A Ellabib
email: ellabib@fstg-marrakech.ac.ma

A Nachaoui
email: nachaoui@math.univ-nantes.fr

An iterative approach to the solution of an inverse problem in linear elasticity

Keywords: Boundary elements, Inverse Problem, Elasticity equations, LU decomposition, Iterative methods

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An iterative approach to the solution of an inverse problem in linear elasticity

Abdellatif Ellabib, Abdeljalil Nachaoui

Introduction

A vast body of engineering experience shows that the theory of linear elasticity allows an accurate modeling of many natural or manufactured solid materials (civil engineering structures, transportation vehicles, machines, the Earth's mantle, rocks mechanics [START_REF] Loret | An effective stress elastic-plastic model for unsaturated porous media[END_REF]) and provides an essential tool for analysis and design. When the governing system of partial differential equations, i.e. the equilibrium, constitutive and kinematics equations, have to be solved with the appropriate initial and boundary conditions for the displacement and/or traction vectors, i.e. Dirichlet, Neumann or mixed boundary conditions the associated problems are called direct problems and their existence and uniqueness have been well established. When one or more of the conditions for solving the direct problem are partially or entirely

D =      ∂ ∂x 0 ∂ ∂y 0 ∂ ∂y ∂ ∂x      , E =          ∂ ∂x 0 0 ∂ ∂y ∂ 2∂y ∂ 2∂x          , C =        1 -ν ν 0 ν 1 -ν 0 0 0 1 -2ν        (1)
Then the strain vector ε = (ε 11 , ε 22 , ε 12 ) T is given by

ε = Ew (2)
The strain tensor ε is related to the stress vector σ = (σ 11 , σ 22 , σ 12 ) T by the constitutive law

σ = 2G 1 -2ν Cε (3)
where G and ν are respectively the Shear modulus and Poisson ratio.

The equilibrium equations are given by

Dσ = b (4)
If we now substitute the constitutive law (3) into the equilibrium equation ( 4), and use the kinematic relations (2) of the elasticity tensor for an isotropic linear elastic material, we obtain the following Lamé system or the Navier equations

           G∆u + G 1 -2ν ∂ 2 u ∂x 2 + ∂ 2 v ∂x∂y = b 1 in Ω G∆v + G 1 -2ν ∂ 2 u ∂x∂y + ∂ 2 v ∂y 2 = b 2 in Ω (5)
The solution of Eqs. [START_REF] Huang | An inverse problem in estimating interfacial cracks in bimaterials by boundary element technique[END_REF] must satisfy prescribed boundary conditions on the boundary Γ of the body, which are based either on the displacements u and v, or the boundary traction t and s. The boundary conditions can be written into the following types u(X) = ũ(X), v(X) = ṽ(X) for X ∈ Γ 1 [START_REF] Jourhmane | A relaxation algorithm for solving a Cauchy problem[END_REF] and t(X) = t(X), s(X) = s(X) for X ∈ Γ 2 [START_REF] Jourhmane | Convergence of an alternating method to solve Cauchy problem for Poisson's equation[END_REF] where (t(X), s(X)) is the traction vector at a point X ∈ Γ 2 with ũ, ṽ, t and s prescribed quantities.

Reconstruction inverse problem

The knowledge of the geometry (the domain Ω) of the problem, the material constants G and ν and the prescribed quantities ũ, ṽ, t and s enable us to determine the displacement vector w(x) and the strain and the stress tensors in the domain Ω. In this case the problem is called direct problem. Different inverse problems can be considered for this direct problem. In all cases, part of the data which is known for the well posed direct problem is not known. In order to find this unknown data, supplementary information have to be provided.

In this work, we are interested by a reconstruction inverse problem where the geometry of the problem and the material constants are determined, but the boundary conditions are not completely known. This problem arises in cases where a portion of the boundary is exposed to environmental conditions which can not be assessed due to physical difficulties or geometrical inaccessibility. The aim in the reconstruction inverse problem is to find the unknown boundary conditions based on the supplementary data provided on the boundary and/or the domain.

Consider the problem where no conditions are prescribed on Γ 2 and assume that it is possible to measure the traction vector on Γ 1 . This gives arise to the supplementary boundary conditions

t(X) = t(X) and s(X) = s(X) for X ∈ Γ 1 ( 8
)
where t and s are given functions.

In the next section, we describe an iterative method to solve numerically the reconstruction problem ( 5), ( 6) and [START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF], which is ill-posed and cannot be solved efficiently by a direct approach.

Description of the alternating algorithm

An alternating algorithm for solving Cauchy problems for elliptic equations was introduced by Kozlov et al. [START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF]. This algorithm was the subject of several studies which addressed various numerical and theoretical aspects (see for example [START_REF] Baumeister | Iterative methods for ill-posed problems modeled by partial differential equations, Journal of Inverse and ill-posed problems[END_REF][START_REF] Jourhmane | A relaxation algorithm for solving a Cauchy problem[END_REF][START_REF] Jourhmane | Convergence of an alternating method to solve Cauchy problem for Poisson's equation[END_REF][START_REF] Nachaoui | Numerical linear algebra for reconstruction inverse problems[END_REF]). We extend here this procedure to the reconstruction problem described above. The iterative algorithm investigated is based on reducing this ill-posed problem to a sequence of mixed well-posed boundary value problems and consists of the following steps.

Giving ω 0 and z 0 , initial approximation of the solution on Γ 2 , we construct a sequence of approximation u k , v k by solving alternately the following mixed well-posed direct problems until a prescribed stopping criterion is satisfied.

u 2k and v 2k are obtained as the solution of

                 G∆u 2k + G 1 -2ν ∂ 2 u 2k ∂x 2 + ∂ 2 v 2k ∂x∂y = b 1 in Ω G∆v 2k + G 1 -2ν ∂ 2 u 2k ∂x∂y + ∂ 2 v 2k ∂y 2 = b 2 in Ω t 2k = t, s 2k = s on Γ 1 and u 2k = ω k , v 2k = z k on Γ 2 (9)
Having constructed u 2k and v 2k we can obtain u 2k+1 and v 2k+1 by solving the problem

                 G∆u 2k+1 + G 1 -2ν ∂ 2 u 2k+1 ∂x 2 + ∂ 2 v 2k+1 ∂x∂y = b 1 in Ω G∆v 2k+1 + G 1 -2ν ∂ 2 u 2k+1 ∂x∂y + ∂ 2 v 2k+1 ∂y 2 = b 2 in Ω u 2k+1 = ũ, v 2k+1 = ṽ on Γ 1 and t 2k+1 = t 2k , s 2k+1 = s 2k on Γ 2 (10)
The sequence ω k and z k are constructed as follows

ω k = F 1 (ω k-1 ) and z k = F 2 (z k-1 ) (11)
where F 1 and F 2 are two relaxation operators that will be determined in order to ensure and possibly accelerate the convergence of the iterative procedure. Note that the similar Kozlov-Maz'ya-Fomin's schemes [START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF] for elasticity problem is obtained by taking

F 1 (ω k-1 ) = u 2k-1 and F 2 (z k-1 ) = v 2k-1 .

Selection criteria for the relaxation factor based on convex combination

To solve Cauchy Problems for Poisson equation Nachaoui et al. [START_REF] Jourhmane | Convergence of an alternating method to solve Cauchy problem for Poisson's equation[END_REF] established a relaxation algorithm by the use of a convex combination of the successive solutions on Γ 2 which produces a convergent and stable numerical solution.

We extend this idea to the reconstructing algorithm (9), [START_REF] Nachaoui | Iterative solution of the drift-diffusion equations[END_REF]. This can be done by defining F 1 and F 2 in (11) as follows:

F 1 (ω k-1 ) = θ 1 u 2k-1 | Γ 2 + (1 -θ 1 )ω k-1 and F 2 (z k-1 ) = θ 2 v 2k-1 | Γ 2 + (1 -θ 2 )z k-1 (12)
where θ 1 and θ 2 are two parameters that will be determined in order to ensure and possibly accelerate the convergence of the iterative scheme. Note that the equivalent of Kozlov-Maz'ya-Fomin's schemes [START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF] for elasticity problem is obtained by taking θ 1 = 1 and θ 2 = 1.

As in [START_REF] Jourhmane | Convergence of an alternating method to solve Cauchy problem for Poisson's equation[END_REF] the numerical tests performed revealed that the algorithm with constants relaxation factors θ 1 end θ 2 is convergent but there are large variation in the rate of convergence. Therefore we developed selection criteria for the relaxation factors.

Consider that constant relaxation factors are applied in the relaxation algorithm associated to [START_REF] Sonneveld | a fast Lanczos-type solver for non-symmetric linear systems[END_REF] and let ω k defined by [START_REF] Nachaoui | Numerical linear algebra for reconstruction inverse problems[END_REF]. Note that a good indicator of the level of accuracy achieved is given by the functions

ϕ 1 (θ 1 ) = w k -w k-1 L 2 (Γ 2 ) and ϕ 2 (θ 2 ) = z k -z k-1 L 2 (Γ 2 )
, since ϕ 1 and ϕ 2 tend to zero as the convergence of the iterative algorithm is achieved. Therefore the relaxation factors are selected such that the functions ϕ 1 and ϕ 2 are minimized. For this we require that ∂ϕ 1 ∂θ 1 = 0 and ∂ϕ 2 ∂θ 2 = 0 which yield

θ k+1 1 = e 2k 1 ,
∀ k ≥ 1, (13)
where

e 2k 1 = u 2k | Γ 2 -u 2k-2 | Γ 2 , e 2k+1 1 = u 2k+1 | Γ 2 -u 2k-1 | Γ 2 , e 2k 2 = v 2k | Γ 2 -v 2k-2 | Γ 2 , e 2k+1 2 = v 2k+1 | Γ 2 -v 2k-1 | Γ 2
, and •, • denotes the inner product in L 2 (Γ 2 ).

Note that the automatic selection of the relaxation factors given in (13) requires 2 inner products at each iteration which are equivalent in discreet form to a number of operations of order O(N) and this is negligible compared to the number of operations needed to solve the two direct problem ( 9) and (10).

