TECHNISCHE UNIVERSIT¨
AT BERLIN
On the homogenization of microstructured surfaces
Adrien Semin Kersten Schmidt
Preprint 2016/13
Preprint-Reihe des Instituts f¨ur Mathematik
Technische Universit¨at Berlin
Preprint 2016/13
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Paper_Helmholtz_Numerik_m3as
Mathematical Models and Methods in Applied Sciences
c
�World Scientific Publishing Company
ON THE HOMOGENIZATION OF MICROSTRUCTURED
SURFACES
ADRIEN SEMIN
Institut für Mathematik, Technische Universität Berlin,
Sekretariat MA 6-4, Straße des 17. Juni 136,
Berlin, 10555, Germany
[email protected]erlin.de
KERSTEN SCHMIDT
Institut für Mathematik, Technische Universität Berlin,
Sekretariat MA 6-4, Straße des 17. Juni 136,
Berlin, 10555, Germany
[email protected]erlin.de
Received (Day Month Year)
Revised (Day Month Year)
Communicated by (xxxxxxxxxx)
The present work deals with the resolution of an elliptic partial differential equation
in a bounded domain made of a thin and periodic layer of finite length.We provide a
method to derive an efficient macroscopic representation of the solution which takes into
account the boundary layer effect occurring in the vicinity of the periodic layer as well
as the corner singularities appearing in the neighborhood of the extremities of the layer.
Our approach combines the method of matched asymptotic expansions and the method
of surface homogenization . This method is shown with the example of the Helmholtz
equation.
Keywords: Asymptotic analysis; periodic surface homogenization; singular asymptotic
expansions; stress intensity factor; numerical methods.
AMS Subject Classification: 32S05, 35C20, 35J05, 35J20, 41A60, 65D15
Introduction
Surfaces with a microstructures show effective properties like an absorption of acous-
tic waves or an impedance for electric fields where much less needs of material or
volume of air is needed as if solutions without a microstructure are used. In many
engineering applications microstructured surfaces are used to create and tailor such
effective properties. Most prominently are microperforated absorbers and liners (see
Fig. 1, Ref. 5 and Ref. 23) for the reduction of acoustic noise of vehicles or aircrafts
or for optimal acoustics in conference or lecture halls. These plates with an array of
perforations above a chamber or an array of chambers each of little volume, where
1
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2Adrien Semin, Kersten Schmidt
(a) (b)
Fig. 1: Illustration of (a) the bias flow liner in an acoustic channel with circular
cross-section (DUCT-C) at the institute of Institute of Propulsion Technology at
DLR Berlin and (b) liners of different porosity (courtesy of F. Bake, DLR Berlin).
the size and distance of the holes are much smaller than the wavelength of the
acoustic waves, lead to a damping of waves in a broad or narrow frequency range.
Probably equally known is the Faraday cage where a mesh of thin conductors leads
to an effective electric shielding. Various examples are shown in Fig. 2, so a channel
that is connected to a side chamber by a perforated wall, a channel with a perfo-
rated wall in its cross-section and the cross-section of a channel in 3D including a
circular wall where a part of is multiperforated. Direct numerical simulations are
exorbitantly expensive for a high porosity as for an accurate computation, e. g.,
with the finite element or finite difference method the size of (at least some) mesh
cells or the grid size have to be at the order of the small scale or even smaller.
Even so the nature of each of these effects is different due to the different phys-
ical phenomena on the microscopic level they all can be modelled in a similar way
by a homogenization procedure along the surface. Exactly as the homogenization of
volumic microstructures 1this surface homogenization leads to models with effective
parameters representing the microstructure, which can be resolved numerically with
Θ
(a) Wave-guide that is connected
to a chamber by a perforated
wall.
Θ
(b) Wave-guide with a
perforated wall in its
cross-section.
Θ
(c) Cross-section
of a cylindrical
wave-guide in 3D
with a partly per-
forated wall.
Fig. 2: Illustration of configurations of multiperforated absorbers. The end-points
of the multiperforated walls meet the domain boundary at different angles Θ.
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On the homogenization of microstructured surfaces 3
a computational effect independently of the ratio of macroscopic and microscopic
scales. The procedure of the surface homogenization differs much from the original
volumic one, and we expect numerical methods based on this asymptotic solution
representation to differ from the numerical methods for the volumic microstructures
4,15. The surface homogenization leads to effective boundary or transmission con-
ditions 2,6,9,18,30, which have to be corrected at the end-points points to represent
the interaction of the microstructure and the singularities correctly (this has been
done for the Poisson problem10,11 and for the Helmholtz problem12). Especially
the interaction with the singular behaviour, that is macroscopically measurable, is
mathematically involved. It is based on an extension of the singularity theory by
Kondratev is needed22. This theory is due to Nazarov in 1991 in Ref. 27 who has
introduced the theory for oscillating boundaries ending at a corner (see also Ref. 29
and Section 17 of Ref. 26).
In this article the surface homogenization is presented as a general methodology
for an effective description and numerical modelling of microstructured surfaces
incorporating the interacting with the singular behaviour at its end-points.
The outline of the paper is as follows. Section 1 is dedicated to the major ideas
of the surface homogenization in presence of singularities. Based on the solution
representation consisting of its macroscopic part, the boundary layer and its near
field part effective transmission condition and corner conditions for the macroscopic
solution at the limit interface or limit end-points of the microstructured layer, re-
spectively, are introduced. How the nature of the transmission conditions is result of
the existence properties of solutions of cell problems for one period of the microstruc-
tured layer (see Fig. 4a) and how its parameters are obtained by pre-computations
of these solutions is explained in Section 2. Then, in Section 3 the relation of the
singular behaviour of the macroscopic part of the solution and the near field part
close to the layer end-points is explained. Finally, in Section 4 the accuracy of the
surface homogenization is illustrated on numerical experiments.
1. Surface homogenization for microstructured layers with
singularities
The obstacles are taken into account either through boundary conditions on its
boundary or through some variation of the coefficients of the differential equation
in the microstructured layer.
Let δbe the characteristic distance between two consecutive holes or two con-
secutive obstacles of the microstructured layer (see e. g. Fig. 5a). Let D(ξ)be a
complex (m×1) vector linearly dependent on the variable ξ= (ξ1,ξ2). Further-
more, let Aδ, Bδbe two functions with values in the space of complex (m×m)
matrices, (Aδ, Bδ)∈C∞(Ωδ)m×m. We assume that the matrices Aδ(x)and Bδ(x)
differ of limit matrices A0(x)and B0(x)in a vicinity of the microstructured layer.
