M. StupazziniM. Stupazzini

Grenoble 3D benchmarkGrenoble 3D benchmark

23rd of July July 20020066Kinsale – Ireland (3rd SPICE workshop)Kinsale – Ireland (3rd SPICE workshop)

Politecnico di MilanoPolitecnico di Milano Dep. of Structural EngineeringDep. of Structural Engineering

Ludwig Maximilians UniversityLudwig Maximilians University DepDep.. of Earth and Environmental Sciences - Geophysics of Earth and Environmental Sciences - Geophysics

Center for Advanced ResearchCenter for Advanced Researchand Studies in Sardiniaand Studies in Sardinia

Seismic wave Propagation and Imaging in

Complex media: a European network MARCO STUPAZZINI

Experienced Researcher

Host Institution: LMU Munich

Date of Birth: 26 / 10 / 1974

Place of Origin: Milano, Italy

Key Words: Computational Seismology, Spectral Element Method, Visco Plasticity, Soil-Structure Interaction

Appointment Time: May 2004

Task Groups: TG 2: Numerical Methods

MARCO STUPAZZINI

Experienced Researcher

Host Institution: LMU Munich

Date of Birth: 26 / 10 / 1974

Place of Origin: Milano, Italy

Key Words: Computational Seismology, Spectral Element Method, Visco Plasticity, Soil-Structure Interaction

Appointment Time: May 2004

Task Groups: TG 2: Numerical Methods

3D numerical simulation of seismic wave propagation in the Grenoble valley (M6 earthquake)

3D numerical simulation of seismic wave propagation in the Grenoble valley (M6 earthquake)

36 km36 km

30 km30 km

SDElements

#Nodes

#Memory

[Mb]tsimulation

[sec.]

time steps

#

Total time simulation

[s.]

Total CPU time (10 CPUs) [min.]

3 216972 6.05·106 5282 0.246E-03 121951 30 3070.45

4 216972 14.2·106 11211 0.154E-03 195187 30 11177.35

Synthetic seismograms

Peak ground velocities

Spectral ratios for the fault parallel component

Max. Displacement

E. Faccioli and A. Rovelli: Project S5 of “DPC-INGV” "Definizione dell’input sismico sulla base degli spostamenti attesi"

(1 giugno 2005 - 30 giugno 2006)

Bedrock

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 24.00

Hypocentral distance [km]

Max D

isp

l. [

m]

Strong 1

Strong 2

Attenaution relationship

Max. Displacement

E. Faccioli and A. Rovelli: Project S5 of “DPC-INGV” "Definizione dell’input sismico sulla base degli spostamenti attesi"

(1 giugno 2005 - 30 giugno 2006)

Alluvial

0.00

0.05

0.10

0.15

0.20

0.25

0.30

6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00

Hypocentral distance [km]

Max D

isp

l. [

m]

Strong 1

Strong 2

Attenaution relationship

ConclusionsConclusions

www.spice-rtn.org

www.stru.polimi.it/Ccosmm/ccosmm.htm

1st

2nd

3rd

SubsidenceLiquefactionLandslides

General problemGeneral problem

A sub-structuring method : the Domain Reduction Method(Bielak et al. 2003)

Local geological feature

Pe(t)

Soil-Structure interaction

Inner Inner regionregion

External regionExternal region

EFFECTIVE NODAL FORCESEFFECTIVE NODAL FORCES PP

Boundary regionBoundary region

Method for the simulation of seismic wave propagation from a half space containing the seismic source to a localized region of interest, characterized by strong geological and/or topographic heterogeneities or soil-structure interaction.

• The free field displacement u0 may be calculated by different methods

Step I ( AUXILIARY PROBLEM )

• The auxiliary problem simulates the seismic source and propagation path effects encompassing the source and a background structure from which the localized feature has been removed.

