non-conforming high order approximations of the ... high order approximations of the elastodynamics...

Non-conforming high order approximationsof the elastodynamics equation

Paola Antonietti

MOX, Politecnico di Milano

Joint work with: I. Mazzieri (PoliMi), A.Quarteroni (EPFL & PoliMi),F.Rapetti (Universite de Nice Sophia-Antipolis ).

Journees Lions-Magenes

P. Antonietti (MOX-PoliMi) High order DG methods for elastodynamics Journees Lions-Magenes 1 / 26

Plan of the talk

1 Introduction and motivations

2 Non-conforming high order methods

3 Theoretical analysis

4 Some applications


Introduction: what is a seismic wave?

The wiggles in a seismogram indicate that theground is being (or was) vibrated by seismic waves.

Seismic waves are propagating vibrations that carryenergy from the source of the shaking outward in alldirections (e.g., stone is thrown into the water).

Seismic waves that are set up during an earthquakeare more complex than those on the pond.

ChristChurch (NZ),2011-02-22 at 8.06.48 AM

(magnitude 6.3)

Circles on water European earthquake potential hazard


Introduction: what is a seismic wave? (cont’d)Prof. Ammon’s course “An Introduction to Earthquakes”, PSU (http://eqseis.geosc.psu.edu/∼cammon/)

The are many different seismic waves.The most important ones are

Compressional or P (primary)Transverse or S (secondary)LoveRayleigh

An earthquake radiates P and Swaves in all directions.

The interaction of the P and S waveswith Earth’s surface → surface waves.

P and S waves travel at differentspeeds (used to locate earthquakes)

http://eqseis.geosc.psu.edu/∼cammon/


Introduction: what is a seismic wave? (cont’d)Prof. Ammon’s course “An Introduction to Earthquakes”, PSU (http://eqseis.geosc.psu.edu/∼cammon/)

P-Waves (Compressional)

The ground is vibrated in thedirection the wave is propagating.

Travel through all types of media.

CP =√

(λ+ 2µ)/ρ (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.

CS =√µ/ρ (S-wave velocity)

Typical speed: ∼ 1→ 8 km/sec

Much of the damage close to an earthquakeis the result of strong shaking caused byS-waves.


The mathematical problemEquilibrium equations for an elastic medium subjected to an external force fext read

ρ ∂ttu−∇ · σ(u) = fext in Ω× (0,T )

u displacement of the medium

ρ material density

ε(u) = 12(∇u +∇u>) strain tensor

σ(u) = λ∇ · u I + 2µ ε(u) = D : ε(u) stress tensor (Hooke’s Law)

λ, µ Lame elastic coefficients

suitable boundary conditions on ∂Ω

suitable initial conditions

Variational Formulation

For any time t ∈ (0,T ] find u = u(t) ∈ V = H1ΓD

(Ω) such that

(ρ ∂ttu, v)Ω + (σ(u), ε(v))Ω = L(v) ∀ v ∈ V

In applications, fvisc = −2ρζ ∂tu− ρζ2u is usually added (ζ decay factor).


Numerical solution of (seismic) wave propagation problems

Waves are typically propagated over many periods=⇒ control the numerical dispersion and dissipation errors

Accurate description of the behaviour at interfaces=⇒ high-order accuracy

The size of the bodies excited is large relative to the wavelengths of interest=⇒ efficient and scalable implementation

Complex geometries and strong contrasts in the media=⇒ methods that accommodate non-conformities

Non-Conforming Spectral Element Methods

Discontinuous Galerkin Spectral Element Method (DGSEM)[Reed & Hill, 1973], [LeSaint & Raviart, 1974]

Mortar Spectral Element Method (MSEM)[Bernardi, Maday & Patera, 1994]

=⇒ Weak continuity across interfaces

=⇒ Non-conforming meshes

=⇒ Non-uniform polynomial approximation degrees


DGSEMs: decompositions

1 - Macro Level (this is usually provided by Engineers)

Subdivide Ω into R non-overlappingmacro-regions Ωk : i.e., Ω =

⋃Rk=1 Ωk .

For each Ωk , define a polynomial approximationorder Nk .

2 - Meso Level

For each Ωk , introduce a conforming partition

Thk , i.e., Ωk =⋃Jk

j=1 Ωjk .

Subdivide the skeleton in M elementarycomponents γm (edges).


DGSEMs: approximation spaceFor each Ωk , and for each Ωj

k ⊂ Ωk

F jk : Ω = (−1, 1)d −→ Ωj

k .

xi : set of of n = (Nk + 1)d Gauss–Legendre-Lobatto (GLL) points.

