Second HPC-GA Workshop Marco Cuﬀaro, Edie Miglio, Mattia ... · Tectonic evolution at mid-ocean ridges: geodynamics and numerical modeling. Second HPC-GA Workshop Marco Cuﬀaro,

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Tectonic evolution at mid-ocean ridges:geodynamics and numerical modeling.

Second HPC-GA Workshop

Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò

Politecnico di Milano - MOX, Dipartimento di Matematica “F. Brioschi”

Bilbao, March 11th 2013

MODELLISTICA E CALCOLO SCIENTIFICO

MODELING AND SCIENTIFIC COMPUTING

M XMILANO

POLITECNIC

O

Marco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 1/ 55

Aims and motivations




Continental riftAnalogue models

The analogue models have the following disadvantages:they lack the thermo-mechanical description of phenomena,

they are very expensive to setup and perform,they are not general, but specific for a given region,they are not replicable.



The analogue models have the following disadvantages:they lack the thermo-mechanical description of phenomena,they are very expensive to setup and perform,

they are not general, but specific for a given region,they are not replicable.



The analogue models have the following disadvantages:they lack the thermo-mechanical description of phenomena,they are very expensive to setup and perform,they are not general, but specific for a given region,

they are not replicable.



The analogue models have the following disadvantages:they lack the thermo-mechanical description of phenomena,they are very expensive to setup and perform,they are not general, but specific for a given region,they are not replicable.



the development of the mathematical and numerical model ofmid-oceanic ridges:

improvement of the boundary conditions to model the ridgemigration;a better thermo-mechanical model (melting);

development of the mathematical tools for the defition of anumerical sandbox useful to reproduce the continental-riftevolution;development and application of meshfree methods andvariational integrators for the simulation of the numericalsandbox.










development of the mathematical tools for the defition of anumerical sandbox useful to reproduce the continental-riftevolution;

development and application of meshfree methods andvariational integrators for the simulation of the numericalsandbox.







Mid-ocean ridgesMathematical model

We can treat the litosphere andasthenosphere as higly viscousincompressible fluid

div(2ηD) −∇p + ρg = 0div v = 0∂θ

∂t + v · ∇θ = div (κ∇θ)η = η0e−Cθ

The Frank-Kamenetskii linearization isuse to approximate the viscosity law

η =1

2A

(µ

I2

)n−1(hb

)me

E+pVRT


Mid-ocean ridgesThe initial condition for the temperature field is the mean oceanicgeotherm

θ− θ0θM − θ0

= erf(

y2√κτ

)


Mid-ocean ridges

Results of numerical simulations in a steady-state regime.

Results of numerical simulations with ridge migration in act.


Mid-ocean ridgesBoundary conditions

The previous boundary conditions have to be improved, sincethey do not depend upon the rheology, (in particular on theviscosity);the simulation showed they can lead to incorrect results nearboundaries.


Mid-ocean ridgesBoundary conditions

A different strategy has been developed; these are the mainhypothesis:

the boundaries are far enough from the ridge,the upwelling motion is negligible compared to shearing.

.. 0.

−h

.x

.y

.vextension.

vmigration

u(y) = u0 + s∫ y

0

dqη(q) p(y) = p0 +

∫ y

0f(q) dq


Mid-ocean ridges: Mathematical model

div(2ηε(u)) −∇p + f = 0 in Ωdivu = 0 in Ω−div(k∇T) + u · ∇T = 0 in Ωu = u on Γ1((2ηε(u) − pI)n = g on Γ2T = T on Γ3−k∇T · n = s on Γ4

whereΓ1 ∩ Γ2 = ∅, Γ3 ∩ Γ4 = ∅,

andΓ1 ∪ Γ2 = ∂Ω, Γ3 ∪ Γ4 = ∂Ω,


Mid-ocean ridges: Numerical Discretization (1)

A(T) BT 0B 0 00 0 C(u)

upT

=

f0g

expanding the nonlinear terms at the first order we get Newton’smethodA(T) BT A ′(T)

B 0 0C ′(u) 0 C(u)

δuδpδT

=

f0g

−

A(T) BT 0B 0 00 0 C(u)

upT


Mid-ocean ridges: Numerical Discretization (1)The Newton step can be solved using the block LU decomposition,denoting

A =

[A(T) BT

B 0

]B1 =

[A ′(T) 0

]B2 =

[C ′(T) 0

],

so we get the following linear system[I 0

B2A−1 I

] [A BT

10 S

] [δxδy

]=

[fg

]where S is the Schur complement,

S = C −B2A−1BT

1

se we needa good Stokes solver,a good preconditioner for S


Mid-ocean ridges: Preconditioners

the Stokes problem is solved using the Schur complement;

the preconditioners for A and the (Stokes-)Schur complementare, respectively,

the algebraic multigrid (AMG) preconditioner (Trilinos);the viscosity-scaled pressure mass-matrix.