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4Adrien Semin, Kersten Schmidt
In the domain Ωδ, we consider the general problem
Lδ(x,∇x)u=f, in Ωδ,
Nδ(x,∇x)u=g, on ∂Ωδ,(1.1)
where the operators Lδand Nδare defined by
Lδ(x,ξ),=D(−ξ)Aδ(x)D(ξ),Nδ(x,ξ),=D(n)Bδ(x)D(ξ),(1.2)
nbeing the unit exterior normal vector on ∂Ωδ. We introduce in a similar way the
operators L0and N0associated to the matrices A0and B0. We assume moreover
that problem (1.1) is well-posed for any δand admits a solution in a variational
space Vδ(Ωδ).
We are going to present the surface homogenization with singularities in a gen-
eral setting, which we will illustrate on the following example.
Example 1.1. We consider for illustration the Helmholtz problem with homoge-
neous wave-number k0in a wave-guide that is connected to a chamber by a multi-
perforated wall with holes of distance δand opening width η(δ). The computational
domain Ωδ=Ω\Ωδ
hole with the periodic array of obstacles Ωδ
hole ⊂(−L, L)×(−δ,δ)
and the limit domain Ω\Γand limit interface Γare illustrated in Figure 5. This
Helmholtz problem can be stated as
Δuδ+k2
0uδ= 0,in Ωδ,
∇uδ·n= 0,on ∂Ωδ\ΓR,
∇(uδ−uinc)·n−ık0(uδ−uinc)=0,on ΓR,
(P)
where uinc is an incoming wave (from left or right), which can be assumed to
solve the homogeneous Helmholtz equation in an infinite wave-guide with Neu-
mann boundary conditions. In the transparent boundary condition on ΓR=
{−L�, L�} × (0, W),L�> L is a first-order approximation of Robin type. As in-
coming wave we consider for example the plane wave uinc = exp(ık0(x1−L�)) on
the left side of ΓRand uinc = 0 on its right side. This example corresponds to the
operators
D(∇x) =
∂x1
∂x2
1
, Aδ(x) =
1
1
−k2
0
=A0(x), Bδ(x) =
1
1
−ık01ΓR(x)
=B0(x).
and the source terms
f= 0, g =1ΓR(x)(∇uinc ·n−ık0uinc).
The natural spaces associated to that problem are Vδ(Ωδ) = H1(Ωδ)and V0(Ω) =
H1(Ω).
In most cases the solution away from the layer of obstacles and the end-points,
i. e. the macroscopic part of the solution, is of practical interest. For example, in
a wave-guide with part of the boundary that is multiperforated (see Fig. 5a) the
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On the homogenization of microstructured surfaces 5
transmission coefficients are of importance, which are macroscopic quantities and,
more precisely, functions of the macroscopic solution31. However, the macroscopic
part is interacting with the solution close to the layer, which we called the boundary
layer part, and the solution close to the end-points, known as the near-field part (see
Fig. 3). The macroscopic solution that is defined only in some distance away from
the microstructured layer can be smoothly extended, however, to the mid-line of the
layer Γ(see Fig. 5b) including the end-points. On the interface Γthe extensions do
not match necessarily as well as their derivatives, but satisfy (effective) transmission
conditions. If the macroscopic part of the solution is extended in a smooth way to
the end-points, the extension is not necessarily regular, e. g. it may tend to infinity
at the end-points of the interface Γ11,12. A similar behaviour has been observed for
the macroscopic solution for problems with oscillating boundaries with corners28 or
a domain with rounded corners8.
Solution representation To obtain an effective description of the macroscopic
part up to the interface Γand its end-points the solution is analyzed asymptotically
for δbased on suitable expansions for the macroscopic part, the boundary layer part
and the near field part (see again Fig. 3). More precisely:
•The macroscopic part of the solution can be written as a modification of
its limit term u0,0by correctors un,q which are weighted with powers of δ,
where the power is a combination of an integer and multiples of π
Θ, where
Θis the opening angle at the macroscopic corner (see Figs. 2 and. 5):
uδ(x)∼u0,0(x) + �
(n,q)�=0
δπ
Θn+quδ
n,q(x).(1.3)
The macroscopic terms uδ
n,q are defined in the limit domain Ω\Γof Ωδfor
δ→0(see Fig. 5b), i. e., up to the corners and the limit interface Γ, where
they might be two-sided.
•The boundary layer part of the solution corrects its macroscopic part in
the neighbourhood on the microstructured layer and each macroscopic term
uδ
n,q is corrected by a boundary layer term Πδ
n,q(xΓ,X)depending on the
nearest point xΓof a point xin this neighbourhood on the interface Γ
and the scaled coordinate X= (x−xΓ)/δ(see diagonally hatched area in
Fig. 3) and lead to transmission conditions (see Section 2). The boundary
layer terms Πδ
n,q are defined in canonical periodicity cells (see, e. g., Fig. 4a
or 4b).
•The near field part of the solution corrects its macroscopic part in the
neighbourhood on the end-points of the microstructured layer and each
macroscopic term uδ
n,q is corrected by a near field term Uδ
n,q(X±)close to
the end-point x±
Odepending on the scaled coordinate X±= (x−x±
O)/δ(see
vertically hatched area in Fig. 3). The near field terms Uδ
n,q are defined in
canonical domains of the vicinity of one end-point (see, e. g., Fig. 4d or 4e)
and lead to corner conditions (see Section 3).
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6Adrien Semin, Kersten Schmidt
Fig. 3: Schematic representation of the overlapping subdomains for the asymptotic
expansion. The macroscopic area (dark gray), away from the corners x±
Oand from
the limit interface Γ, the boundary layer area (light blue vertically hatched) and the
near field areas (light green diagonally hatched) are overlapping each other.
Numerical computation of an effective macroscopic approximation In this
paper, we show how to compute the terms of the solution representation numerically
after each other. In this way an approximation to the macroscopic part of the
solution is obtained that is computable with an effort that is independent of the
number of obstacles or its characteristic size δ. For this some pre-computations are
performed in domains, which are canonical to the boundary layer part and the near
field part of the solution (see Section 2). This is first a domain (see Fig. 4a), which
is obtained by taking a zoom around one obstacle, where the end-points of the layer
and all the other boundaries are relegated towards infinity. The interaction with the
other obstacles are taking into account by regarding a periodicity cell of the now
infinite array of obstacles. Second, pre-computations are performed on a domain
which is obtained by taking a zoom to the end-points of the array obstacles, where
the part of boundary that is not touching the end-point, including the other end-
point, is relegated to infinity (see Section 3). In this way, a conical domain with a
semi-infinite array of obstacles of size and distance of order 1as shown in Fig. 4d
is obtained. This domain has still an infinite number of obstacles and we propose
to approximate the near field solution on a truncated sub-domain with well-chosen
boundary conditions based on its properties towards infinity.
