Mohammad Amin Bozorgmanesh
email: mohammadamin.bozorgmanesh@gmail.com

Fatemeh Kowkabi

The Effect of Mobile Phone Electromagnetic Waves on Honey Bee

Keywords: Electromagnetic waves, Electric field, CCD phenomenon, Bee

With the advancement of technology, the development of telecommunication industries, the construction of new electrical devices, and the increasing use of electromagnetic field generators in modern industrial societies, the study of the biological effects of these waves on the growth and development of living organisms are of interest to many societies. The scientific world is placed. Today, one of the ecological problems is electromagnetic pollution in the environment. The use of mobile phones has increased in the world in recent years, which in turn adds to this pollution. Some bee species worldwide are becoming extinct, called the CCD phenomenon. Our goal in this article is to investigate the effect of electromagnetic waves and their relationship with the new phenomenon of CCD and to offer a proposal to guide bees to return to the hive by fluorescent blue light.

INTRODUCTION

Electromagnetism is a branch of physics that studies electrical and magnetic phenomena and their relationship [START_REF] Cheng | Field and wave electromagnetics[END_REF][START_REF] Bozorgmanesh | Thermal Effects of Electromagnetic Waves on the Human Body[END_REF]. Electromagnetic phenomena in classical physics are studied under Maxwell's laws. According to Maxwell's theory, every moving charged particle emits energy. For the first time, Maxwell [START_REF] Theerman | A Biography[END_REF], obtained the magnetic wave equations and showed that electric and magnetic fields could have a wave-like behavior; According to his theory, electric fields that change with time causes the creation of magnetic fields. The increasing use of radio, television, radar, and telecommunications masts and high pressure has increased the amount of electromagnetic radiation in the environment [START_REF] Kundi | Electromagnetic fields and the precautionary principle[END_REF]. A large amount of magnetic radiation in the environment is produced by mobile phone towers. Insects and small animals are the first living organisms affected by these radiations. If the number of masts and cell phones increases, honey bees may disappear within the next few years. Investigations have shown that the radiation of 900 MHz (MHZ) waves is very bioactive and causes changes in the physiological function of living organisms [START_REF] Pattazhy | Electromagnetic radiation (EMR) clashes with honeybees[END_REF]. The honey bee is a branch of arthropods and in the category of insects, which are key species in the ecosystem that pollinates more than 90 commercial products around the world from honey bees [START_REF] Sivani | Impacts of radio-frequency electromagnetic field (RF-EMF) from cell phone towers and wireless devices on biosystem and ecosystem-a review[END_REF]. Quoted from Albert Einstein: "If the bee disappears from the face of the earth, mankind will not survive more than four years!". The increase in losses in bee colonies worldwide has caused a phenomenon in which the number of bees in the hive decreases quickly without any signs of disease [START_REF] Gallai | Economic valuation of the vulnerability of world agriculture confronted with pollinator decline[END_REF]. Many studies have reported that radio frequency and electromagnetic radiation (EMR) create misleading biological effects that disrupt the functioning of the biological systems of living organisms [START_REF] Blank | Electromagnetic fields stress living cells[END_REF]. This article suggests using fluorescent blue light in the bee colony to guide the bees back to the hive.

Magnetic orientation system in the honey bee:

Sometimes honey bees move more than 12 km away from the hive to collect food [START_REF] Liang | Magnetic sensing through the abdomen of the honey bee[END_REF][START_REF] Ratnieks | How far do bees forage[END_REF]. Using their sense of sight and smell, they mark the surroundings of the hive and use them to find their way back to the hive [START_REF] Collett | Memory use in insect visual navigation[END_REF]. In addition to marking by the sense of sight and smell, they can detect the earth's electromagnetic fields and use them for orientation [START_REF] Gould | Bees have magnetic remanence[END_REF][START_REF] Hsu | Magnetoreception system in honeybees (Apis mellifera)[END_REF]. While flying, honeybees can detect weak magnetic field fluctuations of as much as 26 nT [START_REF] Kirschvink | Is geomagnetic sensitivity real? Replication of the Walker-Bitterman magnetic conditioning experiment in honey bees[END_REF]. There are two types of magnetic orientation systems in animals:

2.1) Magneto-chemical orientation system [START_REF] Johnsen | The physics and neurobiology of magnetoreception[END_REF]: The magneto-chemical orientation system is also known as light-dependent magnetic sense and is carried out by cryptochrome (a type of light receptor in plants and animals), blue ultraviolet radiation receptor [START_REF] Gegear | Cryptochrome mediates light-dependent magnetosensitivity in Drosophila[END_REF].

