Singular Functions and Characterizations of Field Concentrations:a Survey

Hyeonbae Kangand Sanghyeon Yu

1 Department of Mathematics,Inha University,Incheon 22212,South Korea

2 Department of Mathematics,Korea University,Seoul 02841,South Korea

Abstract.In the presence of closely located inclusions of the extreme material property,the physical fields,such as the electric field and the stress tensor,may be concentrated and arbitrarily large in the narrow region between two inclusions.Recently there has been significant progress on the quantitative characterization of the field concentration in the contexts of electrostatics(Laplace equation),linear elasticity(Lam′e system),and viscous flow(Stokes system).This paper is to review such progress in a coherent way.

Key Words:Field concentration,gradient blow-up,closely spaced inclusions,extreme inclusions,Laplace equation,Lam′e system,Stokes system.

1 Introduction

When two inclusions of the extreme material property are located closely to each other,the physical field may be concentrated and arbitrarily large in the narrow region between the inclusions.An inclusion of the extreme material property means a perfectly conducting or insulating inclusion(the conductivity being∞or 0)in the electrostatic case and a hard inclusion and a hole in the elastostatic case,and the corresponding physical fields are the electric field and the stress tensor.Such field concentration may occur in fiberreinforced composites causing failure of the composites[6],and the electric field can be greatly enhanced and utilized to achieve subwavelength imaging and sensitive spectroscopy[35].In this respect it is quite important to understand the field concentration in a quantitatively precise manner.It is also quite important to come up with an efficient numerical scheme to compute the fields in such cases since numerical computation of the field is known to be very hard in the presence of closely located inclusions.

In response to such importance and mathematical challenges involved in this problem,there has been much progress in understanding the field concentration in the last 20 years or more.In the context of electrostatics(or anti-plane elasticity in two dimensions),the field is the gradient of a solution to the Laplace equation and the precise estimates of the gradient were obtained when the conductivity of the inclusions is∞:the blow-up rate of the gradient is?in two dimensions[5,37],where?is the distance between two inclusions,and it is|?ln?|in three dimensions[7].There is a long list of literature in this direction of research among which we mention[3,4,9,13,21,27,28,33,34,38].We also mention for related works[10,12,14,22–24].If the conductivity of the inclusions is 0(the insulating case),the two-dimensional problem is dual to the perfectly conducting case(by means of the harmonic conjugation),and hence the blow-up rate of the insulating case is also?.But the three-dimensional case requires further investigation.In this respect,we mention the paper of Yun[39]where a rather unexpected blow-up rate of the gradient has been found when the inclusions are balls.If the conductivity is away from∞and 0,then the gradient stays bounded no matter how closely located the inclusions are[11,30,31].

While most of the work mentioned above focus on the estimates from above and below of the blow-up rate of the gradient,there is another important direction of research which is to characterize the singular behavior of the gradient.The characterization of the singular behavior means,roughly speaking,the decomposition of the solution u into the form u=s+b where s carries the information of singularity of the gradient?u and b is a regular function in the sense that?b is bounded(or less singular)regardless of the distance between two inclusions.One important feature of such decompositions is that the singular part s is explicitly given and satisfies the governing equation,e.g.,the conductivity equation,the elasticity equation,and so on.It has a significant implication on the numerical computation of the solution in presence of closely located inclusions.Such a computation is known to be a difficult problem because very fine meshes are required since the gradient becomes arbitrarily large in the narrow region.The decomposition enables us to compute the solution u numerically using standard meshes,not refined ones since s is explicit and b is regular.Such a characterization is reminiscent of that related to the corner singularity of elliptic equations which are utilized for computation of the solution to the(interior or exterior)boundary value problem when the domain has a corner[15,25,26].

Characterizations of the field concentration are obtained for the conductivity equation in[1,16–18,32]and for the Lam′e system of the linear elasticity in two dimensions in[19]when inclusions are locally strictly convex.These result has been further extended to the two-dimensional Stokes system for circular inclusions[2].The singular parts of the decomposition are represented by explicit building blocks,which we call singular functions.It is the purpose of this paper to summarize these results on the singularity characterization in a coherent way.

2 Geometry of two inclusions

Suppose that there are unique points z∈?Dand z∈?Dsuch that

We assume that Dis strictly convex near z,namely,there is a common neighborhood U of zand zsuch that D∩U is strictly convex for j=1,2.Moreover,we assume that

dist(D,DU)≥C and dist(D,DU)≥C

for some positive constant C independent of?.This assumption says that points in Dand Dother than neighborhoods of zand zare at some distance to each other.Note that strictly convex domains satisfy all the assumptions.

Letκbe the curvature of?Dat z.Let Bbe the disk osculating to Dat z(j=1,2).Then the radius rof Bis given by r=1/κ.Let Rbe the inversion with respect to?Band let pand pbe the unique fixed points of the combined reflections R°Rand R°R,respectively.We emphasize that?Band?Bare circles of Apollonius of pand p2.

After rotation and translation,we assume that p=(p+p)/2 is at the origin and the x-axis is parallel to the vector p?p.Then one can see(cf.[5])that pand pare written as

where the constant a satisfies

The geometry of Dand Dis depicted in Fig.1 which is taken from[19].

3 Conductivity equation

The conductivity problem with two conducting inclusions is modelled as follows:

Figure 1:Geometry of the two inclusions and osculating circles[19].

where h is a given function harmonic in R.The fact that the solution u takes a constant value on?Dindicates that Dis a perfect conductor meaning that its conductivity is∞.The constantsλare not prescribed and the problem(3.1)is not an exterior Dirichlet problem.The constants are rather determined by the conditions

Here and throughout this paper,n denotes the outward normal on?D(j=1,2).It is worth emphasizing that the constantsλandλmay or may not be the same depending on the given h;When they are different,there occurs a sharp gradient if the distance between Dand Dis short.