Selection criteria for the relaxation factor based on fixed point operator

In this section we develop a second relaxation scheme based on some least-residual strategy. Let F 1 and F 2 be the mappings from L 2 (Γ 2 ) to L 2 (Γ 2 ) defined by solving successively the problems (9), [START_REF] Nachaoui | Iterative solution of the drift-diffusion equations[END_REF] and taking

F 1 (ω k ) = u 2k+1 | Γ 2 , F 2 (z k ) = v 2k+1 | Γ 2 .
Let L 1 and L 2 be two linear operator from L 2 (Γ 2 ) to L 2 (Γ 2 ) defined as follows. For any w 1 , w 2 ∈ L 2 (Γ 2 ), L 1 w 1 and L 2 w 2 are the solutions respectively of the two well posed linear problems ( 9) and [START_REF] Nachaoui | Iterative solution of the drift-diffusion equations[END_REF] where w 1 and w 2 play respectively the role of ω k and z k but with the volume force equal to zero and homogeneous boundary conditions on Γ 1 . Let w n and w d computed by solving respectively the two well posed linear problems ( 9) and ( 10) with homogeneous boundary conditions on Γ 2 . This implies that F 1 and F 2 can be written as:

F 1 (ω) = L 1 ω + w n and F 2 (z) = L 2 z + w d . ( 14
)
Then the marching condition for the displacements on Γ 2 in ( 9) can be relaxed as follows

ω k+1 = ω k + θ 1 (L 1 ω k + w n -ω k ) and z k+1 = z k + θ 2 (L 2 z k + w d -z k ). ( 15
)
Let us define the following vectors

r k 1 = L 1 ω k + w n -ω k , r k 2 = L 2 z k + w d -z k , ( 16
) ω k (θ 1 ) = ω k + θ 1 r k 1 , z k (θ 2 ) = z k + θ 2 r k 2 , (17) r k 1 (θ 1 ) = L 1 ω k (θ 1 ) + w n -ω k (θ 1 ), r k 2 (θ 2 ) = L 2 z k (θ 2 ) + w d -z k (θ 2 ). (18)
Note that here a good indicator of the level of accuracy achieved is given by the functions

φ 1 (θ 1 ) = r k 1 (θ 1 ) L 2 (Γ 2 ) and φ 2 (θ 2 ) = r k 2 (θ 2 ) L 2 (Γ 2 ) (19)
since φ 1 and φ 2 tend to zero as the convergence of the iterative algorithm is achieved. Therefore the relaxation factors are selected such that the functions φ 1 and φ 2 are minimized.

From ( 16), ( 17) and ( 18) we obtain

r k 1 (θ 1 ) = r k 1 + θ 1 (L 1 r k 1 -r k 1 ) and r k 2 (θ 2 ) = r k 2 + θ 2 (L 2 r k 2 -r k 2 ). ( 20
)
Then φ 1 and φ 2 can be written as follows

φ 1 (θ 1 ) = r k 1 2 L 2 (Γ 2 ) + 2θ 1 r k 1 , L 1 r k 1 -r k 1 + θ 2 1 L 1 r k 1 -r k 1 2 L 2 (Γ 2 ) (21) φ 1 (θ 1 ) = r k 2 2 L 2 (Γ 2 ) + 2θ 2 r k 2 , L 1 r k 2 -r k 2 + θ 2 2 L 2 r k 2 -r k 2 2 L 2 (Γ 2 ) (22)
For the functions φ 1 and φ 2 to be minimized we require that ∂φ 1 ∂θ 1 = 0 and ∂φ 2 ∂θ 2 = 0 which yield

θ k 1 = r k 1 , r k 1 -L 1 r k 1 L 1 r k 1 -r k 1 2 L 2 (Γ 2 )
and

θ k 2 = r k 2 , r k 2 -L 2 r k 2 L 2 r k 2 -r k 2 2 L 2 (Γ 2 ) . (23)
Thus the automatic adjustment of ω k+1 and z k+1 is obtained as follows

ω k+1 = ω k + θ k 1 r k 1 and z k+1 = z k + θ k 2 r k 2 , (24)
where θ k 1 and θ k 2 are given by (23).

Note that this scheme requires the solution of the two well posed problems ( 9) and ( 10) and the computation of L 1 r k 1 and L 2 r k 2 which are equivalent in the discreet form to matrix-vector product.

The boundary element method is a very apt tool to solve the auxiliary problems ( 9) and ( 10), since the boundary conditions are the main unknown of the problem and the statement of these problems in term of boundary integral equation reduces the modeling effort to a minimum. Moreover, the BEM determines simultaneously the boundary displacement u, v and its traction t, s, this allows us to solve problem [START_REF] Nachaoui | Iterative solution of the drift-diffusion equations[END_REF] without the need or further finite difference, as one would employ if using the finite element or the finite difference method.

We describe the boundary element method in the next section.

4 Integral equation formulation and boundary element

Integral equation formulation

The linear elasticity problem [START_REF] Huang | An inverse problem in estimating interfacial cracks in bimaterials by boundary element technique[END_REF] in two-dimensional case can be formulated in integral form [START_REF] Brebbia | Boundary Elements An Introductory course[END_REF] as follows

Γ U ij (P, Q){T } j (Q) dΓ - Γ T ij (P, Q){U} j (Q) dΓ + Ω U ij (P, Q)b j (Q)dΩ =      {U} i (P ) if P ∈ Ω 1 2 {U} i (P ) if P ∈ Γ ( 25
)
for i, j = 1, 2, where U ij and T ij denote the fundamental displacements and traction for the twodimensional isotropic linear elasticity [START_REF] Brebbia | Boundary Elements An Introductory course[END_REF] and they are given by

U ij (P, Q) = 1 8πG(1 -ν) δ ij (4ν -3) ln r + r ,i r ,j r 2 T ij (P, Q) = 1 4π(1 -ν)r ∂r ∂n (2ν -1)δ ij - 2r ,i r ,j r 2 + (2ν -1) n i r ,j -n j r ,i r ( 26
)
where r is the distance between the source point P and field point Q, n = (n 1 , n 2 ) denotes the outer normal vector to Γ, r ,i = Q i -P i .

Boundary element method for elasticity equations

The boundary integral equations (25) are solved using boundary element method with constant boundary elements. The boundary is divided into N constant elements. Thus the distribution of the displacements and traction are taken constant on each element and equal to their value at the nodal point, which lies at the midpoint of the element. Denoting by {U} i = {u i , v i } T and {T } i = {t i , s i } T the displacements and traction at the i th node and taking into account that the boundary is smooth at the nodal point of the constant element. Then, the discretized form of Eq. ( 25) can be written as

1 2 {U} i + N j=1 Ĥij {U} j = N j=1 G ij {T } j + F ( 27
)
where G ij and Ĥij are 2 × 2 matrices such that

(G ij ) lm = Γ j U lm (P i , Q) dΓ(Q) and ( Ĥij ) lm = Γ j T lm (P i , Q) dΓ(Q) for l, m = 1, 2. ( 28
)
Eq. ( 27) relates the displacements of the i th node to the displacements and the traction of all the nodes including the i th node. Applying this equation to all the boundary nodal points yields 2N equations, which can be set in matrix form as H U = G T + F where H = Ĥ + 1 2 I and I is 2N × 2N identity matrix. The dimension of matrices Ĥ and G is 2N × 2N, and those the vectors U and T is 2N. They are defined as It should be noted that rearrangement of the unknowns is necessary for mixed boundary conditions. After doing so the following system of 2N linear equations is obtained AX = R + F where A is a square coefficient matrix having dimensions 2N × 2N, R is a vector resulting as the sum of the columns of the matrices G and H multiplied by the respective known boundary values. The main inconvenient with integral formulations is that its time consuming. To make the proposed method more appealing, effort must be devoted to the implementation of algorithm ( 9)-( 10) with ( 12) or (24) and in particular to solving large dense linear systems efficiently. In general, the following factors are considered in choosing an implementation technique :

G = (G ij ) 1≤i,j≤N , Ĥ = ( Ĥij ) 1≤i,j≤N and U = {U} 1 , • • • , {U} N , T = {T } 1 , • • • , {T } N .
1. The accuracy of the calculation, or the quality of the approximations; 2. The computational effort involved, or the efficiency of the method; and 3. The ease-of-implementation

We have tested iterative Krylov methods, method based on LU factorization and the exploitation of the nature of algorithm ( 9)-( 10).

Implementation

The resulting systems of equations, when the boundary element discretization is applied to the reconstructing algorithm ( 9)- [START_REF] Nachaoui | Iterative solution of the drift-diffusion equations[END_REF], will be of the form

A k 1 X 2k = B k 1 (29) A k 2 X 2k+1 = B k 2 ( 30
)
where A k i and B k i , i = 1, 2 are constructed from the discreet form of ( 9) and ( 10) by boundary element method. Note that, from Eq. (28), A k 1 and A k 2 are geometry dependent matrices and depend on the type of the boundary conditions, but not on their values. Therefore A k 1 = A 0 1 and A k 2 = A 0 2 . These matrices can have the factored form :

A 0 1 = L 0 1 R 0 1 , A 0 2 = L 0 2 R 0 2
where L 0 1 , L 0 2 are lower triangular matrices and R 0 1 , R 0 2 are upper triangular matrices. Now from (29) and (30), X 2k and X 2k+1 can be obtained by backward followed by forward substitutions requiring only 4N 2 operation. This gives arise to the following algorithm : The efficiency of the proposed method is illustrated for different orders of system. The efficiency is compared based on the CPU times carried out using the Gaussian elimination method at each iteration.

Due to the properties of matrices A 1 and A 2 (are nonsingular and non-symmetric), one might consider solving AX = B by applying CG to the normal equations A T AX = A T B. This approach is called CGNR [START_REF] Greenbaum | Iterative methods for solving linear system[END_REF].

Alternatively, one could solve AA T Y = B and then set X = A T Y to solve AX = B. This approach is now called CGNE [START_REF] Greenbaum | Iterative methods for solving linear system[END_REF].

There are three disadvantages that may or may not be serious. The first is that the condition number of the coefficient matrix A T A is the square of that of A. The second is that two matrix-vector products are needed for each CG iterate. The third, more important, disadvantage is that one must compute the action of A T on a vector as part of the matrix-vector product involving A T A.