2.2) Magnetic orientation system based on magnetic structure [START_REF] Kirschvink | Magnetite-based magnetoreception[END_REF]: Magnetic orientation system based on magnetic structure, the presence of a type of bacteria with magnetic properties called Fe3O4 in the bee's abdomen that reacts to the magnetic field [START_REF] Kirschvink | Magnetite-based magnetoreception[END_REF][START_REF] Kuterbach | Iron-containing cells in the honey bee (Apis mellifera)[END_REF][START_REF] El-Jaick | Electron paramagnetic resonance study of honeybee Apis mellifera abdomens[END_REF]. The Fe3O4 in the bee's abdomen allows it to sense slight changes in the Earth's field lines. If these delicate structures are affected by even the slightest magnetic pollution, it will disrupt the bee's orientation system [START_REF] Favre | Mobile phone-induced honeybee worker piping[END_REF]. [START_REF] Kuterbach | Iron-containing cells in the honey bee (Apis mellifera)[END_REF] EMF electromagnetic fields, called electric smog pollution, are also known as one of the hidden and silent forms of pollution. Electromagnetic radiation (EMR) is widely used in modern technology and communication [START_REF] Sarkar | Wildlife around communication towers[END_REF]. Radiofrequency and microwave pollution, caused by various sources of mobile phones such as GSM, and UMTS/3G, is an essential factor in reducing population. Insects are like honey bees [START_REF] Everaert | A possible effect of electromagnetic radiation from mobile phone base stations on the number of breeding house sparrows (Passer domesticus)[END_REF][START_REF] Vanengelsdorp | A survey of honey bee colony losses in the US, fall 2007 to spring 2008[END_REF]. When worker bees leave the hive to collect food and never return to the hive, the phenomenon of CCD (Colony Collapse Disorder) occurs; this event occurs when the hive population suddenly disappears [START_REF] Vanengelsdorp | Colony collapse disorder: a descriptive study[END_REF]. Some scientists believe that the new phenomenon of CCD is caused by electromagnetic radiation. Recent studies show that the death of many animals and insects, such as honey bees, birds, and amphibians, is caused by EMR field exposure [START_REF] Nicholls | Bats avoid radar installations: could electromagnetic fields deter bats from colliding with wind turbines?[END_REF][START_REF] Sahib | Impact of mobile phone on the density of Honey Bees[END_REF][START_REF] Sharma | Changes in honeybee behaviour and biology under the influence of cellphone radiations[END_REF]. Balmori's 2009 research on the effect of electromagnetic pollution from cell phone antennas on wildlife shows that microwave and radio frequency pollution has the potential to reduce animal and plant populations near cell phone towers. It becomes [START_REF] Balmori | Electromagnetic pollution from phone masts. Effects on wildlife[END_REF]. Scientists have measured and shown the signal transmitted by the bee in the standby and active modes of the mobile phone. Figure 2 shows that the mobile phone is in standby mode during pointer 1. The bees rested for approximately 10 minutes and produced sounds between 400 Hz and 500 Hz at standard frequencies. When indicator two is set to Active mode, the bees start to produce sounds in high frequencies. The frequencies reached the maximum frequency of 1520 Hz after 1150 seconds (19.16 minutes). At marker 3, the contact was interrupted, but the bees returned to a relaxed state for about 3.3 min after the contact was interrupted. At marker 4, the frequency transmitted by the bees begins to decrease. At marker 5, the bees were completely back to their resting state, and their frequencies ranged between 400 Hz and 500 Hz. The studies conducted by Sainudeen Sahib.S in 2011 on the effect of electromagnetic radiation (EMR) on bees show, to investigate this issue, six colonies of honey bees, three colonies as experimental colonies (T1, T2, T3), and the rest of the control sample. (C1, C2, C3) is selected [START_REF] Sahib | Impact of mobile phone on the density of Honey Bees[END_REF]. Test colonies were exposed to mobile phones operating at 900 MHz for 10 minutes over ten days. The control colonies were not in contact with the cell phone. Research shows that worker bees in test colonies never returned to the hive after ten days. The results of this study are presented in Table 1.

Figure 1. A view of the opened abdomen of a bee with cells with magnetic properties

3-Fluorescent blue light and its effects on honey bees:

Like human eyes, honeybees have three types of light receptors in their compound eyes. Both share blue and green light receptors, but the third type is red for humans and ultraviolet (UV) for honey bees.

Figure 3.

Comparing the spectral range of bee and human vision [START_REF] Peitsch | The spectral input systems of hymenopteran insects and their receptor-based colour vision[END_REF][START_REF] Avarguès-Weber | New vistas on honey bee vision[END_REF] Based on the research of Dr. Oksana Ostroverkhova, a professor in the Department of Physics at Oregon State University, blue fluorescent light is particularly attractive to honeybees. They also found that honey bees prefer fluorescent blue light in the 430-480 nm range compared to 400-420 nm [START_REF] Ostroverkhova | Understanding innate preferences of wild bee species: responses to wavelength-dependent selective excitation of blue and green photoreceptor types[END_REF]. This finding is considered a powerful tool for beekeepers and farmers. For example, if farmers need to pollinate their crops, they can use fluorescent blue light to attract bees for pollination; it can also be used as a tool to bring bees back to the hive. The mentioned point can be considered and used in various artificial intelligence-based sciences and industries [START_REF] Mobasseri | Impact of driving style on fuel consumption[END_REF][START_REF] Mobasseri | Traffic noise and it's measurement methods[END_REF][START_REF] Mobasseri | A Comparative Study Between ABS and Disc Brake System Using Finite Element Method[END_REF][START_REF] Mobasseri | Intelligent Circuit Application for Detecting the amount of Air in Automobile Tires[END_REF][START_REF] Mobasseri | Intelligent Circuit Application for Monitoring the Employees Attendance[END_REF].

4-Conclusion:

The increase in telecommunication towers and mobile phones is a serious threat to reducing honey bees' population. The findings show that the drastic reduction in the honey bee population is due to environmental electromagnetic radiation (EMR), which is a warning for living organisms. The electromagnetic radiation produced by mobile phone towers disrupts the navigation system of honey bees, so they cannot find their way back to the hive. Our proposal to control the bee colony population is to use fluorescent blue light in the bee colony location.

Figure 2 .

 2 Figure 2. The signal transmitted by the bee in Standby and Active mode of the mobile phone[START_REF] Halabi | The effect of cell phone radiations on the life cycle of honeybees[END_REF]

Table 1 .

 1 Changes in the state of the bee colony exposed to mobile phones[START_REF] Chaleshtari | Pollution is caused by electromagnetic fields and their effect on birds[END_REF][START_REF] Sahib | Electromagnetic radiation (EMR) clashes with honey bees[END_REF]

		Parameter	Control group (mean ±	Experimental group for
			standard deviation)	10 days
			40.7	38.2
		Before calling	±	±
			15	12
	Number of worker bees that left the hive/min	During the call	41.5 ±	18.5 ±
			14	13
			42.5
		After the call	±	………….
			14

Funding information:

The authors received no financial support for this research, authorship, and/or publication. Declaration of interests: The authors declares that there is no conflict of interest in this work. Ethical approval: This article does not contain any studies with human participants or animals performed by the authors.


