4D Variational Data Assimilation with Incompact3d using Large Eddy Models

(1)

HAL Id: hal-01764622

https://hal.inria.fr/hal-01764622

Submitted on 12 Apr 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

using Large Eddy Models

Pranav Chandramouli

To cite this version:

Pranav Chandramouli. 4D Variational Data Assimilation with Incompact3d using Large Eddy Models

. Incompact3d Meeting 2018, Mar 2018, London, United Kingdom. �hal-01764622�

(2)

4D Variational Data Assimilation with Incompact3d using Large Eddy Models

Pranav CHANDRAMOULI

Supervised by: Etienne MEMIN & Dominique HEITZ

INRIA - Rennes (Fluminance) Incompact3d Meeting - London, 2018

28/03/2018

Pranav CHANDRAMOULI 4D-Var with LES - Incompact3d Meeting 1 / 29

(3)

Plan

1 Large Eddy Simulation SGS Models

LES - Results

2 Data Assimilation

Introduction to Data Assimilation Code Formulation

VDA with LES - Results

(4)

Objective

To Identify an apt Sub-Grid Scale (SGS) model for coarse LES in terms of statistical accuracy and numerical stability.

(5)

LES - SGS Models

Smagorinsky Variants :

Classic : ν _t = (C _s ∗ ∆) ² S ¯ , (1) Dynamic : C _s ² = − 1

2 L ij M ¯ ij

M _kl M ¯ _kl . (2) where,

L ij = T ij − τ ˜ ij (3) M _ij = ˜ ∆ ² | S ˜ ¯ | S ˜ ¯ _ij − ∆ ¯ ² (| ^ S ¯ | S ¯ _ij ), (4) WALE :

ν _t = (C _w ∆) ² (ς _ij ^d ς _ij ^d ) ^3/2

( ¯ S _ij S ¯ _ij ) ^5/2 + (ς _ij ^d ς _ij ^d ) ^5/4 (5)

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LES - Models under Location Uncertainty (MULU)

dX _t

dt = w (X _t , t) + σ(X _t , t) ˙ B (6) w (X, t) is the large scale drift

σ(X, t)d B ˙ t stands for small scales represented by a noise

M´emin, Etienne. ”Fluid flow dynamics under location uncertainty.” Geophysical & Astrophysical Fluid Dynamics 108.2 (2014) : 119-146.

(7)

MULU

NS formulation as derived in Memin [2014] : Mass conservation :

d t ρ t + ∇·(ρ w ˜ )dt + ∇ρ ·σ d B _t = 1

2 ∇· (a∇q)dt, (7)

˜

w = w − 1

2 ∇ · a (8)

For an incompressible fluid :

∇· (σd B _t ) = 0, ∇ · w ˜ = 0, (9) Momentum conservation :

∂ t w +w ∇ ^T (w − 1

2 ∇ · a)− 1 2

X

ij

∂ x

i

(a ij ∂ x

j

w )

ρ = ρg −∇p+µ∆w .

(10)

(8)

MULU

Modelling of a :

Stochastic Smagorinsky model (StSm) :

a(x , t) = C ||S|| I 3 , (11) Local variance based models (StSp / StTe) :

a(x , nδt ) = 1

|Γ| − 1 X

x

i

∈η(x)

(w(x i , nδt )− w ¯ (x, nδt))(w (x i , nδt)− w(x, ¯ nδt )) ^T C st ,

(12)

(9)

Channel Flow

0 0.5 1 1.5 2 2.5 3 3.5

0 50 100 150 200 250 300 350

<u0>,<v0>,<w0> uτ

y+

Smag Wale StSm StSp StTe Ref u v w

0 0.5 1 1.5 2 2.5 3 3.5

0 50 100 150 200 250 300 350

,< v 0 >,< w 0 > u τ

y +

Smag Wale StSm StSp StTe Ref u v w

Velocity fluctuation profiles for turbulent channel flow at Re

τ

= 395

n

x

× n

y

× n

z

l

x

× l

y

× l

z

∆x ∆y ∆z ∆t

cLES 48×81×48 6.28×2×3.14 0.13 0.005-0.12 0.065 0.002 Ref 256×257×256 6.28×2×3.14 0.024 0.0077 0.012 -

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Data Assimilation LES - Results

Wake Flow

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.5 1 1.5 2 2.5 3 3.5 4 4.5

<u0u0> U2c

x/D PIV - Parn

DNS

Smag StSm

DSmag StSp

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.5 1 1.5 2 2.5 3 3.5 4 4.5

U 2 c

x /D

< u

⁰

u

⁰

> profile in the streamwise direction on the centre-line behind the cylinder.

