M. Stupazzini, C. Zambelli, L. Massidda, L. Scandella M. Stupazzini, C. Zambelli, L. Massidda, L. Scandella R. Paolucci, F. Maggio, C. di Prisco THE SPECTRAL

M. Stupazzini, C. Zambelli, L. Massidda, L. ScandellaM. Stupazzini, C. Zambelli, L. Massidda, L. ScandellaR. Paolucci, F. Maggio, C. di PriscoR. Paolucci, F. Maggio, C. di Prisco

THE SPECTRAL ELEMENT METHODTHE SPECTRAL ELEMENT METHODAS AN EFFECTIVE TOOL FOR SOLVINGAS AN EFFECTIVE TOOL FOR SOLVING

LARGE SCALE DYNAMIC SOIL-STRUCTURE LARGE SCALE DYNAMIC SOIL-STRUCTURE INTERACTION PROBLEMSINTERACTION PROBLEMS

20th of April April 20020066San FranciscoSan Francisco

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

1st

2nd

3rd

SubsidenceLiquefactionLandslides

General problemGeneral problem

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..

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)

Analitical 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 NODEL FORCESEFFECTIVE NODEL FORCES

ui

we

ub

Inner regionInner region



iL o

b be eL o

e eb b

0P

P P K u

P K u

Inner regionInner region



Step II ( REDUCED PROBLEM )


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



[Mb][Mb]


[sec.][sec.]




2790 15 1.177 10-5 570 620 190.0

DRMDRM

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



[Mb][Mb]


[sec.][sec.]




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

Wave propagation in 3D complex domain

1756 1756 mm

2160 m2160 m

891 891 mm

Fault 1Fault 1

Fault 2Fault 2

T = 0.5sT = 0.5s

T = 1.0sT = 1.0s

T = 1.5sT = 1.5s

T = 2.0sT = 2.0sSnapshots of Displacement

ConclusionsConclusions

• GeoELSE is capable to handle „source to structure“ wave propagation problem.

• Thanks to DRM we acchieve:• reduced computational time• dialog between numerical codes oriented for different purposes

• WEB SITE:

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

www.spice-rtn.org

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


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


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


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


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 : 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


-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


-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


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


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?).

Documents

M. Stupazzini, C. Zambelli, L. Massidda, L. Scandella M. Stupazzini, C. Zambelli, L. Massidda, L. Scandella R. Paolucci, F. Maggio, C. di Prisco THE SPECTRAL