The singular function for the conductivity problem is given by

where p(j=1,2)are the fixed points of the combined reflections as defined in section 2.The function q has important properties including the following:

(i)It is harmonic in Rexcept at pand p.

(ii)It takes constant values on osculating circles?B.It is because?Band?Bare circles of Apollonius of pand p.

(ii)It holds that

This function appears in the bipolar coordinate system for?Band?Bas will be seen in section 5.It was used for analysis of the field concentration for the first time in[37].

The singular behaviour of the solution u to(3.1)can be characterized in terms of the function q.In fact,the following decomposition formula is obtained in[16],which is an improvement upon the corresponding decomposition in[1]:

A way to compute Cnumerically is proposed in[16].

The decomposition formula(3.5)has some important consequences.Since?q is bounded from below and above by?(up to constant multiples),the blow-up estimates for?u can be obtained from(3.5).As mentioned before,it can be used to compute u numerically.Since(3.5)extracts the major singular term in an explicit way,it suffices to compute the residual term b for which only regular meshes are required.This idea appeared and was exploited in[17]in the special case when D’s are disks.There the decomposition formula was derived when D’s are disks(of the radius r)with the constant Creplace with

The decomposition formula of the kind(3.5)when Dand Dare three-dimensional balls of the same radii has been derived in[18](see[29]for the case of different radii).In this case the singular function is given as an infinite sum of point charges.

4 Lam′e system

To describe the elasticity problem with two hard inclusions,let(λ,μ)be the pair of Lam′e constants of Dwhich satisfies the strong ellipticity conditions:μ>0 andλ+μ>0(we only consider the two-dimensional case).Then the problem is given as follows in terms of the Lam′e system of equations

Lu:=μ?u+(λ+μ)??·u,

here,u=(u,u)is a vector-valued function:

where H=(h,h)is a given function satisfying LH=0 in R.Here,Ψare the displacement fields of the rigid motions defined by

The boundary conditions to be satisfied by the displacement u on?D(the second line in(4.1))indicate that Dand Dare hard inclusions,and the constants care not given but determined by the conditions

Hereσ[u]denotes the stress tensor corresponding to the displacement vector u defined by

The singular functions for the problem(4.1)are obtained in[19]as linear combinations of point source functions in linear elasticity called nuclei of strain.The following nuclei of strain are used

where

e=(1,0), e=(0,1)and (x,y)=(?y,x).

It turns out that those singular functions can be expressed in the simple forms using the functionζ:

where q is the singular function for the conductivity problem given in(3.3).One can easily see that qare solutions to the Lam′e system,namely,

It is shown in[19]that qtakes‘a(chǎn)lmost’constant values on the osculating circles?B(i=1,2).In fact,there are constantαandβ(which depends on?)such that

Another function related with the boundary valueΨon?Band?Bis constructed in the same paper.But this function has nothing to do with the singular behavior of the field,so we omit it here.It is worth mentioning that the singular functions qand qare effectively utilized to prove the Flaherty-Keller formula on the effective property of densely packed elastic composites[20].

Using the singular functions qand q,it is proved that the solution u to(4.1)admits the following decomposition:

where Cand Care constants depending on?,but bounded independently of?,and b is a function whose gradient is bounded on any bounded subset of D.This decomposition formula enables us to prove that?is an upper bound of?u,and it is also a lower bound in some cases.The fact that?is an upper bound of?u was proved in[8].

We mention that the constants Cand Cappearing in the formula(4.10)are not explicit.Thus further investigation on how to determine them(or compute them numerically)is desired.

5 Stokes system

We also consider the Stokes system in the exterior domain D.Let(U,P)is a given background solution to the homogeneous Stokes system in R,namely,

We consider the following problem of the Stokes system:

with the conditions

(u?U)(x)=O(|x|), ?(u?U)(x)=O(|x|), (p?P)(x)=O(|x|),

as|x|→∞.Hereμrepresents the constant viscosity of the fluid,Ψare the functions given in(4.2),and dare constants to be determined from the equilibrium conditions

Here,σ[u,p]is the stress field induced by the velocity-pressure pair(u,p),namely,

where I is the identity matrix.

The singular functions(h,p),j=1,2,for the problem(5.2)is the solution to the following problem:

with the conditions

h(x)=C+O(|x|), ?h(x)=O(|x|), p(x)=O(|x|),

for some constant Cas|x|→∞.

In[2]singular functions(h,p)are constructed using the stream function formulation for which the bipolar coordinate system is used.The bipolar coordinates(ζ,θ)are defined by

where a is the number appeared in(2.3).It is worth mentioning that the singular function q in(3.3)is nothing but q=ζ/2π.

Suppose that?Band?Bhave the same radius R for convenience.Let

and

Define two constants Aand Bby

Then,the velocity his given by h=he+hewhere

and the pressure pis given by

The formulas for(h,p)are quite involved.But it is proved in[2]that

where(h,p)is a solution whose gradient is bounded regardless of?,and Ais the constant defined by

It is proved in the same paper that if the background velocity field U is given by

for some constantsα,βandγand the background pressure P=0,and if Dand Dare disks of the same radius R(so that D=Bfor j=1,2),then the solution(u,p)admits a decomposition of the following form:

where(u,p)is a solution to the Stokes problem whose stress tensor is bounded.Thus we have

Since

as proved in[2],we have

which says that the stress always blows up at the rate of?provided that U is linear as given in(5.13)and inclusions are circular.It is quite interesting and challenging to extend this result to the non-circular case.

Acknowledgements

This work is supported by NRF 2019R1A2B5B01069967 and 2020R1C1C1A01010882.