discontinuous galerkin spectral element methods …...discontinuous galerkin spectral element...
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Discontinuous Galerkin spectral element methods
for earthquake simulations
Paola F. Antonietti
MOX, Dipartimento di Matematica, Politecnico di Milano
Seminaire du Laboratoire Jacques-Louis LionsUniversite Pierre et Marie Curie, Paris, September, 23 2016
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 1 / 43
Goals
Simulation of large-scale seismic events at regional scale:from far-field to near-field including soil-structure interaction e↵ects
Seismic hazard analysisDynamic analysis of structural systemsPlan of protection strategies and damage loss
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 1 / 43
People
A. Quarteroni(MOX, PoliMi &EPFL)
C. Smerzini(DICA, PoliMi)
R. Paolucci(DICA, PoliMi)
A. Ozcebe(DICA, PoliMi)
M. Stupazzini(MunichRE)
K. Hashemi(DICA, PoliMi)
I. Mazzieri(MOX, PoliMi)
R. Guidotti(U. Illinois)
A. Ferroni(MOX, PoliMi)
F. Gatti(DICA, PoliMi &Ecole Centrale ParisSuplec)
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 2 / 43
Seismic waves
Seismic waves are propagating vibrations that carry energy from the source of theshaking outward in all directions
M6.3 - L’aquila, 2009-04-06 01:32:39(source: earthquake.usgs.gov)
Italian earthquake potential hazard, 2015(source: Italian Civil Defence)
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 3 / 43
Seismic waves
Seismic wavesCompressional or P (primary)Transverse or S (secondary)LoveRayleigh
An earthquake radiates P and S waves inall directions.
The interaction of the P and S waves withEarth’s surface !surface waves.
P and S waves travel at di↵erent speeds(used to locate earthquakes)
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 4 / 43
Seismic waves (cont’d)
P-Waves (Compressional)
The ground is vibrated in thedirection the wave is propagating.
Travel through all types of media.
C
P
=p
(�+ 2µ)/⇢ (P-wavevelocity)
Typical speed: ⇠ 1 ! 14 km/sec
S-Waves (Transverse)
The ground is vibrated in theperpendicular direction to that thewave is propagating.
Travel only through solid media.
C
S
=p
µ/⇢ (S-wavevelocity)
Typical speed: ⇠ 1 ! 8 km/sec
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 5 / 43
SPEED: the computational code
SPEED: SPectral Elements in Elastodynamics with Discontinuous Galerkin
Open-source Fortran code for 3D seismic wave propagation
Discontinuous Galerkin Spectral Element Code
Parallel kernel with hybrid OpenMP-MPI programming
Emerging application within the European project PRACE-2IP WP 9.3
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 6 / 43
Applications
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 7 / 43
Requirements on the numerical scheme
FLEXIBILITYHighly heterogeneous materials and complexgeometries
ACCURACYAvoid dispersive e↵ects on the propagating wavefieldReproduce topographic e↵ects and soilamplifications
EFFICIENCYScalable implementation in parallel machines
������������������
Discontinuous
Galerkin
Spectral
Element
methods
| [Kaser, Dumbser, 2006], [Chung, Enquist, 2006], [Riviere, Shaw, Whiteman, 2007],
[de Basabe, Sen, Wheeler, 2008], [Grote, Diaz, 2009], [Wilcox, Stadler, Burstedde, Ghattas, 2010],
[A., Mazzieri, Rapetti, Quarteroni, 2012], [Etienne, Chaljub,Virieux, Glinsky,2010],
[Mazzieri, Stupazzini, Smerzini, Guidotti, 2013], [Peyrusse, Glinsky, Gelis, Lanteri, 2014],
[A., Ayuso, Mazzieri, Quarteroni, 2015], [A., Marcati, Mazzieri, Quarteroni, 2015]
[Paolucci, Mazzieri, Smerzini, 2015], [A., Dal Santo, Mazzieri, Quarteroni, 2016], [Ferroni, A., Mazzieri,
Quarteroni, 2016]P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 8 / 43
The mathematical model
Let ⌦ ⇢ Rd , d = 2, 3 with boundary @⌦ = �D
[ �N
s.t. |�D
| > 0.The mathematical model of linear elastodynamics reads:
⇢(x)utt
(x, t)�r · �(x, t) = f(x, t), in ⌦⇥ (0,T ],
u(x, t) = 0, on �D
⇥ (0,T ],
�(x, t)n(x) = g(x, t), on �N
⇥ (0,T ],
ut
(x, 0) = u1(x), in ⌦⇥ {0},u(x, 0) = u0(x), in ⌦⇥ {0},
Visco-elastic forces can be introduced in term of f⇠(u,ut) = �2⇢⇠ut
� ⇢⇠2u, (⇠decay factor).