Re n

x×

n

y×

n

z

l

x

/D

×

l

y

/D

×

l

z

/D ∆x/D ∆y/D ∆z/D U∆t/D cLES 3900 241×241×48 20×20×3.14 0.083 0.024-0.289 0.065 0.003

PIV - Parn 3900 160×128×1 3.6×2.9×- 0.023 0.023 - 0.01

DNS 3900 1537×1025×96 20×20×3.14 0.013 0.0056-0.068 0.033 0.00075

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Plan

1 Large Eddy Simulation SGS Models

LES - Results

2 Data Assimilation

Introduction to Data Assimilation Code Formulation

VDA with LES - Results

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Objective

An efficient method to incorporate Experimental Fluid Dynamics (EFD) as a guiding mechanism within Computational Fluid Dynamics (CFD) (Data Assimilation) - within the realm of Large Eddy Simulations (LES).

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Data Assimilation - Types

Sequential approach : Cuzol and M´ emin (2009), Colburn et al.

(2011), Kato and Obayashi (2013), Combes et al. (2015) Variational approach (VDA) : Papadakis and M´ emin (2009), Suzuki et al. (2009), Heitz et al. (2010), Lemke and Sesterhenne (2013), Gronskis et al. (2013), Dovetta et al.

(2014)

Hybrid approach : Yang (2014)

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VDA - General Formulation

Objective : Estimate the unknown true state of interest x ^t (t , x) Formulation :

∂ _t x (t, x) + M (x (t, x)) = q(t, x), (13) x(t ₀ , x) = x ^b ₀ + η(x) (14) Y (t, x) = H (x (t, x )) + (t, x) (15) q(t, x) - model error (Covariance matrix Q)

η(x) - background error (Covariance matrix B) (t, x) - observation error (Covariance matrix R)

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VDA - General Formulation

Cost Function : J(x 0 ) = 1

2 (x 0 − x ^b ₀ ) ^T B ⁻¹ (x 0 − x ^b ₀ )+

1 2

Z t

_f

t

₀

( H (x t ) − Y(t)) ^T R ⁻¹ ( H (x t ) − Y(t)).

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4DVar - Adjoint Method

Evolution of the state of interest (x

0

) as a function of time

∂J

∂η = −λ(t

o

) + B

⁻¹

(∂x(t

0

) − ∂ x

0

). (17)

Le Dimet, Francois-Xavier, and Olivier Talagrand. ”Variational algorithms for analysis and assimilation of meteorological observations : theoretical aspects.” Tellus A : Dynamic Meteorology and Oceanography 38.2

(1986) : 97-110.

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4DVar - Incremental Optimisation

Evolution of the cost function (J(x

0

)) as a function of time

Courtier, P., J. N. Th´epaut, and Anthony Hollingsworth. ”A strategy for operational implementation of 4D-Var, using an incremental approach.” Quarterly Journal of the Royal Meteorological Society 120.519 (1994) : 1367-1387.

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Additional Control

Minimisation of J with respect to additional incremental control parameter δu : δγ ⁽ⁱ⁾ = {δx ⁽ⁱ⁾ ₀ , δu ⁽ⁱ⁾ }

J(δγ ⁽ⁱ⁾ ) = 1

2 ||δx ⁽ⁱ⁾ ₀ + x ⁽ⁱ⁾ ₀ − x ^(b) ₀ || ² _B

−1

+ 1 2

Z t

f

t

0

||δu ⁽ⁱ⁾ + u ⁽ⁱ⁾ − u ^(b) || ² _B

c

+ ...

(18)

constrained by :

∂ t δx ⁽ⁱ⁾ + ∂ x M (x ⁽ⁱ⁾ ) · ∂x ⁽ⁱ⁾ + ∂ u M (u ⁽ⁱ⁾ ) · ∂u ⁽ⁱ⁾ = 0 (19)

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4DVar - Flow Chart

4DVar incremental Data Assimilation using Adjoint methodology.