Pe(t)

Analytical solutions(e.g.: Inclined incident waves)

Numerical method(e.g.: FD, SEM, BEM, ADER-DG)

DRM : 2 steps method

• The reduced problem simulates the local site effects of the region of interest

• The input is a set of equivalent localized forces derived from step I

• The effective forces act only within a single layer of elements adjacent to the interface between the external and internal regions where the coupled term of stiff matrix does not vanish

EFFECTIVE NODAL FORCESEFFECTIVE NODAL FORCES

ui

we

ub

Inner regionInner region

External regionExternal region

Boundary regionBoundary region

iL o

b be eL o

e eb b

0P

P P K u

P K u

Inner regionInner region

Boundary regionBoundary region

External regionExternal region

Step II ( REDUCED PROBLEM )

DRM : 2 steps method

Study case

railway bridgerailway bridge

Wave propagation in 2D

“ Site effects “ & “ Soil Structures Interactions “

“Source“ &

“ Deep propagation“

Fault

zoomzoom

zoomzoom

Computational comparison:

SimulationSimulation # elem.# elem. MemoryMemory

[Mb][Mb]

ttsimulationsimulation

[sec.][sec.]

# time # time stepssteps

Tot. CPU time Tot. CPU time [min.][min.]

Single Single modelmodel

2790 15 1.177 10-5 570 620 190.0

Computational comparison:

SimulationSimulation # elem.# elem. MemoryMemory

[Mb][Mb]

ttsimulationsimulation

[sec.][sec.]

# time # time stepssteps

Tot. CPU time Tot. CPU time [min.][min.]

Single Single modelmodel

2790 15 1.177 10-5 570 620 190.0

DRMDRM

11stst step step2370 14 5.5 10-4 18 362 5.5

Computational comparison:

SimulationSimulation # elem.# elem. MemoryMemory

[Mb][Mb]

ttsimulationsimulation

[sec.][sec.]

# time # time stepssteps

Tot. CPU time Tot. CPU time [min.][min.]

Single Single modelmodel

2790 15 1.177 10-5 570 620 190

DRMDRM

11stst step step2370 14 5.5 10-4 18 362 5

DRMDRM

22ndnd step step585 18 1.177 10-5 570 620 64

++

The computationThe computationwith DRM iswith DRM is

2.8 times faster2.8 times faster

Kinematic source:Kinematic source:Seismic moment tensor density

(Aki and Richards, 1980):

MW = 4.2, slip = 50 cm

Dynamic rupture modellingDynamic rupture modelling(Festa G., IPGP)(Festa G., IPGP)Interface behavior via frictionSlip weakening law + Stress distribution

Initial Principal stresses :4.0 107 Pa 1

1.8 108 Pa 3

100° Orientation0.67 Static friction0.525 Dynamic friction0.4 m DC

150-300m Cohesive zone thickness

Comparison

Comparison

OutlookOutlook

DRMDRM

Study Study casecase

GeoELSE

GeoELSE (GEO-ELasticity by Spectral

Elements)• GeoELSE is a Spectral Elements code for the

study of wave propagation phenomena in 2D or 3D complex domain

• Developers:- CRS4 (Center for Advanced, Research and Studies

in Sardinia)- Politecnico di Milano, DIS (Department of Structural

Engineering)

• Native parallel implementation

• Naturally oriented to large scale applications ( > at least 106 grid points)

Formulation of the elastodynamic Formulation of the elastodynamic problemproblem

Dynamic equilibrium in the weak form:Dynamic equilibrium in the weak form:

iiiiijijii vfvtddvut

2

2

wherewhere u uii = unknown displacement function= unknown displacement function

vvii = generic admissible displacement function (test function)= generic admissible displacement function (test function)

ttii = prescribed tractions at the boundary = prescribed tractions at the boundary

ffii = prescribed body force distribution in = prescribed body force distribution in

Time advancing schemeTime advancing scheme

Finite difference 2Finite difference 2ndnd order (LF2 – LF2) order (LF2 – LF2)

