity and pressure)

Standard finite element space discretization

Forward Euler scheme for time integration


Stabilization

to avoid interpolation from mesh to mesh linear shape functions are

used for velocity and pressure

LBB compatibility condition is not satisfied

Pressure Stabilizing Petrov Galerkin stabilization*

*Tezduyar (1991)


Impact on a rigid object

Koshizuka et al. (1995)



t=0,2 s

t=0,3 s

t=0,4 s



t=0,2 s

Fre

e-s

urf

ace p

ositio

n

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

experimental

numerical



Equations of motion


Numerical tool





Boundary conditions

• fluid-structure interaction

• fluid-fluid interaction


Fluid-structure interaction

two distinct analyses are performed:

- a fluid analysis in the fluid domain;

- a structural analysis in the solid domain.

a coupled analysis is performed

Cremonesi M., Frangi A., Perego U., (2010) A Lagrangian finite element approach for the analysis of fluid-structure

interaction problems, International Journal for Numerical Methods in Engineering, vol 84, pp. 610–630


Fluid-structure interaction

coupled analysis:

Dirichlet-Neumann algorithm

• Strong coupling

• Compatible interfaces

• Same time step for fluid and solid phases (implicit/implicit coupling)

Coupling using the domain decomposition approach (Gravouil-Comberscure*)

• Loose coupling

• Different discretization of the fluid/solid interface (no compatible meshes)

• Different time step for fluid and solid phases (explicit/explicit coupling)

(see Meduri’s PhD thesis)

*A. Gravouil , A. Combescure, “Multi-time-step explicit-implicit method for non-linear structural dynamics.” International Journal for NumericalMethods in Engineering, 50: 199-225, 2001.


Impact on a elastic object

L = 14.6cm

h = 1.2 cm

d = 20/3 h

rho =2500 Kg/m3

E = 106 Kg/s2m

nu = 0

Walhorn et al. (2005); Idelsohn et al. (2008)


Impact on a elastic object

* E. Walhorn, A. Kolke, B. Hubner, D. Dinkler, Fluid–structure coupling with monolithic model involving free surface flows. Comp.

Struct, 83 (2005) 2100–2111

** S.R. Idelsohn, J. Marti, A. Limache, E. Oñate, Unified Lagrangian formulation for elastic solids and incompressible fluids:

Application to fluid–structure interaction problems via the PFEM. Comput. Methods Appl. Mech. Engrg., 197 (2008) 1762–1776.

-4.00E-02

-3.00E-02

-2.00E-02

-1.00E-02

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

0.00E+00 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

Def

lect

ion

[m

]

Time [s]

Walhorn et al. *

Idelsohn et al. **

present method


Deformation of an elastic gate

Antoci et al. (2007)

A = 0.1 m

H = 0.14 m

B = 0.1 m

L = 0.079 m

s = 0.005 m

rho = 1100 Kg/m3

E = 1,2 x107 N/m2



t=0,24s

t=0,40 s

t=0,08 s


0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Dis

pla

cem

ent

[m]

Time [s]

x-disp exp

x-disp computed

x-disp SPH

y-disp exp

y-disp computed

y-disp SPH


* C. Antoci, G. Gallati, S. Sibilla, Numerical simulation of fluid–structure interaction by SPH. Computers and Structures 85

(2007) 879–890

*

*


Filling of an elastic container

A.Franci, E. Onate, J.M. Carbonell, "Unified Lagrangian formulation for solid and mechanics and FSI problems". Computer Methods in Applied mechanics and Engineering, 298(2016) 520-547

Geometry

h 2.5 𝑚

H 3.75 𝑚

R 2.25 𝑚

B 1.3 𝑚

B 4.8714 𝑚

s 0.2 𝑚

Fluid

Density 1000 𝑘𝑔/𝑚3

Viscosity 50/100 𝑃𝑎 ∙ 𝑠

Bulk Modulus 1.75 ∙ 107 𝑃𝑎

# of Nodes 15039

Structure

Density 20 𝑘𝑔/𝑚3

Young's Modulus 2.1 ∙ 107 𝑃𝑎

Poisson Ratio 0.3

# of Nodes 5248

CPU TIME 25 ℎ


Filling of an elastic container

(see Meduri’s PhD thesis)

* A.Franci, E. Onate, J.M. Carbonell, "Unified Lagrangian formulation for solid and mechanics and FSI problems". Computer Methods in Applied mechanics and Engineering, 298(2016) 520-547

present approach

Franci et al. (2016)



Equations of motion


Numerical tool





Boundary conditions


Landslide-substrate boundary conditions

• Standard fluid mechanics boundary conditions, based on macroscopic observation of physical interaction phenomena:

perfect coupling between fluid and containing wall => u = 0 on the boundary

• Molecular dynamics simulations show relative slip/layers between fluid and containment wall, depending of relative density, wall rugosity etc

• Experiments on macroscopic granular material avalanches => increasing relative slip is observed for increasing grain size

(Schaefer, M., Bugnion, L., Kern, M. & Bartelt, P. Granular Matter 12, 327-336, 2010)


Landslide-substrate boundary conditions

Slip in a Couette flow

tangential component of the traction acting on the surface of normal n

a parameter defining the amount of slip, a threshold stress

Concept of “slip length hslip“ in Couette flow (P.A.Thompson, M.O.Robbins, Phys. Rev. A, 1990)

hslip >0 => slip

hslip = 0 => perfect coupling

t

slipuS

on S

slip

p

t I n I n n

u u I n n

t = tangential traction acting on landslide material at basal interface

uslip = landslide sliding velocity at basal interface

Classical Navier boundary conditions: with bulk viscosityslip

sliph

t u

3D extension:


Slip boundary conditions

Extended Navier-type b.c. with frictional pressure-dependent threshold

tan with

tan

1tan for tan 0

tan1 for tan 0 for tan

slip

slip basal

slip basal

slip basal basal

slip

slip

basalbasal slip b

slip

slip

slip

hp

p

p p

pp p

tu t

t

u t

t u t

uu t

tt t u t

utt u

u

0asal

regularized Navier-type slip law

no interpenetration condition

on

0

slip

S

ut

u n

tan1slipNslip basal

slip

h pe

u

u has the dimensions of a length over a viscosity

Exponential regularization of the slip coefficient:

(see Ferri’s PhD thesis)

tan basalp


Slip boundary condition at corners

• Problem with enforcing slip boundary conditions at corner boundary nodes in discretized boundaries

jump in normal directions => locking in case of strong enforcement of b.c.

no slip

Behr, M. On the application of slip boundary condition on curved boundaries. Int. J. Numer. Meth. Fluids 45, 43-51 (2004).

Engelman, M. S., Sani, R. L. & Gresho, P. M. The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow. Int. J. Numer. Meth. Fluids 2, 225-238 (1982)

Slip condition + no-interpenetration condition = LOCKING!

1

0C

d

v n u n u n

• No-interpenetration condition enforced in weak form through penalization

Dione, I., Tibirna, C. & Urquiza, J. Stokes equations with penalised slip boundary conditions. International Journal of Computational Fluid Dynamics 27, 283-296 (2013)

=> addition of a penalization term to the weak form (v = trial function, = penalization parameter)

Cremonesi M., Ferri F., Perego U. (2017) A basal slip model for Lagrangian finite element simulations of 3D landslides,

International Journal for Numerical and Analytical Methods in Geomechanics, vol 41, pp 30-43


Slip nodes

slip nodes

Lagrangian nodes

(c = 0)

Lagrangian formulation (c = 0) is adopted everywhere with the only exception of

boundary nodes at landslide-substrate interface, to be able to enforce slip boundary

conditions.

(fixed and c = u)

+ stabilization terms

T

slip c

D

Dt

uM x K x K x K x u D x P B

D x U 0


Granular material flowing on inclined planes

Huston sand

total volume = 301 cm3 density = 1280 Kg/m3

friction angle = 34° basal friction angle = 28°

mesh nodes = 72499

• I. Manzella, Dry rock avalache propagation: unconstrained flow experiments with granular materials and blocks at

small scale. EPFL Lausanne, 4032, 2008.

• M. Pastor, B. Haddad, G. Sorbino, S. Cuomo, V. Drempetic, A depth-integrated, coupled SPH model for flow-like

landslides and related phenomena International Journal for Numerical and Analytical Methods in Geomechanics ,

Vol: 33(2), 2009


Granular flow: 1 plane


Granular flow: 1 plane


Granular flow: 2 planes


Granular flow: 2 planes


Granular flow: 2 planes (final deposit)


Granular mass on erodible substrate

Crosta, G.B, De Blasio, F.V., De Caro, M., Volpi,

G, Imposimato, S, Roddeman, D. Modes of

propagation and deposition of granular flows onto

an erodible substrate: experimental, analytical,

and numerical study. Landslides (2016):1–22.


The Frank Slide, Alberta, Canada 1903

Over 90 million tons of limestone rock slid down TurteMountain within 100 seconds, destroying the easternedge of Frank.

Between 70 and 90 of the town's residents were killed.

Native tribes were used to call Turtle Mountain "themountain that moves“ for its well-known instability.Coal mining operations may have further weakened themountain's internal structure.

The Frank Slide of 1903 was the deadliest landslide disaster of Canada.

The rock detached from the ridge of Turtle Mountain (700 m wide x 400 m).

The deposit is about 1.7 km wide, almost 2 km long and 18 m thick on average.

The Frank Slide buried part of the mining town of Frank, Northwest Territories, Canada, at 4:10 am of April 29, 1903.

Pastor, M., Blanc, T., Pastor, M.J.: A depth-integrated viscoplastic model for dilatant saturated cohesive-frictional fluidized

mixtures: Application to fast catastrophic landslides. J. Nonnewton. Fluid Mech. 158, 142–153 (2009)






Waves induced by landslides: Vajont tragedy (1963)


Waves induced by landslides: Vajont tragedy (1963)

Town of Longarone before the wave Town of Longarone after the wave


Vajont landslide


0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

900.00

0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00 3500.00WAVE HEIGHT- RIGHT BANK

In situ records

Numerical Simulation

= 0,01 Pa s

= 024°

basal = 0°

hslip = 10000 m

typical elementlength = 20 m

Vajont landslide


Conclusions & Future Developments

• Lagrangian Finite Element technique for landslides simulations;

• landslide modeled as non-Newtonian Bingham-like fluid;

• Navier-like slip boundary conditions with frictional pressure dependent

threshold to better represent interaction between landslide and slope;

• approach validated with numerical test against experimental results and

real records of real landslides.

Perspectives:

• more realistic constitutive law for landslide material, allowing for initial

static equilibrium and instability triggering

Redaelli, I., di Prisco, C. & Vescovi, D. A visco-elasto-plastic model for granular materials under simple shear

conditions. Int. J. Numer. Anal. Meth. Geomech. (2015)

• further validation on real landslides;

• speed-up calculation to allow for practical engineering applications.

Titolo presentazione

sottotitolo

Milano, XX mese 20XX

Thank you for your attention


Department of Civil and Environmental Engineering

Politecnico di Milano

[email protected]

Documents

Titolo presentazione Massimiliano Cremonesi sottotitolo · A.Franci, E. Onate, J.M. Carbonell, "Unified Lagrangian formulation for solid and mechanics and FSI problems". Computer