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Incompact3d (Laizet et al. 2010) - DNS

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Incompact3d (Laizet et al. 2010) - LES

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Synthetic Assimilation at Re 3900

Optimisation Parameter - U(x, y, z, t

0

) Control Parameters - U

in

(1, y, z, t)

C

st

(x , y , z)

Re n

x

× n

y

× n

z

l

x

/D × l

y

/D × l

z

/D U∆t/D Duration Full Domain 3900 361×361

^s

×48 20×20×3.14 0.003 40100∆t Observation 3900 145

ⁱ

×145

ⁱ

×48 6×6×3.14 0.003 100∆t

4DVAR 3900 145×145×48 6×6×3.14 0.003 100∆t

s - Stretched ; i - Interpolated

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Background Velocity - Biased

+ sin y ³ noise

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Assimilated Velocity

Background Reference

——

4DVAR Algorithm

——

Assimilated

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Collated Results - Initial Condition Correction

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LES Coefficient Estimation - Preliminary Results

(27)

Conclusions

Efficient cost reduction is possible using LES with VDA An improvement on the background is obtained using LES based VDA

An estimation of the LES coefficient can be obtained using

VDA

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Future Work

Perform LES DA with experimental data Use sliding windows in VDA

Include outlet condition in optimisation algorithm Assimilate over a larger temporal domain

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References

P. Chandramouli, E. Memin, D. Heitz, and S. Laizet. Coarse large-eddy simulations in a transitional wake flow with flow models under location uncertainty. Comput. Fluids, 168 :170-189, 2018.

P. Chandramouli, E. Memin, D. Heitz, and S. Laizet. A Comparative Study of LES Models Under Location Uncertainty. In Proceedings of the Congr´ es Fran¸ cais de M´ ecanique., 2017b.

P. Chandramouli, E. Memin, and D. Heitz. 4d Turbulent Wake Reconstruction using Large Eddy Simulation based Variational Data Assimilation. Delft, Netherlands, 2017a.

E. Memin. Fluid flow dynamics under location uncertainty. Geophys.

Astro. Fluid, 108(2) : 119-146, March 2014.

A. Gronskis, D. Heitz, and E. Memin. Inflow and initial conditions for direct numerical simulation based on adjoint data assimilation. J.

Comput. Phys., 242 :480-497, June 2013.

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Thanks for your attention

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Reference - Synthetic Velocity Fields

(32)

Models under Location Uncertainty

Modelling of a :

Stochastic Smagorinsky model (StSm) :

a(x , t) = C ||S|| I 3 , (20) Local variance based models (StSp / StTe) :

a(x , nδt ) = 1

|Γ| − 1 X

x

i

∈η(x)

(w(x i , nδt )− w ¯ (x, nδt))(w (x i , nδt)− w(x, ¯ nδt )) ^T C sp ,

(21)

(33)

Plan

3 Channel Flow

4 Wake Flow

(34)

Flow Solver - Incompact3d

Fortran based parallelised flow solver - 2Decomp & FFT (Laizet S. & Lamballais E. (2009))

Simulates the incompressible Navier Stokes equations

Spectral pressure treatment with collocated or staggered grid formulation

(35)

Flow Specifications

Instantaneous streamwise velocity contour

n _x × n _y × n _z l _x × l _y × l _z ∆x ∆y ∆z ∆t

cLES 48×81×48 6.28×2×3.14 0.13 0.005-0.12 0.065 0.002

Ref 256×257×256 6.28×2×3.14 0.024 0.0077 0.012 -

(36)

Channel Flow Statistics

Mean Velocity Profile :

0 5 10 15 20 25

0.01 0.1 1 10 100 1000

u+

y+

Smag Wale StSm StSp StTe Ref

Mean velocity profile for turbulent channel flow at Re

τ

= 395

(37)

Channel Flow

0 5 10 15 20 25

0.01 0.1 1 10 100 1000

u+

y+

Smag Wale StSm StSp StTe Ref

0 5 10 15 20 25

0.01 0.1 1 10 100 1000

u +

y +

Smag Wale StSm StSp StTe Ref

(38)

Channel Flow

0 0.5 1 1.5 2 2.5 3 3.5

0 50 100 150 200 250 300 350

<u0>,<v0>,<w0> uτ

y+

Velocity fluctuation profiles for turbulent channel flow at Re

τ

= 395

(39)

Channel Flow

0 0.5 1 1.5 2 2.5 3 3.5

0 50 100 150 200 250 300 350

<u0>,<v0>,<w0> uτ

y+

0 0.5 1 1.5 2 2.5 3 3.5

0 50 100 150 200 250 300 350

,< v 0 >,< w 0 > u τ

y +

Smag Wale StSm StSp StTe Ref u v w

(40)

