High order homogenization of the Stokes system in a periodic porous medium

(1)

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High order homogenization of the Stokes system in a periodic porous medium

Florian Feppon

To cite this version:

Florian Feppon. High order homogenization of the Stokes system in a periodic porous medium. 2021.

�hal-02880030v2�

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A PERIODIC POROUS MEDIUM

2

FLORIAN FEPPON^† 3

Abstract. We derive high order homogenized models for the incompressible Stokes system in 4

a cubic domain filled with periodic obstacles. These models have the potential to unify the three 5

classical limit problems (namely the “unchanged” Stokes system, the Brinkman model, and the 6

Darcy’s law) corresponding to various asymptotic regimes of the ratioη≡aε/εbetween the radius 7

aεof the holes and the sizeεof the periodic cell. What is more, a novel, rather surprising feature 8

of our higher order effective equations is the occurrence of odd order differential operators when the 9

obstacles are not symmetric. Our derivation relies on the method of two-scale power series expansions 10

and on the existence of a “criminal” ansatz, which allows to reconstruct the oscillating velocity and 11

pressure (uε, pε) as a linear combination of the derivatives of their formal average (u^∗_ε, p^∗_ε) weighted 12

by suitable corrector tensors. The formal average (u^∗_ε, p^∗_ε) is itself the solution to a formal, infinite 13

order homogenized equation, whose truncation at any finite order is in general ill-posed. Inspired 14

by the variational truncation method of [53,27], we derive, for anyK∈N, a well-posed model of 15

order 2K+ 2 which yields approximations of the original solutions with an error of orderO(ε^K+3) 16

in theL²norm. Furthermore, the error improves up to the orderO(ε^2K+4) if a slight modification 17

of this model remains well-posed. Finally, we find asymptotics of all homogenized tensors in the low 18

volume fraction limitη →0 and in dimensiond≥3. This allows us to obtain that our effective 19

equations converge coefficient-wise to either of the Brinkman or Darcy regimes which arise whenη 20

is respectively equivalent, or greater than the critical scalingηcrit∼ε^2/(d−2). 21

Key words. Homogenization, higher order models, porous media, Stokes system, strange term.

22

AMS subject classifications. 35B27, 76M50, 35330 23

1. Introduction. This article is concerned with the high order homogenization

24

of the Stokes system in a periodic porous medium. Let D := (0, L)

^d

be a d-dimensional

25

box filled with periodic obstacles ω

ε

:= ε( Z

^d

+ ηT ) ∩ D obtained by rescaling and

26

repeating periodically a unit hole T (the setting is illustrated on Figure 1). The

27

parameter ε denotes the size of the periodic cell, it is equal to ε := L/N where N ∈ N

28

is a large integer and L is the length of the box. The parameter η is the scaling ratio

29

between the radius a

_ε

:= ηε of the obstacles and the length ε of the cells. The total

30

fluid domain is denoted by D

_ε

:= D\ω

_ε

and it is assumed to be connected. P = (0, 1)

^d

31

is the unit cell and Y = P \ηT denotes its fluid component.

32

We consider (u

_ε

, p

_ε

) ∈ H

¹

(D

_ε

, R

^d

) × L

²

(D

_ε

)/ R the solution to the Stokes system

33

(1.1)



 

 

 

 

−∆u

ε

+ ∇p

ε

= f in D

ε

div(u

_ε

) = 0 u

ε

= 0 on ∂ω

ε

u

ε

is D–periodic,

34

where f ∈ C

_per^∞

(D, R

^d

) (and all its derivatives) is a smooth, D–periodic right hand-

35

side. The goal of this paper is to derive high order effective models for (1.1); i.e. a

36

family of well-posed partial differential equations posed in the homogeneous domain

37

∗Submitted to the editors DATE.

Funding:This work was supported by the Association Nationale de la Recherche et de la Tech- nologie (ANRT) [grant number CIFRE 2017/0024] and by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR).

† Centre de Math´ematiques Appliqu´ees, Ecole´ Polytechnique, Palaiseau, France (florian.feppon@polytechnique.edu).

1

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Y ηT

P = (0, 1)

^d

D

_ε

= D\ω

ε

D ω

ε

Fig. 1.The perforated domainDε=D\ω_ε and the unit cellY =P\(ηT).

D (without the holes) and whose solutions approximate the macroscopic behavior of

38

(u

ε

, p

ε

) at any desired order of accuracy in ε as ε → 0.

39

The literature [52, 47, 29, 5, 7, 4, 8] describes the occurrence of different asymp-

40

totic regimes depending on how the size a

ε

= ηε of the holes compares to the critical

41

size σ

ε

:= ε

^d/(d−2)

in dimension d ≥ 3 (if d = 2, then these regimes depend on

42

how log(a

ε

) compares to −ε

⁻²

, see [7]). In loose mathematical terms, these can be

43

summarized as follows (see e.g. [5, 7] for the precise statements):

44

• if a

ε

= o(σ

ε

), then the holes have no effect and (u

ε

, p

ε

) converges as ε → 0

45

to the solution (u, p) of the Stokes equation in the homogeneous domain D:

46

(1.2)



 

 

−∆u + ∇p = f in D div(u) = 0

u is D–periodic.

47

• if a

ε

= cσ

ε

for a constant c > 0, then (u

ε

, p

ε

) converges as ε → 0 to the

48

solution (u, p) of the Brinkman equation

49

(1.3)



 

 

−∆u + cF u + ∇p = f in D div(u) = 0

u is D–periodic,

50

where the so-called strange term cF u involves a symmetric positive definite

51

d × d matrix F which can be computed by means of an exterior problem in

52

R

^d

\T (see [4] and section 5).

53

• if σ

ε

= o(a

ε

) and a

ε

= ηε with η → 0 as ε → 0, then the holes are “large”

54

and (a

^d−2_ε

ε

^−d

u

ε

, p

ε

) converges to the solution (u, p) of the Darcy problem

55

(1.4)



 

 

F u + ∇p = f in D div(u) = 0 in D

u is D–periodic,

56

where F is the same symmetric positive definite d × d matrix as in (1.3).

57

• if a

ε

= ηε with the ratio η fixed, then (ε

⁻²

u

ε

, p

ε

) converges to the solution

58

(u, p) of the Darcy problem

59

(1.5)



 

 

M

⁰

u + ∇p = f in D div(u) = 0 in D

u is D–periodic,

60

(4)

where M

⁰

is another positive symmetric d × d matrix (which depends on η).

