Finite element approximation of Maxwell's equations with unfitted meshes for borehole simulations

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Finite element approximation of Maxwell’s equations

with unfitted meshes for borehole simulations

Théophile Chaumont-Frelet, Serge Nicaise, David Pardo

To cite this version:

Théophile Chaumont-Frelet, Serge Nicaise, David Pardo. Finite element approximation of Maxwell’s

equations with unfitted meshes for borehole simulations. ICIAM 2019 - International Congress on

Industrial and Applied Mathematics, Jul 2019, Valencia, Spain. �hal-02321135�

Finite element approximation of Maxwell’s equations with

unfitted meshes for borehole simulations

T. Chaumont-Frelet1 _{S. Nicaise}2 _{D. Pardo}3

1_{Inria project-team Nachos}

2_{Univ. Valenciennes, LAMAV}

3_{UPV/EHU, BCAM and Ikerbasque}

ICIAM - July 17, 2019

Novel computational methods for electromagnetic problems in complex nonlinear materials

Finite element approximation of Maxwell’s equations with

unfitted meshes for borehole simulations

T. Chaumont-Frelet1 _{S. Nicaise}2 _{D. Pardo}3

1_{Inria project-team Nachos}

2_{Univ. Valenciennes, LAMAV}

3_{UPV/EHU, BCAM and Ikerbasque}

ICIAM - July 17, 2019

Novel computational methods for electromagnetic problems in complex

Finite element approximation of Maxwell’s equations with

unfitted meshes for borehole simulations

T. Chaumont-Frelet1 _{S. Nicaise}2 _{D. Pardo}3

1_{Inria project-team Nachos}

2_{Univ. Valenciennes, LAMAV}

3_{UPV/EHU, BCAM and Ikerbasque}

ICIAM - July 17, 2019

Novel computational methods for electromagnetic problems in

Finite element approximation of Maxwell’s equations with

unfitted meshes for borehole simulations

T. Chaumont-Frelet1 _{S. Nicaise}2 _{D. Pardo}3

1_{Inria project-team Nachos}

2_{Univ. Valenciennes, LAMAV}

3_{UPV/EHU, BCAM and Ikerbasque}

ICIAM - July 17, 2019

Novel computational methods for electromagnetic problems

in

Geosteering

Guide the drilling trajectory using “real-time” magnetic measurements

Geosteering

Reproduce the measurements using finite element discretizations

Non-fitting meshes (aka unfitted meshes)

Finite element discretizations are build on a mesh of the domain.

A mesh is fitting if the physical interfaces are aligned with faces. The physical coefficients are constant (or smooth) inside each element.

A non-fitting mesh is generated independently of the physical coefficients.

Fitting mesh vs non-fitting mesh

Fitting mesh

Non-fitting mesh

Non-fitting mesh for geosteering

(a) Sliding mesh (b) Zoom around the logging tool

(c) Conductivity model and well trajectory

(d) Position 1 (e) Position 2 (f) Position 3

Pros and cons of non-fitting meshes

Main advantages of non-fitting meshes:

simplified mesh generation (Cartesian grids) minimize remeshing

larger elements

Possible drawbacks of non-fitting meshes:

quadrature schemes to integrate the linear system coefficients possible accuracy loss

Pros and cons of non-fitting meshes

Main advantages of non-fitting meshes:

simplified mesh generation (Cartesian grids) minimize remeshing

larger elements

Possible drawbacks of non-fitting meshes:

quadrature schemes to integrate the linear system coefficients possible accuracy loss

Outline

1

Maxwell’s equations

2

The magnetic formulation

3

The electric formulation

4

Numerical results

Outline

1

Maxwell’s equations

2

The magnetic formulation

3

The electric formulation

4

Numerical results

Maxwell’s equations

M, J are given load terms (electric and/or magnetic dipoles), ω is the operating frequency,

µ0= 4π × 10−7 NA−2 is the vacuum permeability, σ : Ω → R is the conductivity model.

The medium is strongly dissipative (i ω + σ ' σ). The conductivity σ is piecewise constant.

