The Mimetic Finite Difference Methodarturo.imati.cnr.it/~marco/resources/Slides/talk-MFD-FVCA-2008.pdf · The Mimetic Finite Difference Method Gianmarco Manzini1 Istituto di Matematica

Outline

The Mimetic Finite Difference Method

Gianmarco Manzini1

Istituto di Matematica Applicata e Tecnologie Informatiche(IMATI) C.N.R., Pavia, Italy

FVCA5 - June 08-13, 2008 Aussois, France

Manzini, G. The Mimetic Finite Difference Method

Outline

Outline

1 MFD method for Darcy’s problemFormal constructionMimetic conservation equationMimetic constitutive equation

2 Theoretical results and applicationsA priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation

3 Summary


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3 Summary


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3 Summary


Mimetic FormulationTheoretical results and applications

Summary

Formal constructionMimetic conservation equationMimetic constitutive equation

“Mimetic (mathematics)”From Wikipedia, the free encyclopedia

(i) The goal of numerical analysis is toapproximate the continuum, soinstead of solving a partialdifferential equation one aims insolve a discrete version of thecontinuum problem.

(ii) A numerical method is called mimetic when it mimics (orimitates) some properties of the continuum vector calculus.

An example: a mixed finite element method applied to Darcyflows strictly conserves the mass of the flowing fluid.



Summary


Some literature. . .

Mimetic schemes were first proposed in the early eighties:

Samarskii-Tishkin-Favorskii-Shashkov, OperationalFinite-Difference Schemes, Differential Equations, 1981;

many papers were published after this one. . .

Some recent joint work from Los Alamos-Pavia:

Brezzi-Lipnikov-Shashkov,SINUM,2005 (a priori estimates)Brezzi-Lipnikov-Simoncini, M3AS, 2005 (a family of MFDs). . .

Extensions:

Cangiani-M. CMAME, 2008 (post-processing)Beirao da Veiga-M., NME, 2008 (mesh adaptivity). . .



Summary


. . . people and topics (in Pavia)

Main features:family of schemes based on mixed formulation;grids formed by elements of general shape (polygons,polyhedra);

people currently working in Pavia:Beirao da Veiga, Boffi, Brezzi, Buffa, Cangiani, M.,A. Russo, . . .

some topics under investigation:diffusion and convection-diffusion modelsa posteriori estimates and mesh adaptivityelectromagnetismStokes equations

2-D software implementation



Summary


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3 Summary



Summary


Linear diffusion in mixed form

Consider

div (−K∇p) = b, Ω ⊂ IRd , d = 2, 3

+boundary conditions

Let−→F be the flux vector variable:

−→F = −K∇p constitutive equation

div−→F = b conservation equation

(1)

Model problem:

solve (1) for p and−→F with suitable boundary conditions



Summary


Formally

FIRST,

(let Th be a family of partitions of Ω formed by polygonalelements, h being the mesh size);

(i) degrees of freedom for− scalar fields −→ discrete scalars , Qh;− vector fields −→ discrete vectors , Xh;

Qh and Xh are not functions, but vectors of numbers!

(ii) “discrete” operators:− the discrete divergence DIVh : Xh → Qh;− the discrete flux (or gradient) Gh : Qh → Xh;

satisfying a duality relationship (discrete Gauss-Green formula).



Summary


Formally

FIRST,








Summary


Formally

FIRST,








Summary


Formally

THEN,

mimic the continuous differential equations by using discreteoperators acting on the discrete scalar and flux unknownsph ∈ Qh and Fh ∈ Xh:

constitutive equation:

−→F = −K∇p −→ Fh = Ghph

conservation equation:

div−→F = b −→ DIVhFh = bI

(where bI is a suitable interpolation of b in Qh)Manzini, G. The Mimetic Finite Difference Method


Summary


Qh, degrees of freedom for scalar fields

Ω, Th

EqE

• q ∈ Qh means q =˘

qE¯

E∈Th

(equivalent to a piecewise

constant function)

• dim(Qh) = number of elements

of the mesh.

• ”interpolation” operator:

(pI)E =1|E |

Z

Ep dV .



Summary



Ω, Th

EqE


qE¯

E∈Th


constant function)


of the mesh.