After this pre-computations, we compute the terms of the macroscopic expan-
sion step-by-step (see Section 4). Each term of the macroscopic expansion depends
only on the previous terms. However, each corrector term of the limit solution is
singular at the end-points of Γ. More precisely, they increase towards infinity when
approaching the end-point. For this each macroscopic term to be computed is de-
composed into a regular part and a singular part. The unbounded singular part
is given analytically as a function of the previous terms, whereas the regular part
lives in an usual Sobolev space like H1(Ω\Γ)and can be computed with classical
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On the homogenization of microstructured surfaces 7
adaptive finite element methods32, where the data depends on the terms of lower
order.
10
X2
X1
η(δ)/δ
(a) The periodicity cell
�
Ω.
10
X2
X1
(b) The limit of periodicity
cell
�
Ωwhen η(δ) = o(δ).
1
(c) The scaled domain
around one hole, when
η(δ) = o(δ).
θ=π−Θ
η(δ)/δ
θ=π
X1
X2
(d) The domain
�
Ω−.
θ=π
X1
X2
θ=π−Θ
(e) The limit of domain
�
Ω−when η(δ) =
o(δ).
Fig. 4: The periodicity cell �
Ωand the normalized domain �
Ω−.
Justification with error estimates The asymptotic solution representation can
be verified theoretically, which has been done in Ref. 10 for the Poisson problem in
a wave-guide with Dirichlet boundary conditions connected to chamber by multi-
perforated wall and in Ref. 12 for the Helmholtz problem with Neumann boundary
conditions. The error estimates are based on the above mentioned theory of the
solutions of the near field problems in the conical domain with the semi-infinite
array of obstacles (see Fig. 4d) in special weighted Sobolev spaces and a matching
procedure of the different expansions.
In general, one expects an optimal macroscopic modelling error in a subdomain
Ωαof Ωδof fixed distance α>0away from the microstructured layer that is of the
order of the first neglected term, i. e., for any s > 0it holds in the energy norm of
the problem
�uδ−�
π
Θn+q<s
δπ
Θn+quδ
n,q�Ωα=O(δs|ln δ|κ(s)),(1.4)
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8Adrien Semin, Kersten Schmidt
where κ(s)∈Ndepends on s. An optimal error also in the vicinity of the microstruc-
tured layer can be expected if combinations of the macroscopic terms and near field
terms multiplied with well-suited cut-offfunctions and respective boundary layer
terms are added.
Example of a wave-guide connected to chamber by multiperforated wall
Example 1.2. We consider for numerical illustration of the example 1.1 the domain
Ωδ
hole as a thin plate of length 1(i. e. L= 0.5) and width 0.075, containing 1/δholes
periodically spaced. For this domain the periodic cell domain is given by Fig. 4a,
and the near-field domain close to the end-point x−
Ois given by Fig. 4d. We denote
by ρthe porosity of the thin plate, i. e. the characteristic size of a hole is η(δ) = ρδ.
The value of the angle at the end-points is Θ= 3π/2. The width of the chamber
and of the wave-guide are both equal to W= 0.5. The length of the wave-guide is
L�= 2.5and the wave number in (P) is k0= 5π.
2. The periodic layer of obstacles and transmission conditions
As it was told in the introduction, one seeks for an effective description for the
macroscopic part taking into account the interaction with the periodic layer and
the corner singularities. This section focuses on the interaction with the periodic
layer. For the effective description the macroscopic solution is extended to the mid-
line Γof the layer (see Fig. 5b), however, only away from end-points, where we
postpone the analysis to Sec. 3. On the mid-line Γthese extensions do not match
necessarily, and the macroscopic solution as well as its derivative can become dis-
continuous and fulfill transmission conditions which compromise the periodic layer
and its impedance in an effective way.
To expose this effective behaviour of a macroscopic solution, it is expanded in
powers of δ, the distance between the size of the holes. This becomes
uδ(x) = uδ
0(x) + δuδ
1(x) + δ2uδ
2(x) + . . . , (2.1)
where the dependence of the terms uδ
qon δis due to the end-points, which we will
suspend at this moment, and to the possibly smaller scale of the geometry (e. g.
η(δ) = o(δ)in example (P)).
It has been already widely spread in the literature that the transmission condi-
tions for the limit solution uδ
0take then the general form
(BΓuδ
0)(x)=0 on Γ,(2.2)
where BΓis an operator taking the two limits of uδ
0and its normal derivative on Γ.
For our example (P), we have
BΓv=�[v]Γ
[∇v·n]Γ�(2.3)
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On the homogenization of microstructured surfaces 9
Θ
x−
O
ΓR
x+
O
Ωδ
hole
η(δ)
ΓR
δ
(a)
Θ
x−
O
ΓR
x+
O
Γ
ΓR
(b)
Fig. 5: Illustration of (a) the computational domain Ωδ=Ω\Ωδ
hole based on a
polygonal domain Ωfor Example 1.1, an acoustic wave-guide problem, and (b) its
limit domain Ω\Γwith the limit interface Γ. The microstructured layer is formed
by a thin rigid wall, on which Neumann boundary conditions are posed, with an
periodic array of holes of size η(δ)and distance δ.
for η(δ)∼δ, which corresponds for the acoustics in the limit vanishing layer, and
BΓv= ([v]Γ−Z�∇v·n�Γ,[∇v·n]Γ)�for δlog η(δ)∼1(i. e.,η(δ)=β1/δfor some
β∈(0,1)) corresponding to an impedance boundary condition in the limit (see also
Ref. 30). If Dirichlet conditions on the boundary of the obstacles of (P) are taken,
then for η(δ)∼δone obtains BΓv= ([v]Γ,�v�Γ)�corresponding to a closed wall
(see also Ref. 19 for the electromagnetic scattering on a cylindrical Faraday cage).
The first corrector satisfies similar transmission conditions, with a source term
depending on the limit solution
(BΓuδ
1)(x)=(B1
Γuδ
0)(x)on Γ(2.4)
as well as the higher order correctors with a source term depending on all previous
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10 Adrien Semin, Kersten Schmidt
terms
(BΓuδ
q)(x) =
q−1
�
p=0
(Bq−p
Γuδ
p)(x)on Γ.(2.5)
For our example for η(δ)∼δwe have
B1
Γv=�2D∞�∇v·n�Γ
N0(∂2
Γ+k2
0)�v�Γ�(2.6)
with two parameters D∞,N0∈R.