Channel Flow Wake Flow

Channel Flow

0 0.5 1 1.5 2 2.5 3 3.5

0 50 100 150 200 250 300 350

<u0>,<v0>,<w0> uτ

y+

0 0.5 1 1.5 2 2.5 3

0 50 100 150 200 250 300 350

,< v 0 >,< w 0 > u τ

y +

Smag Wale StSm StSp StTe Ref u v w

(41)

Plan

3 Channel Flow

4 Wake Flow

(42)

Flow Parameters

Re n

x

× n

y

× n

z

l

x

/D × l

y

/D × l

z

/D ∆x/D ∆y/D ∆z /D U∆t/D cLES 3900 241×241×48 20×20×3.14 0.083 0.024-0.289 0.065 0.003

PIV - Parn 3900 160×128×1 3.6×2.9×- 0.023 0.023 - 0.01

LES - Parn 3900 961×961×48 20×20×3.14 0.021 0.021 0.065 0.003 DNS 3900 1537×1025×96 20×20×3.14 0.013 0.0056-0.068 0.033 0.00075

(43)

Flow Statistics

Centerline statistical comparison :

−0.4

−0.2 0 0.2 0.4 0.6 0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Uc

x/D PIV - Parn

DNS Smag

StSm StSp

Mean streamwise velocity in the streamwise direction along the centerline

behind the cylinder.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.5 1 1.5 2 2.5 3 3.5 4 4.5

<u0u0>U2c

x/D PIV - Parn

DNS

Smag StSm

DSmag StSp

Fluctuating streamwise velocity in the streamwise direction along the centerline

behind the cylinder.

(44)

Channel Flow Wake Flow

Flow Statistics

−0.4

−0.2 0 0.2 0.4 0.6 0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Uc

x/D PIV - Parn

DNS

Smag StSm

StSp

−0.4

−0.2 0 0.2 0.4 0.6 0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5

U c

x /D

DNS StSm

(45)

Flow Statistics

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.5 1 1.5 2 2.5 3 3.5 4 4.5

<u0u0>U2c

x/D PIV - Parn

DNS

Smag StSm

DSmag StSp

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.5 1 1.5 2 2.5 3 3.5 4 4.5

U 2 c

x /D

DNS StSm StSp

(46)

Vorticity Structures

Smagorinsky Stochastic Smagorinsky

Stochastic Spatial DNS

(47)

Vorticity Structures

Smagorinsky Stochastic Smagorinsky

Stochastic Spatial DNS

(48)

Simulation Cost

Model w/o model Smag StSm StSp

Computational Cost [s/iteration] 1.34 2.877 3.654 5.711

Table – Computational cost [s] per iteration for each model.

Cost of StSp under coarse resolution = 0.46% cost of DNS

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4D Variational Data Assimilation with Incompact3d using Large Eddy Models

HAL Id: hal-01764622

https://hal.inria.fr/hal-01764622

Submitted on 12 Apr 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

using Large Eddy Models

Pranav Chandramouli

To cite this version:

Pranav Chandramouli. 4D Variational Data Assimilation with Incompact3d using Large Eddy Models

. Incompact3d Meeting 2018, Mar 2018, London, United Kingdom. �hal-01764622�

4D Variational Data Assimilation with Incompact3d using Large Eddy Models

Pranav CHANDRAMOULI

Supervised by: Etienne MEMIN & Dominique HEITZ

INRIA - Rennes (Fluminance) Incompact3d Meeting - London, 2018

28/03/2018

Plan

1 Large Eddy Simulation SGS Models

LES - Results

2 Data Assimilation

Introduction to Data Assimilation Code Formulation

VDA with LES - Results

Objective

To Identify an apt Sub-Grid Scale (SGS) model for coarse LES in terms of statistical accuracy and numerical stability.