We used the bi-conjugate gradient stabilized (BICGSTAB) [START_REF] Van Der | BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems[END_REF], which was developed to have the same convergence rate as the conjugate gradient squared (CGS) [START_REF] Sonneveld | a fast Lanczos-type solver for non-symmetric linear systems[END_REF] at its best, without having the same difficulties (irregular convergence behavior). An advantage of BI-CGSTAB over other Krylov method such as the generalized minimal residual method (GMRES) [START_REF] Saad | GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems[END_REF] is that it has limited computation and storage requirement in each iteration step. Comparison of these methods can be found in [START_REF] Nachaoui | Iterative solution of the drift-diffusion equations[END_REF][START_REF] Van Der | Modern methods for the iterative solution of large systems of linear equations[END_REF].

An alternative to Algorithm 1 is an implementation of the reconstructing algorithm ( 9) and [START_REF] Nachaoui | Iterative solution of the drift-diffusion equations[END_REF] where all the linear system are solved by an iterative solver for each iteration. The proposed method summarized in the next algorithm : Algorithm 2 1. set k = 0 and choose the initial estimate (ω 0 , z 0 ) and a tolerance for the iterative solver 2. Compute H and G 3. Compute A 0 1 , B 0 1 , A 0 2 and B 0 2 4. Solve systems (29) and (30) using an iterative solver 5.

k = k + 1 6. compute w k , z k , B k 1 7. solve A 0 1 X 2k = B k
1 using an iterative solver 8. compute B k 2 and solve A 0 2 X 2k+1 = B k 2 using an iterative solver 9. repeat steps 5-8 until convergence 10. End do.

The following stopping criterion can be used for iterative solvers of linear systems

B -AX 2 B 2 < α (31)
Note that to compare Algorithm 2 with Algorithm 1, we take the same tolerance α for all k but this is not necessary for the convergence of Algorithm 2. Note also that Algorithm 2 converges when α is not very small.

Numerical results

In order to illustrate the performance of the numerical method described above, we solve the inverse elasticity problem ( 5), ( 6) and ( 8) in two-dimensional annular domain Ω given by

Ω = (x, y) ∈ R 2 , 1 < x 2 + y 2 < 16 .
We assume that the boundary is split into two parts

Γ 1 = (x, y) ∈ R 2 , x 2 + y 2 = 16 and Γ 2 = (x, y) ∈ R 2 , x 2 + y 2 = 1 .
The exact solution of the direct problem is given by

u(x, y) = 1 2G(1 + ν) V (1 -ν)x -W (1 + ν) x x 2 + y 2 v(x, y) = 1 2G(1 + ν) V (1 -ν)y -W (1 + ν) y x 2 + y 2 (32)
where V = -σ 0 -16σ 1 15 and W = 16(σ 0σ 1 ) 15 and stress tensor is given by

σ 11 (x, y) = V + W x 2 -y 2 (x 2 + y 2 ) 2 , σ 22 (x, y) = V -W x 2 -y 2 (x 2 + y 2 ) 2 and σ 12 (x, y) = 2W xy (x 2 + y 2 ) 2 (33)
with σ 1 = 10 10 , σ 0 = 2 × 10 10 , G = 3.35 × 10 10 and ν = 0.34.

The auxiliary problems ( 9) and (10) corresponding to this example are discretized by boundary element method using a piecewise constant polynomial interpolation. The number of boundary elements used for the discretization of the boundary Γ is taken to be N ∈ {80, 160, 320, 640, 1280}. We denote by • 0,Γ 2 the discreet L 2 norm defined on Γ 2 .

The convergence of the algorithm may be investigated by evaluating at every iteration the error

G u k = u 2k -u an 0,Γ 2 , G v k = v 2k -v an 0,Γ 2 (34)
where u 2k and v 2k are the numerical displacements on the boundary Γ 2 obtained after k iterations and u an , v an are the exact displacements of the problem given by (32). In a similar way we may evaluate the errors in retrieving the traction on the boundary Γ 2 given by

G t k = t 2k+1 -t an 0,Γ 2 , G s k = s 2k+1 -s an 0,Γ 2 (35)
The behavior of the method is investigated by evaluation the difference between two consecutive approximations for the displacements solutions and its traction on the boundary Γ 2 given by

E u k = u 2k -u 2k-2 2 0,Γ 2 /N 2 , E v k = v 2k -v 2k-2 2 0,Γ 2 /N 2 (36) E t k = t 2k+1 -t 2k-1 2 0,Γ 2 /N 2 , E s k = s 2k+1 -s 2k-1 2 0,Γ 2 /N 2 (37)
Based on absolute errors the following stopping criterion is considered

max(E u k , E v k ) < η ( 38
)
where η is a small prescribed positive quantity. Note that (38) express that the sequence (u 2k , v 2k ) converge in Sobolev spaces H

1 2 (Γ 2 ) × H 1 2 (Γ 2 )
. For all numerical experiments, we take η = 10 -7 .

This test is used to analyze the behavior with respect to accuracy and efficiency of the techniques considered in this work when applied to the approximation solution of boundary element method systems of algebraic equations.

Comparative result for various parameter relaxation

Table 1 presents results obtained based on Algorithm 1 for various number of boundary elements. We denote by k 1 , k 2 and k 3 respectively the number of iterations required to achieve the convergence using θ 1 = θ 2 = 1, θ 1 , θ 2 computed by ( 13) and θ 1 , θ 2 computed by ( 23) respectively. We denote by CP U 1 , CP U 2 , CP U 3 the CPU time required for the convergence in the three cases.

We observe from Table 1 that the reconstructing algorithm is very efficient when used with the automatic adjustment of θ 1 , θ 2 given by ( 13) or (23). The behavior of numerical solution of the first components of the displacement and traction vectors (u and t) are similar to that of the second components (v and s respectively) and therefore they have not been presented here. k and G t k for automatic selection of θ 1 , θ 2 given by ( 13) or (23) as functions of the number of iterations k for various number of boundary elements N. We can see easily that the quantities G u k and G t k decrease when N increases and they remain constant after a few iteration. Therefore the method are stable with respect to the number of boundary element. According to these results, we observe a weak oscillation of E u k and E t k due to the automatic search of the relaxation parameter at each iteration, see Fig. For N = 320, we observe from Fig. 4 (b), (c), 5 (a), (b) that when the algorithm is implemented with the dynamically estimated relaxation parameter given by ( 13) or (23), the prescribed criterion (36) is satisfied after 6 iterations for automatic selection given by (13), after 5 iterations for automatic selection given by (23), while it is required 15 iterations for θ 1 = θ 2 = 1. With the fixed optimal parameter the convergence is obtained after 13 iterations. We notice that at the convergence, the same precision is reached for the displacements solutions and traction with the different relaxation parameters, see Fig. 5 (c) and Fig. 6. These figures show the numerical solutions and their tractions obtained on the boundary Γ 2 for four relaxation parameters, namely the two dynamically estimated parameter given by ( 13) or (23), the fixed optimal parameter and the fixed parameter θ 1 = θ 2 = 1.

As it can be seen from Fig. 5 (c) -6 the displacement and traction solutions corresponding to 320 of boundary elements is in good agreement with the analytical solution.

Noisy on the tractions data

We examine the algorithm stability when t and s data are noisy. The noisy data on tractions are presented in Fig. 8 (b), (c) and the corresponding numerical solutions are shown in Fig. 9.

Theta/(2PI) u on Gamma_2 For p ≥ 3, the numerical results obtained with perturbed data coincide with the numerical results obtained with no perturbed data. Therefore this case is not presented here.

It can be seen that as p increases, the numerical solution better approximates the exact solution, whilst remaining stable. For p = 1, the obtained results are to be considered more than reasonable, keeping in mind that the problem ( 5), ( 6) and [START_REF] Kozlov | An iterative method for solving the Cauchy problem for elliptic equations[END_REF], is an ill-posed problem with a very oscillatory data.

Comparison of LU decomposition and linear iterative solvers

To compare the efficiency for different iterative solvers, five computational meshes are used to generate five systems of 160, 320, 640, 1280, 2560 linear equations.

The iterative solvers BI-CGSTAB, CGNE, CGNR, CGR and CGS, are considered. as convergence tolerance in (31).

All the Krylov methods worked well, some, BI-CGSTAB and CGS, performed very well, yielding the solution in a small number of iterations. CGNE and CGNR revealed some difficulty in achieving a good convergence. The results show that BI-CGSTAB and CGS are more efficient than LU decomposition procedure, but BI-CGSTAB is by far the fast solver. The normalizing technique corresponding to the CGNE method is penalized due to the worsening of the condition number of their matrix relative to the condition number of the matrix of the original linear system.

Conclusion

In this paper a numerical iterative boundary element method for solving Cauchy problem in linear elasticity equations has been developed. It can be concluded that the iterative approach produces an accurate, convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise. The stability of method was also investigated by perturbing the given data with various levels of noise. Following are the main conclusions drawn from the numerical experiments :

• The accuracy of iterative approach is improved by the use of automatic selection of relaxation parameter.

• The proposed iterative approach could be improved by using Krylov methods as linear solver.

Algorithm 1 1 .A 0 2 and B 0 2 4. Compute L 1 , L 2 , R 1 and R 2 5 .

 12125 set k = 0 and choose the initial estimate (ω 0 , z 0 ) 2. Compute H and G 3. Compute A 0 1 , B 0 1 , Solve systems (29) and (30) replacing A 0 1 and A 0 2 by their factorized forms 6. until convergence do 7. k = k + 1, compute ω k , z k and B k 1 8. replace A k 1 by L 1 R 1 and solve (29) 9. compute B k 2 , replace A k 2 by L 2 R 2 and solve (30) 10. End do.

Fig. 1 .

 1 Fig. 1. E u k (a), E t k (b) and G u k (c) of first dynamical choice of relaxation parameter for different N .

Fig. 2 .

 2 Fig. 2. E u k (a), E t k (b) and G u k (c) of second dynamical choice of relaxation parameter for different N .

Fig. 1 (

 1 Fig.1(a),(b)-2(a),(b) show, on a semi-log scale, the corresponding successive difference E uk and E t k for automatic selection of θ 1 , θ 2 given by (13) or (23) as functions of the number of iterations k.

Fig. 1 (

 1 Fig.1 (c), 2 (c), 3 (a)-(b) show, on a semi-log scale, the corresponding sequences G uk and G t k for automatic selection of θ 1 , θ 2 given by (13) or (23) as functions of the number of iterations k for various number of boundary elements N. We can see easily that the quantities G u k and G t k decrease when N increases and they remain constant after a few iteration. Therefore the method are stable with respect to the number of boundary element.