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 9 / 43
The mathematical model: notation
u : ⌦⇥ [0,T ] �! Rd is the displacement vector field.
"(u) = 12 (ru+ru>) is the strain tensor
� : ⌦⇥ [0,T ] ! S = Rd⇥d
sym is the Cauchy stress tensor (Hooke’s law)
�(x, t) = D"(u(x, t))
D : S �! S is the sti↵ness tensor (spd, uniformly bounded)
D⌧ = 2µ⌧ + �tr(⌧ )I 8 ⌧ 2 S
�, µ 2 L
1(⌦) are the Lame parameters
⇢ 2 L
1(⌦) is the mass density
C
P
=p(�+ 2µ)/⇢ (P-wave velocity), C
S
=pµ/⇢ (S-wave velocity)
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 10 / 43
Weak form
For any t 2 (0,T ] find u = u(t) 2 H10,�
D
(⌦) such that:
(⇢utt
, v)⌦ + (D"(u), "(v))⌦ = (f, v)⌦ + (g, v)�N
8 v 2 H10,�
D
(⌦).
The above problem is well posed, and its unique solution satisfies a stabilityestimate in the following energy norm
ku(s)k2E = k⇢1/2ut
(s)k2L
2(⌦) + kD1/2"(u)(s)k2L
2(⌦) 8 s 2 [0,T ].
For the proof see [Raviart, Thomas, 1983], [Duvaut, Lions, 1976].
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 11 / 43
SPEED paradigm
1st - LEVEL
Subdivide ⌦ intomacroregions ⌦
k
’s.
Define a polynomialapproximation degreep
k
� 1 in each ⌦k
.
2nd- LEVEL
Introduce a (hexahe-dral/tetrahedral)conforming partitionT k
h
inside each ⌦k
(non-matching at theinterfaces)
3rd - LEVEL
Spectral ElementMethod (SEM) /Tetrahedral SpectralElement Method(TSEM) within ineach ⌦
k
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 12 / 43
SPEED paradigm
v 2 Vhp
is
discontinuous across the skeleton Fh
continuous inside each ⌦k
a (suitable) polynomial of degree p
k
when restricted to a mesh element 2 T k
h
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 12 / 43
SPEED paradigm
SEM: classical Lagrangian shape functions associated to the GLL points
TSEM: C 0 boundary adapted basis [Sherwin et al., 2005] (modification ofthe basis of [Dubiner, 1991])
If the first two levels coincide ! elementwise DG approach
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 12 / 43
Weighted average and jump trace operators
For (regular-enough) vector-valued andtensor-valued functions v and ⌧ define on eachF 2 F
h
on the skeleton shared by elements± 2 ⌦±
{v}� = �v+ + (1� �)v�, {⌧}� = �⌧+ + (1� �)⌧�, � 2 [0, 1]
[[v]] = v+ � n+ + v� � n�, [[⌧ ]] = ⌧+ n+ + ⌧� n�,
Here v � n = (vnT + nvT )/2
With the above definitions [[v]] 2 S = Rd⇥d
sym
If boundary conditions are enforced ”a la DG”, the above definition modifiesaccordingly on Dirichlet boundary faces
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 13 / 43
Displacement DG formulations
For any t 2 (0,T ], find uh = uh(t) 2 Vhp
such that
(⇢uhtt
, v)Th
+A(uh, v) = (f, v)Th
+ hg, viFN
h
8 v 2 Vhp
A(w, v) =X
k
("(w),D"(v))⌦k
� h{D"(w)}�, [[v]]iFh
� h[[w]], {D"(v)}�iFh
+ h�[[w]], [[v]]iFh
.