(pI)E =1|E |

Z

Ep dV .



Summary



Ω, Th

EqE


qE¯

E∈Th


constant function)


of the mesh.


(pI)E =1|E |

Z

Ep dV .



Summary


Xh, degrees of freedom for vector fields

Ω, Th

E

E ′

e

• G ∈ Xh means G =˘

GeE

¯

e is an edge of E

• GeE + Ge

E′ = 0 ∀e ⊆ E ∩ E ′

dim(Xh) = number of edges

of the mesh.

• ”interpolation” operator:`−→

F Ié

E=

1|e|

Z

e

−→n eE ·

−→F dV

−→n eE

E



Summary



Ω, Th

E

E ′

e


GeE

¯

e is an edge of E

• GeE + Ge

E′ = 0 ∀e ⊆ E ∩ E ′


of the mesh.


F Ié

E=

1|e|

Z

e

−→n eE ·

−→F dV

−→n eE

E



Summary



Ω, Th

E

E ′

e


GeE

¯

e is an edge of E

• GeE + Ge

E′ = 0 ∀e ⊆ E ∩ E ′


of the mesh.


F Ié

E=

1|e|

Z

e

−→n eE ·

−→F dV

−→n eE

E



Summary


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3 Summary



Summary


Discrete divergence operator DIVh : Xh → Qh

Let−→G be a true vector field ; take the cell average of div

−→G

over E and use Gauss Theorem:

1|E |

∫

Ediv

−→G dV =

1|E |

∫

∂E

−→n E ·−→G dS =

1|E |

∑

e∈∂E

|e|(−→

GI)e

E

Let G ∈ Xh; the discrete divergence of G in Qh is definedelement by element as the constant value

(DIVhG)E =

1|E |

∑

e∈∂E

|e|GeE



Summary


Discrete divergence operator DIVh : Xh → Qh

Let−→G be a true vector field ; take the cell average of div

−→G

over E and use Gauss Theorem:

1|E |

∫

Ediv

−→G dV =

1|E |

∫

∂E

−→n E ·−→G dS =

1|E |

∑

e∈∂E

|e|(−→

GI)e

E

Let G ∈ Xh; the discrete divergence of G in Qh is definedelement by element as the constant value

(DIVhG)E =

1|E |

∑

e∈∂E

|e|GeE



Summary


The mimetic conservation equation: DIVhFh = bI

Take the cell average of the conservation equation:

div−→F = b −→

1|E |

∫

Ediv

−→F dV =

1|E |

∫

Eb dV = bI |E

and define the flux Fh as the solution to

(DIVhFh)E = bI |E ≡

1|E |

∫

Ediv

−→F dV

using the inner product[p, q

]Qh

:=∑

E |E | pEqE =

∫

Ω

p q dV ,

variational form:[DIVhFh, q

]Qh

=[bI , q

]Qh

∀q ∈ Qh.



Summary


The mimetic conservation equation: DIVhFh = bI

Take the cell average of the conservation equation:

div−→F = b −→

1|E |

∫

Ediv

−→F dV =

1|E |

∫

Eb dV = bI |E

and define the flux Fh as the solution to

(DIVhFh)E = bI |E ≡

1|E |

∫

Ediv

−→F dV

using the inner product[p, q

]Qh

:=∑

E |E | pEqE =

∫

Ω

p q dV ,

variational form:[DIVhFh, q

]Qh

=[bI , q

]Qh

∀q ∈ Qh.



Summary


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3 Summary



Summary


The mimetic constitutive equation: Fh = Ghph

Now, we discretize the constitutive equation :

−→F = −K∇p

Note that the conservation equation

1|E |

∑

e∈∂E

|e| (Fh)eE = (bI)E ∀E ∈ Th

holds for• many finite volume schemes;• the RT0−P0 mixed finite element method

. . . So let us first have a look at these approaches.



Summary


The mimetic constitutive equation: Fh = Ghph

Now, we discretize the constitutive equation :

−→F = −K∇p

Note that the conservation equation

1|E |

∑

e∈∂E

|e| (Fh)eE = (bI)E ∀E ∈ Th

holds for• many finite volume schemes;• the RT0−P0 mixed finite element method

. . . So let us first have a look at these approaches.



