M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland...
-
Upload
sibyl-stone -
Category
Documents
-
view
214 -
download
0
Transcript of M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland...
![Page 1: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/1.jpg)
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
![Page 2: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/2.jpg)
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
![Page 3: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/3.jpg)
3D numerical simulation of seismic wave propagation in the Grenoble valley (M6 earthquake)
![Page 4: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/4.jpg)
3D numerical simulation of seismic wave propagation in the Grenoble valley (M6 earthquake)
36 km36 km
30 km30 km
![Page 5: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/5.jpg)
![Page 6: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/6.jpg)
![Page 7: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/7.jpg)
![Page 8: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/8.jpg)
![Page 9: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/9.jpg)
![Page 10: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/10.jpg)
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
![Page 11: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/11.jpg)
![Page 12: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/12.jpg)
![Page 13: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/13.jpg)
Synthetic seismograms
![Page 14: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/14.jpg)
Peak ground velocities
![Page 15: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/15.jpg)
Spectral ratios for the fault parallel component
![Page 16: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/16.jpg)
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
![Page 17: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/17.jpg)
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
![Page 18: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/18.jpg)
ConclusionsConclusions
www.spice-rtn.org
www.stru.polimi.it/Ccosmm/ccosmm.htm
![Page 19: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/19.jpg)
1st
2nd
3rd
SubsidenceLiquefactionLandslides
General problemGeneral problem
![Page 20: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/20.jpg)
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.
![Page 21: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/21.jpg)
• 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
![Page 22: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/22.jpg)
• 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
![Page 23: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/23.jpg)
Study case
railway bridgerailway bridge
![Page 24: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/24.jpg)
Wave propagation in 2D
“ Site effects “ & “ Soil Structures Interactions “
“Source“ &
“ Deep propagation“
Fault
zoomzoom
zoomzoom
![Page 25: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/25.jpg)
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
![Page 26: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/26.jpg)
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
![Page 27: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/27.jpg)
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
![Page 28: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/28.jpg)
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
![Page 29: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/29.jpg)
Comparison
![Page 30: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/30.jpg)
Comparison
![Page 31: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/31.jpg)
OutlookOutlook
DRMDRM
Study Study casecase
GeoELSE
![Page 32: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/32.jpg)
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)
![Page 33: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/33.jpg)
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
![Page 34: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/34.jpg)
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
![Page 35: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/35.jpg)
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 ,
![Page 36: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/36.jpg)
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..
![Page 37: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/37.jpg)
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
![Page 38: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/38.jpg)
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
![Page 39: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/39.jpg)
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
![Page 40: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/40.jpg)
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
![Page 41: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/41.jpg)
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
![Page 42: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/42.jpg)
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
![Page 43: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/43.jpg)
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
![Page 44: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/44.jpg)
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)
![Page 45: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/45.jpg)
-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)
![Page 46: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/46.jpg)
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
![Page 47: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/47.jpg)
Comparison
![Page 48: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/48.jpg)
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?).
![Page 49: M. Stupazzini M. Stupazzini Grenoble 3D benchmark July 2006 23 rd of July 2006 Kinsale – Ireland (3rd SPICE workshop) Politecnico di Milano Dep. of Structural.](https://reader036.fdocuments.in/reader036/viewer/2022062806/56649eab5503460f94bb0542/html5/thumbnails/49.jpg)