2

2 2

2

2

t t

t t

n+1 n n-1

n+1 n-1

u u uu

u uu

Spatial discretizationSpatial discretization

Spectral element method SEM Spectral element method SEM (Faccioli et al., 1997)(Faccioli et al., 1997)

min

c

xt

Courant-Friedrichs-Levy (CFL) stability conditionCourant-Friedrichs-Levy (CFL) stability condition

Suitable for modelling a variety of physical problems Suitable for modelling a variety of physical problems (acoustic and elastic wave propagation, thermo elasticity, (acoustic and elastic wave propagation, thermo elasticity, fluid dynamics)fluid dynamics)

Accuracy of high-order methodsAccuracy of high-order methods

Suitable for implementation in parallel architecturesSuitable for implementation in parallel architectures

Great advantages from last generation of hexahedral Great advantages from last generation of hexahedral mesh creation program (e.g.: CUBIT, Sandia Lab.)mesh creation program (e.g.: CUBIT, Sandia Lab.)

Why using spectral elements Why using spectral elements ??

NNNh ehCuu ,

Why using spectral elements ? acoustic problem

n=1n=1

Acoustic wave propagation through an irregular domain. Acoustic wave propagation through an irregular domain. Simulation with spectral degree 1 Simulation with spectral degree 1 (left) (left) exhibits numerical exhibits numerical dispersion due to poor accuracy.dispersion due to poor accuracy.

n=2n=2

Simulation with spectral degree 2 Simulation with spectral degree 2 (right) (right) provides better provides better results.results. Change of spectral degree is done at Change of spectral degree is done at run timerun time..

Navier’s equation:

bPbu

eu

eP

+Γ

Γ

+Fault

iu

bu

Γb-P

Internal domain

External domain

0i iii ib ii ib

b b bbi bb bi bb

u uM M K Kin

u u PM M K K

Internal domain:

bb be b bb be b b

e e eeb ee eb ee

M M u K K u P

u u PM M K Kin

External domain:

0

0

0 0

0 0

bb

ii ib ii ibi i

bi bb bibe bb be

e e eeb ee eb e

bb

e

b b

M M K Ku u

M M KM M K K

u u PM M K

u K

K

u

DRM : 2 steps method

ujo = vector of nodal displacements j = i, b, e

Pbo

= forces from domain + to 0

AUSILIARY PROBLEM (Step I)

0bP

bu0

eu0

eP

+Γ

Γ

+Faglia

iu0

bu0

Γ0b-P

0 0 0 0 0b bb b be e bb b be eP M u M u K u K u

Internal domain (0)

External domain (0)

Mass and stiffness matrices do not change because properties in + do not change

0 0 0

0 0

bb be bb beb b b

e e eeb ee eb ee

M M K Ku u P

u u Pin

M M K K

External domain (0):

Change of variables :0

0

b b b

e e e

i iu w

u

u

u

u

w

w

DRM : 2 steps method

0

0bb be b bb be b b b

e eeb ee eb ee

M M w K K w P P

w wMin

M K K

0b b bb b be e bb b be eP P M w M w K w K w

External domain - External domain (0):

0i iii ib ii ib

b b bbi bb bi bb

u uM M K Kin

u u PM M K K

Dominio interno:

00

0 0

0 00

0 0

0

bb be bb be

e eeb ee eb ee

bb bbbb

e

ii ib ii ibi i

bi bb b bi b

b e

b b

b

M M K K

w wM M K K

M KuP

M

M M K Ku u

M M u K K u

K

0bu

0 0 0 0 0b bb b be e bb b be eP M u M u K u K u

DRM : 2 steps method

0 0

0 0

0

0

0 0

0

bb be bb b

ii ib ii ibi i

bi bb b b e

e eeb ee eb ee

be e be e

eb b eb

i bb

b

bM M K K

w

M M K Ku u

M M u K

wM M K K

M u K u

M u K u

K u

• M and K matrices of the original problem• P localization within a single layer of elements in + adjacent to