Nicolas Moesa

J Tinsley Oden

Kumar Vemagantia

Jean-Fran�ois Remacleb

Simplified methods and a posteriori error estimation for the homogenization of representative volume elements (RVE)

Keywords:

Homogenization techniques usually rely on solving a boundary value problem on the representative element volume (RVE). This problem is generally complex to solve when the micro-structure is realistic, especially in three dimensions. In this paper, we develop two simplified methods providing approximate micro-fields over the RVE. These fields yield upper and lower bounds to the exact homogenized property.

The a posteriori estimation of the modeling error introduced by the simplified methods is thus straightforward. Both simplified methods are based on a two-scale strategy. The RVE is decomposed into subdomains over which the solution is sought as a smooth part (meso-scale) plus a correction (micro-scale). The correction is expressed in terms of smooth part through a prolongation operator. This operation is performed independently on each subdomain and is thus readily parallelizable. Then, the smooth part of the solution is obtained by solving a 'meso' problem involving all the subdomains. In the numerical experiments, we consider 2-D linear scalar diffusion problems with periodic boundary conditions on the RVE. The RVE is made of a two-phase material consisting of a matrix in which circular or elliptical inclusions are distributed randomly. Numerical examples are computed in a parallel computation done on a cluster of 16 Intel P.C.

Introduction

We are interested in the linear multi-scale problems of the type suggested in the sketch in Fig. 1. The structure (macro-scale) is assumed to be obtained by the periodic repetition of the representative volume element (meso-scale). The RYE is considered to be complex so that it may decomposed into micro-scale entities. Roughly speaking, the meso-scale involves the inclusions interactions whereas the micro-scale involves the inclusion-matrix interactions.

The size of the RYE is assumed to be small compared to the macro-scale so that the 'asymptotic' homogenization theory applies [l]. The homogenized properties are obtained by solving a periodic boundary value problems on the RYE. This problem is difficult when the micro-structure is complex so that simplified techniques have been proposed in the literature [START_REF] Suquet | A simplified method for prediction of homogenized elastic properties of composites with a periodic structure[END_REF][START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF][START_REF] Ghosh | Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and voronoi cell finite element model[END_REF].

In this paper, we introduce a new method in which the solution over the RYE is modeled as the sum of two components: meso-scale and micro-scale components. One of the main concerns in the development of the method is to be able to obtain an a posteriori estimate of the modeling error introduced. In this sense, the present work is related to previous work on hierarchical modeling of heterogeneous bodies by Oden et al. [START_REF] Zohdi | Hierarchical modeling of heterogeneous bodies[END_REF][START_REF] Oden | Analysis and adaptive modeling of highly heterogeneous elastic structures[END_REF][START_REF] Moes | Investigation of the interactions between the numerical and the modeling errors in the homogenized dirichlet projection method[END_REF]. We also note that a similar approach to homogenization techniques for porous media is being developed by Arbogast [8] using mixed finite element techniques. The proposed approach consists of five steps: (I) The RVE is decomposed into subdomains whose sizes are approximately the size of the micro features in the RVE, Fig. 3.

(2) On each subdomain, the solution is expressed as the sum of a meso-scale and a micro-scale component.

(3) Still on each subdomain, the micro-scale part of the solution is expressed in terms of the meso-scale part through a boundary value problem (homogenization step). ( 4) The subdomains are assembled to define a meso-scale problem to be solved on the RVE.

(5) The meso-scale being known, one recovers the micro-part of the solution on each subdomain using the results of the third step above (localization step). By properly choosing the type of boundary conditions used in step 3 and the way the subdomains are assembled to create the meso-problem in step 4, it is possible to obtain lower and upper bounds to the exact homogenized properties of the RVE. The modeling error is thus easy to estimate.

Concerning the practical implementation of the methods, parallel computing resources are used, especially for steps 3 and 5. Also, for flexibility, finite elements are used in step 3.

The plan of the paper is as follows. In Section 2, the problem to be solved on the RVE is detailed, first in the strong form and then in two equivalent variational forms: primal and hybrid. Section 3 introduces the two simplified methods to model the reference problem and the bounds on the homogenized property are derived. Then, Section 4 is devoted to the practical case of the use of the finite-element method to solve the boundary value problems appearing in the simplified strategies. Numerical experiments are carried out in Section 5 and the major points of the paper are summarized in the conclusion.

Description of the problem

Scalars will be italic and vectors bold. For instance, the temperature will be denoted by u and the heat flux and the temperature gradient will be denoted by u and E, respectively. More complex operators will be uppercase (e.g. E for the conductivity tensor). We consider a cell n = (-1, l)n, n =I, 2, 3, occupied by a heterogeneous linearly-conductive material, Fig. 2. We assume that n contains no holes or cracks. We denote by an the boundary of n and n the outward unit normal along an. We define an# as the set of pairs of point facing each other on the boundary of fJ: a!J#={(x, x'):x, x'Eu!J, x#x', (x-x')An n outward normal at x}

The average over n of a field a defined on n will be denoted by (a).

Assuming periodic boundary conditions, the homogenization problem to be solved on il is to find a triple (u, E, u) such that the temperature constraints, the conservation law and the constitutive law hold: @ @ @ $ @ @ @ @ @ @ @ @ i--n @ ® e @ e @ $ @ � @ � ® @@ ® $ ® C2Si!5:> Fig. V (x,x')EiJ!J #

(1)

u(x)•n = u(x')-n' V (x,x')EiJ!J# u=EE on n
Either E or u is given. In the sequel we shall consider E as given. The conductivity_lensor E is a function of position x i.e. E = E( x) EL "'(!J)nxn and E is symmetric and eUiptic; i.e. for a.e. x E!J, there are constants a0, a 1 >0, such that a0a1 a�a1E(x)a�a 1 a' a for any aEIRn. Problem (1) corresponds to a heat transfer homogenization procedure problem ifu is understood as the heat flux.