61

Furthermore M

⁰

/| log(η)| → F if d = 2, and M

⁰

/η

^d−2

→ F (if d ≥ 3) when

62

η → 0, so that there is a continuous transition from (1.5) to (1.4), see [6].

63

One of the long-term motivations driving this work is the need to lay down theo-

64

retical material that would allow to optimize the design of fluid systems by homog-

65

enization methods similar to those available in the context of mechanical structures

66

[21, 20, 10, 50, 14]. To date, the Brinkman [24, 25, 30] and the Darcy models [56, 51]

67

are commonly used by topology optimization algorithms in order to conveniently in-

68

terpolate the physics of the fluid at intermediate “gray” regions featuring locally a

69

mixture of fluid and solid. However, the above conclusions imply that these models

70

are consistent only in specific ranges of obstacle sizes a

_ε

: the Brinkman model (1.3)

71

is relevant when there are none or tiny obstacles, while the Darcy models (1.4) and

72

(1.5) should be used at locations where the obstacles are large enough. The arising

73

of these different regimes (1.2)–(1.5) is consequently a major obstacle towards the de-

74

velopment of ‘de-homogenization’ methods [14, 37, 50, 39, 40] for the optimal design

75

of fluid systems, which would enable to interpret “gray” designs as locally periodic

76

“black and white” microstructures (featuring for instance many small tubes or thin

77

plates).

78

It turns out that there is a continuous transition between these regimes which

79

can be captured by higher order homogenized equations, which is the object of the

80

present article. These higher order models are obtained by adding corrective terms

81

scaled by increasing powers of ε to the Darcy equation (1.5); they yield more accurate

82

approximations of (u

ε

, p

ε

) when ε is “not so small”. For a desired order K ∈ N , the

83

homogenized model of order 2K + 2 reads

84

(1.6)



 



 



2K+2

X

k=0

ε

^k−2

D

^kK

· ∇

^k

v

^∗_ε,K

+ ∇q

^∗_ε,K

= f , div(v

^∗_ε,K

) = 0,

v

^∗_ε,K

is D–periodic,

85

where (v

_ε,K^∗

, q

_ε,K^∗

) is a high order homogenized approximation of (u

ε

, p

ε

). The coeffi-

86

cient D

^kK

is a k-th order matrix valued tensor which can be computed by a procedure

87

involving the resolution of cell problems; it makes D

^kK

· ∇

^k

a differential operator of

88

order k (the notation is defined in section 2 below). Finally, the high order equation

89

(1.6) encompasses at least the Brinkman and the Darcy regimes in the sense that it

90

converges coefficient-wise to either of (1.3) and (1.4) for the corresponding asymptotic

91

regime of the scaling η (see Remarks 5.6 and 5.7) (the analysis of the subcritical case

92

leading to the Stokes regime (1.2) requires more sophisticated arguments which are

93

to be investigated in future works).

94

A rather striking feature of (1.6) is the arising of odd order differential operators

95

despite the symmetry of the Laplace operator −∆ (these vanish, however, in case

96

the obstacle ηT is symmetric with respect to the cell axes; see Corollary 3.16). This

97

fact is closely related to the vectorial nature of the Stokes system (1.1): the tensors

98

D

^kK

are symmetric and antisymmetric valued matrices for respectively even and odd

99

values of k. This property ensures that D

^kK

· ∇

^k

is a symmetric operator for any

100

0 ≤ k ≤ 2K + 2 (see Remark 3.12). To our knowledge, such terms have so far not

101

been proposed in the literature seeking similar higher order corrections for the Stokes

102

system, although these have been observed in other vectorial contexts [27, 28, 53].

103

Most of the available works have focused on situations with low regularity for f , T

104

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and D (see [52, 5]), where the homogenization process can be justified only for the

105

approximation at the leading order in ε. Error bounds for higher order approximations

106

of (u

ε

, p

ε

) (namely for the truncation of the ansatz (1.7) below) have been obtained

107

in [46, 26], without relating these to effective models. A few additional works have

108

sought corrector terms from physical modelling considerations [35, 18, 17], without

109

considering odd order operators.

110

Our derivation is inspired from the works [19, 53, 15]; it is based on (non-standard)

111

two-scale asymptotic expansions and formal operations on related power series which

112

give rise to several families of tensors and homogenized equations at any order. We

113

extend our previous works [34, 33] where we investigated the cases of the perforated

114

Poisson problem and of the perforated elasticity system. Expectedly, the major dif-

115

ficulty in extending the analysis to (1.6) is the treatment of the pressure variable p

ε 116

and of the incompressibility constraint div(u

ε

) = 0. Note that the D–periodicity

117

assumption on f and u

ε

is made in order to eliminate additional difficulties related

118

to the arising of boundary layers (see [43, 22, 23, 11]).

119

The starting point of the method of two-scale expansions is to postulate an ansatz

120

for the velocity and pressure solution (u

ε

, p

ε

):

121

(1.7) u

_ε

(x) =

+∞

X

i=0

ε

ⁱ⁺²

u

_i

(x, x/ε), p

_ε

(x) =

+∞

X

i=0

ε

ⁱ

(p

^∗_i

(x) + εp

_i

(x, x/ε)), x ∈ D

_ε

,

122

where the functions u

_i

(x, y) and p

_i

(x, y) are P –periodic with respect to y ∈ P , and D–periodic with respect to x ∈ D. In (1.7), the oscillating function p

_i

(x, y) is required to be of zero average with respect to y:

Z

Y

p

i

(x, y)dy = 0, ∀i ≥ 0.

The aim of the homogenization process is to obtain effective equations for the formal

123

“infinite order” homogenized averages u

^∗_ε

and p

^∗_ε

defined by

124

(1.8) u

^∗_ε

(x) :=

+∞

X

i=0

ε

ⁱ⁺²

Z

Y

u

_i

(x, y)dy, p

^∗_ε

(x) :=

+∞

X

i=0

ε

ⁱ

p

^∗_i

(x), x ∈ D.

