discontinuous galerkin spectral element methods …...discontinuous galerkin spectral element...

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

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

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)

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)

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)

Seismic waves (cont’d)

P-Waves (Compressional)

The ground is vibrated in thedirection the wave is propagating.

Travel through all types of media.

(�+ 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.

µ/⇢ (S-wavevelocity)

Typical speed: ⇠ 1 ! 8 km/sec

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

Applications

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 ],

(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).

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

=p(�+ 2µ)/⇢ (P-wave velocity), C

=pµ/⇢ (S-wave velocity)

Weak form

For any t 2 (0,T ] find u = u(t) 2 H10,�

(⌦) such that:

(⇢utt

, v)⌦ + (D"(u), "(v))⌦ = (f, v)⌦ + (g, v)�N

8 v 2 H10,�

(⌦).

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].

SPEED paradigm

1st - LEVEL

Subdivide ⌦ intomacroregions ⌦

Define a polynomialapproximation degreep

� 1 in each ⌦k

2nd- LEVEL

Introduce a (hexahe-dral/tetrahedral)conforming partitionT k

inside each ⌦k

(non-matching at theinterfaces)

3rd - LEVEL

Spectral ElementMethod (SEM) /Tetrahedral SpectralElement Method(TSEM) within ineach ⌦

SPEED paradigm

v 2 Vhp

discontinuous across the skeleton Fh

continuous inside each ⌦k

a (suitable) polynomial of degree p

when restricted to a mesh element 2 T k

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

Weighted average and jump trace operators

For (regular-enough) vector-valued andtensor-valued functions v and ⌧ define on eachF 2 F

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

If boundary conditions are enforced ”a la DG”, the above definition modifiesaccordingly on Dirichlet boundary faces

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

8 v 2 Vhp

A(w, v) =X

("(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], [....]

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

= (f, v)Th

+ hg, viFN

(A�h � "(uh), ⌧ )Th

� h{uh}(1��) � {uh}, [[⌧ ]]iF I

+ hc22[[�h]], [[⌧ ]]iF I

+ h[[uh]], {⌧}iFh

+ hc22�hn, ⌧ niFN

= hc22g, ⌧ niFN

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]

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 +

⇢�1⇤ kf(⌧)k

2(⌦) d⌧ 0 < t T .

(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 �

= ;)X In the absence of external forces, displacement-stress DG methods are fully

conservative

| [A., Ayuso, Mazzieri, Quarteroni, 2015], [A., Ferroni, Mazzieri, Quarteroni, 2016]

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

k�3/2

where m = min(p + 1, k) with k > 1 + d/2. The hidden constant depends onC (u,u

,T ,material properties).

k|u(s)k|2E = k⇢1/2ut

(s)k2L

2(⌦) + ku(s)k2DG + k��1/2{D"(u)}k2L

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]

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 .

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.

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

8 v 2 Vhp

The bilinear form associated to the interior penalty DG method reads as before

A(w, v) =X

("(w),D"(v))⌦k

� h{D"(w)}�, [[v]]iFh

� h[[w]], {D"(v)}�iFh

+ h�[[w]], [[v]]iFh

�(x) = ↵{D 12 } max

2{+,�}

2|F |||

, x 2 F , F 2 F

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]

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 +

0⇢�1⇤ kf(⌧)k

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

k�3/2

where m = min(p + 1, k) with k > 1 + d/2. The hidden constant depends onC (u,u

,T ,material properties).

(s)k2L

2(⌦) + ku(s)k2DG

| [A., Mazzieri, Nicolo’, in prep.]

Time integration

U(0) = u0

U(0) = u1

For time discretizations we set �t = T/N, and t

= 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

Stability of the leap-frog scheme

The leap-frog time integration scheme is stable provided that

�t C

where c

(�+ 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)

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)

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

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]

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

[m/s] c

[m/s] ⇢[Kg/m3]L 0-1000 4000 2000 2600HS 1000-17000 6000 3464 2700

| [Day et al. 2001]

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

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

2Radial

E = 0.034813

0 1 2 3 4 5 6 7 8 9-2

/s] Transverse E = 0.034407

0 1 2 3 4 5 6 7 8 9

time [s]

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

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

| [Mazzieri et al, 2015]

Comparison between recorded (black line) and simulated (redline) three-component velocity waveforms for a representative

subset of ten stations

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

Map of permanent ground uplift simulated by SPEED (left) and observed by COSMO-SkymedInSAR processing (right)

Domain size: 74⇥ 51⇥ 20 km

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]

New Zealand Canterbury Plains Christchurch (CBD)

Geological map of Christchurch provided by GNS (New Zealand research institute) [Forsythet al. 2008]

| [Mazzieri et al. 2013]

Soft-soil h ⇡ 25/50m, p = 2

Central Business District h ⇡ 5m, p = 1

Layer c

[m/s] c

[m/s] ⇢[Kg/m3] Q(⇠)Foundation 400 650 2400 200Building 100 163 2400 200

Mechanical parameters [Bielak et al. 2010]

West-East (W-E) and South-North (S-N) PeakGround Velocity

Displacements on the buildings of the CBD

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

SEM 38569 3 ⇡ 3 mln 2.3 %SIP 20602 4-3-2 ⇡ 1.8 mln 6.3 %

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

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

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)

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

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 !!

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