Fig. 3 .

 3 Fig. 3. G t k of two dynamical choice of relaxation parameter (a), (b) for different N and variation of first dynamical estimated relaxation parameter (c) as a function of number of iteration.

  3(c), 4 (a). This slight instability in E u k and E t k does not affects the behavior of the errors G u k and G t k as it is illustrated in Fig. 1 (c), 2 (c), 3 (a) -(b).

Fig. 4 .

 4 Fig. 4. Variation of second dynamical estimated relaxation parameter (a) as a function of number of iteration and E u k (b), G u k (c) of four relaxation parameter.

Fig. 5 .

 5 Fig. 5. E t k (a), G u k (b) of four relaxation parameter and numerical, exact solution for displacement u (c) with N = 320.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Numerical and exact solutions for displacement v (a) and tractions t (b), s (c) for N = 320

Fig. 8 .

 8 Fig. 8. Numerical traction t (a) results with perturbed displacements data with N = 320 and perturbed tractions t (b), s (c) data on Γ 1 .

Fig. 9 .

 9 Fig. 9. Numerical displacement u (left) traction t (right) results obtained for perturbed tractions data and N = 320.

  The 2N equations within the matrix Eq. (27) contain 4N boundary values, that is 2N values of displacements and 2N values of traction. However, a total of 2N values are known from the boundary conditions. Consequently, Eq. (27) can be used to determine the 2N unknown boundary values.

Table 1

 1 CPU time for fixed parameter, dynamically estimated parameter by (13) and (23) using Algorithm 1.

	N	CP U 1	CP U 2	CP U 3	k 1 k 2 k 3
	80	0.07199003 0.05799199 0.06399098 18 6	6
	160 0.48792499 0.44793200 0.47592701 16 6	6
	320 14.6957663 13.953879 14.0178692 15 6	5
	640 224.173124 122.282414 122.070447 14 6	5
	1280 1386.6820 907.992332 913.368518 13 6	5

Table 2 -

 2 3 summarizes the results obtained for these solvers. The times shown are the average time in CPU times. The average number of iterative solver iterations is given in brackets. We used α = 10 -07

Table 2

 2 CPU time for different solvers (first iteration) for dynamical 1

	N	LU	BI-CGSTAB	CGNE	CGNR	CGR	CGS
	80	0.0579	0.0619(83)	0.0709(99)	0.0689(92) 0.1549(93) 0.0699(97)
	160 0.4479	0.2899(75)	0.3239(101)	0.3239(94) 0.6499(77) 0.3549(98)
	320 13.953	4.4043(83)	13.510(432) 7.7138(243) 13.053(89) 5.0732(93)
	640 122.28	21.042(85)	85.438(611) 32.835(227) 50.697(78) 23.798(97)
	1280 907.99	84.896(90)	1054.1(1964) 388.05(719) 221.24(85) 88.798(94)
	Table 3
	CPU time for different solvers (first iteration) for dynamical 2
	N	LU BI-CGSTAB	CGNE	CGNR	CGR	CGS
	80	0.06 0.12(99;177)	0.15(98;163)	0.12(91;108) 0.28(89;84) 0.11(90;96)
	160 0.47 0.61(79;237)	1.10(93;330)	0.66(93;146) 1.33(76;80) 0.64(86;106)
	320 14.0 7.13(72;83)	19.1(305;297)	10.7(115;230) 18.2(72;58) 7.66(81;71)
	640	122	29.2(68;55)	149(440;636)	50.4(88;267) 84.3(69;58) 36.5(77;75)
	1280 913	129(69;87)	1117(718;1362) 448(120;712) 321(69;58)	137(73;61)

Acknowledgements

This work was carried out in the framework of the project 'Inverse problems and shape optimization' when the first author was a visiting Professor at Laboratoire de Mathématiques Jean Leray, Université de Nantes. The research of the first author was supported in part by 'Action Intégrée' France-Maroc MA/05/116.


The Missing Share

The ritual language of sharing as a 'total social fact' in the eastern Himalayas (Northwest Yunnan, China)

Stéphane Gros

Northwest Yunnan is where the eastern flank of the Himalayas, in a spectacular bend, turns abruptly south. This is the heart of what is called the Hengduan Mountains, where the main rivers of Asia flow southward in parallel gorges separated by high altitude mountain ranges. Here, the administrative boundaries of the Tibetan Autonomous Region, the provinces of Yunnan and Sichuan, and the country of Burma (Myanmar) meet. It is the home of Tibetans, but also of several Tibeto-Burman speaking peoples, such as the Naxi,1 the Lisu, 2 the Nung and the Drung, amongst a few others. 3 From the mid-nineteenth century until the mid-twentieth century, only a few explorers, geographers, botanists and scholars had crossed these mountains. 4 Despite the increasing amount of research done during the last decade, we still know very little of Northwest Yunnan's history and ethnology, especially of its more remote parts, such as the upper reaches of the Salween river (Nujiang) and the easternmost source of the Irrawaddy, where the Drung people live. 5 Historically, the northwestern borders of Yunnan province lay at the juncture of both Chinese and Tibetan expansion. This remote area was coveted by Tibetans, Naxi and Chinese, and has been a theatre of unceasing conflicts. For a long time, several political legitimacies coexisted, with the empowered parties exercising quite freely their rights on this territory. To this configuration was then added the progressive influence of the Qing dynasty, and later on that of the Republicans with their respective colonial policies.

Nevertheless, until the first half of the twentieth century, this patch of land between Tibet, China and Burma was like a free zone, where only local chieftains retained any real authority.

As with some other Tibeto-Burman speaking people still presently living in the area, the Drung, long known under the Chinese name of Qiuzi (Kiutzu), were caught between their more powerful neighbours despite the isolation of their mountainous valley.

While very little was known about these people, their valley soon became a pawn in imperial rivalries. As the British botanist, F. Kingdon Ward once wrote: "It is certainly curious that no less than three empires should have laid claim to this wild valley. China has claimed it on the rather shabby plea that she was the first to oppress the Kiutzu [Qiuzi]-taxation without compensation is a sheer oppression.

Tibet claimed it on the ground that she had long been accustomed to extract slaves from that region. Britain's claim rests, in part, on the perfectly absurd ground that the inhabitants want her to take it over". (1924: 190) Since the end of the nineteenth century up the the 1930s, this area was considered an "un-delimited frontier." China and Britain were not the only two rivals. Local rulers, Tibetans but also Naxi, added to the complex layering of spheres of power over this area.

There, interethnic relationships were framed by power relations between these neighbours, a configuration in which the Drung people could only occupy the lower end of the political hierarchy. Though not reported with much detail in historical documents, some Drung women and men were indeed at times taken away, or exchanged for oxen as we will see, to become slaves in an alien land. I will here argue that such facts can better be understood in this context by taking into consideration the whole system of goods exchange and political relationships, which reveals not only a social hierarchy, but also cultural values that are important for our understanding of the socio-political specificities of this area.

In Northwest Yunnan and its adjacent regions (upper Burma, eastern Tibet), I believe there is something like a "ritual language" that is common to these otherwise dissimilar groups. It seems that the way locals, whether Tibetan, Naxi or less politically centralised groups such as the Drung, understood the workings of political action was intimately linked with that of ritual action. This echoes Leach's view of highland Burma and refers to his oft-cited formulation that "people may speak different languages, wear different kinds of clothes, live in different kinds of houses, but they understand each other's ritual. Ritual acts are ways of 'saying things' about social status, and the 'language' in which these things are said is common to the whole Kachin area" (1997 [1954]: 279).

From this point of view, I will reflect upon political relations between the local populations and try to shed some light on their underlying principle. In this chapter, I will analyse the workings of political relationships and exchange networks in a regional system involving the circulation of both goods and people, but also values regarding wealth and status differences. Using an ethno-historical approach, I will show how exchange, partaking and debt are for the local societies of this very specific area, where China, Tibet and Burma meet, key notions that articulate a more general politico-religious system. I will do so by combining ethnographic data collected mainly among the Drung with historiographical sources in western and Chinese languages. My aim is twofold. On the one hand, I would like to try to characterize the notion of ritual language used by Leach in his book Political Systems and discuss its heuristic value. By so doing, I hope to contribute to a clarification of the notion so that we can better appreciate its place in Leach's work. It is my conviction that it lies at the very heart of his anthropology. On the other hand, presenting my own analysis based on an ethno-historical approach to the relationships between the Drung and their neighbours on the Sino-Burmese border, I would like to make use of this notion of ritual language in an attempt to elaborate it further.

The notion of "ritual language"

Ritual was an important focus of Leach's later anthropological work, but, before this, a good deal of his approach in Political Systems depended upon his understanding of ritual. In fact, Leach did not attempt to define ritual but to reconcile divergent views in an approach to ritual broadly conceived as a system of symbolic communication. I will here recapitulate the essential aspects of Leach's "unorthodox" treatment of ritual -as he labelled it himself in the introductory note of the 1964 reprint of the book -and underline what makes the basic characteristics of his notion of ritual language. 6 For the members of a group, the meaning of their actions is expressed in what Leach calls "ritual," giving to this notion an extended meaning. Ritual, Leach stated, "serves to express the individual's status as a social person in the structural system in which he finds himself for the time being" (1997 [1954]: 10-11). It could be said that ritual is the home-made model which refers directly to the more abstract order of reality. Leach 6 It should be clear here that my aim is not to elaborate on the concept of ritual in anthropological thinking in general, but to frame the present discussion within Leach's own definition. It is equally important to understand that Leach's use of the notion of "ritual language" covers a broader range of social actions than what is generally implied by the ritual use of language, or religious language. This later and more exclusive acceptation of the notion (see for example Sadan, this volume) should not be confused with the meaning it takes in the present case.

views ritual in its cultural context as "a pattern of symbols" that "makes explicit the social structure" (ibid.: 15). The structure that is symbolised in ritual is "the system of socially approved 'proper' relations between individuals and groups. These relations are not formally recognised at all times [… and] this neglect of formal structure is essential if ordinary informal social activities are to be pursued at all" (ibid.: 15-16). "My thesis," wrote Leach, "is that in ritual action and in myth the actor is 'making statements' concerning the same abstract order of reality as that with which the anthropologist is concerned when he uses technical jargon to describe some feature of social structure" (ibid.: 86).