The permeability µ0is constant.

We are mostly interested in magnetic measurements.

Regularity of the solution

In the following, we focus on the case where: Ω is convex,

the medium is layered (σ ' σ(z)).

Magnetic field regularity

H ∈H1(Ω) 3

Electric field regularity

E|Ωl ∈  H1(Ωl) 3 E ∈H1/2−ε(Ω)3 8/41

Approximation by polynomials (quick analysis)

Approximation of H

kH− πhHk0,Ω' h

Approximation of E with fitting meshes

kE− πhEk0,Ω' h

Approximation of E with non-fitting meshes

kE− πhEk0,Ω' h1/2−ε

The key question

Standard approximation theory (C´ea’s Lemma) yields:

Fitting meshes

kE−Ehk0,Ω+ kH−Hhk0,Ω' h

Non-fitting meshes

kE−Ehk0,Ω+ kH−Hhk0,Ω' h1/2

Given thatH is globally regular, is the convergence rate ofHhsharp?

Remark on the general case

In general, the regularity ofE depends on σ

Regularity of E E ∈Hτ (σ)(Ω)3 where 0 < τ (σ) < 1/2. Interpolation errors kH− πhHk0,Ω' h kE− πhEk0,Ω' hτ (σ) Interpolation errors kE− πhEk0,Ω+ kH− πhHk0,Ω' hτ (σ) 11/41

Outline

1

Maxwell’s equations

2

The magnetic formulation

3

The electric formulation

4

Numerical results

The magnetic formulation

We recast Maxwell’s equation into a second-order system:  −i ωµ0H+ ∇ × σ−1∇ ×H
= M in Ω, σ−1∇ ×H
× n = 0 on ∂Ω. We obtainE' ∇ ×H by post-processing. Variational formulation

FindH∈ H(curl, Ω) such that

b(H, v ) = (M, v ) for all v ∈ H(curl, Ω).

b(H, v ) = −i ωµ0(H, v ) + (σ −1

∇ ×H, ∇ × v )

Finite element discretization

We consider first-order N´ed´elec elements on a mesh Th= {K }: Vh=

n

vh∈ H(curl, Ω) | vh|K= a × x + b, a, b ∈ C3 o

.

Discrete variational formulation

FindHh∈ Vh such that

b(Hh, vh) = (M, vh) for all vh∈ Vh.

Error estimates

Assuming that M ∈ H(div, Ω):

Regularity

H ∈H1(Ω)3 ∇ ×H∈H1/2−ε(Ω)3

Interpolation error

kH− πhHk0,Ω. hkMkdiv,Ω k∇ × (H− πhH)k0,Ω. h1/2−εkMk0,Ω

Error estimate (C´ea’s Lemma)

kH−Hhk0,Ω+ k∇ × (H−Hh)k0,Ω. h1/2−εkMkdiv,Ω

Error estimates

Can we improve this error estimate? Yes!

The Poisson equation

Consider the simpler problem 

u− ∆u = f in Ω, ∇u· n = 0 on ∂Ω, where f ∈ L2(Ω).

Using Lagrange elements, we are in the similar situation where:

Interpolation error

ku− πhuk0,Ω. h2kf k0,Ω

Error estimate (C´ea’s lemma)

ku−uhk0,Ω. hkf k0,Ω

Duality technique: the “Aubin-Nitsche trick”

Introduceξas the unique solution to

a(w ,ξ) = (w ,u−uh), ∀w ∈ H1(Ω). Then, selecting w =u−uh

ku−uhk20,Ω= a(u−uh,ξ) = a(u−uh,ξ− πhξ) . ku−uhk1,Ωkξ− πhξk1,Ω. We observe thatξis solution to

 ξ− ∆ξ = u−uh in Ω, ∇ξ· n = 0 on ∂Ω, andξ∈ H2 (Ω) with |ξ|2,Ω. ku−uhk0,Ω sinceu−uh∈ L2(Ω). 17/41

Duality technique: the “Aubin-Nitsche trick”

Application to Maxwell’s equations

Direct application fails. Indeed, if we letξsolve

b(w ,ξ) = (w ,H−Hh), ∀w ∈ H(curl, Ω), thenξ∈ H/ 1

(Ω)
3

becauseH−Hh∈ H(div, Ω)./

The key idea is to introduce the Helmhotlz-Hodge decomposition

H−Hh= φ + ∇p. with ∇ · φ = 0.