125

In Proposition 3.7 below, we obtain that (u

^∗_ε

, p

^∗_ε

) solves the following formal “infinite-

126

order” homogenized equation,

127

(1.9)



 



 



+∞

X

k=0

ε

^k−2

M

^k

· ∇

^k

u

^∗_ε

+ ∇p

^∗_ε

= f , div(u

^∗

) = 0,

u

^∗_ε

is D–periodic,

128

which involves a family of constant matrix-valued tensors (M

^k

)

_k∈_N

. Classically, trun-

129

cating directly (1.9) yields, in general, an ill-posed model [12]. Several methods have

130

been proposed to address this issue in order to obtain nonetheless well-posed higher

131

order equations [16, 13, 1, 2, 15]. In our case, we adapt an idea from [53], whereby

132

the coefficients D

^kK

are obtained thanks to a minimization principle (described in sec-

133

tion 4) which makes indeed (1.6) well-posed. It is based on the existence of remarkable

134

(6)

identities which relate the oscillating solution (u

ε

, p

ε

) to its formal average (u

^∗_ε

, p

^∗_ε

):

135

(1.10)



 

 

 

 

u

ε

(x) =

+∞

X

i=0

ε

ⁱ

N

ⁱ

(x/ε) · ∇

ⁱ

u

^∗_ε

(x)

p

_ε

(x) = p

^∗_ε

(x) +

+∞

X

i=0

ε

ⁱ⁻¹

β

ⁱ

(x/ε) · ∇

ⁱ

u

^∗_ε

(x),

∀x ∈ D

ε

,

136

where (N

ⁱ

(y))

_i∈_N

and (β

ⁱ

(y))

_i∈_N

are different families of respectively matrix valued

137

and vector valued P –periodic tensors (of order i). The ansatz (1.10) is substantially

138

different from (1.7); following [15], we call it “criminal” because the expansions of

139

(1.10) depend on u

^∗_ε

which is itself a formal power series in ε (eqn. (1.8)).

140

The order of accuracy at which the solution (v

^∗_ε,K

, p

^∗_ε,K

) yields an approximation

141

of the original solution (u

ε

, p

ε

) is determined by how many leading coefficients of (1.6)

142

and (1.9) coincide (Proposition 4.5). In Proposition 4.10, we show that D

^kK

= M

^k

for

143

0 ≤ k ≤ K, which allows to infer error estimates of order O(ε

^K+3

) in the L

²

(D) norm.

144

It may seem disappointing that one needs to solve an equation of order 2K +2 in order

145

to obtain approximations of order O(ε

^K+3

) “only”. This shortcoming is related to the

146

zero-divergence constraint: in the scalar and elasticity cases considered in [34, 33], it

147

turns out that K + 1 extra coefficients coincide, namely D

^kK

= M

^k

for 0 ≤ k ≤ 2K + 1,

148

which yields error estimates of order O(ε

^2K+4

). In the present context devoted to the

149

Stokes system (1.1), the equation obtained by substituting D

^kK

with M

^k

in (1.6) for

150

K + 1 ≤ k ≤ 2K + 1,

151

(1.11)



 



 



ε

^2K

D

^2K+2_K

· ∇

^2K+2

v b

_ε,K^∗

+

2K+1

X

k=0

ε

^k−2

M

^k

· ∇

^k

v

^∗_ε,K

+ ∇ q b

_ε,K^∗

= f div( v b

_ε,K^∗

) = 0

b v

_ε,K^∗

is D–periodic,

152

corresponds to applying the truncation method of [53] to the mixed variational for-

153

mulation rather than to the minimization problem associated with (1.1) (see Re-

154

mark 4.11). While the minimization principle ensures that (1.6) is well-posed, we do

155

not know whether this is the case for (1.11). However if it is, then Proposition 4.5

156

implies that (1.11) improves the approximation accuracy up to the order O(ε

^2K+4

).

157

The article outlines as follows. Notation conventions related to tensors and tech-

158

nical assumptions are exposed in section 2.

159

In section 3, we introduce cell problems and their solution tensors (X

^k

, α

^k

) which

160

allow to identify the functions u

_i

, p

^∗_i

and p

_i

in the ansatz (1.7). We show that the

161

formal average (u

^∗_ε

, p

^∗_ε

) solves the infinite order homogenized equation (1.9) involving

162

the tensors M

^k

. After defining the tensors N

^k

(y) and β

^k

(y), we derive the “criminal”

163

ansatz (1.10) expressing (u

_ε

, p

_ε

) in terms of p

^∗_ε

and of the derivatives of u

^∗_ε

. Through-

164

out this section, a number of algebraic properties are stated for the various tensors

165

coming at play, such as the symmetry and the antisymmetry of the matrix valued

166

tensors M

^k

for respectively even and odd values of k, and the simplifications taking

167

place in case the obstacle ηT is symmetric with respect to the cell axes.

168

Section 4 details the truncation process of the infinite order equation (1.9) leading

169

to the well-posed model (1.6). We then provide an error analysis of the homogenized

170

approximations of (u

ε

, p

ε

) generated by our procedure: our main result is stated in

171

(7)

Corollary 4.15 where we show that the solution (v

^∗_ε,K

, q

^∗_ε,K

) of (1.6) yield approxima-

172

tions of (u

ε

, p

ε

) in the L

²

(D

ε

) norm of order K + 3 and K + 1 for the velocity and

173

the pressure respectively. We establish explicit formulas relating the coefficients D

^kK 174

to the coefficients M

^k

and we briefly discuss the improvement provided by (1.11) in

175

case it is well-posed.

176

The last section 5 investigates asymptotics of the tensors M

^k

in the low volume

177

fraction limit where the scaling of the obstacle η converges to zero. Our main result is

178

Corollary 5.5 where we obtain the “coefficient-wise” convergence of the infinite order

179

homogenized equation as well as the one of (1.6) towards either of the Brinkman or

180

Darcy regimes (1.4) and (1.5) when η is respectively equivalent or greater than the

181

critical size η

_crit

∼ ε

^2/(d−2)

, and towards the Stokes regime (1.3) for η = o(ε

^2/(d−2)

)

182

in the case K = 0. Although our error estimates for (1.6), are a priori not uniform

183

in η, this suggests that our higher order model (1.6) has the potential to yield valid

184

approximations in any regime of size of holes (at least for K = 0 or above the critical

185

scale). Note that our analysis is unfortunately unsufficient to establish the convergence

186

of the high order coefficients ε

^k−2

M

^k

with k > 2 towards 0 as η → 0. Future works

187

will investigate higher order asymptotics of the tensors M

^k

in the subcritical regime

188

η = o(ε

^2/(d−2)

) which are required to establish or invalidate such a claim.