� = ↵{D}max(p2+ , p2�)
min(h+ , h�)↵ (large enough) positive constant
Symmetric Interior Penalty method (SIP): � = 0.5 [Wheeler, 1978], [Arnold, 1982]
Weigthed Symmetric Interior Penalty method (wSIP): � 6= 0.5 [Stenberg, 1998]
| [Perugia, Schotzau, 2001], [Houston, Schwab, Suli, 2002], [....]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 14 / 43
Displacement-stress DG formulations
Let ⌃h
= [Vhp
]d . Find (uh,�h) 2 C
2((0,T ];Vhp
)⇥ C
0((0,T ];⌃h
) such that
8>>>>>><
>>>>>>:
(⇢uhtt
, v)Th
+ (�h, "(v))Th
� h{�h}� � c11[[uh]], [[v]]iF
h
= (f, v)Th
+ hg, viFN
h
,
(A�h � "(uh), ⌧ )Th
� h{uh}(1��) � {uh}, [[⌧ ]]iF I
h
+ hc22[[�h]], [[⌧ ]]iF I
h
+ h[[uh]], {⌧}iFh
+ hc22�hn, ⌧ niFN
h
= hc22g, ⌧ niFN
h
.
for all v 2 Vhp
and ⌧ 2 ⌃h
Full discontinuous Galerkin (FDG) method: c22 6= 0, c11 6= 0, � = 0.5 [Castillo, Cockburn,
Perugia, Schotzau, 2000]
Local discontinuous Galerkin (LDG) method: c22 = 0, � = 0.5 [Cockburn, Shu, 1989]
Alternating choice of fluxes (ALT) method: c22 = c11 = 0 and � = 1 or � = 0 [Cheng, Shu,
2008], [Xu, Shu, 2012]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 15 / 43
Stability analysis
Let ↵ be large enough and let uh 2 Vhp
be the approximate solution obtained with adisplacement DG method. Then
kuh(t)kE . kuh(0)kE +
Zt
0
⇢�1⇤ kf(⌧)k
L
2(⌦) d⌧ 0 < t T .
ku(s)k2E = k⇢1/2ut
(s)k2L
2(⌦) + ku(s)k2DG
X Unified derivation and analysis for the elementwise DG approach using the flux
formulation [Arnold, Brezzi, Cockurn, Marini, 2001/02]
X Here, g = 0, for simplicity
X Analogous stability bounds hold for displacement-stress DG methods (for ALTmethod provided that �
N
= ;)X In the absence of external forces, displacement-stress DG methods are fully
conservative
| [A., Ayuso, Mazzieri, Quarteroni, 2015], [A., Ferroni, Mazzieri, Quarteroni, 2016]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 16 / 43
Semi-discrete a-priori error estimates
Assume that ↵ is large enough. Let uh be the approximated solution of u,assumed su�ciently regular. Then
sup0<sT
ku(s)� uh(s)kE . h
m�1
p
k�3/2
where m = min(p + 1, k) with k > 1 + d/2. The hidden constant depends onC (u,u
t
,utt
,T ,material properties).
k|u(s)k|2E = k⇢1/2ut
(s)k2L
2(⌦) + ku(s)k2DG + k��1/2{D"(u)}k2L
2(Fh
)
X No need to penalize the jumps of ut
X Optimal rate of convergence with respect to h
X The same result holds for displacement-stress DG formulations (in a suitableenergy norm)
| [A., Ayuso, Mazzieri, Quarteroni, 2015], [A., Ferroni, Mazzieri, Quarteroni, 2016]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 17 / 43
From standard-shaped elements to polytopic grids
Polyhedral grids provide extreme flexibility incomplex geometries
DG methods are naturally well suited tohandle polytopic grids
Employ an elementwise DG approach in theregion where polytopic elements appear
Assume (for simplicity) that p = p � 1, forany .