We employ the “Aubin-Nitsche trick” to estimate kφk0,Ω.

Additional “orthogonality” arguments are used to estimate k∇pk0,Ω.

Details of the proof

T. Chaumont-Frelet, S. Nicaise and D. Pardo SIAM J. Numerical Analysis 2018

For regular solution τ (σ) > 1/2:

L. Zhong, S. Shu, G. Wittum and J. Xu J. Computational Mathematics 2009

General case (done in the same time): A. Ern and J.L. Guermond

Computer & Mathematics with Applications 2018

Error estimate

Error estimate

kH−Hhk0,Ω. h1−2εkMkdiv,Ω

We can remove the “−2ε” if we employ “non-Sobolev” interpolation spaces.

The convergence rate is linear, as in the case of fitting meshes.

Outline

1

Maxwell’s equations

2

The magnetic formulation

3

The electric formulation

4

Numerical results

The electric formulation

Another option is to consider the second-order system: 

−i ωµ0σE+ ∇ × ∇ ×E = J in Ω,

E× n = 0 on ∂Ω,

and post-processH' ∇ ×E.

Variational formulation

FindE∈ H0(curl, Ω) such that

b(E, v ) = (J, v ) for all v ∈ H0(curl, Ω).

b(E, v ) = −i ωµ0(σE, v ) + (∇ ×E, ∇ × v )

Finite element discretization

We consider first-order N´ed´elec elements on a mesh Th= {K }: Vh=

n

vh∈ H0(curl, Ω) | vh|K= a × x + b, a, b ∈ C3 o

.

Discrete variational formulation

FindEh∈ Vhsuch that

b(Eh, vh) = (J, vh) for all vh∈ Vh.

Error estimate

Assuming that J ∈ H(div, Ω)

Regularity

E ∈H1/2−ε(Ω)3 ∇ ×E ∈H1(Ω)3

Interpolation error

kE− πhEk0,Ω. h1/2−εkJkdiv,Ω k∇ × (E− πhE)k0,Ω. hkJk0,Ω

Error estimate (C´ea’s Lemma)

kE−Ehk0,Ω+ k∇ × (E−Eh)k0,Ω. h1/2−εkJkdiv,Ω

Error estimate

Can we improve this error estimate? Yes!

A modified “Aubin-Nitsche trick” (Step 1)

Letφ∈ H0(curl, Ω) solve

b(w ,φ) = (∇ × w , ∇ × (E−Eh)), ∀w ∈ H0(curl, Ω), so that

k∇ × (E−Eh)k20,Ω= b(E−Eh,φ) = b(E−Eh,φ− πhφ).

Intermediate error estimate

k∇ × (E−Eh)k20,Ω. h 1/2−ε

(kφ− πhφk0,Ω+ k∇ × (φh− πhφ)k0,Ω) kJkdiv,Ω.

It remains to estimate the interpolation errors in the right-hand side.

A modified “Aubin-Nitsche trick” (Step 2)

Sinceφ∈ H0(curl, Ω) is solution to

b(w ,φ) = (∇ × w , ∇ × (E−Eh)), ∀w ∈ H0(curl, Ω), we can show thatφ∈H1/2−ε(Ω)

3 with

|φ|1/2−ε,Ω. k∇ × (E−Eh)k0,Ω.

First interpolation error estimate

kφ− πhφk0,Ω. h1/2−εk∇ × (E−Eh)k0,Ω

A modified “Aubin-Nitsche trick” (Step 3)

We have

k∇ ×φk0,Ω. k∇ × (E−Eh)k0,Ω, but we cannot extend this result to

|∇ ×φ|s,Ω for s > 0, because ∇ × (E−Eh) /∈ H(curl, Ω). We cannot estimate

k∇ × (φ− πhφ)k0,Ω directly.