REMARK. Let (u, E) be a heat flux, temperature gradient pair satisfying the first four equations of the problem ( 1 ) . This pair satisfies Hill's macrohomogeneity equality (see [9)):

(2)

This equality expresses the equality of virtual work between the microscopic scale and the macroscopic scale. The remainder of this Section is devoted to the description of two variational formulations associated to the strong form ( 1 ). The first is the classical primal formulation which corresponds to the minimization of the potential energy. The second is the primal hybrid formulation which involves a saddle-point problem.

Primal formulation

Using the convenient notation, u = E • x, and eliminating E and u in ( 1), we obtain a formulation involving only the temperature field:

di v(EV(u+u) )=O } u(x)::::u(x') V(x,x')E ail# ((EV(u +u))-n') Ix +((EV(u +u))•n')lx' =0 V (x, x')EiJil# A weak fonnulation of problem (3) is to find uE r# such that B(u +u,v)=O VvE"fl' # where B(u, v)= L EVu•Vv dx
Owing to the assumed properties of E, B( •, • ) is symmetric coercive and induces the energy norm on '11':

llvllEW) ={B(v, v) }1 12
(3)

(4) (5)
Problem ( 4) possesses a unique solution for given u and, as is well known, this solution also minimizes the total potential energy J(v) =tB(v +u, v +u)

J(u)= inf J(v) ver,,. (6)
REMARK. The minimization problem (6) defines the solution u as a function of the data E: u = u('E). There exists a linear symmetric and elliptic operator EHoM such that

J(u(i)) 1 H O M - E E•E,
and the homogenized constitutive relation associated to the RYE is u=EH O M E (see [l]).

Primal hybrid formulation

This type of fonnulation has been introduced in [START_REF] Jones | A generalization of the direct-stiffness method of structural analysis[END_REF][START_REF] Pian | The basis of finite element methods for solid continua[END_REF] and mathematically studied in [START_REF] Raviart | Primal hybrid finite element methods for 2nd order elliptic equations[END_REF][START_REF] Babuska | Mixed-hybrid finite element approximations of second-order elliptic boundary value problems[END_REF]. We need a partition [!/' of the domain il into N subdomains ilk, 1 �k�N:

k#l ( 8
)
For any ilk C[J, we denote by ank the boundary and nk the unit outward nonnal along ilk. iJf;, is assumed to be Lipschitz. Each ilk is assumed to be the image of a master element fJ under an affine invertible map Fk, 1 �k�N. We need three spaces: the space °If' of non-confonning temperatures

(9)
where w* is the restriction of w to Ilk, the space .Ji of continuous heat fluxes across the boundary of the subdomains and the subspace .Ji# C.Ji of anti-periodic fluxes on ail .Ji# ={µE.Ji : µ(x)+ µ(x')=O V (x, x')E af;#}

The primal hybrid fonnulation is to find a pair (w, A)E "WX.Ji# such that

{A(w+u,v)+C(v, A)=O VvE"W C(w, µ)=0 V µE.Ji# (10) (11)
where

A(w, v) = ± f EVw •Vv dx, k =I Jnk C(v, µ)= �1 in* µv ds (12)
Here 'fan* µv ds' denotes the duality pairing on H-112 (iJ{}k) and H 112 (iJ{}k). Using the results in [12], one can show that

• problem [START_REF] Pian | The basis of finite element methods for solid continua[END_REF] has a unique solution (w, A); the solution w belongs to "If'# and is equal to the solution u of (4);

• problem ( 11) is equivalent to the saddle point problem

L(w, A.)= inf. sup L(v, µ)= sup inf L(v, µ) uE '# µ, E .!ll !I' µ, E .Al;;; uE 'W where L(v, µ)=J(v) + C (v, µ)
• the solution (w, A.) to [START_REF] Pian | The basis of finite element methods for solid continua[END_REF] and the solution u to ( 4) satisfy the equality

L(w, A)=J(u) (13)
(14) (15)
It is clear that if w is sufficiently smooth, the multipliers A are the boundary fluxes A=(EV(w +u))•n on subdomain boundaries.

Two-scale modeling strategy

The goal is to replace the spaces "If'# and .;f,f,# by the subspaces f;}# C "If'# and ;{.{,# C .;f,f,# so that the approximate solutions ii and (w, A) defined by

J(ii)= inf J(v) (16
)

u E Y# L(w, A)= inf sup L(v, µ) uE 'W µ, E .it # (17)
yield upper and lower bounds to the exact energy

L(w, A)�L(w, A)=J(u)�J(ii) ( 18
)
and thus upper and lower bounds to the homogenized properties owing to [START_REF] Moes | Investigation of the interactions between the numerical and the modeling errors in the homogenized dirichlet projection method[END_REF].

The construction of the subspaces uses a two-scale representation of the solution: a meso-scale component will take into account the interaction between the subdomains and a micro-scale component will depend on the micro-structure of each subdomain. In the sequel, the superscript 'M' shall denote meso-scale related quantities and the superscript 'm' the related micro-scale quantities.

Upper bound

We particularize the partition [!f> by assuming that it is a triangulation of the set {} and that each subdomain {}k is a polyhedron. As indicated earlier, each {}k C {} is the image of fl through an affine invertible mapping We need to build the subspace f;# C Ad# where Ad# is recalled below:

.4t# = {µ E Ad: µ ( x) + µ ( x') = 0 'rt ( x, x') E an #}

Ad = { µ E fi H-112( aflk); 3T E H ( div; fl) s.t. T • nk =µon an*' I �k �N} k =l (27) (28) (29) (30)
This is done using the ideas developed for hybrid finite elements, ( see [START_REF] Raviart | Primal hybrid finite element methods for 2nd order elliptic equations[END_REF][START_REF] Babuska | Mixed-hybrid finite element approximations of second-order elliptic boundary value problems[END_REF]). First, we design a subspace Ad� CH-112( af;|k). We denote Sm(iJ!J) the set of all functions defined over an whose restrictions to any (n -1 )-dimensional face n1 of an are polynomials of degree � m.