189

2. Setting and notation conventions related to tensors. In the sequel, we

190

consider the following two classical assumptions for the distributions of the holes ω

ε 191

(we recall the schematic of Figure 1), following [5]:

192

(H1) Y = P \(ηT ) ⊂ P , as a subset of the unit torus (opposite matching faces

193

of (0, 1)

^d

are identified) is a smooth connected set with non-empty interior.

194

(H2) The fluid component D

ε

= D\ω

ε

is a smooth connected set.

195

Remark 2.1. Assumption (H1) does not necessarily imply (H2), see [3] for a coun-

196

terexample. Assumption (H1) is not very restrictive and can easily be generalized to

197

the case where the subset Y has m connected components with m ∈ N (see Appendix

198

7.5.6 in [33]). Assumption (H2) is stronger, but is also more connected to physical

199

applications. It forbids the existence of isolated fluid inclusions. Most of our deri-

200

vations only assume (H1). However, we rely on both assumptions (H1) and (H2) in

201

order to obtain error bounds section 4, because we use some technical results of [5].

202

Below and further on, we consider scalar and vectorial functions such as

203

(2.1) u : D × P → R

(x, y) 7→ u(x, y) , u : D × P → R

^d

(x, y) 7→ u(x, y)

204

which are both D and P –periodic with respect to respectively the first and the second

205

variable, and which vanish on the hole D ×(ηT ). The arguments x and y of u(x, y) are

206

respectively called the “slow” and the “fast” or “oscillating” variable. With a small

207

abuse of notation, the partial derivative with respect to the variable y

j

(respectively

208

x

j

) is simply written ∂

j

instead of ∂

y_j

(respectively ∂

x_j

) when the context is clear.

209

The star–“∗”– symbol is used to indicate that a quantity is “macroscopic” in the sense that it does not depend on the fast variable x/ε; e.g. (v

_ε,K^∗

, q

_ε,K^∗

) or (u

^∗_ε

, p

^∗_ε

) in (1.6) and (1.9). In the particular case where a two-variable quantity u(x, y) is given such as (2.1), u

^∗

(x) always denotes the average of y 7→ u(x, y) with respect to the y variable:

u

^∗

(x) :=

Z

P

u(x, y)dy = Z

Y

u(x, y)dy, x ∈ D,

(8)

where the last equality is a consequence of u vanishing on P\Y = ηT . When a function X : P → R depends only on the y variable, we find occasionally more convenient to write its cell average with the usual angle bracket symbols:

hX i :=

Z

P

X (y)dy.

In all what follows, unless otherwise specified, the Einstein summation convention

210

over repeated subscript indices is assumed (but never on superscript indices). Vectors

211

b ∈ R

^d

are written in bold face notation.

212

The notation conventions used for tensor related operations are summarized in the

213

nomenclature below. Some of them are not standard; they allow to avoid to system-

214

atically write partial derivative indices (e.g. 1 ≤ i

₁

. . . i

_k

≤ d) and to distinguish them

215

from spatial indices (e.g. 1 ≤ l, m ≤ d) associated with vector or matrix components.

216

Scalar, vector, and matrix valued tensors and their coordinates

217

b Vector of R

^d

218

(b

j

)

_1≤j≤d

Coordinates of the vector b

219

b

^k

Scalar valued tensor of order k (b

^k_i

1...i_k

∈ R for 1 ≤ i

1

, . . . , i

k

≤ d)

220

b

^k

Vector valued tensor of order k (b

^k_i₁_...i_k

∈ R

^d

for 1 ≤ i

1

, . . . , i

k

≤ d)

221

B

^k

Matrix valued tensor of order k (B

^k_i

1...i_k

∈ R

^d×d

for 1 ≤ i

₁

, . . . , i

_k

≤ d)

222

(b

^k_j

)

_1≤j≤d

Coordinates of the vector valued tensor b

^k

(b

^k_j

is a scalar tensor of order

223

k).

224

(B

_lm^k

)

_1≤l,m≤d

Coefficients of the matrix valued tensor B

^k

(B

_lm^k

is a scalar tensors of

225

order k).

226

b

^k_i₁_...i_k_,j

Coefficient of the vector valued tensor b

^k

(1 ≤ i

1

, . . . i

k

, j ≤ d)

227

B

_i^k

1...i_k,lm

Coefficients of the matrix valued tensor B

^k

(1 ≤ i

1

, . . . i

k

, l, m ≤ d)

228

Tensor products

229

b

^p

⊗ c

^k−p

Tensor product of scalar tensors b

^p

and c

^k−p

:

230

(2.2) (b

^p

⊗ c

^k−p

)

_i₁_...i_k

:= b

^p_i

1...ip

c

^k−p_i

p+1...ik

.

231

a

^p

⊗ b

^k−p

Tensor product of a scalar tensors a

^p

and a vector valued tensor b

^k−p

:

232

(2.3) (a

^p

⊗ b

^k−p

)

_i₁_...i_k

:= a

^p_i

1...ip

b

^k−p_i

p+1...ik

.

233

B

^p

⊗ C

^k−p

Tensor product of matrix valued tensors B

^p

and C

^k−p

:

234

(2.4) (B

^p

⊗ C

^k−p

)

_i₁_...i_k_,lm

:= B

_i^p

1...ip,lj

C

_i^k−p

p+1...ik,jm

.

235

Hence a matrix product is implicitly assumed in the notation B

^p

⊗C

^k−p

.

236

B

^p

: C

^k−p

Tensor product and Frobenius product of matrix tensors B

^p

and C

^k−p

:

237

(2.5) (B

^p

: C

^k−p

)

i1...i_k

:= B

^p_i

1...i_p,lm

C

_i^k−p

p+1...i_k,lm

.

238

b

^p

· c

^k−p

Tensor product and inner product of vector valued tensors b

^p

and c

^k−p

:

239

(2.6) (b

^p

· c

^k−p

)

i₁...i_k

:= b

^p_i

1...i_p,m

c

^k−p_i

p+1...i_k,m

.

240

(9)

B

^p

· c

^k−p

Tensor product of a matrix tensor B

^p

and a vector tensors c

^k−p

:

241

(2.7) (B

^p

· c

^k−p

)

i₁...i_k,l

:= B

_i^p

1...i_p,lm

c

^k−p_i_p+1_...i

k,m

.