A modified “Aubin-Nitsche trick” (Step 3)

We introduceψ=φ− (E−Eh). We see that b(v ,ψ) = b(v ,φ) − b(v ,E−Eh)

= (∇ × v , ∇ × (E−Eh)) − b(v ,E−Eh) = −i ωµ0(v , σ(E−Eh)).

We can show that ∇ ×ψ∈ (H1_(Ω))3_with

|∇ ×ψ|1,Ω. kE−Ehk0,Ω.

Intermediate interpolation result

k∇ × (ψ− πhψ)k0,Ω. hkE−Ehk0,Ω. h3/2−εkJkdiv,Ω.

A modified “Aubin-Nitsche trick” (Step 3)

Second interpolation error estimate

k∇ × (φ− πhφ)k0,Ω. 

h + h3/2−εkJkdiv,Ω.

A modified “Aubin-Nitsche trick” (Step 4)

We regroup the intermediate results: k∇ × (E−Eh)k0,Ω. h1/2−ε



h1/2−ε+ h + h3/2−εkJkdiv,Ω.

Error estimate

k∇ × (E−Eh)k0,Ω. h1−2εkJkdiv,Ω

We can remove the “−2ε” if we employ “non-Sobolev” interpolation spaces. The convergence rate is linear, as in the case of fitting meshes.

Surprisingly ∇ ×Ehconverges faster thanEhin L2(Ω)-norm!

Outline

1

Maxwell’s equations

2

The magnetic formulation

3

The electric formulation

4

Numerical results

2.5D Maxwell’s equations

Ha(x , y , z)e−i ξydy , Hˆ ξ = ( ˆHxξ, ˆH ξ y) 2.5D formulation Find ( ˆHξy, ˆH ξ ) ∈ H1( ˆΩ) × H(curl, ˆΩ) ( (i ωµ0+ ξ2σ−1) ˆH ξ + curlσ−1curl ˆHξ− i ξσ−1_∇ Hξy = Mξ, i ωµ0Hξy− ∇ · σ −1 ∇Hξ y
+ iξ∇ · σ −1 Hξ
= Mξ y. Ha(x , y , z) ' N X j =1 wjHˆa(x , z)ei ξjy 32/41

Logging tool

dRX

dTX

Transmitter 1 Receiver 1 Receiver 2 Transmitter 2

Logging tool (dTX= 0.6 m, dRX= 0.1 m) Plog= 1 2Im  logH 11 zz H12 zz + logH 22 zz H21 zz  Alog= 1 2Re  logH 11 zz H12 zz + logH 22 zz H21 zz  33/41

Benchmark

θ

θ

Fitting (left) and non-fitting (right) meshes

Two layers: Apparent resistivity (P-log)

Two layers: Apparent resistivity (A-log)

Thin layer: Apparent resistivity (P-log)

Thin layer: Apparent resistivity (A-log)

Outline

1

Maxwell’s equations

2

The magnetic formulation

3

The electric formulation

4

Numerical results

Main abstract results

Assume that µ = µ0is constant.

Assume that σ represents a layered media. Assume that J and M are sufficiently smooth.

Assume that the linear system coefficients are exactly integrated.

Error estimates for fitting meshes

kE−EhkL2(Ω)= O(h) kH−HhkL2(Ω)= O(h)

Error estimates for non-fitting meshes

kE−EhkL2_(Ω)= O(h1/2) kH−HhkL2_(Ω)= O(h)

Main abstract results

The use of non-fitting meshes deteriorates the convergence rate ofEh. The convergence rate ofHhis preserved.

T. Chaumont-Frelet, S. Nicaise and D. Pardo SIAM J. Numerical Analysis 2018

Non-fitting meshes can simulate tools based on magnetic field measurements.

Main numerical results

We have tested different quadrature schemes.

Non-fitting meshes provide accurate results with an appropriate scheme.

T. Chaumont-Frelet, D. Pardo and A. Rodr`ıguez-Roz`as Computational Geosciences 2018