We define on each subdomain the homogenized functional

L hom (v M M ) -i • nf L (v M + v m M ) '-' (v k M' " k M ) E OfH' k M X f; k M k k ' µk - k k k ' µ k ' v

Finite element approximation

We now consider the practical case in which the subdomains il k , I � k � N are meshed using finite elements, defining local finite element spaces W� C 'W',, = H 1 (il k ). We assume that the meshes are conforming from subdomain to subdomain and that the periodicity condition may be exactly imposed. More precisely, the local finite element spaces w;, 1 � k � N are assumed to be such that 'Y� = {v E C 0 (il): v k E WZ, v(x) = v(x') 'rJ (x, x' ) E ail,.. }-::/' {O}

The primal reference problem (2.6) now reads

J(u h ) = inf J(v) vE'Yi
whereas the primal hybrid formulation ( 13) is now

L(w h , A h )= inf sup L(v, µ) vEW h J.LEAI.#
where wh is defined by f;«f;¬ h ={w E L 2(il); w k E w:, 1 � k �N}

We have the properties

(46) (47) (48) (49)
• the saddle point problem (48) has a unique solution w h . This solution belongs to 'Y� and is equal to the solution u h of ( 47); the solution A h is not necessarily unique. • the following equality holds: L (w h , A h )= J(u h ).

As before, the goal is to compute approximate solutions (w h , A1) and f; h providing lower and upper bound to the true energy i.e.

Numerical experiments

We consider two 2-D numerical experiments. In both experiments, the domain is decomposed into quadrilateral subdomains over which a bilinear finite element type representation of the meso-scale temperature is used (in both primal and hybrid approach). For the hybrid approach, the micro temperature space is defined by (60) and the fluxes along the edges of the quadrilateral are constant.

RVE with circular inclusion

As a first simple problem, we consider the RVE shown in Fig. 4 containing 7 circular inclusions. The matrix and the inclusions are made of an isotropic heat conductor material whose conductivities are 1 and I 00, respectively. The imposed temperature gradient is E = (1, 0). The computation on the full mesh shown Fig. 5 gives the reference solution, Fig. 6. The reference homogenized heat conductivity in the x direction is K= 1.4399.

The RVE is decomposed into 9 subdomains along the lines shown Fig. 4. The computation needed for the conforming and hybrid approach is split among 9 processors of a cluster of Intel machines running Linux. The conforming and hybrid approaches give the solutions shown in Figs. 7 and8, respectively. The bounds obtained on the homogenized conductivity are K _ = 1.3903 ::;:; K = 1.4399 ::;:; K + = 1.4787 (61)

whereas the Reuss and Voigt bounds are 1.09 and 18.92! The error on the average, (K)= (K+ + K_ /2) 1.4345, is bounded by £K = (K+ -K_)/(K+ + K_) = 3.08%. In Figs. 9 and 10, we observe, as expected, that the 'conforming' temperature field is continuous whereas the 'hybrid' temperature field is not. On the other hand, the hybrid approach yields continuous heat fluxes but the conforming approach does not.

Concerning the quality of the micro-field, the energy error in the difference of the conforming and hybrid temperature field is The h index is omitted for the sake of simplicity of notation. The quantity €rn /2 12.02%, gives the exact error on the average in the two solutions:

(63)

The proof may be found in [START_REF] Moes | A two-scale strategy and a posteriori error estimation for modeling heterogeneous structures[END_REF] and uses the Prager-Synge hypercircle theorem [START_REF] Prager | Approximation in elasticity based on the concept of functions space[END_REF].

The errors discussed above are modeling type errors. They come from simplified assumptions on the fluxes or temperature on the boundary of each subdomain. Table 1 gives the influence of the quality of the mesh on these modeling error. We observe a strong stability of the modeling errors when the mesh is refined.

Table l

Influence of the number of degrees of freedom in the mesh on the modeling errors (c0 = ((K) -K)I K).

RVE with elliptic inclusions

The second example involves a more complex micro-structure, Fig. 11. The heat conductivity properties of the inclusions and the matrix are the same as in the previous example. The imposed temperature gradient is (64)

whereas the Reuss and Voigt bounds are 1.03 and 3.97. The error on the average is bounded by EK 0.29% and the error on the average of the micro-field is €� = 3.87%.

The Figs. 12 and 13 shows the iso-contour of the confonning temperature solution. Fig. 14 gives the heat fluxes lines (lines tangent to the heat flux at each point). They tend to concentrate in the inclusions. Fig. 15 shows the x component of the flux in one subdomain. The parallel heat flux lines in the inclusion indicates that the flux is constant in the inclusion.

Conclusion

Two simplified methods to analyze linear representative volume elements have been introduced. They are based on a two-scale decomposition of the sought approximate solution and yield upper and lower bounds to the exact homogenized properties. Due to this bound property, the a posteriori error estimation of the modeling error is straightforward. Moreover, the exact energy error associated to the average of the two approximate micro-fields may be computed using the Prager-Synge theorem.

The implementation of the methods uses finite elements for flexibility and parallel resources for speed. The interaction between the modeling error (the main concern in this paper) and the numerical errors should be investigated in future work. In a fi rst example, a strong stability of the modeling error with respect to the numerical error is observed.

Fig. I .

 I Fig. I. The different scales associated to the model problem.

Fig. 3 .

 3 Fig. 3. Domain decomposition.

Fk. 2 : 2 .