The parameters in the transmission conditions and its nature depend on exis-
tence and uniqueness of problems with suitable boundary conditions in the peri-
odicity cell domain �
Ωgiven in Fig. 4a. To defined such a domain, one has to scale
around one hole with respect to δ, take an appropriate ansatz and plug this ansatz
in the rescaled problem. If a smaller scale is involved (e. g. η(δ)=o(δ)), then these
periodicity cell problems will contain a point contribution (see Fig. 4b), coming
from resolution of another problem in a geometry scaled with η(δ)around one hole
(see Fig. 4c)24.
To obtain the parameters, in general, the solution of a cell problem has to be
computed, but sometimes they appear just as a function of geometrical parameters
or are even simple constants. For example, the impedance parameter Zin BΓfor
the periodic wall with asymptotically small openings is a simple constant in 2D and
depends on the geometry of the holes in 3D 25,30. In the example (P) of this article,
the parameters D∞has to be computed by such a cell problem, where N0=|�
Ωhole|
is just the size of the opening in scaled coordinates.
More specifically, for this example, the condition �uδ
0�Γ= 0 comes from the
solution of �−ΔΠ = 0,in �
Ω,
∇Π·n= 0,on ∂�
Ωhole ∩�
Ω,(2.7)
where Πand its derivative are 1-periodic and Πis bounded. The boundary ∂�
Ωhole∩�
Ω
is the blue boundary on Fig. 4a. The condition �∇uδ
0·n�Γorigins from the existence
of the blockage function D
�−ΔD= 0,in �
Ω,
∇D · n= 0,on ∂�
Ωhole ∩�
Ω,(2.8)
where Dand its derivative are 1-periodic and D−X2is bounded. This problems
defines Dup to an additive constant coming from problem (2.7). This constant is
set up, choosing D∞corresponding to the limit of ±(D−X2)for X2→±∞. Then,
N1in B1
γis no other than ��
Ω∂X1DdX.
The problem (2.8) can be solved numerically on a truncated periodicity cell
�
ΩBfor given B�2using Dirichlet-to-Neumann (DtN) boundary operators Λ±
B
based on a Fourier expansion in X1in the spirit of Ref. 20 and Ref. 21, using
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On the homogenization of microstructured surfaces 11
a spectral decomposition of Dwith the theory of self-adjoint compact operators
(Theorem VI.11 of Ref. 3), for X1∈(0,1):
Λ±
BD(X1,±B):=−�
n�=0
2π|n|��1
0
D(˜
X1,±B)e−2ıπn˜
X1d˜
X1�e2ıπnX1.(2.9)
With this DtN boundary operator, problem (2.8) can be truncated on �
ΩB, adding
the condition
∇D · n+Λ±
BD=±1,on Γ±
B= (0,1) × {±B},(2.10)
and we look for a periodic solution D∈H1(�
ΩB). We complete this problem taking
the limit condition
lim
X2→±∞exp(π|X2|)�D(X1, X2)−X2∓D∞�= 0, X1∈(0,1).(2.11)
into account. Using again the spectral decomposition of D, we have
(i)�Γ+
B
D+�Γ−
B
D= 0,(ii)�Γ+
B
D−�Γ−
B
D−2B= 2D∞.(2.12)
Computations of Dand D∞are illustrated for the example 1.2. The DtN oper-
ators are truncated using 8modes (i. e. −4�n�4in (2.9)).
In Fig. 6 the blockage function Dis plotted for different values of porosity, when
the plate has four holes, i. e. δ= 1/4(resp. the plate has eight holes, i. e. δ= 1/8).
In Fig. 7, the blockage coefficient D∞is plotted with respect to the characteristic
size ρof the obstacle.
3. The end-point of the periodic layer and corner conditions
In the previous section, we derived an effective description for the macroscopic part
extended in the mid-line Γ, through the description of transmission conditions (2.5).
The derivation of these transmission conditions is effective away from the end-points
x±
O, then one can ask if these transmission conditions are still valid when they are
to these end-points. Equivalently, one can ask himself what would be the correct
singular behaviour of uδ
qclose to the end-points.
To expose this effective behaviour of a macroscopic solution, it is expanded in
powers of δπ/Θ, where Θis the opening angle of the end-points. This becomes
uδ
q(x) = uδ
0,q(x) + δπ/Θuδ
1,q(x) + δ2π/Θuδ
2,q(x) + . . . (3.1)
where the dependence of the terms uδ
n,q on δmay be logarithmic (i. e. in ln(δ))
and is due to the end-point singularities, and to the possibly smaller scale (e. g.
η(δ) = o(δ)in example (P)).
For each macroscopic term uδ
n,q, we seek for a given stress intensity factor sδ
n,q �∈
V0(Ω\Γ), such that uδ
n,q −sδ
n,q ∈V0(Ω\Γ). Here, the set V0(Ω\Γ)is the set of
functions such that their restriction to any connected domain K⊂Ωbelongs to
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12 Adrien Semin, Kersten Schmidt
(a) H= 0.3,ρ= 0.9,D∞=
0.038775.
(b) H= 0.3,ρ= 0.5,D∞=
0.37172.
(c) H= 0.3,ρ= 0.1,D∞=
3.2658.
(d) H= 0.6,ρ= 0.9,D∞=
0.022052.
(e) H= 0.6,ρ= 0.5,D∞=
0.273737.
(f) H= 0.6,ρ= 0.1,D∞=
1.91781.
Fig. 6: Plot of D−X2for different values of porosity ρand relative wall thickness
H= 0.3or H= 0.6. The periodicity cell �
Ωis obtained by identification with δ= 1/4
for H= 0.3and δ= 1/8for H= 0.6.
V0(K). Such a function has been studied e. g. in Ref. 13 for a domain containing a
crack (i. e. Θ= 2π). These stress intensity factors are separated in two cases:
(i) due to the transmission conditions (2.4) with a source term depending on the
limit solution, we obtain a singular behaviour for the first corrector close to the
end-points, which is consistent with the matching with the near field. Numerical
pre-computations in a neighborhood of the end-points of the periodic layer are
not necessary. This point will be more deeply studied in Sec. 3.1,
(ii) in addition higher-order correctors exhibit a singular behavior that is not caused
by the source term in the transmission condition only, since this singular behav-
ior is in the kernels of L0and BΓclose to the end-points, but can be explained
only with the matching to the near field. For this, we need pre-computations of
singular enhancement functions S±and singularity enhancement factors L(S±)
in a neighborhood of the end-points of the periodic layer. This point will be more
deeply studied in Sec. 3.2,
(iii) in general, a part of the singularity is correctly obtained studying the behaviour
due to the source terms in the transmission conditions and a part is not correctly
obtained and one needs to study the matching with the near field and pre-
compute singularity enhancement functions / factors.