LES - SGS Models

Smagorinsky Variants :

Classic : ν t = (C s ∗ ∆) 2 S ¯ , (1) Dynamic : C s 2 = − 1

2 L ij M ¯ ij

M kl M ¯ kl . (2) where,

L ij = T ij − τ ˜ ij (3) M ij = ˜ ∆ 2 | S ˜ ¯ | S ˜ ¯ ij − ∆ ¯ 2 (| ^ S ¯ | S ¯ ij ), (4) WALE :

ν t = (C w ∆) 2 (ς ij d ς ij d ) 3/2

( ¯ S ij S ¯ ij ) 5/2 + (ς ij d ς ij d ) 5/4 (5)

LES - Models under Location Uncertainty (MULU)

dX t

dt = w (X t , t) + σ(X t , t) ˙ B (6) w (X, t) is the large scale drift

σ(X, t)d B ˙ t stands for small scales represented by a noise

MULU

NS formulation as derived in Memin [2014] : Mass conservation :

d t ρ t + ∇·(ρ w ˜ )dt + ∇ρ ·σ d B t = 1

2 ∇· (a∇q)dt, (7)

˜

w = w − 1

2 ∇ · a (8)

For an incompressible fluid :

∇· (σd B t ) = 0, ∇ · w ˜ = 0, (9) Momentum conservation :

∂ t w +w ∇ T (w − 1

2 ∇ · a)− 1 2

X

ij

∂ x

(a ij ∂ x

w )

ρ = ρg −∇p+µ∆w .

(10)

MULU

Modelling of a :

Stochastic Smagorinsky model (StSm) :

a(x , t) = C ||S|| I 3 , (11) Local variance based models (StSp / StTe) :

a(x , nδt ) = 1

|Γ| − 1 X

x

∈η(x)

(w(x i , nδt )− w ¯ (x, nδt))(w (x i , nδt)− w(x, ¯ nδt )) T C st ,

(12)

Channel Flow

0 0.5 1 1.5 2 2.5 3 3.5

0 50 100 150 200 250 300 350

< u 0 >,< v 0 >,< w 0 > u τ

y +

Smag Wale StSm StSp StTe Ref u v w

Velocity fluctuation profiles for turbulent channel flow at Re

= 395

n

× n

× n

l

× l

× l

∆x ∆y ∆z ∆t

cLES 48×81×48 6.28×2×3.14 0.13 0.005-0.12 0.065 0.002 Ref 256×257×256 6.28×2×3.14 0.024 0.0077 0.012 -

Classic : ν _t = (C _s ∗ ∆) ² S ¯ , (1) Dynamic : C _s ² = − 1

M _kl M ¯ _kl . (2) where,

L ij = T ij − τ ˜ ij (3) M _ij = ˜ ∆ ² | S ˜ ¯ | S ˜ ¯ _ij − ∆ ¯ ² (| ^ S ¯ | S ¯ _ij ), (4) WALE :

ν _t = (C _w ∆) ² (ς _ij ^d ς _ij ^d ) ^3/2

( ¯ S _ij S ¯ _ij ) ^5/2 + (ς _ij ^d ς _ij ^d ) ^5/4 (5)

dX _t

dt = w (X _t , t) + σ(X _t , t) ˙ B (6) w (X, t) is the large scale drift

d t ρ t + ∇·(ρ w ˜ )dt + ∇ρ ·σ d B _t = 1

∇· (σd B _t ) = 0, ∇ · w ˜ = 0, (9) Momentum conservation :

∂ t w +w ∇ ^T (w − 1

(w(x i , nδt )− w ¯ (x, nδt))(w (x i , nδt)− w(x, ¯ nδt )) ^T C st ,

Objective : Estimate the unknown true state of interest x ^t (t , x) Formulation :

∂ _t x (t, x) + M (x (t, x)) = q(t, x), (13) x(t ₀ , x) = x ^b ₀ + η(x) (14) Y (t, x) = H (x (t, x )) + (t, x) (15) q(t, x) - model error (Covariance matrix Q)

2 (x 0 − x ^b ₀ ) ^T B ⁻¹ (x 0 − x ^b ₀ )+

( H (x t ) − Y(t)) ^T R ⁻¹ ( H (x t ) − Y(t)).

Minimisation of J with respect to additional incremental control parameter δu : δγ ⁽ⁱ⁾ = {δx ⁽ⁱ⁾ ₀ , δu ⁽ⁱ⁾ }

J(δγ ⁽ⁱ⁾ ) = 1

2 ||δx ⁽ⁱ⁾ ₀ + x ⁽ⁱ⁾ ₀ − x ^(b) ₀ || ² _B

||δu ⁽ⁱ⁾ + u ⁽ⁱ⁾ − u ^(b) || ² _B

∂ t δx ⁽ⁱ⁾ + ∂ x M (x ⁽ⁱ⁾ ) · ∂x ⁽ⁱ⁾ + ∂ u M (u ⁽ⁱ⁾ ) · ∂u ⁽ⁱ⁾ = 0 (19)