these preconditioners are almost optimal.The matrix S is preconditioned with the iterative inverse of C(this last problem is preconditioned with AMG preconditioner).this preconditioner seems to scale well with the number ofdegrees of freedom (for each numerical experiments we getabout 10 − 20 iterations to solve the temperature).



the Stokes problem is solved using the Schur complement;the preconditioners for A and the (Stokes-)Schur complementare, respectively,

the algebraic multigrid (AMG) preconditioner (Trilinos);

the viscosity-scaled pressure mass-matrix.
















these preconditioners are almost optimal.

The matrix S is preconditioned with the iterative inverse of C(this last problem is preconditioned with AMG preconditioner).this preconditioner seems to scale well with the number ofdegrees of freedom (for each numerical experiments we getabout 10 − 20 iterations to solve the temperature).





these preconditioners are almost optimal.The matrix S is preconditioned with the iterative inverse of C(this last problem is preconditioned with AMG preconditioner).

this preconditioner seems to scale well with the number ofdegrees of freedom (for each numerical experiments we getabout 10 − 20 iterations to solve the temperature).







Mid-ocean ridges: Numerical treatment of stress BC

To compute the stress boundary conditions we use the fixed pointiteration method, with the following scheme

given a solution (u, p,T) the boundary condition g iscomputedfrom g a new solution (u, p,T) is computed


Mid-ocean ridges

Results of numerical simulations in a steady-state regime.


Mid-ocean ridges

Results of numerical simulations with migration


Mid-ocean ridges

Scalability results: 250000 dofMarco Cuffaro, Edie Miglio, Mattia Penati, Marco Viganò Tectonic at mid-ocean ridge 18/ 55

Mid-ocean ridges

Scalability results: 1000000 dof


Mid-ocean ridges


Mid-ocean ridges


Mid-ocean ridges


Geomod example

Figure : Extension: sylicon on all the base

Figure : Sylicon only in the central part


Drucker Prager: Conf. 1 (MI, SI, V)


Drucker Prager: Conf. 2 (MI, SI, V)


Von Mises: Conf. 1 (MI, SI, V)


Von Mises: Conf. 2 (MI, SI, V)


Continental riftMathematical model

The previous model is extended to include the elastic behavior oflithosphere (upper-convected Maxwell model)

div T −∇p + ρg = 0div v = 0

T + λOT = 2ηD

ρdhdt =

dpdt + T : D + ρr − div q

from the dimensional analysis we get the following estimates forthe upwelling

ρdhdt ≈ dp

dt ρr ρdhdt div q ρ

dhdt


Optimal transportation mesh-free method

Continuum mechanic problems which involve largedeformations or which are formulated with respect to thereference configuration need a continuous remeshing or someother strategies to overcome the difficulties arising from thecontinuous mesh update.

Mesh-free methods are an alternative, since they give up themesh and use only a point set to discretize the problem.the OTM is designed for the continuum mechanic problems,since it inherits from the continuous problem the symmetriesand conservation properties, avoiding some issues of othermesh-free methods.



Continuum mechanic problems which involve largedeformations or which are formulated with respect to thereference configuration need a continuous remeshing or someother strategies to overcome the difficulties arising from thecontinuous mesh update.Mesh-free methods are an alternative, since they give up themesh and use only a point set to discretize the problem.

the OTM is designed for the continuum mechanic problems,since it inherits from the continuous problem the symmetriesand conservation properties, avoiding some issues of othermesh-free methods.



Continuum mechanic problems which involve largedeformations or which are formulated with respect to thereference configuration need a continuous remeshing or someother strategies to overcome the difficulties arising from thecontinuous mesh update.Mesh-free methods are an alternative, since they give up themesh and use only a point set to discretize the problem.the OTM is designed for the continuum mechanic problems,since it inherits from the continuous problem the symmetriesand conservation properties, avoiding some issues of othermesh-free methods.


Local maximum-entropy shape functions

Set of nodesX = xa ∈ Rd, a = 1, . . . ,N

Non-negative shape functions

pa : conv X → R>0

First order consistency conditions∑a

pa(x) = 1∑

apa(x)xa = x



.Local maximum-entropy problem..

.

For x fixed, minimize βUx(p) − H(p)subject to pa > 0∑

apa = 1∑

apaxa = x

where Ux(p) =∑

apa ‖x − xa‖2 and H(p) = −

∑a

pa log pa



.Local maximum-entropy problem..

.