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 18 / 43
Polytopic grids
– Element interfaces F with arbitarily small measure
– On polytopic elements the discrete space of polynomials of degree p is built on thephysical frame.
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 19 / 43
DG formulation on polytopic grids
For any t 2 (0,T ), find uh = uh(t) 2 Vhp
such that
(⇢uhtt
, v)Th
+A(uh, v) = (f, v)Th
+ hg, viFN
h
8 v 2 Vhp
The bilinear form associated to the interior penalty DG method reads as before
A(w, v) =X
k
("(w),D"(v))⌦k
� h{D"(w)}�, [[v]]iFh
� h[[w]], {D"(v)}�iFh
+ h�[[w]], [[v]]iFh
�(x) = ↵{D 12 } max
2{+,�}
�C
INV
p
2|F |||
, x 2 F , F 2 F
h
with ↵ > 0 large enough (independent of p, |F | and ||).
| [A. Brezzi, Marini, 2009], [Bassi et al, 2014, 2014], [A., Giani, Houston, 2013], [Cangiani, Georgoulis,
Houston, 2014], [Cangiani, Dong, Georgoulis, Houston, M2AN, 2015], [A., Houston, Sarti, Verani, 2015],
[A., Cangiani, Collis, Dong, Georgoulis, Giani, Houston, 2016], [Cangiani, Dong, Geourgolis, 2016]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 20 / 43
Stability and semi-discrete error estimates
Let ↵ be large enough and let uh 2 Vhp
be the approximate solution obtainedwith a displacement DG formulation. Then,
kuh(t)k2E . kuh(0)kE +
Zt
0⇢�1⇤ kf(⌧)k
L
2(⌦) d⌧ 0 < t T
Assume that ↵ is large enough. Let uh be the approximated solution of u,assumed su�ciently regular. Then
sup0<sT
ku(s)� uh(s)kE . h
m�1
p
k�3/2
where m = min(p + 1, k) with k > 1 + d/2. The hidden constant depends onC (u,u
t
,utt
,T ,material properties).
ku(s)k2E = k⇢1/2ut
(s)k2L
2(⌦) + ku(s)k2DG
| [A., Mazzieri, Nicolo’, in prep.]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 21 / 43
Time integration
8><
>:
MU+ A
DG
U = F
U(0) = u0
U(0) = u1
For time discretizations we set �t = T/N, and t
n
= n�t, for n = 0, ...,N
leap-frog schemeexplicit, second-order accurate
Runge-Kutta methods: RK4 schemeexplicit, fourth-order accurate, �t� adaptive
DG method of lines [A., dal Santo, Mazzieri, Quarteroni, 2016]
implicit, arbitrary high-order accurate, �p/�t� adaptive
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 22 / 43
Stability of the leap-frog scheme
The leap-frog time integration scheme is stable provided that
�t C
CFL
(p2)
c
P
h
where c
P
=p
(�+ 2µ)/⇢ is the P-wave velocity.
Computed upper bound for CCFL
p SEM SIP NIP2 0.3376 0.2621 0.21633 0.1967 0.1368 0.10454 0.1206 0.0795 0.06075 0.0827 0.0530 0.04006 0.0596 0.0374 0.02817 0.0449 0.0280 0.02108 0.0351 0.0216 0.01629 0.0281 0.0172 0.012910 0.0231 0.0140 0.0105
p-rate -1.8463 -1.9247 -1.9360
The constants for the SIP arearound 70% with respect the SEM
The NIP has constants always morerestrictive than those of SIP
| [A., Mazzieri, Quarteroni, Rapetti, 2012]P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 23 / 43
Convergence test (3D, hexahedral grids)
⌦ = (0, 1)3, � = µ = ⇢ = 1, time interval (0,T ] = (0, 10].
