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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
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
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
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?).