Defined the partition function

Z(x,λ) =∑

ae−β‖x−xa‖2+λ·(x−xa).

For β(x) ∈ R>0 and x ∈ int conv X. Then the unique solution ofthe problem is

pa(x) =1

Z(x,λ∗)e−β‖x−xa‖2+λ∗·(x−xa),

where λ∗ is the unique solution of

λ∗ = arg minλ∈Rd

Z(x,λ).



Spatial smoothness

β ∈ Cr =⇒ pa ∈ Cr

Smoothness with respect β

p(β) ∈ C0([0,∞)), p(β) ∈ C∞((0,∞))

Limits of shape functions as β→ ∞pa converge to a Delaunay convex approximants as β→ ∞



Spatial smoothness



p(β) ∈ C0([0,∞)), p(β) ∈ C∞((0,∞))




Spatial smoothness



p(β) ∈ C0([0,∞)), p(β) ∈ C∞((0,∞))




0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0


Convergence

Let us consider the following problem

−∆u = 2 − x2 − y2, inΩ

u = 0 on∂Ω

Ω = (−1, 1)x(−1, 1).γ = βh2


Convergence: L2 norm

2−20

2−15

2−10

22 23 24 25 26 27

Nodes

Rel

ativ

e er

ror

γ

1

2

3

4

5

L2 error


Convergence: H1 norm

2−12

2−10

2−8

2−6

2−4

2−2

22 23 24 25 26 27

Nodes

Rel

ativ

e er

ror

γ

1

2

3

4

5

H1 error


Optimal transportation

The motion of an inviscid fluid of non-interacting particles in Rd isgoverned by the equations

ρ+ div (ρu) = 0˙(ρu) + div (ρu ⊗ u) = 0

This problem can be recasted as an optimal transportationproblem, that admits a variational formulation

minimize J(ρ,u) =∫b

a

∫Rd

12ρ ‖u‖2 dx dt

subject to ρ+ div (ρu) = 0


Optimal transportation

The problem can be recasted in term of deformation map φ hencethe velocity and the density are given by

u(x, t) = φt(φ−1t (x)) ρ(x, t) = ρa(φ

−1t (x))

det∇φt(φ−1t (x))

and Benamou and Brenier (1999) showed that

inf J(ρ,u) = inf 1b − a

∫Rdρa(x) ‖φb(x) − x‖2 dx


Optimal transportation - discretizationGiven a finite set of timestep t0 = a, t1, t2, . . . , tN = b, thesemidiscrete problem is given by

J =∑

k

1tk+1 − tk

∫ 12ρk(x) ‖φk→k+1(x) − x‖2 dx,

then using a quadrature rule

J =∑

k

∑n

1tk+1 − tk

12ρk(xn,k)vn ‖φk→k+1(xn,k) − xn,k‖2

denoting xn,k+1 = φk→k+1(xn,k) and mn,k = ρk(xn,k)vn, then thesolution of this problem isxn,k+1 = xn,k + (tk+1 − tk)

xn,k − xn,k−1tk − tk−1

mn,k = mn,0


Optimal transportation - solid and fluid flows

This framework can be extended for inviscid fluids and elasticsolids, adding a free energy term to the functional J

J(ρ,u,C) =∫b

a

∫Rd

12ρ ‖u‖2 − ρU(ρ,C) dx dt

The kinetic energy term is discretized in same way, but for the freeenergy the trapezoidal quadrature rule is used

J =∑

k

1tk+1 − tk

∫ 12ρk ‖φk→k+1(x) − x‖2 dx+

+tk+1 − tk

2

[∫ρkU(ρk,Ck) dx +

∫ρk+1U(ρk+1,Ck+1) dx

]



Two points sets are introduced:nodal set xa, used to construct the shape functionsmaterial set xn, used as quadrature points

So the displacement map can be written as a linear combination ofshape functions

φk→k+1(x) − x =∑

apa(x)da,

and the problem can be completely discretized

J =∑

k

∑n

1tk+1 − tk

12mn,k ‖φk→k+1(xn,k) − xn,k‖2 +

+tk+1 − tk

2 [mn,kU(ρn,k,Cn,k) + mn,k+1U(ρn,k+1,Cn,k+1)]



The solution of this problem isdk = (tk+1 − tk)M−1

k

(lk +

tk+1 − tk−12 fk

)xn,k+1 = φk→k+1(xn,k), xa,k+1 = φk→k+1(xa,k)

vn,k+1 = vn,k det∇φk→k+1(xa,k)

where

la,k =∑

nmn,k

xn,k − xn,k−1tk − tk−1

pa,k(xn,k)