242

Hence a matrix-vector product is implicitly assumed in B

^p

· c

^k−p

.

243

Contraction with partial derivatives

244

b

^k

· ∇

^k

Differential operator of order k associated with a scalar tensor b

^k

:

245

(2.8) b

^k

· ∇

^k

:= b

^k_i

1...i_k

∂

_i^k

1...i_k

.

246

b

^k

· ∇

^k

Differential operator of order k associated with a vector tensor b

^k

: for

247

any smooth vector field v ∈ C

_per^∞

(D, R

^d

),

248

(2.9) b

^k

· ∇

^k

v = b

^k_i₁_...i_k_,l

∂

_i^k₁_...i_k

v

l

.

249

B

^k

· ∇

^k

Differential operator of order k associated with a matrix valued tensor

250

B

^k

: for any smooth vector field v ∈ C

_per^∞

(D, R

^d

),

251

(2.10) (B

^k

· ∇

^k

v)

_l

= B

_i^k

1...i_k,lm

∂

^k_i

1...i_k

v

_m

.

252

Special tensors

253

(e

j

)

1≤j≤d

Vectors of the canonical basis of R

^d

.

254

e

j

Scalar valued tensor of order 1 given by e

j,i₁

:= δ

i₁j

(with 1 ≤ j ≤ d).

255

δ

ij

Kronecker symbol: δ

ij

= 1 if i = j and δ

ij

= 0 if i 6= j.

256

I Identity tensor of order 2:

I

i₁i₂

= δ

i₁i₂

.

The identity tensor is another notation for the Kronecker tensor and it

257

holds I = e

_j

⊗ e

_j

with summation on the index 1 ≤ j ≤ d.

258

J

^2k

Tensor of order 2k defined by:

J

^2k

:=

ktimes

z }| { I ⊗ I ⊗ · · · ⊗ I .

259

With a small abuse of notation, we consider zeroth order tensors b

⁰

to be constants

260

(i.e. b

⁰

∈ R if b

⁰

is scalar) and we still denote by b

⁰

⊗ c

^k

:= b

⁰

c

^k

the tensor product

261

with a k-th order tensor c

^k

. The same convention also applies to vector valued and

262

matrix valued tensors.

263

In all what follows, a k-th order tensor b

^k

(scalar, vector or matrix valued) truly makes sense when contracted with k partial derivatives, as in (2.8)–(2.10). Therefore all the tensors considered throughout this work are identified to their symmetrization:

b

^k_i₁_...i_k

≡ 1 k!

X

σ∈S_k

b

i_σ(1)...i_σ(k)

,

where S

_k

is the permutation group of order k. Consequently, the order in which the

264

(derivative) indices i

₁

, . . . , i

_k

are written in b

^k_i

1...i_k

does not matter.

265

Finally, in the whole work, we write C, C

_K

or C

_K

(f ) to denote universal constants

266

that do not depend on ε but whose values may change from lines to lines (and which

267

may depend on η or on the obstacle T ).

268

Remark 2.2. In a limited number of places, the superscript or subscript indices

269

p, q ∈ N are used. Naturally, these are not to be confused with the pressure variables

270

p

ε

or q

ε

introduced in (1.1).

271

(10)

3. Infinite order homogenized equation and criminal ansatz. We start

272

by identifying the two-scale structure of (u

ε

, p

ε

) which arise in the form of the ansatz

273

(1.7). Because it helps emphasizing the arising of Cauchy products, we assume, in

274

this section only, that the right-hand side f can be formally decomposed into a power

275

series in ε:

276

(3.1) ∀x ∈ D, f (x) =

+∞

X

i=0

ε

ⁱ

f

ⁱ

(x).

277

3.1. Identification of the “classical” ansatz: tensors (X

^k

, α

^k

). Inserting

278

(1.7) into the Stokes system (1.1) yields the following cascade of equations:

279

(3.2)



 



 



−∆

yy

u

i+2

+ ∇

y

p

i+2

= f

i+2

− ∇

x

p

^∗_i+2

− ∇

x

p

i+1

+ ∆

xy

u

i+1

+ ∆

xx

u

i

, div

_y

(u

_i+2

) = −div

x

(u

_i+1

),

u

₋₂

= u

₋₁

= 0, p

₋₁

= 0, u

_i

(x, ·) = 0 on ∂(ηT )

u

i

(x, ·) is P –periodic for any x ∈ D, u

_i

(·, y) is D–periodic for any y ∈ P ,

280

for any i ≥ −2, where the operators −∆

yy

, −∆

xy

, −∆

yy

are defined by

−∆

xx

= −div

x

(∇

x

·), −∆

xy

= −div

x

(∇

y

·) − div

y

(∇

x

·), −∆

yy

:= −div

y

(∇

y

·).

In order to solve (3.2), we introduce a family of respectively vector valued tensors

281

(X

^k_j

(y))

_1≤j≤d

and scalar valued tensors (α

^k_j

(y))

_1≤j≤d

defined by induction as the

282

unique solutions in H

_per¹

(Y, R

^d

) × L

²

(Y )/ R to the following cell problems:

283

( −∆

yy

X

⁰_j

+ ∇

y

α

⁰_j

= e

j

in Y, div

y

(X

⁰_j

) = 0 in Y (3.3)

284

( −∆

yy

X

¹_j

+ ∇

y

α

¹_j

= (2∂

l

X

⁰_j

− α

⁰_j

e

l

) ⊗ e

l

in Y div

y

(X

¹_j

) = −(X

⁰_j

− hX

⁰_j

i) · e

l

⊗ e

l

in Y, (3.4)

285

( −∆

yy

X

^k+2_j

+ ∇

y

α

^k+2_j

= (2∂

l

X

^k+1_j

− α

^k+1_j

e

l

) ⊗ e

l

+ X

^k_j

⊗ I in Y

div

y

(X

^k+2_j

) = −(X

^k+1_j

− hX

^k+1_j

i) · e

l

⊗ e

l

in Y ∀k ≥ 0.

(3.5)

286 287

Equations (3.3)–(3.5) are supplemented with the following boundary conditions:

288

(3.6)



 

 

 

  Z

Y

α

^k_j

dy = 0

X

^k_j

= 0 on ∂(ηT) (X

^k_j

, α

^k_j

) is P –periodic

∀k ≥ 0.