Summary


Finite volume discretization

We use a direct formula for the flux:

−→F = −K∇p (with K = κI)

K

L

A

B

−→n AB

−−→n AB ·

−→F ≈κ

pK − pL∣∣−→x K −−→x L∣∣

K

L

A

B

−−→F ≈κ

(pK − pL∣∣−→x K −

−→x L∣∣−→n AB

γAB+

pB − pA∣∣−→x B −−→x A∣∣−→n KL

γKL

)



Summary


Mixed finite elements discretization

In mixed finite element the constitutive equation is discretized byusing an explicit representation of the flux field from the fluxdegrees of freedom inside each element.

Let E be a triangle and take the Raviart-Thomas space:

RT0(E) :=

−→v (x , y) =

(α

β

)+ γ

(xy

)(x , y) ∈ E

.

Reconstruct RE (G) inside E by using the degrees of freedomGe

E and the canonical basis functions of RT0(E).



Summary



In mixed finite element the constitutive equation is discretized byusing an explicit representation of the flux field from the fluxdegrees of freedom inside each element.

Let E be a triangle and take the Raviart-Thomas space:

RT0(E) :=

−→v (x , y) =

(α

β

)+ γ

(xy

)(x , y) ∈ E

.

Reconstruct RE (G) inside E by using the degrees of freedomGe

E and the canonical basis functions of RT0(E).



Summary



The Raviart-Thomas field RE (G) preserves:

1. the degrees of freedom: −→n eE · RE

(−→G)

= GeE

2. the elemental divergence: divRE(G)

= DIVh,EG

3. constant vector fields: P0-compatible;let G

−→c = (−→c )I with −→c constant inside E ; then,

RE(G

−→c ) =−→c .



Summary





(−→G)

= GeE


= DIVh,EG



RE(G

−→c ) =−→c .



Summary





(−→G)

= GeE


= DIVh,EG



RE(G

−→c ) =−→c .



Summary



Given Fh, G in Xh (degrees of freedom):

1. reconstruct RE (Fh), RE (G) inside each element E ;

2. define the “RT0 inner product ”:

(Fh, G

)RT0

:=∑

E

∫

EK−1RE (Fh) · RE (G) dV

3. rewrite the RT0 − P0 variational form of K−1−→F = −∇pas:

(Fh, G

)RT0

=∑

E

∫

Eph divRE (G) dV ∀G ∈ Xh



Summary






(Fh, G

)RT0

:=∑

E

∫



(Fh, G

)RT0

=∑

E

∫




Summary






(Fh, G

)RT0

:=∑

E

∫



(Fh, G

)RT0

=∑

E

∫




Summary


MFD: mimicking Mixed Finite Elements

FIRST, let RE(·)

be a reconstruction for E with the propertiesof mixed finite elements:


(−→G)

= GeE ;


= DIVh,EG;



RE(G

−→c ) =−→c .

We can build many operators: we do not have uniqueness!



Summary



FIRST, let RE(·)



(−→G)

= GeE ;


= DIVh,EG;



RE(G

−→c ) =−→c .




Summary



FIRST, let RE(·)



(−→G)

= GeE ;


= DIVh,EG;



RE(G

−→c ) =−→c .




Summary



FIRST, let RE(·)



(−→G)

= GeE ;


= DIVh,EG;



RE(G

−→c ) =−→c .




Summary



THEN, given Fh, G in Xh (degrees of freedom):


2. define the mimetic inner product:

[Fh, G

]Xh

:=∑

E

∫


3. mimetic discretization of K−1−→F = −∇p:

[Fh, G

]Xh

=∑

E

∫

EphdivRE (G)dV =

[ph,DIVhG

]Qh

∀G ∈ Xh



Summary






[Fh, G

]Xh

:=∑

E

∫



[Fh, G

]Xh

=∑

E

∫

EphdivRE (G)dV =

[ph,DIVhG

]Qh

∀G ∈ Xh



Summary






[Fh, G

]Xh

:=∑

E

∫



[Fh, G

]Xh

=∑

E

∫

EphdivRE (G)dV =

[ph,DIVhG

]Qh

∀G ∈ Xh



Summary



Scheme formulation:[Fh, G

]Xh

−[ph,DIVhG

]Qh

= 0 ∀G ∈ Xh[DIVhFh, q

]Qh

=[bI, q

]Qh

∀q ∈ Qh.