As the French anthropologist Jean Pouillon noted in his afterword to the French edition of Political Systems [1972], according to Leach, structure is a matter of concepts, not facts. And between the facts and the model elaborated by the ethnologist lies what we can call the "indigenous model." This model reflects the underlying order that is supposed to guide people's social activities: "Ritual performances have this function [of reminding about this underlying order] for the participating group as a whole; they momentarily make explicit what is otherwise a fiction" (Leach 1997 [1954]: 16). In other words, "ritual" as an aspect of culturally defined behaviour is attributed by Leach the qualities of a language, and such "verbal categories" as ritual actions and myth refer to an as if system of ideas, a "structure."

The example of the Hpalang community that Leach studied in detail can help in presenting Leach's approach in more concrete terms. This community comprised several ethnicities, such as Jinghpaw, Atsi, Maru, Lisu and Chinese. Each of these groupsapproached by Leach as "sectors" within the social structure-had its own language, religious practices and customary behaviour. But in Leach's discussion of the principles of Kachin social structure common to all of them, such differences can only be understood as internal variations.

"Since the members of the Hpalang community were all part of one political system, they had to be able to communicate with one another about political status, and they did so in the language of ritual action. Precisely because Kachin society as a whole is made up of numerous sub-groups speaking diverse spoken languages we may expect that at a ritual level there is a rather simple stereotyped ritual 'language' which is understood by all sub-groups and in which issues of status are constantly represented in much the same way. Kachin ritual expression is relatively simple precisely because Kachin culture is complex." (ibid.: 102) Being part of a same political system, people communicate about their social identity (status) through ritual means. There are therefore two associated aspects which deserve further discussion: that of communication and that of identity.

Ritual and communication

In Leach's view, ritual was not limited to its religious (or transcendental) aspects.

He clearly expressed his rebuttal of the classic Durkheimian dichotomy between "sacred" and "profane":

"… [T]echnique and ritual, profane and sacred, do not denote types of action but aspects of almost any kind of action. Technique has economic material consequences which are measurable and predictable; ritual on the other hand is a symbolic statement which 'says' something about the individuals involved in the action." (ibid.: 13) Ritual is therefore a category that encompasses different types of activities, and schematically all types of action can have a ritual aspect. It can be true of the way people dress, speak (and the language in which they choose to speak), etc. But myth and ritual (in the restricted sense of ritual performances) are the two main types of action that are of primary importance: "Myth, in my terminology, is the counterpart of ritual; myth implies ritual, ritual implies myth, they are one and the same. […] As I see it, myth regarded as a statement in words 'says' the same thing as ritual regarded as a statement in action." (ibid.: 13) "…[R]itual action and belief are alike to be understood as forms of symbolic statement about the social order." (ibid.: 14) "… [M]yth and ritual are one and the same. Both are modes of making statements about structural relationships" (ibid.: 264).

Leach's approach is limited to myth as something that reflects patterns of social relations. Ultimately, the only thing "rituals" have in common is that they are actions that communicate meanings, or, in some cases could create the very meanings they communicate. Therefore, it appears in all these formulations, that the roots for Leach's later exploration of the concept of ritual (1968, 1971 [1966], 1976) were already there as the theoretical background of his Political Systems. Ritual, as he understood it, was basically a "symbolic communicative performance;" myths were a matter of verbal communication, whereas ritual performances were a matter of symbolic communication. 7 He proposes that we look for similarities in patterns of communication rather than try to arrive at universally valid definitions of ritual.

Ritual, identity and ambiguity

Leach's main goal was to offer an analysis of the Kachin system, and his formulation of the use of ritual to express status is embedded in the context of empirical political behaviour. However, ritual also appears as a means to deal with specific systems of relationships and to express an individual's identity in the system. This is why, for example, everyday actions fall into the "ritual" sphere that Leach had in mind, and language change expresses the social positioning of actors: "The two sides of this paradox [in some cases Kachins seem conservative about language but others seem almost willing to change their language as a man might change a suit of clothes] both exemplify the same social fact, namely that, in my terminology, for a man to speak one language rather then another is a ritual act, it is a statement about one's personal status; to speak the same language as one's neighbours expresses solidarity with those neighbours, to speak a different language from one's neighbours expresses social distance or even hostility." (ibid.:

49)

In Leach's work, ritual appears as a means of expression in a more general politics of identity. This politics of identity encompasses specific claims of status position in the socio-political hierarchy; ritual elements can be integrated into both religious and secular settings that affect the identity of participants.

When Leach demonstrates the process by which Kachins have become Shans and Shans have become Kachins, he proceeds to show that this has been possible because Shans and Kachins share a common language of ritual expression, and can thus be thought of as members of "one society." One could say that there is a ritual language of ethnic relations: in multi-ethnic societies, some "lingua franca" is needed to designate status positions between groups and individuals. This is a prerequisite for the contextual 7 The general discussion on the implications and limitations of such an approach goes far beyond the scope of this paper. Leach's understanding of ritual action has been discussed in a stimulating way by Tambiah (2002: 350-356), pointing out his two complementary perspectives, i.e. special behaviour in ritual occasions and behaviour in everyday life (presentation of self). definition of ethnic identities and Leach's discussion of the fluidity of identities is one of his important and well-known contributions to theories of ethnicity.8

Another important aspect underpinning part of Leach's interpretation of the dynamic of the system is the ambiguity of the ritual language.

"Myth and ritual is a language of signs in terms of which claims to rights and status are expressed, but it is a language of argument, not a chorus of harmony. If ritual is sometimes a mechanism of integration, one could as well argue that it is often a mechanism of disintegration." (ibid.: 278) "There are, as we have seen, a large number of 'ritual acts' which can be said to have the same meaning whether the actor is a Shan, a gumsa Kachin or a gumlao Kachin, but the inferences that are to be drawn from such acts will be entirely different in each case. That such ambiguity does not lead to intolerable misunderstandings is due to the essential vagueness of all ritual statements. Ritual and mythology 'represent' an ideal version of social structure. It is a model of how people suppose their society to be organised, but it is not necessarily the goal towards which they strive." (ibid.: 286) As a "language of argument," the "ritual language" allows manipulation by protagonists. Its potential ambiguity is essential to the social dynamic that Leach aimed at describing. In his approach to myth and ritual, Leach rejected the functionalist view of a coherent whole composed of one mutually consistent set of myths that served as a charter for ritual action. Myth and ritual are both ways in which Kachin and Shan represent to themselves their social structure in its changing nature and with its contradictions.

For Leach, there is no significant difference between so-called magical rites and expressive political or technological action. Ritual is no more than a symbolic expression of status and is analysed in terms of the logic of linguistics. As he later on wrote in Culture and Communication (1976), we engage in rituals in order to transmit collective messages to ourselves, and these messages, Leach asserted, are always about the social order. For Leach, as he further developed in this book, ritual is a medium for the expression of cultural ideals and models that, in turn, serves to orient other forms of social behavior.

But, on what grounds can one argue that there is one system of ritual expression?

And if this language is common and "'represents' an ideal version of social structure," as Leach has it, what is this common structure? Or, to put it another way, what is the model, the underlying common order? Ritual theory has evolved since the publication of Leach's Political Systems, and there has been criticism of the tendency to analyse ritual as a language as well as criticism of the attempt to analyse ritual as being communicative. After all, Leach himself pointed out that ritual is a category scholars have invented, and in large measure, theories of ritual depend upon the phenomena to which one chooses to assign the ritual label. But I shall now address these questions through an analysis of the social and political situation of the borderlands of northwest Yunnan, with special reference to the first half of the twentieth century. My intent here is to follow on from Leach's intuition about the existence of a common ritual language and approach the interactions between the Drung people and their neighbours in much the same fashion. I propose to study the way in which the Drung were integrated into the multi-ethnic environment of the eastern Himalayas, and how, through their interactions with their neighbours, they produced as well as were ascribed their differences. In this context, the debt seems to be an essential matrix of the construction and the expression of the political relationship between the groups, and it contributed to establish or abolish the frontiers between them. In this sense, the logic of debt and its ritual, political, and economical expressions constitutes a common language that enables all the groups to produce their differences, and which, in the process, also includes them in a common social system.

Debt and hierarchy

In Leach's manner, my perspective here is to consider the Drung in terms of their relations with their neighbours, not as an isolated social entity. Considering their geographical location and their socio-cultural characteristics, they could potentially be considered part of the Kachin world, in the sense Leach used that category. The Drung (Dulong) people are a Tibeto-Burman speaking group officially recognized as one of the fifty-five "minority nationalities" (shaoshu minzu) of the People's Republic of China. They are relatively isolated in the small valley through which the easternmost source of the Irrawaddy meanders, today an administrative division of the Dulong and Nu Nationalities Autonomous County of Gongshan (map 1). 9 As they were officially recognized as the separate "Dulong minzu," they were by this process distinguished from the Nung (Nu minzu) to the east, and from the Rawang that were on the other side of the national border.

But, even up to the present, Drung have familial, cultural and linguistic links with these close neighbours that transcend natural and political barriers (see Gros, 2004).

However, their inclusion into the Kachin social system is not a given. For most of their history, their relationships with political powers have been oriented to the East, with both Tibetan chieftains and the Chinese empire, first indirectly through the office of tusi (indigenous chief) granted to the Naxi rulers, and later on by means of direct administration. For the Drung people, power and authority have always been linked to the east, since mythical times, and they nowadays often refer to this configuration as the time of the "Chiefs of the East."

It is these political relations with Tibetans, Naxi and Chinese that I am taking into consideration in the following analysis of the Drung people's place in a regional system.

The political situation of this area bordering Tibet and Burma was, by the mid-nineteenth century, under the authority of a Naxi indigenous chief (tusi), who was at this time himself under the higher authority of a Chinese official. To the north, Tibetans of Tsarong also extended their political influence, and at some point, the Naxi chief gave away to the Tibetans, nominally at least, his right to receive taxes from the Drung and the Nung. Therefore, for the population of the upper Salween valley and of the Drung valley, this resulted in overlapping spheres of authority. Nung as well as Drung were at the meeting point of the colonial expansion of their more powerful neighbours, whom they would sometimes resist by making ephemeral alliances with each other. More often, debt or even confiscation of land would cause the families to move away. In this area, the Drung and the Nung occupied the lower end of the political hierarchy.