289

290

Remark 3.1. In view of the notation conventions of section 2, the non bold symbols ⊗e

_l

and ⊗I indicate the arising of extra partial derivatives indices. For instance, the first line of (3.5) must be understood as

−∆

yy

X

^k+2_j,i

1,...i_k+2

+∇α

^k+2_j,i

1...i_k+2

= 2∂

i_k+2

X

^k+1_j,i

1...i_k+1

−α

^k+1_j,i

1...i_k+1

e

i_k+2

+X

^k_j,i₁_...i_k

δ

i_k+1i_k+2

.

(11)

We introduce the k-th order matrix valued tensors X

^k

whose columns are the vector valued tensors (X

^k_j

):

(X

_ij^k

(y))

_1≤i,j≤d

:=

X

^k₁

(y) . . . X

^k_d

(y)

, ∀y ∈ Y, ∀k ≥ 0.

We also denote by α

^k

the k-th order vector valued tensor whose coordinates are the scalar tensors α

^k_j

:

α

^k

(y) := (α

^k_j

(y))

_1≤j≤d

, ∀y ∈ Y, ∀k ≥ 0.

Following the conventions of section 2, we use a star notation to denote the average

291

of respectively the tensor X

^k

and of the vector fields u

i

:

292

(3.7) X

^k∗

:=

Z

Y

X

^k

(y)dy, ∀k ≥ 0, u

^∗_i

(x) :=

Z

Y

u

_i

(x, y)dy, ∀x ∈ D, ∀i ≥ 0.

293

The tensors X

^k

and α

^k

enable to solve the cascade of equations (3.2):

294

Proposition 3.2. Assume (H1). The solutions u

i

(x, y), p

i

(x, y) of the cascade

295

of equations (3.2) are given by

296

(3.8)

u

i

(x, y) =

i

X

k=0

X

^k

(y) · ∇

^k

(f

_i−k

(x) − ∇p

^∗_i−k

(x))

p

_i

(x, y) =

i

X

k=0

α

^k

(y) · ∇

^k

(f

_i−k

(x) − ∇p

^∗_i−k

(x)),

297

where the functions p

^∗_i

are uniquely determined recursively as the solutions to the

298

following elliptic system: for any i ≥ 0,

299

(3.9)



 

 

 

 

−div

x

(X

^0∗

∇

x

p

^∗_i

) = −div

x

(X

^0∗

f

i

)

−

i

X

k=1

div(X

^k∗

· ∇

^k

(f

i−k

− ∇

x

p

^∗_i−k

)) in D

ε

, Z

D

p

^∗_i

dx = 0

p

^∗_i

is D–periodic.

300

Recognizing Cauchy products, the identities (3.8) and (3.9) rewrite formally in terms

301

of equality of formal power series:

302

u

ε

(x) =

+∞

X

i=0

ε

ⁱ⁺²

X

ⁱ

(x/ε) · ∇

ⁱ

(f (x) − ∇p

^∗_ε

(x)), (3.10)

303

p

ε

(x) = p

^∗_ε

(x) +

+∞

X

i=0

ε

ⁱ⁺¹

α

ⁱ

(x/ε) · ∇

ⁱ

(f (x) − ∇p

^∗_ε

(x)), (3.11)

304

div(u

^∗_ε

(x)) = 0 where u

^∗_ε

(x) =

+∞

X

i=0

ε

ⁱ⁺²

X

^i∗

· ∇

ⁱ

(f (x) − ∇p

^∗_ε

(x)).

(3.12)

305 306

Proof. The result is proved by induction. The case i = −1 is straightforward

307

thanks to the convention u

₋₁

= p

₋₁

= 0. In this proof we use the short-hand

308

(12)

notation h

i

(x) = f

i

(x) − ∇p

^∗_i

(x). Assuming (3.8) and (3.9) hold till rank i + 1 with

309

i ≥ −2, we compute, substituting (3.8) into (3.2):

310

(3.13)



 



 



(−∆

yy

u

i+2

+ ∇

y

p

i+2

)(x, y)

= h

_i+2,j

(x)e

_j

+ (2∂

_l

X

⁰_j

(y) − α

⁰_j

(y)e

_l

) ⊗ e

_l

) · ∇h

i+1,j

(x) +

i

X

k=0

((2∂

_l

X

^k+1_j

(y) − α

^k+1_j

(y)e

_l

) ⊗ e

_l

+ X

^k_j

(y) ⊗ I) · ∇

^k+2

h

_i−k,j

(x)

div

y

(u

i+2

)(x, y) = −

i+1

X

k=0

(X

^k_j

(y) · e

l

⊗ e

l

) · ∇

^k+1

h

i+1−k,j

(x).

311

The system (3.13) admits a unique solution (u

i+2

, p

i+2

) with R

Y

p

i+2

(x, y)dy = 0 if and only if the following compatibility condition (the so-called “Fredholm alterna- tive”) holds (for any i ≥ −1):

Z

Y

div

y

(u

i+2

)(x, y)dy = −

i+1

X

k=0

[hX

^k_j

i · e

l

⊗ e

l

] · ∇

^k+1

h

i+1−k,j

(x) = 0.

The above equation determines p

^∗_i+1

given the values of p

^∗_k

for 0 ≤ k ≤ i:

(hX

⁰_j

i · e

_l

)∂

_l

(f

_i+1,j

− ∂

_j

p

^∗_i+1

) = −

i+1

X

k=1

[hX

^k_j

i · e

_l

⊗ e

_l

] · ∇

^k+1

(f

_i+1−k,j

− ∂

_j

p

^∗_i+1−k

), which is (3.9) at order i + 1. This identity allows to rewrite div

_y

(u

_i+2

) as

312

(3.14) div

y

(u

i+2

)(x, y) = −

i+1

X

k=0

[(X

^k_j

(y) − hX

^k_j

i) · e

l

⊗ e

l

] · ∇

^k+1

h

_i+1−k,j

(x).

313

By linearity, (3.13) and (3.14) and the definitions of (X

^k_j

, α

^k_j

) through the cell problems

314

(3.3)–(3.5) imply the result at rank i + 2.

315

Remark 3.3. The truncation of the series (3.12) at first order yields the well-

316

known Darcy’s law [52]. The next terms of the series have been obtained in [46, 26],

317

at least up to the order i = 1.

318

Remark 3.4. The ansatz (3.10) is already non-standard (when compared to (1.7))

319

because it features p

^∗_ε

which is a formal power series in ε (recall (1.8)); in other words

320

it is not an ansatz but rather a compact way of writing the Cauchy products (3.8).