Li: set of the Lagrangian polynomials based on the approximation points xi .

(1,1)

(-1,-1)

x

y

Approximation space

Xδ(Ωk) = vδ ∈ C0(Ωk) : vδ|Ωj

k F j

k ∈ QNk (Ω) ∀ Ωjk.

Vδ = vδ ∈[L2(Ω)

]d: vδ|Ωk

∈ [Xδ(Ωk)]d ∀Ωk and vδ|ΓD = 0,


Semi-discrete DG formulation

∀t ∈ (0,T ] find uδ = uδ(t) ∈ Vδ such that

(ρ ∂ttuδ, v)Ω +ADG(uδ, v) = L(v) ∀v ∈ Vδ

where ADG(·, ·) = A(·, ·) + B(·, ·)

A(u, v) = (σ(u), ε(v))Ω ∀u, v ∈ Vδ

B(·, ·) controls the jumps of the discrete solution, and

L(v) = (fext, v)Ω + BCs ∀v ∈ Vδ


Trace operators and definition of B(·, ·)On each edge γ, we define the average andjump operators for v and σ

v =1

2(vi + vj), [[v]] = vi ⊗ ni + vj ⊗ nj ,

σ =1

2(σi + σj), [[σ]] = σi · ni + σj · nj ,

B(u, v) =∑γ

− (σ(u), [[v]])γ + θ ([[u]], σ(v))γ + ηγ ([[u]], [[v]])γ

θ = −1, 1, 0→ SIPG,NIPG,IIPG [Arnold, Brezzi, Cockburn, Marini, SINUM 2001/02].

ηγ = αλ+ 2µAN2γ

hγ, with α > 0 at our disposal

where qA = harmonic average, Nγ = maxNi ,Nj, hγ = minhi , hjP. Antonietti (MOX-PoliMi) High order DG methods for elastodynamics Journees Lions-Magenes 10 / 26

Boundedness and coercivity of ADG(·, ·)Let V (δ) = Vδ ⊕ (H2(Ω) ∩H1

ΓD(Ω)), and define

||vδ||2DG = ||D1/2 ε(vδ)||20,Ω +∑γ

ηγ ||[[vδ]]||20,γ ∀vδ ∈ Vδ,

|||v|||2DG = ||v||2DG +K∑

k=1

(hk

N2k

)2

|v|22,Ωk∀v ∈ V (δ).

There exist M, κ > 0 such that:

ADG(u, v) ≤ M |||u|||DG|||v|||DG ∀u, v ∈ V (δ)

ADG(uδ,uδ) ≥ κ ||uδ||2DG ∀uδ ∈ Vδ.

For SIPG/IIPG methods coercivity holds provided the stability parameter αis chosen sufficiently large.


Semi-discrete error estimates

Semi-discrete error estimates?

Provided that, for any t ∈ [0,T ], the exact solution u = u(t) ∈ Hs(Ω), s ≥ 2

supt∈[0,T ]

|||(u− uδ)(t)|||DG .hr−1

Ns−3/2r = minN + 1, s

Remarks:

Optimal in h, suboptimal in N by a factor of N1/2

Know for DG methods (cf. [Houston, Schwab, Suli, 2000], Perugia and Schotzau, 2002].)

A similar method was introduced and analyzed by [Riviere, Wheeler, 2003]and [Riviere, Shaw, Whiteman, 2007] but the bilinear form contains theadditional term ∑

γ

ηγ([[ ∂tuδ]], [[vδ]])γ

See also [Grote et al., 2006]

? Antonietti, Mazzieri, Quarteroni, Rapetti, CMAME, to appearP. Antonietti (MOX-PoliMi) High order DG methods for elastodynamics Journees Lions-Magenes 12 / 26

Time integration (leap-frog)

MUδ + ADGUδ = F

Uδ(0) = u0,δ

Uδ(0) = u1,δ

Set ∆t = T/N, and tn = n∆t, for n = 0, ...,N

leap-frog scheme:

MUnδ = [2M− (∆t)2ADG]Un−1

δ −MUn−2δ + (∆t)2Fn−1 n ≥ 2

MU1δ = [M− (∆t)2

2 ADG]u0,δ −∆tMu1,δ + (∆t)2

2 F0

Stability


Fully-discrete error estimates

Fully-discrete error estimates?

It holds

||u(tn)− unδ||DG . ∆t2 +hr−1

Ns−3/2

where r = min(N + 1, s).

Remarks:

Optimal in h, suboptimal in N by a factor of N1/2

Proof: follows as in [Riviere, Shaw, Whiteman, 2003].