An historically informed study of this area enables us to discover some aspects of the processes of political integration. Yunnan's borderlands had been the object of rivalry between Naxi, Tibetans and Chinese, so that the interests of several political authorities were simultaneously focused there. At the same time, the principal actors profited handsomely from the development of large-scale commercial activities (cf. Gros 1996). Debt appears to have been an institutional means for gaining political ascendancy and control over land and it allowed the layering of various spheres of power.

By the very means of debt relations there was a possibility of land appropriation and the debt system, therefore, played a role in land distribution. In numerous villages in the upper reaches of the Salween, the Nung people had several creditors to whom they had to cede their land while remaining on the land as tenant farmers. As guarantee for rising debts, rights over land were ultimately transferred. Land could pass through different hands and debt was one of the expressions of this mobility, implying changes in social relations.

The creditor, having obtained the economic dependency of part of the population, was potentially a chief. Loans with interest were granted to locals by Naxi tusi, Buddhist temples, and some Tibetan chiefs as well. Commercial activities, like the selling of salt, several kinds of metal tools, cloth and wool, were also in their hands.

The enslaved and the indebted represented two different degrees in a relationship of dependence. The relationship between the debtor and his creditor gave rise to a stable relationship of political dependency. This was the basis for a stratification of power and social organisation, thus revealing the strong link between political and economic relationships. In other words, the initial relationship based on exchange was transformed into an asymmetrical relationship, characteristic of dependency. As such, the creditordebtor relationship becomes a central paradigm for understanding the grounding of relationships of dependence.10 Tibetan and indigenous chiefs (tusi) exploited this very same process. Indeed, in this remote area up to the first half of the twentieth century, impoverished peasants could not escape from the debt system. This was especially the case for the less powerful Drung and Nung who became potential slaves. However, it is important to specify to what extent, and in which cases, it might be possible to speak of slavery, for it seems that diverse modes of servitude coexisted.

There were cases of external slavery: abduction, raids, all of which had as a target others, people external to the community of reference. Yet, there were also forms of internal slavery, in particular for debt. Debt slavery was described by missionaries and travelers in this part of the Sino-Tibetan finges as an issue that concerned a great part of the population. It was present in eastern Tibet (Tsarong, Dzayul) as well as in the Tibetanized part of northwest Yunnan.11 Debt was closely associated with service labor and, in its extreme form, slavery.

One of the missionaries who resided for many years at the border of southeastern Tibet specified that, amongst approximately six thousand families one could find in Tsarong at that time: "270 only are tributary [i.e. "taxes payers", khral-pa]; the others are farmers for the lamaseries [monasteries] or slaves of some rich owner" (Goré, 1923: 377).

The farmers who did not have any means of paying their debts could become the slaves of their creditor or settle their debt by giving one of their children. Those enslaved for debt could, in theory, free themselves; but rare were those who had the means to pay for their freedom. The creditor could transfer the debt to a purchaser, and the debtor thus changed creditor and his status was transmitted to his descendants. In these Tibetan areas, many slaves were Drung or originated from the regions at the edges of the Himalayas (Upper Burma, Arunachal, Assam), but the debt phenomenon concerned the population as a whole, Tibetan or not. 12 As mentioned previously, it also happened that Drung people were captured when victims of raids. However, some Drung also played an active part in a relatively important trade, geographically if not numerically, in which people were exchanged against goods, more often oxen. According to some oral accounts, the inhabitants of the south of the Drung valley took part in this trade more actively. 13 These accounts accord with information provided by Father Fage who mentioned that: "Dijoux 14 […] are robbers, plunderers, and the trade of the slaves is almost entirely in their hands." He continued by specifying that the Drung (and those who were at that time included in the same Qiuzi category) "like the ox flesh passionately, and would not renounce any sacrifice in order to get some. The Thibetans [sic] exploit this disproportionate craving and send to them each year hundreds of oxen; they receive in exchange the young slaves who receive our solicitude." 15 There is probably a degree of exaggeration in the remarks of the missionary because, at a rate of hundreds of oxen per annum, not many Drung would have remained.

12 See Gros, 1996, 2001. One of the most precise descriptions of the situation in Tsarong is that provided by Rockhill (1881: 285-286), in which the ambiguity there can be between indebtedness and slavery and the possibility of a transformation from one to the other is rather clear: "… While speaking of the Tsarong it is proper to note that the slavery exists there in a more aggravated form than in any other portion of Tibet. While now and then a poor Tibetan pilgrim, on his way through the Tsarong to the Dokéla [Dokerla], or to some other famous sanctuary, may become indebted to some one for the amount of his board or the like, and be obliged to work out by four or five years of labor his little indebtedness, he at least eventually gains freedom; but such is not the case with most of the slaves in the Tsarong, who are taken from among the Lissus and other non-Tibetan tribes inhabiting the country. When one of these is in debt to a Tibetan and unable to meet the demands of his creditor, he becomes his "life servant" (ts'é yo [tshe-gyog]). The master has the right to sell, kill, or otherwise dispose of him; he is given a wife, or a share in a woman, and all the children born to him are slaves. Even if he should be able to get together enough to pay off his debt, the master may refuse it and count his labor as only a set-off for the interest of the sum due. The missionaries have bought and freed a number of these ts'é yo, but usually they have been able to buy only slave children; this class of person formed the nucleous of several of their little Christian communities." 13 Due to space limitation I cannot here all the details regarding this situation as it figures in Gros, 2005: chap. 4. 14 The Father Fage is here using his own transcription of the name of a valley to the west of Dulongjiang in todays Burma. 15 Letter to M. Legrégeois, Bonga (Thibet), July 23, 1857, Archives de la Société des Missions Étrangères de Paris, vol. 556 (2) [839].

Nevertheless, it should be recalled that not only Drung people were exchanged. The slave trade also targeted adults or children who originated from other areas to the west. The French missionaries partly constituted their small community by purchasing slaves. These communities, therefore, included children, (both boys and girls), generally non-Tibetans who were orphans, victims of abduction or who had been sold by their indebted parents. 16 According to missionaries as well as the accounts of travelers at the turn of the twentieth century, the locality of Menkong in Tsarong was a center of the slave trade. In a short note, J. H. Edgar wrote that "Menkong before 1911 was an important slave market where men and women were bought and sold openly," and according to the French Tibetologist J. Bacot, fathers, mothers and children could be sold separately (1912: 273).

Most of the slaves were non-Tibetans and were most probably from the Drung and related groups. 17

The living conditions of the slave and the rights that the master had over him could vary somewhat, but it is clear that slaves could be sold in Tsarong and its surroundings.

The market extended beyond Tsarong, since we know that some Lisu took part in it, and that the Naxi indigenous chiefs (tusi) of the Mekong valley owned a number of Qiuzi slaves who were freed on the order of the Chinese authorities during first half of the eighteenth century.

All together, the origin and status of the "slaves" varied from one place to another and included people living at the margins of society or who were in some ways excluded from it such as orphans, robbers, prisoners of war, people who had been captured or were in debt. Many testimonies indicate that their living and working conditions were not particularly difficult. However, the relative discomfort of work and the living conditions is not a sufficient criterion to identify or distinguish the status of the enslaved. To speak about slavery and slaves in the fullest sense of the term, it is necessary to admit the fundamental fact of exclusion. The status of the slave is marked by his total exclusion from a fundamental dimension of the society: that through which identity and status of member of the community is articulated. As Testart (2001: 24) wrote: "the slave is a man without identity." Within Drung society, for example, this exclusion refers to kinship. For this reason, it is often specified by the Drung themselves that the exchanges for oxen often concerned 16 See Launay, 1902, I: 245. Archives de la Société des Missions Étrangères de Paris, vol. 556 A(2). Letter from M. Renou to Mgr de Sinopolis, Tcha-mou-tong, January 28, 1859 [1379]. 17 See also Bailey, 1912: 338; 1945: 89. orphans, and more significantly-in this patrilineal society-children who did not have a recognized father. By extension, one can consider that for the Tibetans in Tsarong, this exclusion referred to the community of language and culture, as well as religious identity, so that the slaves originated from groups considered "ethnically" different and non-Buddhist.18 Thus, a difference remained according to the origin of the person. In this area, between Tibetans themselves slavery was initially a system by which persons could be pawned (pledged) for debts, and the enslavement was thus only the result of a transformation of the status of the pawned person. In the long term, the pawned person was likely to become slave. 19 What is of importance is that the trafficking of the enslaved proceeded between groups which were distinct by their identity and status, and that the slaves were generally, it seems, children or teenagers. However, the sale alone did not make it possible to ensure that they would always be slaves: in the case of debts, it is possible that we are closer to the workings of a pawn system. But if there was a firm sale and the individual who was sold was cut from his family ties, it would indeed be slavery. However, should they be pawned for debts or slaves, they would likely fall into the networks of the slave trade. In Tsarong, it seems that pawned persons could be sold or transferred, like slaves.

Therefore, if we consider documented cases, slavery is found together with other types of service labor that do not imply a radical exclusion from society. In the case of a person pawned for debt, or of voluntary dependence or abduction (which could, however, mean that the individual would actually be cut off from his familial ties), the bond with the community of origin is preserved, which implies the possibility of a return to that community.

Similar facts are to be found on a regional scale, and a comparative study could be carried out from the borders of Yunnan to Assam. The great diversity found in status complicates such a task, 20 but one can underline at least two significant characteristics regionally: the important role of external slavery (by capture or abduction), and the phenomenon of internal slavery, which corresponds to the possibility of selling a member of one's community or, more precisely, to having rights over a person and being able to transfer these rights.

The sale of slaves has also been documented among the populations of Upper Burma. For example, among the Taron(g), adultery was punished by the sale of the two guilty parties (Mya-Tu, 1966: 33). Further to the south among the Maru, the sale of people to the Kachin (Jinghpaw) seems to have been relatively frequent (Pritchard, 1914: 528). It is probable that, in the first case, the key issue was about selling into slavery, but the second case could relate more to sale for indebtedness, with, at least in theory, the possibility of releasing the pawned person from the enslaved state by repurchasing him/her later on. 21 Nevertheless, there can also be no intention of repurchase, so that the (voluntary) sale becomes, in fact, final.