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On the homogenization of microstructured surfaces 13
10−1100
10−2
10−1
100
101
∼1
ρ
D∞
�
Ωhole ={(0,0.5−0.5ρ)∪(0.5 + 0.5ρ,1)} × (−0.3,0)
�
Ωhole ={(0,0.5−0.5ρ)∪(0.5 + 0.5ρ,1)} × (−0.6,0)
Fig. 7: Plot of D∞with respect to the porosity ρof the obstacle. Close to ρ= 0,
the quantity D∞ρremains constant.
The stress intensity factors and their nature depend on the shape of singularity
enhancement functions in the two conical domains containing an infinite periodic
layer �
Ω±(see Fig. 4d). To define the problem whose the singular enhancement
functions are solution, one has to scale around one corner with respect to δto
obtain the domain �
Ω±, take an appropriate ansatz and plug this ansatz in the
rescaled problem. If a smaller scale is involved (e. g. η(δ) = o(δ)), then these near
field problems will contain an infinite periodic point contribution (see Fig. 4e),
coming from resolution of another problem in a geometry scaled with η(δ)around
one hole (see Fig. 4c).
To obtain these functions, in general, the solution of a near field problem has
to be computed (Sec. 3.2), but sometimes they can be computed analytically, using
the impedance parameters that were computed in the previous section 3.1.
3.1. Singular behaviour due to source terms in the transmission
conditions
In this section, we are interested into the resolution of
L0v= 0,in K±
N0v= 0,on ∂K±
BΓ±v=f, on Γ±,
(3.2)
where f=B1
Γ±u,u∈V±(K±)being an homogeneous solution of (3.2). Here
V±(K±)is the set of functions φsuch that the function φχ±∈V0(Ω), where χ±is
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14 Adrien Semin, Kersten Schmidt
aC∞truncating function compactly supported in a vicinity of x±
O. The operators
BΓ±and B1
Γ±are formally the operators BΓand B1
Γwritten on Γ±instead of Γ.
X1
X2
θ=π−Θ
θ=π
Fig. 8: Semi-infinite conical domain K−with the semi-infinite interface Γ−(plotted
in blue).
More specifically, in the example (P), close to the corner x±
O, the limit term u0,0∈
(H1)±(K±)(i. e. u0,0χ±∈H1(Ω)). Using again the self-adjoint operators for the 1D
Laplace-Beltrami operator ∂2
θwith Neumann boundary conditions at θ−={π−
Θ,π}(resp. θ+={0,Θ}), we can express u0,0close to the end-point x±
Oas a linear
combination of radial Bessel functions of first kind Jnπ
Θ(k0r±)times cosine functions
in θ±. Problem (3.2) with right-hand side f:= Jnπ
Θ(k0r±) cos nπ
Θ(θ±−Θ±
0), with
Θ−
0=πand Θ+
0= 0, for n�= 1, admits a solution v∈(H1)±(K±). However, the
function f:= Jπ
Θ(k0r±) cos π
Θ(θ±−Θ±
0)gives equivalently the condition
BΓ±v=�∓2D∞Jπ
Θ−1(k0r±)π
Θsin π2
Θ
∓N0π
Θ�π
Θ−1�Jπ
Θ−1(k0r±) cos π2
Θ�(3.3)
There exists a particular solution φ±
1of (3.2) under the form
φ±
1(r±,θ±) = Jπ
Θ−1(k0r±)ψ±
1(θ±),(3.4)
where the study of the function ψ±
1(θ±)is postponed in appendix Appendix A.
Then, from u0,0, we extract the contribution corresponding to the Bessel function
Jπ
Θ(k0r±)
�±(u0,0) = 2
ΘJπ
Θ(k0r±)�I±
uδ
0,0(r±,θ±) cos π
Θ(θ±−Θ±
0)dθ±, I−= (π−Θ,π), I+= (0,Θ),
(3.5)
then close to the corner x±
O, the singular behaviour of uδ
0,1is given by the stress
intensity factor k0Θ
2π�±(u0,0)φ±
1(r±,θ±),i. e.
uδ
0,1−�
±
k0Θ
2π�±(u0,0)φ±
1χ±∈H1(ΩT∪ΩB).(3.6)
This last relation means that the restriction of this function to ΩT(resp. ΩB) is in
H1(ΩT)(resp. H1(ΩB)).
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On the homogenization of microstructured surfaces 15
3.2. Singular behaviour coming from the matching with the near
field
As it was already stated in the introduction of this section, there exists two singu-
larity enhancement factors L(S±)such that the singular behaviour of uδ
2,0is given
by the stress intensity factor
−πL(S±)�±(u0,0)(k0/2)2π/Θ
Γ(π/Θ)Γ(π/Θ+1) Yπ
Θ(k0r±) cos π
Θ(θ±−Θ±
0),i. e.
uδ
2,0+�
±
πL(S±)�±(u0,0)(k0/2)2π/Θ
Γ(π/Θ)Γ(π/Θ+ 1)Yπ
Θ(k0r±) cos π
Θ(θ±−Θ±
0)χ±∈H1(Ω).
(3.7)
Note that the function φ±
2(r±,θ±) := (r±,θ±)�→ Yπ
Θ(k0r±) cos π
Θ(θ±−Θ±
0)is an
homogeneous solution of (3.2).
More generally, a higher order macroscopic term has a singular behaviour as a
linear combination of canonical stress intensity factors yk,±(r±)φk,±(θ±)solutions
of (3.2) with factors that are productions of a functional of lower order macroscopic
terms and a related singularity enhancement factor.
These functionals of lower order macroscopic terms are obtained projecting these
terms on their regular part. For example, �±(u0,0)is given by relation (3.5).