Cf. also [Grote, Schotzau 2009]

? Antonietti, Mazzieri, Quarteroni, Rapetti, CMAME, to appearP. Antonietti (MOX-PoliMi) High order DG methods for elastodynamics Journees Lions-Magenes 14 / 26

Numerical results (toy problem):

Ω = (0, 1)2, ΓD ≡ ∂Ω, ρ = λ = µ = 1

u(t, x , y) = sin(√

2πt)[− sin2(πx) sin(2πy); sin(2πx) sin2(πy)

]N1 = N2 = 2

10−1

100

10−5

10−4

10−3

10−2

h1

|| u

− u

δ||

L2(Ω

)

3

MSEM A

MSEM B

SIPG A

SIPG B

SEM

Grid A

2 4 6 8 1010

−10

10−8

10−6

10−4

10−2

N

|| u

− u

δ||

L2(Ω

)

SEMSIPG N

2 = N

1

SIPG N2 = N

1+1

SIPG N2 = N

1+2

Grid A

Grid B

The error is computed at the observation time t∗ = 2 (∆t = 1 · 10−4).


Numerical results: elastic scattering by a circle

Ω = (0, 2000)m × (0, 2000)m

Circular inclusionRadius R = 250mCenter C = (1000, 1000)m

Plane wave (ω = 15Hz).

Case 1 -STIFF INCLUSION

Square: CS = 1155m/s,CP = 2000m/s, ρ = 1Kg/m3

Circle: CS = 1732m/s,CP = 3000m/s, ρ = 2Kg/m3

hSquare ≈ 70m

hCircle ≈ 100m

Case 2 -SOFT INCLUSION

Square: CS = 2310m/s,CP = 4000m/s, ρ = 2Kg/m3

Circle: CS = 600m/s, CP = 1400m/s,ρ = 1Kg/m3

hSquare ≈ 70m

hCircle ≈ 35m


Displacements along the circular inclusion

Conforming SEM, DGSEM (N=4)

0 5 10 151

1.5

2

2.5

3

3.5

4

4.5

5

Tim

e [

s]

Displx [m]

0 5 10 151

1.5

2

2.5

3

3.5

4

4.5

5

Tim

e [

s]

Displx [m]

STIFF vs SOFT inclusion

SOFT inclusion: snapshotst = 1.5, 2.25 s.


Geophysical application I: Gubbio (PG) alluvial basin

Its a strong seismicity region (between Eurasian and African plates).

Seismic events characterized by magnitude up to 7.

Umbria earthquake (26.09.97, h11:40, MW 6.0).

Alluvium is loose, unconsolidated (not cemented together into a solidrock) soil.

Courtesy of R. Paolucci, DIS, PoliMiP. Antonietti (MOX-PoliMi) High order DG methods for elastodynamics Journees Lions-Magenes 18 / 26

Geophysical application I: Gubbio alluvial basin

NON CONFORMING (48091 spectral nodes) CONFORMING (61385 spectral nodes)

Polynomial degree N = 4.

External force: Ricker-type function at xs

External force


Geophysical application II: Time histories


Geophysical application II: Acquasanta bridge, Liguria Italy

The Acquasanta bridge on theGenova-Ovada railway.

Remarkable structure both forthe site features and the localgeological and geomorphologicalconditions.

The foundations of several piersare located on weak rock.

Some instability problems havebeen detected in the past on thevalley slope towards Ovada.



DGSEM (≈ 40000 d.o.f.) SEM (≈ 100000 d.o.f.)

Discretization parameters

Layer cP [m/s] cS [m/s] ρ[Kg/m3] Q(ζ) N (SEM) N (DGSEM)Bridge 1218 716,7 1750 100 4 2

1 1100 635 2400 100 4 22 1100 635 2400 250 4 43 1700 982,5 2600 250 4 44 2300 1330 2800 250 4 45 2300 1330 2800 250 4 4

Seismic moment is given at the point S.



DGSEM (≈ 40000 d.o.f.) SEM (≈ 100000 d.o.f.)