For the Kachin, the mayam, who according to Leach represented almost half of the population, were slaves; they were out of kinship bonds and could be sold [but see Maran, this volume]. 22 Leach had a different perspective and, according to him: "the majority were voluntary serfs -or even adoptive sons -of their Master, rather than movable goods" (Leach, 1968: 145, n. 2). The problem of voluntary dependence also exists in the case of the Drung. Some -without preliminary debts, without being sold -decided to put themselves at the service of a Tibetan chief.

With regard to voluntary dependence in northwest Yunnan, many seem to have preferred the servile condition to freedom. "Slavery is all in all only a life time's domesticity, without pledges, and many slaves, accustomed to this condition, prefer it to freedom" wrote Father Goré (1992 [1939]: 106). Outside of Tsarong, in the Tibetan communities of the Mekong valley, the slaves seemed better off. According to J. Bacot, serfdom would be a more correct term. He gives the example of a Tibetan who had seven wives and about fifty slaves (he himself did not know exactly how many), with whom he shared his harvest. "This appearance of richness does not require at all the possession of a single taël. […] No external difference distinguishes the Master from the slave. They wear the same clothes, eat together and are polite with each other" (1912: 273; emphasis added).

To me, the 'softness' of the living conditions does not change anything with regard to status. Indeed, it is a matter of wealth, as the master is rich because of his dependents. The 21 See for example this case mentioned in Pritchard (1914: 530): "He was sixteen years old, and had been a slave in Kachin country for two years. His people had sold him originally for a coat and cooking-pots. He had nothing but good to say of his Kachin owners. […] Pritchard asked him why he did not run away from his Kachin masters, and he replied that, in the first place, he had no wish to do so, as he was very well treated; and, in the second, his own people who sold him had told him that they would sooner or later buy him back" (also quoted in Leach, 1979 [1954]: 303). In this case, the sale is not definitive and the pawned person is still a member of his kinship group. 22 I follow Testart (2001: 151-152) on this point, based on the data provided in the Appendice III in Leach (1979 [1954]: see also 160-162); See also Nugent, 1982: 519. fact that it could be, like Bacot put it, just an "appearance of richness" is significant. It follows that prestige is acquired thanks to the number of dependents or slaves. 23 The slave, sharing the misfortunes as well as the good fortune of his master, also contributes to the master's prestige.

Ritual and power

By emphasizing the complexity of political and commercial interactions and the dynamic relationship between these contiguous societies, one can point out some fundamental aspects of their relationships. The importance of trade and debt invites us to see in exchange a means, as limited as it is, of access to power. However, it is a power that goes well beyond the economic domain.

By recognizing the value that Drung people granted to some goods that they had secured by trading with their neighbours, one restores the active role of these goods within the exchange system. For example, gongs, which came from Burma, iron utensils (machetes, pots and tripods in particular) were all items of prestige that Drung traded with their neighbours. Indeed, they formerly constituted the principal set of goods that, locally, could enter into the composition of some forms of compensation, in particular those made for matrimonial exchanges. Among these goods, the most prestigious were certainly the oxen. They were bought outside the Dulong valley because Drung people themselves did not raise them. Drung people and some of their neighbours needed oxen as a prestige good for matrimonial exchanges, as well as for communal sacrifices.

Locally, among the Drung, there were sufficiently rich individuals who, for their own prestige, organized on their own behalf the sacrifices of oxen, and would divide the meat that would be partaken. Because Drung people carried out their own sacrificial rituals in their villages, they needed oxen, which they often got by exchanging those excluded by their society. The equivalence (or conversion) between ox and slave in the exchange, tends to show that the slave as well as the ox enter the category of the ritual goods that ensure the prestige of the owner.

From the Drung people's point of view, oxen were the main goal of the trade that sometimes involved the circulation of people, often children and women. The exchange of persons occupied the highest-value level of exchange. Furthermore, if "exchange" may be described by detailing what one person gives to another, there was a higher order of exchange, a larger set of relations that structured exchange itself.

It is worth mentioning here how this structure of exchange and debt relationship was also used on another level by the Drung, by referring to their political allegiance to the Tibetan chieftains of Tsarong. Formerly, the Tibetans chieftains were considered by the Drung of the northern part of the Dulong valley as the "masters of the place" (mvli aqkang), and a tax was annually perceived as a compensation "to eat the land" (amra kai).

This relationship of subordination was initiated in a particular way.

The Drung tell the story that only those, in the north of the valley, who had received their share of the meat of nine oxen that the Tibetans had once killed, became subordinate and had to pay the tax. The meat had been divided in the form of skewers and had been distributed in all the north of the valley. It is told that in the south of the valley, the distribution did not take place because the six porters (Drung people originating from the south of the valley) died on the way. Thus, "the tax was cut" (kri tot) where they died, this place becoming a geographical limit: not having received a share of meat, the inhabitants of the south of the valley would not pay the tax.

The distribution of the meat among the inhabitants of the Drung valley legitimised the political ascendancy of the Tibetans who were recognized as "masters of the land" by the Drung. One could expect the distribution of meat to be carried out by those who had a claim to power. But in that case, the status of the Tibetan chiefs seemed already acquired, and the distribution was but a legitimisation. According to Drung people's accounts, the distribution of meat carried out by the Tibetans chiefs seems to have crystallised the hierarchy by confirming a relation of dependence.

Interestingly, this very same process of meat distribution appears also in the relationship between the Naxi tusi (indigenous chiefs) and their dependents, the Lisu, Nung and Drung people of the Mekong and Salween valleys. Some historical records mention that this chief, one of the main authorities in northwest Yunnan up to the beginning of the twentieth century, organized feasts for his dependents. 24 Every year, all (the representatives of) his subjects were required to come and bring him a present for the New Year, and he was to entertain all of them for three days. Every three years, he also organized feasts during which he distributed a piece of beef.

These feasts seem to correspond to something other than a simple tributary relationship. What must be underlined in this case is that political centralisation was also a ritual process. 25 Through these feasts, the status of the tusi and the political centralisation that he represented were confirmed and his prestige was enhanced. The extent of the festivals given by the Naxi indigenous chiefs seem to have been sufficiently significant for Roux to refer to them in the following terms: "In a few days [at the beginning of August 1895], more than one million of these natives [Lisu, Nu and Qiuzi] will come, like every year at the time of Ho-Pa-Tsié, 26 to bring to him, at a rate of approximately one taël per capita, their tribute in kind, in the form of wax, musk, powder of gold, medicinal plants, etc. On the other hand, he will give each of them a piece of meat, a bowl of rice and a cup of brandy; fifteen oxen are hardly sufficient for these festivals, during which the savages perform in front of him dances and songs, as if adoring him. […] Formerly, people of the Kioukiang [Qiujiang] valley had each year to provide him a slave. Currently this habit is abolished, but, to preserve at least the form and part of the value, they give him in place a statue made out of wax of the size of a twelve years old child" (Roux, 1897: 227). This reference to slavery once again stresses the importance of this institution and the form of dependence that could be established. If a slave is due, one leaves the sphere of exchange to enter an asymmetrical relationship characteristic of dependence. Therefore, power seems above all to be a matter of economic control rather than effective political authority. And control was essentially asserted over ritual goods (such as slaves and oxen) enabling one to assert one's rank or claim for status.

In northwest Yunnan, as in neighboring areas, the role and value of the ox was of primary importance: the ox can be seen as a good that enhanced status and prestige, as did slaves. The sacrifice of the ox and the sharing of meat was an expression of these local conceptions of prestige and wealth, common amongst several societies in the Himalayan border areas as well as in mainland Southeast Asia and often refered to under the label of "Feast of Merit."27 Moreover, the ritual slaughter of animals and the distribution of their flesh appear to be the principle means through which one could acquire (or confirm) power and status. Stevenson (1943) rightly qualified the working of these feasts as a "prestige economy," for one of their main concerns was with prestige and wealth. 28 Such a sacrificial practice remains a cultural characteristic of the Drung people today. Even though they abandoned it under the political constraints of the 1950's, the meaning and cultural value of the practice is still very much alive.

The missing share

One of the many myths the Drung people have in their repertoire is especially important for understanding some essential values shared by the members of their society. This myth also tells us about the logic (and/or the morality) of interpersonal relationships:

One day, after an abundant harvest, a man called Punggrin Pung decided to organise a great feast (dvruq-wa) that lasted for nine days and nights. All types of animals on earth joined the party. Then came the day to end the feast, meat and alcohol where nearly finished. The little that was left, according to custom, had to be shared equally. The number of participants was counted, and the meat divided. 29 But, no matter how they would count, again and again, two shares were always missing. The two who received no meat decided to go hunting in the mountains.

During the hunt, one of the two disappeared and became a spirit, Tsheu-pvlang, who demands offerings.

This man who became a spirit requires his due. He is one of the most feared amongst the Drung people. He appears as an always unsatisfied figure, the prototype of the kind of spirits with whom the relationship is that of debt: no sharing can be done without him potentially coming around, as if people's debts towards him were permanent and could never be paid. This myth is often told in order to account for the origin of this particular spirit and to explain the origin of the sacrificial feast that the Drung people used to perform yearly.

The Drung name for the ritual sacrifice of an ox is called dvruq-wa, literally "groupmaking." Its very name underlines one of its essential aspects: according to the general 28 See for example Russell (ed.), 1989. I find myself much in accordance with Durrenberger (1989: 114) in the way he described the Lisu "ideology of honor": "In highland Southeast Asia, there is an ideology of honor and wealth that can be translated into rank and prestige under certain circumstances. Where wealth and access to valued goods are scarce, hierarchic forms will develop; where they are widespread, egalitarian forms will develop. Both hierarchic and egalitarian forms are based on the same ideology, but the social forms are largely shaped by economic relationships." 29 The animal could differ according to different versions.

principle, the sacrificial victim is cut and its meat shared, the sharing network drawing the contours of the community. 30 Among the Drung, this kind of sacrifice was an opportunity for the one who was rich enough to own an ox to validate his prestige (koksang) and his wealth (karji). Drung people did not raise their own cattle in the past, and oxen were obtained through trade relations outside their valley, especially with the Tibetans who exchanged them against goods or, more likely, individuals who would become slaves. Oxen were representative of one's wealth and such feasts were the principal means of political expression. On the one hand they reinforced the cohesion of the local group as well as alliance relationships, but on the other hand they could only enable one to affirm a provisional dominance, or prestige; this prestige could be challenged, as it was embedded in a permanent process of competition. This individualist competition is totally in opposition to hereditary status.