• Substitute Fh = Ghph in the first equation:[Ghph, G

]Xh

=[ph,DIVhG

]Qh

∀G ∈ Xh

to get the MFD discretization :

K−1−→F = −∇p −→ Fh = Ghph in Xh



Summary



Scheme formulation:[Fh, G

]Xh

−[ph,DIVhG

]Qh

= 0 ∀G ∈ Xh[DIVhFh, q

]Qh

=[bI, q

]Qh

∀q ∈ Qh.

• Substitute Fh = Ghph in the first equation:[Ghph, G

]Xh

=[ph,DIVhG

]Qh

∀G ∈ Xh

to get the MFD discretization :

K−1−→F = −∇p −→ Fh = Ghph in Xh



Summary


Features

Let R(·) = RE (·) be a lifting operator such that RE (·) isP0-compatible on E .

1. R(Xh) ⊂ H(div,Ω); hence R(Xh) − P0 is a conformingmixed discretization that generalizes RT0 − P0;

2. we can use elements of very general shapes, evennon-convex (but at least star-shaped) elements areadmissible;

3. the implementation and analysis of the MFD method aresimilar to those of the Mixed Finite Element method.



Summary


Mimetic scalar product for the flux

The scalar product for fluxes is given by assembling “elemental”scalar products, each one of which can be represented by asymmetric positive definite (SPD) matrix:

∫

EK−1RE (Fh) · RE (G) dV −→ ME = RE

K−1E

|E |RT

E + ME

RE(·)

is not unique −→ family of matrices

ME contains free positive parameters and ensures that ME

is an SPD matrix;

the formulas for RE , ME depend on the shape of E and canbe derived without the explicit knowledge of RE

().



Summary

A priori estimatesPost-processing of solutionAn error estimator and mesh adaptivityConvection-Diffusion-Reaction Equation

Outline



3 Summary



Summary


A priori error estimates for Darcy’s equation[Brezzi-Lipnikov-Shashkov, SINUM 2005]

General assumptions:• Ω polygonal or polyhedral with Lipschitz continuous boundary;• Th is a partition satisfying some mesh regularity assumptions;• the scalar product

[·, ·]

Xhsatisfies local consistency and stability;

1. If p ∈ H2(Ω) then

∣∣∣∣∣∣−→F I − Fh∣∣∣∣∣∣

Xh≤ Ch

∣∣∣∣p∣∣∣∣

H2(Ω)

where∣∣∣∣∣∣ ·∣∣∣∣∣∣2

Xh=[·, ·]

Xh.



Summary



2. if p ∈ H2(Ω) then∣∣∣∣∣∣pI − ph

∣∣∣∣∣∣Qh

≤ Ch∣∣∣∣p∣∣∣∣

H2(Ω)

3. superconvergence: if p∈H2(Ω), b∈H1(Ω), and Ω is convex :

∣∣∣∣∣∣pI − ph

∣∣∣∣∣∣Qh

≤ Ch2(∣∣∣∣p

∣∣∣∣H2(Ω)

+∣∣∣∣b∣∣∣∣

H1(Ω)

)



Summary



2. if p ∈ H2(Ω) then∣∣∣∣∣∣pI − ph

∣∣∣∣∣∣Qh

≤ Ch∣∣∣∣p∣∣∣∣

H2(Ω)

3. superconvergence: if p∈H2(Ω), b∈H1(Ω), and Ω is convex :

∣∣∣∣∣∣pI − ph

∣∣∣∣∣∣Qh

≤ Ch2(∣∣∣∣p

∣∣∣∣H2(Ω)

+∣∣∣∣b∣∣∣∣

H1(Ω)

)



Summary


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3 Summary



Summary


Post-processing of solution[Cangiani-M. CMAME 2008]

Let −→v be a vector field on E . Then,

−→vInterpolate−→

−→v I Reconstruct−→

−→v ∗ ∈(P0(E)

)d

where the elemental reconstructed vector is given by:∫

E

−→v ∗ · ∇q =[−→v I, (KE∇q)I

]E ∀q ∈ P1(E).