To obtain the singular enhancement factor L(S±), one has to compute a partic-
ular near field function (also called singular enhancement function) S±in a stretched
multi-perforated domain around one end-point (see Fig. 4d) solution of a Laplace
equation with a prescribed behavior at infinity away from the perforations, i. e.
for radial coordinate R→ ∞ and for θnot being the angle of the interface. More
precisely, we are looking for S±solution of the system
ΔS±= 0,in �
Ω±,
∇S±·n= 0,on ∂�
Ω±,
S±−Rπ/Θcos π
Θ(θ±−Θ±
0) = o(1), R → ∞,θ�=π−Θ±
0
(3.8)
We can see that the equation and the boundary conditions we have to consider
are no other than the principal symbol of the Helmholtz equation and the Neumann
boundary condition of (P). In the general case linear operators L0,N0given e. g.
by (3.2), denoting respectively L0,N0their principal part, we have to solve
L0V=F, in �
Ω±,
N0V=G, on ∂�
Ω±,(3.9)
with a prescribed behavior towards infinity coming from the expansion of the ho-
mogeneous solutions of (3.2). Such a problem has been studied by Sergei Nazarov
in the case of a periodic boundary with Dirichlet boundary conditions27 and with
Neumann boundary conditions29 for a general linear differential operator and has
been studied by the authors10,12 and relies on the use of Mellin transform, as well
as on the extension of the Kondrat’ev theory22. The possible right hand-sides F
and Gin (3.9) would come from the study of the high-order near field terms.
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16 Adrien Semin, Kersten Schmidt
The standard variational space to solve problem (3.9) in the case of the Laplace
equation if
V±(�
Ω±) = �V∈H1
loc(�
Ω±),∇V∈L2(�
Ω±),V
(1+R) ln(2 + R)∈L2(�
Ω±)�,(3.10)
which, equipped with the norm
�V�2
V±(
�
ٱ)=����
V
(1+R) ln(2 + R)����
2
L2(
�
Ω±)
+�∇V�2
L2(
�
Ω±),(3.11)
is a Hilbert space. However, it is clear that with the requested condition towards
infinity (3.8-iii), the singular enhancement function S±cannot belong to V±(�
Ω±).
Therefore, we shall decompose it into a particular function (also called asymptotic
block) S±that has this prescribed behavior towards infinity, and its remainder
R±=S±−S±belongs to V±(�
Ω±). To write the asymptotic block S±, one starts
from the limit behavior Rπ/Θcos π
Θ(θ±−Θ±
0). To take into account the presence
of the infinite periodic layer, we multiply this limit behaviour by a smooth cut-off
function χ(X2)such that
χ(t) = 0,|t|<1,χ(t) = 0,|t|>2,(3.12)
and we go back and forth between the radial behavior of S±and its behavior close to
the infinite periodic layer, similarly as going back and forth between the macroscopic
part of the solution and its periodic layer corrector in Section 2. Therefore, the radial
part of the asymptotic block S±can be written towards infinity as
S±=Rπ/Θcos π
Θ(θ±−Θ±
0)+Rπ/Θ−1ψ1(θ±)+O(Rπ/Θ−2),(3.13)
the function ψ1in (3.13) being the same as the function defined in (3.4). In the
particular case Θ=π(see e. g. 2a), we need to take into account one additional
term in that expansion. Neglecting the O(Rπ/Θ−2)part and multiplying by χ(R)
to have a regular behavior towards R= 0, the remainder R±satisfies problem
(3.9) with F=−L0S±=−ΔS±and G=−N0S±=−∇S±·n. This problem is
well-posed and admits a unique solution in V±(�
Ω±). It can be shown then that the
leading part of this remainder towards infinity is the same as the leading part of the
problem (3.9) written on the conical domain K±instead on the domain �
Ω±),i. e.
there exists a constant L(S±)independent on the choice of the truncating function
such that
R±∼L(S±)R−π/Θcos π
Θ(θ±−Θ±
0).(3.14)
The problem (3.9) can be solved numerically on a truncated near field domain
�
Ω±
Refor given Re�2using an approximate Robin boundary condition using the
behavior of R±given by (3.14). With this approximate Robin boundary condition,
problem (3.9) can be truncated on �
Ω±
Re, adding the condition
∇R±·n+π
ΘRe
R±= 0,on Γ±
Re,(3.15)
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On the homogenization of microstructured surfaces 17
where the artificial boundary Γ−
Reis given by Γ−
Re={(Recos θ−, Resin θ−),θ−∈
(π−Θ,π)}and the artificial boundary Γ+
Reis given similarly. Additionally, for these
artificial boundaries we choose Resuch that Γ±
Re∈�
Ω±,i. e. they do not intersect
any hole. Using again the behavior of R±(3.14), we have
L(S±)∼2
ΘRπ/Θ
e�Γ±
Re
R±.(3.16)
10−1100
10−3
10−2
10−1
100
∼
1
ρ
L(S±)
�
Ωhole ={(0,0.5−0.5ρ)∪(0.5 + 0.5ρ,1)} × (−0.3,0)
�
Ωhole ={(0,0.5−0.5ρ)∪(0.5 + 0.5ρ,1)} × (−0.6,0)
Fig. 9: Plot of L(S±)with respect to the porosity ρof the obstacle, for the truncat-
ing radius Re= 30.5. The periodicity cell is obtained by identification with δ= 1/4
for H= 0.3and δ= 1/8for H= 0.6. Close to ρ= 0, the quantity L(S±)ρremains
constant.
Computations of L(S±)are illustrated for the example 1.2. In Fig. 9, the sin-
gular enhancement coefficient L(S±)is plotted with respect to the characteristic
size ρof the obstacle for the truncating radius Re= 30.5. In Fig. 10a, the singular
enhancement coefficient L(S±)is plotted with respect to the characteristic trun-
cating radius Reof the near-field domain for the porosity ρ= 0.3. Contrarily to the
computation of the blockage coefficient D∞which exponentially convergences with
respect to the characteristic domain size B(see e. g. Ref. 17), the convergence rate
of the singular enhancement coefficient L(S±)is only polynomial (see Fig. 10b),
and at least two different computations have to be achieved (e. g. for Re= 30.5and
Re= 35.5) to obtain a good approximation of this coefficient.
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18 Adrien Semin, Kersten Schmidt
0 20 40
0.176
0.18
0.184
0.188
Re
L(S±)
(a)
101102
10−2
10−1.8
2
3
Re
L(S±)−0.167
(b) L(S±)�0.167 + 0.098R−2/3
e
Fig. 10: Plot of L(S±)with respect to the truncating radius Re, for the porosity
ρ= 0.3. The periodicity cell is obtained by identification with δ= 1/8for H= 0.6.
4. Computation of the macroscopic solution
This section is dedicated to the computation of each macroscopic term un,q of the
expansion (1.3). These terms solve the problem
L0(x,∇x)uδ
n,q =fn,q,in Ω\Γ,
N0(x,∇x)uδ
n,q =gn,q,on ∂Ω,(4.1)
with the transmission condition given by (2.5)
(BΓuδ
n,q)(x) =
q−1
�
p=0
(Bq−p
Γuδ
n,p)(x)on Γ.(4.2)
and with possibly corner singularities that have been studied in Section 3. One
important point to notice that, as it was already explained in Section 1, the compu-
tational effort of each macroscopic term is independent of the parameter δ, since the
linear differential operators involved in these equations, as well as the computational
domain, are independent of δ.