321

The next proposition establishes the symmetry and antisymmetry of the matrices

322

X

^k∗

(eqn. (3.7)) for respectively odd and even values of k. We note that similar

323

identities have been found for the Poisson [34] or the wave equation [1].

324

Proposition 3.5. For any k ≥ 0 and 0 ≤ p ≤ k, 1 ≤ i, j ≤ d, the following

325

identity holds for the matrix valued tensor X

^k∗

:

326

(3.15) X

_ij^k∗

= (−1)

^p

Z

Y

((−∆

yy

X

^p_i

+∇α

^p_i

)·X

^k−p_j

+∇α

_j^k−p

·X

^p_i

−X

^k−p−1_j

·X

^p−1_i

⊗I)dy

327

with X

⁻¹_i

= 0 by convention. In particular, for any k ≥ 0, X

^2k∗

and X

^2k+1∗

take

328

values respectively in the set of d × d symmetric and antisymmetric matrices:

329

(3.16) X

_ij^2k∗

= (−1)

^k

Z

Y

(∇X

^k_i

: ∇X

^k_j

+ ∇α

^k_i

· X

^k_j

+ ∇α

^k_j

· X

^k_i

− X

^k−1_i

· X

^k−1_j

⊗ I)dy

330

(13)

331

(3.17)

X

_ij^2k+1∗

= (−1)

^k

Z

Y

(X

^k_i

· ∇X

^k_j

− X

^k_j

· ∇X

^k_i

+ α

^k_i

X

^k_j

− α

^k_j

X

^k_i

) · e

l

⊗ e

l

dy + (−1)

^k

Z

Y

(X

^k−1_j

· X

^k_j

− X

^k−1_i

· X

^k_j

)dy.

332

Proof. The result holds for p = 0 because X

_ij^k∗

=

Z

Y

X

^k_j

· e

_i

dy = Z

Y

X

^k_j

· (−∆

_yy

X

⁰_i

+ ∇α

⁰_i

)dy.

Assuming now that (3.15) holds till rank p with k > p ≥ 0, we prove the result at

333

rank p + 1. We write, after an integration by parts and by using (3.3)–(3.5):

334

X

_ij^k∗

= (−1)

^p

Z

Y

[−X

^p_i

· ∆X

^k−p_j

− α

^p_i

div(X

^k−p_j

) − α

^k−p_j

div(X

^p_i

)

335

− X

^k−p−1

· X

^p−1_i

⊗ I]dy

= (−1)

^p

Z

Y

(X

^p_i

· (2∂

l

X

^k−p−1_j

− α

_j^k−p−1

e

l

) ⊗ e

l

+ X

^k−p−2_j

⊗ I − ∇α

^k−p_j

) · X

^p_i

336

+ α

^p_i

X

^k−p−1_j

· e

l

⊗ e

l

+ α

^k−p_j

X

^p−1_i

· e

l

⊗ e

l

− X

^k−p−1_j

· X

^p−1_i

⊗ I dy

337

= (−1)

^p

Z

Y

− X

^k−p−1_j

· ((2∂

_l

X

^p_i

− α

^p_i

e

_l

) ⊗ e

_l

+ X

^p−1_i

⊗ I) + α

^k−p−1_j

div(X

^p+1_i

)

338

− ∇α

^k−p_j

· X

^p_i

− α

^k−p_j

div(X

^p_i

) + X

^k−p−2_j

· X

^p_i

⊗ I dy

339

= (−1)

^p

Z

Y

[−X

^k−p−1_j

· (−∆

yy

X

^p+1_i

+ ∇α

_i^p+1

)

340

− ∇α

^k−p−1_j

· X

^p+1_i

+ X

^k−p−2_j

· X

^p_i

⊗ I]dy,

341342

whence (3.15) at rank p + 1. Finally, the expression (3.16) for X

_ij^2k∗

is obtained by

343

setting k ← 2k and p ← k in (3.15). The expression for X

_ij^2k+1∗

is obtained by setting

344

k ← 2k + 1 and p ← k and performing an integration by parts.

345

3.2. Derivation of the infinite order homogenized equation and of the

346

criminal ansatz. We now proceed on the derivation of the infinite order homogenized

347

equation (1.9). Let us recall the classical positive definiteness of the Darcy tensor X

^0∗

.

348

Corollary 3.6. Assume (H1). The matrix X

^0∗

= (X

_ij^0∗

)

_1≤i,j≤d

(defined in

349

(3.7)) is positive symmetric definite.

350

Proof. See [52] or Corollary 7.8 in [33].

351

Hence, the following definition of the tensors (M

^k

)

_k∈_N

makes sense.

352

Proposition 3.7. Let M

^k

be the tensor of order k defined by induction as fol-

353

lows:

354

(3.18)



 



 



M

⁰

= (X

^0∗

)

⁻¹

M

^k

= −(X

^0∗

)

⁻¹

k−1

X

p=0

X

^k−p∗

⊗ M

^p

, ∀k ≥ 1.

355

(14)

Then the source terms f

i

(eqn. (3.1)) can be expressed in terms of the averaged

356

summands u

^∗_i

(x) and p

^∗_i

(x) ( (1.8) and (3.7)) through the following identity:

357

(3.19) ∀i ≥ 0, f

_i

(x) − ∇p

^∗_i

(x) =

i

X

k=0

M

^k

· ∇

^k

u

^∗_i−k

(x).

358

Recognizing a Cauchy product, (3.19) and (3.12) rewrite formally as the “infinite

359

order” homogenized system (1.9) for the formal average (u

^∗_ε

, p

^∗_ε

) defined in (1.8).

360

Proof. The proof is identical to the one of Proposition 5 in [33], it amounts to

361

average the first line of (3.8) with respect to y and to solve the resulting triangular

362

system determining f

_i−k

− ∇p

^∗_i−k

in terms of u

^∗_i

.

363

The definition (3.18) essentially states that P

+∞

k=0

ε

^k−2

M

^k

· ∇

^k

is the inverse of the

364

formal power series P

+∞

k=0

ε

^k+2

X

^k∗

· ∇

^k

. In this spirit, it is even possible to write a

365

fully explicit formula (see [34], Proposition 6 and Remark 2 for the proof):

366

Proposition 3.8. For any k ≥ 1, the tensor M

^k

is explicitly given by

367

(3.20) M

^k

=

k

X

p=1

(−1)

^p

X

i1+···+ip=k 1≤i1,...,i_p≤k

(X

^0∗

)

⁻¹

⊗ X

ⁱ¹^∗

⊗ · · · ⊗ (X

^0∗

)

⁻¹

⊗ X

ⁱ^p^∗

⊗ (X

^0∗

)

⁻¹

.