In effect,−→v ∗ = RT

E−→v I

where RE is the matrix used in the definition of the elementalscalar product; so, easy to implement and very cheap .



Summary


Post-processing of solution[Cangiani-M. CMAME 2008, L. Beirao da Veiga Numer. Math. 2008]

1. Reconstruct the elemental gradient :

Fh|EReconstruct

−→−→F ∗

h|ECalculate−→ ∇Ep∗

h = −K−1E

−→F ∗

h;

2. reconstruct the piecewise linear pressure field as

ph|∗

E (−→x ) := pE + ∇E p∗

h · (−→x −

−→x E ), ∀−→x ∈ E

(−→x E center of gravity of E)

Under the hypothesis yielding scalar super-convergence:∣∣∣∣p − p∗

h

∣∣∣∣L2(Ω)

+ h∣∣∣∣∣∣p − p∗

h

∣∣∣∣∣∣1,h ≤ Ch2

(∣∣∣∣p∣∣∣∣

H2(Ω)+∣∣∣∣b∣∣∣∣

H1(Ω)

)

with∣∣∣∣∣∣q∣∣∣∣∣∣2

1,h=∑

E

∣∣∣∣∇q∣∣∣∣2

L2(E)+∑

e

h−1e

∣∣∣∣[[q]]∣∣∣∣2

e.



Summary


Outline



3 Summary



Summary


An a posteriori error estimator for MFDs[L. Beirao da Veiga, Numer. Math. 2008]

p∗

h , post-processed pressure; [[q]]e jump of q across edge e.

error estimator: η2 :=∑

E ηE2 and

ηE2 :=

∣∣∣∣∣∣Fh + (KE∇p∗

h)I∣∣∣∣∣∣2

E+

12

∑

e∈∂E

h−1e

∣∣∣∣ [[p∗

h ]]e∣∣∣∣2

L2(e)

target error: err2 =∑

E errE2 with

errE2 :=

∣∣∣∣−→F −R(Fh)∣∣∣∣2

L2(E)+ h2

E

∣∣∣∣div (−→F −R(Fh))

∣∣∣∣2L2(E)

+∣∣∣∣∣∣p − p∗

h

∣∣∣∣∣∣1,E

and∣∣∣∣∣∣q∣∣∣∣∣∣2

1,E=∣∣∣∣∇q

∣∣∣∣2L2(E)

+∑

e∈∂E

h−1e

∣∣∣∣ [[q]]e∣∣∣∣2

L2(e)

efficiency: cηE ≤ errE ;

reliability: err ≤ Cη;.



Summary


An a posteriori error estimator for MFDs[L. Beirao da Veiga, Numer. Math. 2008]

p∗

h , post-processed pressure; [[q]]e jump of q across edge e.

error estimator: η2 :=∑

E ηE2 and

ηE2 :=

∣∣∣∣∣∣Fh + (KE∇p∗

h)I∣∣∣∣∣∣2

E+

12

∑

e∈∂E

h−1e

∣∣∣∣ [[p∗

h ]]e∣∣∣∣2

L2(e)

target error: err2 =∑

E errE2 with

errE2 :=

∣∣∣∣−→F −R(Fh)∣∣∣∣2

L2(E)+ h2

E

∣∣∣∣div (−→F −R(Fh))

∣∣∣∣2L2(E)

+∣∣∣∣∣∣p − p∗

h

∣∣∣∣∣∣1,E

and∣∣∣∣∣∣q∣∣∣∣∣∣2

1,E=∣∣∣∣∇q

∣∣∣∣2L2(E)

+∑

e∈∂E

h−1e

∣∣∣∣ [[q]]e∣∣∣∣2

L2(e)

efficiency: cηE ≤ errE ;

reliability: err ≤ Cη;.



Summary


Some experiments on mesh adaptivity[L. Beirao da Veiga & M., IJNME, 2008]

The indicator ηE allows us to select elements needing refinement:

1. calculate local error estimates ηE ;

2. mark for refinement those elements such that ηE ≥ tol ηmax ;

3. subdivide the marked elements as follows:

No need for special treatment of hanging nodes!