 22 The boundary of each ii will be denoted by ail. The approximate temperature field ii is sought as the sum of a smooth conforming temperature field u M which is polynomial over each subdomain and a 'rough' correction field um having a zero value on the boundary of each subdomain. More precisely, on each subdomain nk, we define the spaces "If'� and r: such that r� ={v EH1(flk): v = 0 °F;1, 0 E P P (i})} r: ={v EH1(flk): vk = 0 on a.q} (19) (20) where P (n), p 2:: I, is the space of polynomials of degree � p in the n variables x 1 1 ••• , xn defined over fJ. p -We also define the corresponding global spaces 'V� and r : and the space of approximation 'V# 'V� ={v E 'V#: vk E 'V�, 1 � k �N} n C 0 (fl) r : = {v E "/!' # : vk E r;, I �k�N}M m M E <¥/" M m cyf" m } v + v : v r # • v E r #It is not difficult to show that the inclusion r# c 'V# holds.Let us now detail the practical treatment of (16). Using the decomposition (23), we obtain inf J( V )inf inf J( VM + V m ) vE'Y# vMer � vmer:;JThe second infimum may be split into a sum of independent infima:N where inf Jk(v� + v;) k =l u}i'E'r}i'In the minimization problem (25), v� is considered as an imposed parametric function belonging to 'V�. For each subdomain, the infimum is given by v; = u:(v�) +u; where u;(v�) is defined for all v� E 'V� by the problem u:(v�) E r;: Bk(v� + u;(v�), v:) = 0 'rt v: E r: and u; is defined by the problem We can introduce the local homogenized potential by J�0 m (v�) = J*(v� + u;(v�) +u;) Noting that "/!'� C f1�= 1 'V�, the minimization problem (16) takes the form N Lower bouf;

,

  f1 w�, k =l the saddle-point problem (3.19) finally takes the form L(w,AM)=L hom (w M, A M )= inf sup L hom ( VM, µ M ) v M E'w M J.L M E.Jt ;;! where N L hom ( VM M ) = "' L hom ( VM M ) The problem (43) is well posed since the problem (34) is well posed and the fact that the direct sum condition (35) is imposed. On the contrary, the following saddle-point problem is not necessarily well-posed (45) The construction of the homogenized functional L horn somehow ensures automatically the satisfaction of the LBB condition.

Fig. 4 .

 4 Fig. 4. RVE with circular inclusions: geometry of the problem and domain decomposition.

Fig. 5 .

 5 Fig. 5. The mesh defining the reference solution (4 141 DOF).

Fig. 6 .

 6 Fig. 6. Reference temperature field.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Temperature field obtained by the conforming approach. Fig. 8. Temperature field obtained by the hybrid approach.

Fig. 9 .Fig. 10 .

 910 Fig. 9. x component of the heat flux obtained by the conforming approach.

EFig. 11 .

 11 Fig. 11. RVE with elliptic inclusions: geometry of the problem and domain decomposition.

Fig. 12 .

 12 Fig. 12. Temperature field obtained by the conforming approach.

Fig. 14 .Fig. 13 .Fig. 15 .

 141315 Fig. 14. Heat flux lines obtained by the hybrid approach.

  2. Geometry of the RVE problem.

Acknowledgments N.M., J.T.O. and K.V. gratefully acknowledge the support of this work by the U.S. Offi ce of Naval Research under Grant N00014-95-l-0401 and the National Science Foundation under grant ECS-9422707 and the Intel Corporation under Grant: Intel Grant Letter dated August 14, 1997 to The University of Texas at Austin. They are also grateful to Dr. Todd Arbogast for helpful discussions.

. ;f;{ M = {µ E D . ;f;{ � ; µ Ian,+ µIan/= 0 on flk n n,, v flk, n, s.t. nk nn, :;i: 0}

The saddle-point problem (3.2) now reads L( w, ,\ M ) = inf sup L(v, µ M ) vE'W ,, M e.M � This problem may be proved to be well posed using the results in [START_REF] Raviart | Primal hybrid finite element methods for 2nd order elliptic equations[END_REF][START_REF] Babuska | Mixed-hybrid finite element approximations of second-order elliptic boundary value problems[END_REF].

As an example of such a decomposition, we may take 'W' � = 'V� defined previously equation ( 19) and w; as the orthogonal complement in the H1(lJ k) inner product, all the constant modes being excluded:

(37) Note that for a given choice of 'W'�, the choice of w: satisfying the conditions (36) is not unique. The choice in the example above is particular since orthogonality is not required.

and the inclusion r:, C r: holds. The practical computation of (51) follows the same step as in Section 3.

Lower bound

The lower bound in (50) is given by

where .f;#= At,� CA(,# was defined in (32). Using the results in [START_REF] Raviart | Primal hybrid finite element methods for 2nd order elliptic equations[END_REF][START_REF] Babuska | Mixed-hybrid finite element approximations of second-order elliptic boundary value problems[END_REF], the solution to the saddle-point problem (57) exists and is unique if the following condition is satisfied for all k, 1.:;;; k.:;;; N {µEAt,�;VvE'W'�f µ vds o}

A way to ensure this condition in practice is to require that number of degrees of freedom charactering the temperature field on the boundary ank is strictly larger than the number of degrees of freedom characterizing the flux on ank.

The practical computation of (57) follows the step detailed in Section 3. We work with the spaces 'W' �, h and w:• h on each subdomain that must satisfy the direct sum condition EXAMPLE. We take 'W' �, h f;IJ ' m ,h


Language endangerment

• approximately 6,000 and 7,000 languages in the world

• over 97% of the world's population speak only 4% of the world's languages; and conversely, 96% of the world's languages are spoken by only 3% of the world's population

• at least 50% of the world's languages are losing speakers

• in most world regions, about 90% of the languages may be replaced by dominant languages by the end of the 21st century https://ich.unesco.org/doc/src/00120-EN.pdf

Why is linguistic diversity important?