Mk,ab =∑

nmn,kpa,k(xn,k)pb,k(xn,k)I

fa,k =∑

n[ρn,kbn,kpa,k(xn,k) + σn,k : ∇pa,k(xn,k)] vn,k


Application to the Stokes flows

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

inf(u,p)

∫Ω

(‖∇u‖2 − p div u

)dx −

∫∂Ω

λ · (u − u) dσ



U =λ

2 [trE]2 + µtrE2

λ = 4 × 102Paµ = 1.2 × 105Pa


Veselov-type discretizationChoosen the discrete set of times

0 = t0 < t1 < t2 < · · · < tN−1 < tN = T

then the motion is described by a sequence of positions

q0, q1, q2, . . . , qN−1, qN

and the discrete action is defined by the sum

Sq =

N−1∑k=0

Ld(qk, qk+1)

where Ld(qk, qk+1) is called discrete Lagrangian, it depends ontwo subsequent positions and it’s a reasonable approximation ofthe action between the configurations qk and qk+1

Ld(qk, qk+1) ≈∫ tk+1

tk

L(q, q) dt


Variational Integrators in time: Störmer-Verlet integrationUsing the trapezoidal quadrature rule and approximating thevelocity with finite difference we get the discrete Lagrangian

Ld(qk, qk+1) = (tk+1 − tk)

[12

(qk+1 − qktk+1 − tk

)2−

V(qk) + V(qk+1)

2

]

This leads to the well known Störmer-Verlet methodqk+1 − qktk+1 − tk

−qk − qk−1tk − tk−1

=tk+1 − tk−1

2 F(qk)

and the related velocity Verlet methodqk+1 − qktk+1 − tk

= pk +tk+1 − tk

2 F(qk)

pk+1 − pktk+1 − tk

=F(qk) + F(qk+1)

2


Midpoint rule

Instead, using the midpoint quadrature rule we get the followingdiscrete Lagrangian

Ld(qk, qk+1) = (tk+1 − tk)

[12

(qk+1 − qktk+1 − tk

)2− V

(qk + qk+1

2

)]

and the related method isqk+1 − qktk+1 − tk

= pk +tk+1 − tk

2 F(

qk+1 + qk2

)pk+1 − pktk+1 − tk

= F(

qk+1 + qk2

)


Other variational integrators

Most of standard symplectic methods can be written as variationalintegrators, among them:

Newmark-β methods, the first proof of symplecticity of thesemethods exploits the framework of variational integrators,

Ld(q0, q1) = h[

12

(ηβ(q1) − ηβ(q0)

h

)2− V(ηβ(q0))

]

symplectic partitioned Runge-KuttaGalerkin methods (Lobatto IIIA-IIIB)

Ld(q0, q1) ≈ ext SNI(q) : q ∈ VN, q(0) = q0, q(h) = q1


Wave equation

Probably the simplest model is the linear wave equation, whichrepresents a linearization of the hyperelastic problem. The relatedLagrangian density in the one dimensional space is

L =12

[(∂φ

∂t

)2−

(c∂φ∂x

)2]

Applying the trapezoidal quadrature rule in time we get thesemi-discrete Störmer-Verlet scheme

∫B

φk+1 − φktk+1 − tk

ψ dX =

∫B

pkψ−tk+1 − tk

2 c2∂φk∂x

∂ψ

∂x dX

∫B

pk+1 − pktk+1 − tk

ψ dX = −

∫B

c2 ∂

∂x

(φk + φk+1

2

)∂ψ

∂x dx


Wave equation

c = 1;P1;∆t = 0.01s;I = [0, 2π];Nx = 100.

0 2 4 6 8 10t

−24

−22

−20

−18

−16

−14

−12

−10

−8

log

2

‖uh−u‖ 2

‖u‖ 2

positionmomentum


Wave equation: linear momentum conservation (=0)

0 2 4 6 8 10t

−58

−56

−54

−52

−50

−48

−46

log

2

∣ ∣ ∣ ∣∫2π

0

pdx

∣ ∣ ∣ ∣


Wave equation: energy

0 2 4 6 8 10t

0

1

2

3

4

|Eh−E|

E×10−9 + 4.9335×10−4


Conclusions and further developments

improvement of BC treatment;preconditioners;preliminary results on OTM.

Future developments of this projectintroduction of the melting;development of a fluid-structure and structure-structuremethod based on OTM;application of OTM and VI to geodynamic problems, couplingthe mechanical problem with the thermodynamic.


Thank you for your attention!


Second HPC-GA Workshop Marco Cuﬀaro, Edie Miglio, Mattia ... · Tectonic evolution at mid-ocean ridges: geodynamics and numerical modeling. Second HPC-GA Workshop Marco Cuﬀaro,

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