368

We now introduce matrix valued tensors N

^k

and vector valued tensors β

^k

which allow

369

to obtain the “criminal ansatz” (1.10) expressing the velocity and pressure (u

ε

, p

ε

) in

370

terms of their formal average (u

^∗_ε

, p

^∗_ε

).

371

Proposition 3.9. Let N

^k

and β

^k

be respectively the k−th order matrix valued and vector valued tensors defined for any k ∈ N by

N

^k

(y) :=

k

X

p=0

X

^k−p

(y) ⊗ M

^p

, β

^k

(y) :=

k

X

p=0

(−1)

^p

M

^p

· α

^k−p

(y), ∀y ∈ Y.

Then the summands u

i

(x, y) and p

i

(x, y) of (3.10) and (3.11) are given for any i ≥ 0

372

in terms of the averages u

^∗_i

(eqn. (3.7)) and p

^∗_i

as follows:

373

(3.21) u

i

(x, y) =

i

X

k=0

N

^k

(y) · ∇

^k

u

^∗_i−k

(x), p

i

(x, y) =

i

X

k=0

β

^k

(y) · ∇

^k

u

^∗_i−k

(x).

374

Recognizing Cauchy products, the identities (3.21) can be rewritten formally as the

375

“criminal ansatz” (1.10).

376

Proof. The result is obtained by substituting (3.19) into (3.8) which yields u

i

(x, y) =

i

X

p=0 i−p

X

q=0

X

^p

(y) ⊗ M

^q

· ∇

^p+q

u

^∗_i−p−q

(x)

=

i

X

k=0 k

X

p=0

(X

^p

(y) ⊗ M

^p−k

) · ∇

^k

u

^∗_i−k

(x) (change of indices k = p + q) from where the identity (3.21) for u

i

(x, y) follows by inverting the summation. Simi- larly, we obtain

p

_i

(x, y) =

i

X

k=0 k

X

p=0

((M

^p−k

)

^T

· α

^p

(y)) · ∇

^k

u

^∗_i−k

(x),

(15)

hence (3.21) by using (M

^p−k

)

^T

= (−1)

^p−k

M

^p−k

(see Corollary 3.11 below).

377

In what follows, we denote by (N

_j^k

)

_1≤j≤d

and by (β

_j^k

)

_1≤j≤d

respectively the column vectors and the coefficients of N

^k

(y) and β

^k

(y):

∀1 ≤ i, j ≤ d, N

_j^k

:= N

^k

e

j

and β

^k_j

:= β

^k

· e

j

.

In addition, the convention N

_j⁻¹

= 0 is assumed. We shall in the sequel use several

378

times the following properties of (N

_j^k

, β

_j^k

) which are dual to those of (X

^k_j

, α

^k_j

).

379

Proposition 3.10. The k-th order tensors N

^k

, (N

_j^k

)

_1≤j≤d

, β

^k

and (β

_j^k

)

_1≤j≤d

380

satisfy:

381

(i) R

Y

N

⁰

(y)dy = I and R

Y

N

^k

(y)dy = 0 for any k ≥ 1;

382

(ii) R

Y

β

^k

(y)dy = 0 for any k ≥ 0;

383

(iii) For any k ≥ −2 and 1 ≤ j ≤ d,

384

(3.22)

( −∆

yy

N

_j^k+2

+ ∇β

_j^k+2

= (2∂

_l

N

_j^k+1

− β

_j^k+1

e

_l

) ⊗ e

_l

+ N

_j^k

⊗ I + M

^k+2

e

_j

, div(N

_j^k+2

) = −(N

_j^k+1

− hN

_j^k+1

i) · e

_l

⊗ e

_l

;

385

Proof. (i) and (ii) are straightforward consequences of (3.18).

386

(iii) is obtained by writing, for k ≥ 0 (implicit summation on the repeated index

387

j assumed):

388

−∆

yy

N

_j^k+2

+ ∇β

_j^k+2

= −∆

yy k+2

X

p=0

X

^k+2−p_i

(y) ⊗ M

_ij^p

! + ∇

k+2

X

p=0

α

^k+2−p_i

(y) ⊗ M

_ij^p

!

=

k

X

p=0

h

(2∂

l

X

^k+1−p_i

− α

^k+1−p_i

e

l

) ⊗ e

l

+ X

^k−p_i

⊗ I i M

_ij^p

+ (2∂

l

X

⁰_i

− α

⁰_i

e

l

)M

_ij^k+1

+ M

_ij^k+2

e

i

= (2∂

l

N

_j^k+1

− β

^k+1_i

e

l

) ⊗ e

l

+ N

_j^k

⊗ I + M

^k+2

e

j

. div(N

_j^k+2

) =

k+2

X

p=0

div(X

^k+2−p_i

)M

_ij^p

= −

k+1

X

p=0

M

_ij^p

(X

^k+1−p_i

− < X

^k+1−p_i

>) · e

_l

⊗ e

_l

. The proof is identical for k = −1 and k = −2.

389

The identity (3.22) allows to infer important properties characterizing the tensors M

^k

390

which are similar to those of Proposition 3.5.

391

Corollary 3.11. For any 1 ≤ p ≤ k − 1, it holds M

_ij^k

= (−1)

^p+1

Z

Y

((−∆

yy

N

_i^p

+ ∇β

_i^p

) · N

_j^k−p

+ ∇β

_j^k−p

· N

_i^p

− N

_i^p−1

· N

_j^k−p−1

⊗ I)dy.

Consequently, for any k ≥ 0,

392

• M

^2k

is a symmetric matrix valued tensor, and the following identities hold:

M

_ij⁰

= Z

Y

∇N

_i⁰

: ∇N

_j⁰

dy,

∀k ≥ 1, M

_ij^2k

= (−1)

^k+1

Z

Y

(∇N

_i^k

: ∇N

_j^k

+∇β

_i^k

·N

_j^k

+∇β

_j^k

·N

_i^k

−N

_i^k−1

·N

_j^k−1

⊗I)dy.