Summary


Mesh adaptivityL-shaped domain, K = II, p(r , θ) = r2/3 sin(2θ/3)

102

103

104

105

10610

-3

10-2

10-1

100

After 3 refinements η(), err(•) vs #elements



Summary


Outline



3 Summary



Summary


Convection-Diffusion-Reaction Equation(In collaboration with A. Cangiani and A. Russo)

• Problem:−→F = −(K∇ p +

−→β p) in Ω

div−→F + c p = b in Ω

p = 0 on ∂Ω

with

• β, K and c smooth fields plus usual coercivity conditions,

• Ω ⊂ IR2 is a polygonal domain.



Summary


Convection-Diffusion-Reaction EquationScheme formulation

discretization of convection-reaction terms

convection:∫

Ω

K−1−→β p · R(G)

dV −→∑

E pE[(−→β )I, G

]E

reaction:∫

Ω

c p q dV −→[cIph, q

]Qh

Scheme formulation[Fh, G

]Xh

−[p,DIVhG

]Qh

+∑

E pE[(−→β )I , G

]E = 0 ∀G ∈ Xh

[DIVhFh, q

]Qh

+[cIph, q

]Qh

=[b, q

]Qh

∀q ∈ Qh



Summary


Convection-Diffusion-Reaction Equation

a priori estimate for the diffusive regime:

∣∣∣∣−→F −R(Fh)∣∣∣∣

H(div,Ω)+∣∣∣∣p − ph

∣∣∣∣L2(Ω)

≤ Ch(∣∣∣∣p

∣∣∣∣H1(Ω)

+ h∣∣∣∣p∣∣∣∣

H2(Ω)+∣∣∣∣b − bI

∣∣∣∣L2(Ω)

),

(with−→β ∈ W 2,∞(Ω), c ∈ W 1,∞(Ω), and coercivity assumptions)

superconvergence for pressure cell averages under sameassumptions of the pure elliptic problem and

−→β ∈ R(Xh):

∣∣∣∣∣∣pI − ph

∣∣∣∣∣∣Qh

≤ Ch2(∣∣∣∣p

∣∣∣∣H2(Ω)

+∣∣∣∣b∣∣∣∣

H1(Ω)

)



Summary


Convection-Diffusion-Reaction EquationConvergence results

p(x , y) = sin(2π x) sin(2π y)

+x2 + y2 + 1,

−→β = (1, 3)T ,

c(x , y) = x y2

n h L2-error Rate Hdiv -error Rate1 9.135 10−2 9.134 10−2 −− 1.823 10−1 −−

2 4.654 10−2 4.630 10−2 1.007 8.572 10−2 1.1183 2.346 10−2 2.315 10−2 1.012 4.168 10−2 1.0524 1.175 10−2 1.164 10−2 0.995 2.039 10−2 1.0345 5.880 10−3 5.841 10−3 0.995 1.007 10−2 1.0186 2.940 10−3 2.927 10−3 0.996 5.027 10−3 1.002



Summary


Convection-Diffusion-Reaction EquationSuperconvergence results

p(x , y) = sin(2π x) sin(2π y)

+x2 + y2 + 1,

−→β = (1, 3)T ,

c(x , y) = x y2

n h Qh-error Rate1 9.135 10−2 3.069 10−2 −−

2 4.654 10−2 1.078 10−2 1.5513 2.346 10−2 2.807 10−3 1.9644 1.175 10−2 7.483 10−4 1.9135 5.880 10−3 1.904 10−4 1.9756 2.940 10−3 4.796 10−5 1.989



Summary

Summary

The MFD method for second-order elliptic problems

mimics properties of continuous operators; e.g. DIVh, Gh

satisfy discrete Green-like formulas ;

works on element of very general shape;

shows a strong connection with the lowest-order mixedfinite element method RT0 − P0, helpful in establishing thetheoretical foundation.


The Mimetic Finite Difference Methodarturo.imati.cnr.it/~marco/resources/Slides/talk-MFD-FVCA-2008.pdf · The Mimetic Finite Difference Method Gianmarco Manzini1 Istituto di Matematica

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