Exact solution
u(t, x , y , z) = sin(3⇡t)
2
4� sin2(⇡x) sin(2⇡y) sin(2⇡z)
sin(2⇡x) sin2(⇡y) sin(2⇡z)sin(2⇡x) sin(2⇡y) sin2(⇡z)
3
5 .
Leap-frog scheme used for the time integration with time step �t = 1 · 10�5
Computed convergence ratesp SIP SEM1 1.1212 0.94922 2.1157 2.06223 2.8478 3.01354 3.7973 3.7973
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 24 / 43
Grid dispersion and dissipation errors
New generalized eigenvalue approach
Analysis for the fully-discrete approximations
Cross-validation with SE discretization
Low grid dispersion errors for p � 4 and � < 0.2 (5 or more points per wavelength)
No-dissipation e↵ects when using the leap-frog scheme
| [De Basabe et al. 2010], [Ainsworth 2006], [Seriani et al. 2008]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 25 / 43
Validation: Layer over halfspace
⌦ = (�30, 30) ⇥ (�30, 30) ⇥ (0, 17) Km
Seismic moment in S = (0, 0, 2) Km
Receiver R = (6, 8, 0) Km
1 of 4 symmetric quadrants Moment time history
Layer Depth [m] c
P
[m/s] c
S
[m/s] ⇢[Kg/m3]L 0-1000 4000 2000 2600HS 1000-17000 6000 3464 2700
| [Day et al. 2001]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 26 / 43
Validation: Layer over halfspace (DGSEM vs SEM)
Model El. p d.o.f. �t
SEM 814833 4 ⇡150 mln 3·10�4
SIP 70228 5 ⇡30 mln 5·10�4
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 27 / 43
Validation: Layer over halfspace (DGTSEM)
Layer: p = 5
Half-space: p = 4
Velocity recorded at point (6,8,0) Km
0 1 2 3 4 5 6 7 8 9-2
0
2Radial
E = 0.034813
0 1 2 3 4 5 6 7 8 9-2
0
2
velo
cit
y[m
/s] Transverse E = 0.034407
0 1 2 3 4 5 6 7 8 9
time [s]
-2
0
2Vertical E = 0.027606P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 28 / 43
Applications
Emilia (IT) earthquake 29.05.2012, magnitude 6.0
Christchurch (NZ) earthquake 22.02.2011, magnitude 6.2
Acquasanta (IT) railway bridge
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 29 / 43
Emilia (IT) earthquake 29.05.2012, magnitude 6.0
Collapse of the San Francesco Church (XVcentury), Mirandola. Courtesy of A. Penna.
Availability of an exceptionalstrong-motion dataset in near-faultconditions and on deep soft soil
Good knowledge of a deep and largesedimentary basin such as the Po Plain
Strong economic impact of the earthquake(⇡13 billion of Euros, Munich RE)
Previously selected as possible site for aNuclear Power Plant in Italy due to itsmoderate seismicity
Structural map of Italy from [Bigi et al., 1992].Epicenters of the two mains hocks the seismic
sequence: May 20 (MW
6.1 ) and May 29(M
W
6.0).
| [Mazzieri et al, 2015]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 30 / 43
Emilia (IT) earthquake 29.05.2012, magnitude 6.0
Comparison between recorded (black line) and simulated (redline) three-component velocity waveforms for a representative
subset of ten stations
p = 3
Dofs ⇡ 350 millions (⇡ 2 millions elements)
�t 0.001 s, T = 30 s
fmax 1.5 Hz
Simulation time ⇡ 5 hours on 4096 cores(FERMI@CINECA)
Computational model
| [Mazzieri et al, 2015]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 31 / 43
Emilia (IT) earthquake 29.05.2012, magnitude 6.0
Map of permanent ground uplift simulated by SPEED (left) and observed by COSMO-SkymedInSAR processing (right)
| [Mazzieri et al, 2015]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 32 / 43
Emilia (IT) earthquake 29.05.2012, magnitude 6.0
Domain size: 74⇥ 51⇥ 20 km
3
Hexahedral elements: ⇡ 2 millions (⇡ 500 millions of unknowns)
Time step: �t = 5 10�4 and final time T = 50 s.