In the following, the different macroscopic terms of the expansion are computed
for the example (P) (Section 4.1) and a finite sum of the expansion is compared
with a reference solution computed by resolving all the obstacles (Section 4.2).
4.1. Computation of the macroscopic term of the expansion
For the example problem (P), the macroscopic term uδ
1,0corresponding to the weight
δπ/Θis solution of (4.1)-(4.2) with right-hand side fδ
0,0=gδ
0,0= 0 and contains no
stress intensity factor (i. e. uδ
1,0∈H1(ΩT∪ΩB)). Then it stands uδ
1,0= 0. In a
similar way, and using the transmission conditions (4.2), the macroscopic term uδ
1,1
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On the homogenization of microstructured surfaces 19
corresponding to the weight δπ/Θ+1 is equal to 0. Therefore the first non-negligible
macroscopic terms are uδ
0,0,uδ
0,1and uδ
2,0.
Numerical simulations are carried using the C++ Finite Elements Library
Concepts7,14. These terms are computed on a mesh generated using GMSH16. This
mesh is refined close to the corners and solves the limit interface Γ(see Fig. 11). In
particular, the interface Γis refined close to the end-points.
Fig. 11: Mesh used for the computation of the macroscopic solution. The blue line is
the limit interface Γ. The red arcs are the domain integration to get the functionals
�±(u0,0). These lines are resolved by this mesh.
Computation of the limit solution In this paragraph, the problem (4.1)-(4.2)
for n=q= 0, with f0,0= 0 and g0,0=1ΓR(∇uinc ·n−ık0uinc)is studied. The
transmission condition (4.2) gives nothing other than no jump conditions, so that
the limit interface Γis transparent for uδ
0,0. This problem admits then a unique
solution u0,0∈H1(Ω)independent of δand is resolved numerically using an hp-
refinement strategy towards the end-points.
From the resolution of this limit problem, we compute the trace operator B1
Γu0,0
on Γusing (2.6) and the values �±(u0,0)that will be used for the determination of
the stress intensity factors of the upcoming terms.
Computation of the corrector uδ
0,1In this paragraph, the problem (4.1)-(4.2)
for n= 0, q = 1, with f0,1=g0,1= 0 and with the transmission operator B1
Γgiven
by (2.6)
B1
Γv=�2D∞�∇v·n�Γ
N0(∂2
Γ+k2
0)�v�Γ�
is studied. Moreover, uδ
0,1admits a prescribed stress intensity factor given by the
relation (3.6)
uδ
0,1−�
±
k0Θ
2π�±(u0,0)φ±
1χ±∈H1(ΩT∪ΩB).
To solve this problem, one has to introduce the function ˜uδ
0,1∈H1(ΩT∪ΩB)
corresponding to the regular part of uδ
0,1, by subtracting its stress intensity factor
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20 Adrien Semin, Kersten Schmidt
from itself. Therefore, the function ˜uδ
0,1has to satisfy the following problem
L0(x,∇x)˜uδ
0,1=−�
±
k0Θ
2π�±(u0,0)L0(x,∇x)(φ±
1χ±),in Ω\Γ,
N0(x,∇x)˜uδ
0,1=−�
±
k0Θ
2π�±(u0,0)N0(x,∇x)(φ±
1χ±),on ∂Ω,
(BΓ˜uδ
0,1)(x)=(B1
Γuδ
0,0)(x)−�
±
k0Θ
2π�±(u0,0)(BΓφ±
1χ±)(x),on Γ.
(4.3)
Problem (4.3) seems a priori as complicated to solve as the problem satisfied by
uδ
0,1, since the right-hand side of the first line, for example, could not possibly belong
to L2(Ω). Hopefully, since the singular enhancement function φ±
1is solution of (3.2),
a suitable choice for the cut-offfunctions χ±would be cut-offfunctions that are
identically equal to 1in a vicinity of the end-points x±
O. To do so, being two suitable
numbers ri< re, the functions χ±depend only on the radius r±=|x−x±
O|and is
chosen such that χ±= 1 for r±< riand χ±= 0 for r±> re. It will ensure then that
the boundary operator N0(x,∇x)can commute with the truncating function, i. e.
N0(x,∇x)(φ±
1χ±) = χ±N0(x,∇x)(φ±
1) = 0. Introducing for a linear operator A
the commutator operator [A,χ±] = Aχ±−χ±Awhich will be compactly supported
in the support of ∇χ±, and using that φ1
±is solution of (3.2), problem (4.3) can be
simplified to
L0(x,∇x)˜uδ
0,1=−�
±
k0Θ
2π�±(u0,0)�L0(x,∇x),χ±�φ±
1,in Ω\Γ,
N0(x,∇x)˜uδ
0,1= 0,on ∂Ω,
(BΓ˜uδ
0,1)(x)=(B1
Γuδ
0,0)(x)−�
±
k0Θ
2π�±(u0,0)(BΓφ±
1χ±)(x),on Γ.
(4.4)
Numerically, the function ˜uδ
0,1is computed using finite elements discontinuous
over the interface Γ, since the jump of ˜uδ
0,1across the interface Γis non-zero. The
Neumann jump of ˜uδ
0,1appears naturally when writing the variational formulation
associated to the problem (4.4), whereas the Dirichlet jump has to be taken into
account e. g. using a penalization method.
Computation of the corrector uδ
2,0In this paragraph, as well as for the com-
putation of the corrector uδ
0,1, the problem (4.1)-(4.2) for n= 2,q= 0, with
f2,0=g2,0= 0, is studied. The transmission condition (4.2) gives nothing other
than no jump conditions, so that the limit interface Γis transparent for uδ
2,0. But,
contrarily to the resolution of the limit solution, the H1-regularity close to the cor-
ners do not hold. Then, one has to introduce the function ˜uδ
2,0corresponding to the
regular part of uδ
2,0, by subtracting its stress intensity factor from itself, using (3.7).
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On the homogenization of microstructured surfaces 21
Therefore, the function ˜uδ
2,0has to satisfy the following problem
L0(x,∇x)˜uδ
2,0=−�
±
πL(S±)�±(u0,0)(k0/2)2π/Θ
Γ(π/Θ)Γ(π/Θ+ 1) �L0(x,∇x),χ±�φ±
2,in Ω\Γ,
N0(x,∇x)˜uδ
2,0= 0,on ∂Ω,
(BΓ˜uδ
2,0)(x)=0,on Γ.