+ΓeP

+

Γ eΓ

bP i

b

e

P

P P

P

(Step II)

iuΓ

eΓ

bu

eP

bP

+̂

+̂

ew

REDUCED PROBLEM (Step II)0 0 0

0 0 0

0

be e be e be e

eb b eb b eb b

M u K u C u

M u K u C u

DRM : 2 steps method

1

2

00

0 0 ( , )

0 ( , )

0 0 0

0

0

0

0

0

bb be bb be

e eeb ee eb ee

bb b

ii ib i i i b

bi bb b b

b

b

b

e

i

M M u u F

M M K

u u

M M K

w wM M

u u F u u

K K

M uP

M

0

0

bb b

eb

K u

K

Non linear properties in the internal domain

The effectiveness of the method depend on the accuracy of the absorbing boundary conditions

iuΓ

eΓ

bu

eP

bP

+̂

+̂

ew

DRM : 2 steps method

DRM : 2D Validations using Spectral Elements (GeoELSE)

Homogeneous valley in a layered half space

Mechanical properties

VS [m/s] VP [m/s] [m/s]

Valley 45 105 1000

Half space

50 100 1200

80 140 1600

100 180 1800

DRM : 2D Validations using Spectral Elements (GeoELSE)

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

0 5 10 15 20 25 30-0.02

00.02

t (s)

DRM w single step

y = 1872

x = 0

x = - 50

x = - 100

x = - 156

x = - 208

x = - 250

x = - 300

x = - 390

y = 1872

x = 0

x = - 50

x = - 100

x = - 156

x = - 208

x = - 250

x = - 300

x = - 390

y = 1872

x = 0

u x (m

)

Relative displacements (w)Total displacements (u=w+uo)

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

-0.020

0.02

0 5 10 15 20 25 30-0.02

00.02

t (s)

DRM u=w+uo

single step

y = 1872

x = 0

x = - 50

x = - 100

x = - 156

x = - 208

x = - 250

x = - 300

x = - 390

u x (m)

Homogeneous valley in a layered half space

Internal points

External points

Canyon in a homogeneous half space

Mechanical properties

VS [m/s] VP [m/s] [m/s]

Canyon 50 100 1200

Half space 80 140 1600

DRM : 2D Validations using Spectral Elements (GeoELSE)

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

0 5 10 15 20 25 30-0.02

0

0.02

t (s)

DRM wsingle step

y = 1872

x = - 80

x = - 100

x = - 156

x = - 208

x = - 250

x = - 300

x = - 390

u x (m

)

Relative displacements (w)Total displacements (u=w+uo)

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

0 5 10 15 20 25 30-0.02

0

0.02

t (s)

DRM u=w+u0

single step

y = 1872

x = - 80

x = - 100

x = - 156

x = - 208

x = - 250

x = - 300

x = - 390

u x (m

)

Internal points

External points

Canyon in a homogeneous half space

DRM : 2D Validations using Spectral Elements (GeoELSE)

Calculation of effective forces Pb and Pe

ORIGINAL PROBLEM

II STEP Analysis of wave propagation inside the

reduced model.Interface elements

Nodes eNodes e

Nodes b

eΓ

Γ

P

Calculation of u0 for a homogeneous modelI STEP

• Analytical solution• Numerical methods (Ex. Hisada, 1994)• Same method used for step II (ex. SE)

Oblique propagation of plane waves inside a valley

DRM : 2 steps method

Comparison

ConclusionsConclusions

•Capabilities of DRM to handle „source to structure“ wave propagation problem with reduced CPU time

• Dialog between numerical codes oriented for different purposes

•Kinematic model are satisfactory to describe the low frequency bahaviour (e.g.: PGD and PGV) while PGA seems to be overestimated (nucleation, constant rupture velocity and instantaneous drop of the slip on the fault boundaries?).