Tests performed on FERMI Bluegene Q, CINECA Italy
| [Dagna, P., Enabling SPEED for near Real-time Earthquakes Simulations, 2013]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 33 / 43
Christchurch (NZ) earthquake 22.02.2011, magnitude 6.2
New Zealand Canterbury Plains Christchurch (CBD)
Geological map of Christchurch provided by GNS (New Zealand research institute) [Forsythet al. 2008]
| [Mazzieri et al. 2013]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 34 / 43
Christchurch (NZ) earthquake 22.02.2011, magnitude 6.2
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 35 / 43
Christchurch (NZ) earthquake 22.02.2011, magnitude 6.2
Soft-soil h ⇡ 25/50m, p = 2
Central Business District h ⇡ 5m, p = 1
Layer c
S
[m/s] c
P
[m/s] ⇢[Kg/m3] Q(⇠)Foundation 400 650 2400 200Building 100 163 2400 200
Mechanical parameters [Bielak et al. 2010]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 36 / 43
Christchurch (NZ) earthquake 22.02.2011, magnitude 6.2
West-East (W-E) and South-North (S-N) PeakGround Velocity
Displacements on the buildings of the CBD
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 37 / 43
Acquasanta (IT) railway bridge
Acquasanta bridge and surrounding valley
Mechanical Parameters
Reference solution: SEM on a fine conforming grid with p = 3.
Model El. p d.o.f. �t/�t
CFL
SEM 38569 3 ⇡ 3 mln 2.3 %SIP 20602 4-3-2 ⇡ 1.8 mln 6.3 %
| [Mazzieri et al. 2013]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 38 / 43
Acquasanta (IT) railway bridge
Non-conforming mesh: di↵erent colorsidentify di↵erent sub-domains.
Good fit for the data: grid dispersionerrors are under 10%.
Horizontal velocity in R
SEM - SIP
| [Mazzieri et al. 2013]
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 39 / 43
SPEED: Built-in capabilities http://speed.mox.polimi.it
1 Models- Elastic/visco-elastic soil models- Non-Linear Elastic soil model
2 Methods- Discontinuos Galerkin Spectral Element method- Leap-frog, Runge-Kutta (RK3 and RK4) time integration schemes- First order absorbing boundaries
3 Seismic excitation modes- plane wave load- Neumann surface load- volume force load
4 Pre and Post processing tools- Matlab GUI for input/output processing of earthquake scenarios- User Guide, Tutorials and documentation
5 Web-repository database of synthetic earthquake scenarios
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 40 / 43
SPEED: Available soon http://speed.mox.polimi.it
1 Models- Elasto-acoustic coupling
2 Methods- SEM-BEM coupling for absorbing boundaries- Discontinuous time integration schemes- Tetrahedral spectral elements- Polytopic grids
SPEED around the world (from 2013)
Academic Institutions (17) Companies (3)
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 41 / 43
Conclusions and further developments
Conclusions
DG methods can be coupled with spectral element approximations in a”domain decomposition fashion” to deal e�ciently with wave propagationphenomena in complex 3D configurations
Exploit the flexibility of DG methods in tuning the discretization parameterswhile keeping limited the proliferation of the unknowns
Further developments
3D implementation of DG methods on polytopic grids and validation on realearthquake scenarios
Fast solution techniques for TSEM
High-order DG methods for time integration
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 42 / 43
Acknowledgments
SIR project n. RBSI14VT0S: PolyPDEs: Non-conforming
polyhedral finite element methods for the approximation
of partial di↵erential equations.
Fondazione Cariplo/RL project n. 2015-0182PolyNum: Polyhedral numerical methods for partial
di↵erential equations
MunichRe project:MRPMII: Spectral Element Methods for earthquake
simulations
(PI: Paolucci)
Thank you for your attention !!
P.F. Antonietti (MOX-PoliMi) DGSE methods for earthquake simulations Seminaire du LJLL 43 / 43