(4.5)
The numerical effort to compute the corrector ˜uδ
2,0is the same as the numerical
effort to compute the limit solution.
4.2. Computation of a reference solution and comparison
Computations are illustrated for the example 1.2. The meshes obtained for a thin
plate perforated with 4holes (Fig. 12a) and for a thin plate perforated with 8holes
(Fig. 12b) are refined close to the holes and resolve them.
(a) δ= 1/4(b) δ= 1/8
Fig. 12: Mesh used for the computation of the reference solution, for the porosity
ρ= 0.3.
(a) δ= 1/4(b) δ= 1/8
Fig. 13: Reference solution, for the porosity ρ= 0.3.
Study of the robustness of the error It was already studied in 12 that the error
estimate (1.4) holds for (s, κ(s)) = (1,0),(s, κ(s)) = (4
3,0) and (s, κ(s)) = (2,1),
with a constant Cdepending on the canonical hole domain �
Ωhole. In particular, C
depends on the porosity ρof the thin plate. As it was already shown on Figures 7
and 9, this constant could possibly degenerate as ρ→0. To study, the robustness
of the error, let δ= 1/8and let the height Hof the canonical obstacle be obtained
by parameter identification (i. e. H= 0.6).
On Figure 14, several plots of the L2macroscopic error (1.4) computed on the
domain Ω0.25 are shown for different values of s:(s, κ(s)) = (1,0) corresponds to
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22 Adrien Semin, Kersten Schmidt
0 0.2 0.4 0.6 0.8 1
10−2
10−1
100
101
ρ
Macroscopic error
macroscopic error uδ−u0,0
macroscopic error uδ−u0,0−δu0,1
macroscopic error uδ−u0,0−δu0,1−δ4
3u2,0
macroscopic error with the three scale
Fig. 14: Computation of the macroscopic error for the problem (P), taking into
account more and more terms of the expansion, with respect to the porosity, and
comparison with the macroscopic error obtained with a three scale strategy.
the macroscopic error between the reference solution uδand the limit macroscopic
term u0,0(blue solid plot), (s, κ(s)) = (4
3,0) corresponds to the macroscopic error in
which we take into account the first order corrector (red dashed plot), and (s, κ(s)) =
(2,1) corresponds to the macroscopic error in which we take into account the second
order corrector (brown dotted plot). For comparison, the plot of the L2macroscopic
error (1.4) computed with ssmall and with a limit term obtained from a tree scale
strategy (i. e. BΓv= ([v]Γ−Z�∇v·n�Γ,[∇v·n]Γ)�) is shown using a green plot
with squares. These different error curves show that, when the porosity ρof the
material is not too small, the application of this method with a two scale strategy
gives a more and more accurate solution, when more and more macroscopic terms
are considered in the expansion. A contrario, when the ρis too small, the correctors
degrade the obtained error, and it would be more appropriate to consider a three
scale strategy.
Acknowledgment
The author would like to thank Robert Gruhlke (TU Berlin) for helpful discussions
and partial implementation of the numerical experiments.
Appendix A. Appendix: definition of the profile function ψ±
1
In this appendix, the construction of the angular function ψ±
1that is used in the
transmission conditions (3.3) and in the stress intensity factor (3.6) is detailed.
Moreover, this function can be seen as the first element of a family of functions that
can be derived by a general behavior of the form
cos nπ
Θ(θ±−Θ±
0), n ∈N∗.
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On the homogenization of microstructured surfaces 23
This function is constructed as
ψ±
1(ln r±,θ±) =
2
�
q=0
ψ±
1,q(θ±)(ln r±)q, w1,1,q,±∈C∞(I±
1)∩C∞(I±
2),(A.1)
where I±
1= (a±,γ±),I±
2= (γ±, b±)with a+= 0,γ+=π,b+=Θ, and, a−=π−Θ,
γ−= 0,b−=π. This function is constructed such that the function
v±
1(r±,θ±):=Jπ
Θ−1(k0r±)ψ±
1(ln r±,θ±)
satisfies
Δv±
1+k2
0v±
1= 0 in K±
1∩K±
2,
∂θv±
1(r±, a±) = 0, r±>0,
∂θv±
1(r±, a±) = 0, r±>0,
[v±
1(r±,γ±)]∂K±
1∩∂K±
2=Jπ
Θ−1(k0r±)a1,1,±, r±>0,
[∂θ±v±
1(r±,γ±)]∂K±
1∩∂K±
2=Jπ
Θ−1(k0r±)b1,1,±, r±>0,
(A.2)
where
K±
j=�(r±cos θ±, r±sin θ±)∈K±, r±∈R∗
+,θ±∈I±
j�, j ={1,2},(A.3)
and,
a1,1,±=∓Dn
1
π
Θsin π2
Θ,(A.4)
b1,1,±=∓Nt
2
π
Θ�π
Θ−1�cos π2
Θ± Nn
2
π
Θ�π
Θ−1�cos π2
Θ.(A.5)
In view of 12 , since λΘ=π−Θis not a multiple of π,sin(π−Θ)�= 0 the functions
ψ±
1,1and ψ±
1,2are identically equal to 0. Therefore, ψ±
1does not depend on ln r±,
and there exists two constants w±
1,+and w±
1,−such that
ψ±
1(θ±) = �w±
1,+cos �π
Θ−1�(θ±−Θ±
0),sin θ±>0,
w±
1,−cos �π
Θ−1�(θ±−Θ±
0∓Θ),sin θ±<0.(A.6)
We insert expression (A.6) in the Dirichlet and Neumann jump conditions, means
the fourth and fifth lines of (A.2) gives, using that the jump is the limit value for
θ±>γ±minus the limit value for θ±<γ±and using that γ±−Θ±
0=±π:
w±
1,+cos �π
Θ−1�π−w±
1,−cos �π
Θ−1�(π−Θ) = ∓a1,1,±,
w±
1,+sin �π
Θ−1�π−w±
1,−sin �π
Θ−1�(π−Θ) = b1,1,±
�π
Θ−1�.(A.7)
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Paper_Helmholtz_Numerik_m3as
24 Adrien Semin, Kersten Schmidt
Determinant of system (A.7) is no other than sin(π−Θ)which is non-zero by
assumption on Θ. Therefore, this system is invertible, and we get
w±
1,+=1
sin(Θ−π)�∓a1,1,±sin �π
Θ−1�(π−Θ) + Θb1,1,±
π−Θcos �π
Θ−1�(π−Θ)�,
w±
1,−=1
sin(Θ−π)�∓a1,1,±sin �π
Θ−1�π+Θb1,1,±
π−Θcos �π
Θ−1�π�.
(A.8)
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