R1 valley (Displ. vertical) R2 bridge (Displ. Horizontal)


Geophysical application III: Christchurch earthquake, 22.02.2011, MW 6.2

Geological map of Christchurch provided by GNS (New Zealandresearch institute) [Forsyth et al. 2008]

Geological cross-section: numbers refer to the stations within a 40 kmradius from the epicenter

Courtesy of R. Paolucci, DIS, PoliMiP. Antonietti (MOX-PoliMi) High order DG methods for elastodynamics Journees Lions-Magenes 23 / 26

Geophysical application III: Christchurch earthquake, 22.02.2011, MW 6.2

DGSEM approach for the CentralBusiness District (CBD)

Not-Honoring approach for the contact ofalluvial and soft sediments [Guidotti etal., 2011]

Soft-soil h ≈ 25m,N = 2

Central Business District h ≈ 5m,N = 1

Layer cS [m/s] cP [m/s] ρ[Kg/m3] Q(ζ)Foundation 400 650 2400 200

Building 100 163 2400 200

Mechanical parameters [Bielak et al. 2010]


Conclusions

Non-Conforming Spectral Element Methods

preserve the spectral accuracy typical of high order methods

have a high accuracy and extremely low dispersion error

improve the geometrical flexibility of spectral discretizations

can naturally handle non-conforming grids and complex geometries

independent local meshesdifferent spectral approximation degrees

allow the study of combined effect of strong lateral variations of soilproperties and topographic amplification in heterogeneous media


Ongoing work

Simulation of 3D complex earthquake scenarios

Extension to tetrahedral meshes

Implicit/high order time-integration schemes

Effective preconditioners for the resulting system of equations

Release of the new code SPEEDSPectral Elements in Elastodynamics with Discontinuous Galerkinjointly developed at MOX and DIS - PoliMI

Ackwoledgements

We are grateful to R. Paolucci, C. Smerzini (DIS, PoliMi), and M.Stupazzini (Munich RE) for providing input for the geophysicalapplications and their help in the analysis of numerical results.

THANK YOU FOR YOUR ATTENTION!


Richter scale: quantify the “size” of an earthquakeML (local magnitude scale): is approximately determined from the logarithm of theamplitude of waves recorded by seismographs.

The energy released ((closely correlated to its destructive power)) scales with the

3/2 power of the shaking amplitude

Example: An earthquake that measures 5.0 on the Richter scale has a

shaking amplitude 10 times larger than one that measures 4.0 (and

corresponds to an energy release of 31.6 = (101)3/2 times greater)

In the 1970s ML scale was replaced by the moment magnitude scale (MMS)

MMS (MW ) measures the size of earthquakes in terms of the energy released:

MW =2

3log10 M0 − 10.7

Seismic moment M0 = µAD

µ is the shear modulus of the rocks involved in the earthquakeA is the area of the rupture along the geologic fault where theearthquake occurredD is the average displacement on A

wikipedia.orgP. Antonietti (MOX-PoliMi) High order DG methods for elastodynamics Journees Lions-Magenes 27 / 26

Stability of the leap-frog integration scheme

AimEstablish the largest time step ∆t such that the solution remains bounded with respectto problems’s data, i.e.,

∆t ≤ CCFLh

CP

where

h=mesh-size

CP =√

(λ+ 2µ)/ρ (=P-wave velocity)

CCFL=Courant number (depends on N)

Generalized eigenvalue problem on Ω to estimate CCFL

The CFL condition is satisfied if there exists a positive constant c∗(λ, µ, α) such that

CCFL ≤c∗(λ, µ, α)

N2.

For the NIPG c∗(λ, µ, α) = c∗(λ, µ). For the SIPG c∗(λ, µ, α) ∼ α−1/2


Numerical results: Sampling ratio δ = 0.2 and CP/CS = 2

Computed upper bound for CCFL as a function of N

N SEM MSEM SIPG NIPG2 0.3376 0.3333 0.2621 0.21633 0.1967 0.1770 0.1368 0.10454 0.1206 0.1118 0.0795 0.06075 0.0827 0.0776 0.0530 0.04006 0.0596 0.0570 0.0374 0.02817 0.0449 0.0434 0.0280 0.02108 0.0351 0.0342 0.0216 0.01629 0.0281 0.0277 0.0172 0.0129

10 0.0231 0.0227 0.0140 0.0105

N-rate -1.8463 -1.8253 -1.9247 -1.9360

Remarks

The constants for the SIPG are around 70% with respect the SEM, while for theMSEM are around 95%

The NIPG has constants always more restrictive than those of SIPG

High-order (possibly implicit) time integration schemes (as Runge-Kutta methods) areunder investigation


Gubbio (PG) alluvial basin: external force

f(x, t) = g(x)h(t)

with

g(x) = δ(x− xs)w h(t) = h0[1− 2β(t − t0)2] exp(−β(t − t0)2)︸︷︷︸Ricker-type function

where

δ = Dirac distribution

xs = Source location

Direction of applied load [cf. Casadei et al, 2002]

h0=scaling factor

t0 = 2 s (time shift)

β = π2ν2maxs

−1

νmax (maximum frequency) = 3 Hz


non-conforming high order approximations of the ... high order approximations of the elastodynamics...

Documents