• each language provides pieces of evidence for understanding patterns in the structure and function of human language and human prehistory

• each language helps to better understand the extent and limits of linguistic diversity

• each language is a unique expression of the human experience of the world, so that the knowledge of any single language may be the key to answering fundamental questions of the future

Initiatives

• since the 1990s the crisis of linguistic diversity has been the subject of growing interest from linguists and the public (e.g. unwritten languages lack of previous attestation and documentation variation and variability (language-internal, across closely related languages) multilingualism and language contact Gains • discovery of previously unreported features and configurations of features

• a better understanding of the extent and limits of linguistic diversity

• a better understanding of patterns in the structure and function of human language

• a better understanding of principles and mechanisms of language change and language history Ersu-Lizu-Duoxu, Shuxi

• Ersu and Xumi: Comparative and Cross-Varietal Documentation of Highly Endangered Languages of South-West China (2013-2017)

• funded by the Endangered Languages Documentation Programme (ELDP)

• audio, video, text, and image files of the project are deposited with the Endangered Languages Archive (elararchive.org)

[ba⁵³] 'tired' vs.

[pa⁵³] 'arrive'

[ɡo³¹] 'cold' vs.

[ko³¹] 'early'

[pa⁵³] 'tired; arrive'; [ko³¹] 'cold; early'

[pa⁵³~ba⁵³] 'tired; arrive'; [ko³¹~ɡo³¹] 'cold; early' Language-internal variation

• distinction between /n/ and /l/ before all vowels

[nje³³] 'year' vs.

[lje³³] 'manure'

[nje³³] 'year; manure'

[nje³³] ~ [lje³³] 'year; manure' Language-internal variation Language-internal variation

• speakers who consistently make the largest overall number of phonological distinctions, also including those that do not exist in Southwestern Mandarin (conservative group)

• speakers who only make phonological distinctions that exist in Southwestern Mandarin (convergence group)

• speakers for whom the reduction of the overall number of phonological distinctions is not yet complete and who employ some distinctions that do not exist in Southwestern Mandarin in ways inappropriate for the more conservative variety of Duoxu (transitional group)

Coarticulation

• the articulatory modification of a given speech sound arising from coproduction or overlap with neighboring sounds in the speech chain (e.g. Recasens 2018)

• fricative-nasal sequences the antagonistic requirements of turbulence generation for the production of fricatives (a tightly closed velum to allow turbulent airflow in the oral tract) and that of nasals (a lowered velum to allow airflow through the nasal cavity)

Fricative-nasal sequences • fricative strengthening: • it has been hypothesized that when the period of oral constriction is very brief and entirely devoiced (as in Shuhi [m ̥ h] and [n ̥ h]), cues for the place of articulation may become weakened. In perceptual terms, that could result in the failure on the part of the hearer to perceive place-of-articulation information in the signal, leaving the velic lowering gesture as the only remaining characteristic gesture of the sound Nasal devoicing

Lizu & Ersu: *F-n > *F-t > t ~ k | *F-n >s • ??: Lizu & Ersu: *FN | N ̥ (m ̥ n ̥ ŋ ̊) ~ h ̃ | x
• *FN | N ̥ (m ̥ n ̥ ŋ ̊) ~ h ̃ | x *FN > N ̥ > h ̃ > x
Within the development of Lizu, we must first posit a period in which original fricative-nasal clusters began to devoice, giving rise to voiceless nasals. Devoicing proceeded until the closure period became too brief to be perceptually effective. The oral closure period then disappeared, leaving the velic lowering gesture as the only remaining gesture, resulting in a voiceless nasal glottal fricative Following the loss of allophonic nasalization of the vowel adjacent to the original nasal segment, the nasalization of the initial (/h ̃/) was likely reinterpreted in Ersu as an unintentional effect and no longer produced

[x] can be considered a realization of /h/; or the non-phonemic sound change [h] > [x] occurred; or that the separate existence of /x/ [x] in the phonological system "absorbed" [h ̃] once it lost its nasalization, causing a merger

Fricative strengthening

• *F-n > *F-t > t (> k); *F-n > s
In Lizu, the original fricative-nasal sequences give rise to a cluster with an epenthetic stop, which further simplifies to a stop. In Kala Lizu, the stop has the same place of articulation as the original nasal, as revealed by cognate forms in Duoxu with the alveolar nasal (that is, *F-n > *F-t > t), as in 'nose': Duoxu /nja²²(ku⁵³)/, Kala Lizu tó(Nbù)

In Ersu, on the other hand, the original fricative-nasal sequences appear to retain the fricative part, strengthened by the addition of an epenthetic vowel, whereas the original nasal is lost, as in (lɛ)sv̩ ́ 'finger', sv̩ (kv̩ ) 'snot'

Coexistence of two pathways of change

• can be explained if we assume a different degree of cohesion among the two elements of the original fricative-nasal sequences

• a tighter cohesion between the fricative and nasal elements of the original sequence (or a closer temporal distance between the articulatory gestures for the fricative and nasal elements of the cluster) may result in coarticulatory overlap of velic and oral gestures, leading to fricative weakening and nasal devoicing (*FN > N ̥ > h ̃ > x)

• an increase in the temporal distance between the articulatory gestures for the fricative and nasal elements of the original sequence may result in a delayed velic lowering and oral closure for the nasal relative to the end of the fricative, leading to an epenthetic stop or a vowel (*F-n > *F-t > t > k or *F-n > s)

Coexistence of two pathways of change

• a tighter cohesion between the two elements of the cluster likely characterizes a fricative-nasal sequence when it is part of one syllable (specifically, occurs in the syllable-initial position)

• a looser cohesion between the two elements of the cluster likely characterizes a fricative-nasal sequence, when its constituent elements are located in adjacent syllables, making them adjacent segments with varying degrees of coarticulatory overlap. The latter situation would be more likely to arise if the fricative were part of a prefix, and the nasal were part of the root: *F-n

  • fricative weakening anticipatory velopharyngeal opening for nasality during the acoustic duration of the fricative diminishes the oral pressure build-up behind the fricative constriction, and attenuates the amplitude of frication anticipatory velopharyngeal opening for nasality and the sharing of a single laryngeal spreading gesture for the fricative and the nasal • fricative strengthening delayed velum lowering typically results in denasalization of the initial portion of the nasal the delayed opening of the velic valve preserves frication, but the air accumulated behind the constriction causes a burst and an intervening stop or a vowel • *FN: variation across closely related languages the degree to which coarticulatory changes are produced and perceived may vary from dialect to dialect, dialects may differ in gestural coordination in consonant sequences in terms of degrees of gestural overlap (Brownan & Goldstein 1991) • fricative weakening: loss of the original fricative in Duoxu: *FN > N (m n ŋ)

Gains

• our data contribute to a better understanding of voiceless nasals and /h ̃/ as a type of sound

• our data shed light on the consecutive stages of nasal devoicing in natural languages, from nasal clusters to voiceless nasals to a voiceless nasal fricative to non-nasal fricatives, hence furthering of our understanding of synchronic and diachronic aspects of the development of fricative-nasal sequences

• the insights gained in regard to Tibeto-Burman languages may well be applicable to other languages and language families in which similar sounds and correspondence patterns are found, helping us to understand and reconstruct historical developments in those languages

Gains

• our data shed light on the language-specific word structure in the common ancestor of Duoxu, Ersu, Lizu

• different pathways of change taken by fricative-nasal sequences within one closely related language cluster provide additional insights into the morphological and syllabic structure of words with contiguous fricative-nasal sequences at particular points in time

• different pathways of change followed by individual languages may have some diagnostic value in relation to subgrouping at higher-level nodes on the Tibeto-Burman family tree


Katia Chirkova Wang

Today on the ELAR blog we are congratulating Wang Dehe (王德和), a native speaker of the Ersu language. Wang is a 2017 ELDP grantee and the winner of the 2018 Linguistic Society of America's (LSA) Excellence in Community Linguistics Award, an award that recognizes outstanding contributions by members of language communities (typically outside the academic sphere of professional linguists) for the benefit of their community's language. Wang has collaborated with ELDP grantee Katia Chirkova (http://www. katiachirkova.info; http://crlao.ehess.fr/index.php?490), who was kind enough to tell us a bit about Wang's story.

Wang's story is inspirational, as he grew from a passionate native speaker and an interested language consultant, to an independent fieldworker and researcher, who puts his heart and soul in the study, preservation, and promotion of his native language.

Wang Dehe (his Chinese name) was born as Njjimar Deho (his name in Ersu) in a small Ersu village in Southwest China. He has always been fascinated by writing, probably because Ersu -an undocumented and endangered Tibeto-Burman language -is unwritten. It has become a life-long dream of Wang to find a way for his native language to acquire a written form. After a career of teaching physics in secondary school, Wang started studying the traditional pictographic writing system used by Ersu ritual priests. Working with those priests he soon became an expert on the Ersu pictographic script. As he started attending conferences and publishing articles in linguistic journals, Wang came in contact with linguists working on the documentation and description of lesserknown Tibeto-Burman languages of the Chinese Southwest. This sparked his interest in linguistic fieldwork and language documentation. When I met Wang Dehe in 2010, I was working on the Lizu language, which is closely related to Ersu. I became interested in learning more about Ersu and started working with Wang as a language consultant. I soon discovered that, even though Wang is not professionally trained in linguistics, he has a remarkable insight in his native language, and great dedication to its study and promotion. As a team we worked on a phonetic/phonological sketch of his native Ersu variety, and on that basis we created a Romanization system for Ersu. Once his dream of having a writing system for Ersu had become a reality, Wang enthusiastically started training members of the Ersu community in the use of our newly created Romanization system. In the past four years, he has organized no fewer than twelve self-sponsored workshops in the use of the Ersu Romanization system for Ersu community members in his native Ganluo County, Sichuan Province, China. He is now planning even more workshops, which are free of charge for all participants! Wang Dehe and I also collaborated in a four-year ELDP Major Documentation Project "Ersu and Xumi: Comparative and Cross-Varietal Documentation of Highly Endangered Languages of South-West China" (2013-2017). Wang did an excellent job working on Ersu, by compiling an impressive corpus of 177 traditional Ersu stories and song lyrics (transcribed in the Ersu Romanization system). He also participated in the documentation of other languages in the project. This includes (a) documentation work on the moribund Duoxu language of Mianning County, spoken by no more than a handful of elderly speakers, (b) the first-ever large-scale sociolinguistic surveys of the Ersu and Xumi languages, and (c) the first-ever survey of the dialects of the Lizu language. Upon the completion of the Ersu and Xumi ELDP documentation project, Wang successfully applied for an independent ELDP project, which starts in December of this year and aims to produce a comprehensive illustrated dictionary of Ersu with audio files.

In addition to his dedicated work on language documentation, Wang Dehe has a genuine talent to spark interest among people in his work, and a contagious enthusiasm for the cause of language preservation and promotion. Through regular Chinese-language postings on his blog (http://wdh51818.blog.163.com), he raises awareness of language endangerment and loss, and successfully promotes his linguistic documentation work, thereby increasing awareness and enhancing the visibility and prestige of Ersu in China and abroad.

In summary, Wang Dehe is a native Ersu linguist, ethnographer, and historian, who makes important contributions to the documentation, preservation, and analysis of Ersu. He has exceptional insight into Ersu and great enthusiasm for the language and culture of his native Ersu community.

We congratulate Wang Dehe and wish him many successful projects in the years to come!