Thus, for the Drung people, it was the language of sacrifice and sharing that prevailed at the political level. It was at the heart of the local politico-religious dynamic, assuring social cohesion, reaffirming power relations and kinship ties. The ritual ensured the maintenance of the ties that linked the sacrifice's giver with his social horizon, his relatives, affines and neighbours. At the same time, the function of sacrifice was crucial for the internal process of competition and redistribution within Drung society.

Similar to sacrificial practices such as "Feasts of Merit" that emphasise sharing and the display of wealth, these feasts could create and/or validate distinctions regarding power and social status by transforming material wealth into social rank. And in some cases, these feasts were more about recognition or legitimisation than the granting of power. 31 Nowadays, the ceremony of the ox sacrifice has disappeared among the Drung people. But its underlying principle still prevails in the Drung people's ideology and representation of power. What is still alive is the idea that any pretension to authority should be validated through generosity. 30 It is interesting to contrast the drung expression, "group making" (dvruq-wa) and the kachin one "nat making" (nat galaw). Nevertheless, Leach (1979 [1954]) description of the Kachin nat galaw could be extended to the drung context: "Thus from a certain point of view a Kachin religious sacrifice may be regarded as a purely technical and economic act. It is a procedure for killing livestock and distributing the meat, and I think there can be little doubt that for most Kachins this seems the most important aspect of the matter. A nat galaw ('nat making', a sacrifice) is almost a synonym for a good feast. But from the observer's point of view there is a great deal that goes on at a sacrifice that is quite irrelevant as far as butchery, cooking and meat distribution are concerned. It is these other aspects which have meaning as symbols of social status, and it is these other aspects which I describe as ritual…" (p. 13; see also p. 172, 174). 31 One could refer for example (among many others) to Fürer-Haimendorf (1967); Lehman (1989); Woodward (1989); Jacobs (1991: Chap. 8); Kammerer and Tannenbaum (1996).

This allows us to understand some aspects of the present day relationship of the Drung to the Chinese state (see Gros, 2005). Constant subsidies and assistance from the communist state, perceived as a welfare state, are an integral part of the validity of its power and authority. Without what can be seen as a constant dispensing of wealth, the state would lose its legitimacy and prestige-a prestige that it had gained, especially for the last generation, through constant assistance and help. This state, seen as a "provider of goods," appears through the lens of the Drung's vision as only a historical manifestation of a type of political power that depends on a principle of redistribution.

Obviously, some fundamental principles of social relationships pertain even if the structure of which they were a part are being altered or destroyed. The "prestige economy"

does not play a central political role anymore; it is relegated to the margins.

In the Drung valley, attempts to effect conversion to Christianity have been relatively successful. "Religion" -Christianity -is being valorised in opposition to "Superstition" -popular religion. Christians gather regularly for the holydays of the Christian calendar, and at these gatherings, pigs or other animals are slaughtered and their meat is shared among all participants. According to the Drung themselves, it is the same as "making the group" (dvruq-wa). These Christian feasts, and the community they contribute to creating are a contextual answer to the present economic, political and cultural situation.

From that perspective, Christianity could constitute a new horizon. Conversion could be a way to go back to the time of sacrifice, to a social cohesion in which a new "being together" becomes possible.32 Macdonald (1980) has pointed out the importance of the conceptual model of "creative dismemberment" among some Himalayan societies and argues that the model could directly be used and exploited by political and economic elites as an instrument of social control. In the context of political centralisation, the sacrifice gives way to relations of domination, as we have seen. One can also argue that, through the sacrifice, it is rather previous conditions that are confirmed and re-established: the ties of reciprocal obligations, the stratified status in a social hierarchy, etc. According to Oppitz (1997), the creativity lies in myth, whereas in reality, sacrificial practices can only re-create. But it could also be argued that through meat division, something else comes into existence, and is, therefore, created.

For the Drung, the logic of sacrifice was central politically; it was a means by which the contours of the society could be redrawn, a means of making and un-making the group. The mythical model is that of "the missing share." That is to say, to paraphrase the myth, that the heart of the matter is a debt relationship. Humans have a debt toward the spirit that each sacrifice can only temporarily appease. But among humans themselves, the feast giver's generosity and the subsequent sharing of the meat, under the appearance of the gift, hides the debt.

We have seen that debt relationships have played an important role in the constitution of local spheres of power in the larger region of northwest Yunnan. These relationships were formalised by ritual means and ritual came to assert the participants' respective positions. In this case, it seems to me that ritual communication is not just an alternative way of expressing something. It is also the expression of things that cannot be expressed in any other way, and maybe, as Gellner (1999: 139) argued, "it is precisely because rituals do not simply say things that they acquire their power to persuade and legitimate."

Conclusion

Because of his broad conception of ritual as communication extended to many aspects of social life, the notion of ritual language used by Leach seems rather fuzzy. As such, the notion of ritual language can prove helpful in dealing with aspects of identity and political and religious relations in a multi-ethnic area, be it the Kachin Hills or elsewhere, such as in northwest Yunnan. Yet, despite its heuristic value, which enables one to point out similarities between otherwise dissimilar groups, it needs further elaboration.

I have tried to show that for the Drung, meat division, and more generally sacrificial feasting that involves the distribution of the flesh of the sacrificed animal, appears to be a model of internal and external power distribution. 33 But it is also, more generally speaking, a model for inclusion and distinction. In other words, there is a double movement: at the same time as it creates or confirms social relationships, it also sets people apart by classifying and attributing social positions. In the case of the Drung, I see debt as a prototypical form of relationship that prevails in both relations between individuals and groups, and between people and supernatural entities. It is this ideology that expresses itself through ritual actions such as meat division. Or to put it another way, the notion of debt is the principle of efficiency that structures the world view and gives ritual performances their meaning at the same time. I am tempted to propose that the notion of debt could aptly characterize some essential aspect of the common structure on which this ritual language of sharing is articulated. I would argue that Leach's notion of ritual language can be reformulated by placing the notion of debt as a central element of what this "language" tells us of the underlying order.

In this specific case, the ideology that the ritual language expresses is in a broad sense an ideology of "power," but authority and control only appear with control over the source of wealth, which includes ritual goods (oxen) and land. This ideology involves interrelated concepts of wealth and power that are central in the dynamic of social relations. This dynamic itself is supported by the logic of the debt that we find in relationships between people, and between people and the other world. To paraphrase Leach then, I would say that a common ritual language exists insofar as what it expresses through symbolical performances is a general ideology of power that reflects not simply social relations, but relations of indebtedness. And this formulation, I believe, find its roots in Leach's work itself, partly in Political Systems and clearly in his concise Social Anthropology (1982).

The practices surrounding debt relationships were analysed in detail by Leach when he dealt with the categories of hka (debt) and hpaga (trade, ritual wealth object). As Leach formulated it, almost any kind of legal obligation existing between two Kachins is likely to be described as a debt, and debts are expressed in terms of hpaga (for a discussion, see Ho this volume). He also underlined the important aspect that "the debt is a kind of credit account which ensures the continuity of the relationship. There is thus a kind of paradox that the existence of a debt may signify not only a state of hostility but also a state of dependence and friendship " (1982: 153).

From Mauss to Lévi-Strauss, following a social approach to exchange, there is more in the exchange than just what is exchanged. What is "more" is the social relationship established in the process. In The Gift, Mauss insists on the imperativeness of the obligation to reciprocate, and even states that "the punishment for failure to reciprocate is slavery for debt " (1990: 42). It seems that Leach's position was very much inspired by Mauss with the nuance that the network of social relationships that one can observe through gift-exchange behaviour is a network of indebtedness. He makes this point clearly in Social Anthropology by writing: "persisting relationships only exist as feelings of indebtedness. From time to time every such persisting debt relationship needs to be made manifest in an actual gift transaction, but the relationship is in the feeling of indebtedness not in the gift" (1982: 154). In other words, the mechanism of gift-exchange is that of debt. [START_REF] Gregory | also approaches the gift economy as a debt economy by underlying that what the giver desires in the gift-exchange are the social relationships themselves and not the things exchanged[END_REF] Though not stated very explicitly, Leach's concern in Political Systems was very much the notion of power, but in the limited sense of the "attribute of 'office holders', that is of social persons who occupy positions to which power attaches" (1997 [1954]: 10). But if power is a result of participation in interpersonal relationships, both the political and economic aspects are embedded in a global system. Leach's analysis in terms of the logic of linguistics applied to ritual can be extended to this global system, and, in his own words:

"the fields of kinship, economy, politics, law, religion etc., are all versions of the same thing. Each mode of expression of a particular relationship is metaphoric of all other" (1982: 158). In forging the notion of ritual language, it seems to me that Leach was formulating his version of Mauss's "total social fact." Both notions refer to the totality of society and its institutions, and to the mechanism by which individual interests combine to make a social system. In a way, Leach's conclusion of his Social Anthropology could, retrospectively, be used fruitfully as an entry point for understanding some important arguments in his Political Systems: "The thesis that runs through this book is that it is only when we come to understand that relationships between man and man and man and god are, at least in a metaphorical sense, the equivalent of economic indebtedness, that we can really appreciate how this transformation of economics into ideology through the mediation of kinship actually occurs. So it is really Chapter 5 [Debt, Relationship, Power], which elaborates a sociological theory of debt, which provides the keystone to my argument " (1982: 223). And it is very much the same argument that lies behind his treatment of his Kachin ethnography and his use of the notion of ritual language.

For recent studies on Naxi (and Moso) see for exampleOppitz and Hsu (eds), 1998;[START_REF] Mathieu | History and Anthropological Study of the Ancient Kingdoms of the Sino-Tibetan Borderland-Naxi and Mosuo[END_REF]

2 For the Lisu people of the upper Salween river valley, see the long introduction inDessaint and Ngwâma, 1994. 3 Nu and Dulong are respectively the standardized form (and official name in Chinese) of the names of these two groups. The Drung are part of a linguistic group together with the Nung and the Rawang of Burma. About the relationships between these groups, seeGros,

 4 The French missionaries of the Foreign Missions Society of Paris were among the first westerners to discover this part of the Sino-Tibetan borders, and because of their long term relationships with the local populations, they also were able to provide detailed information about many aspects of this area, including its geographical, cultural and political particularities. See[START_REF] Gros | Terres de confins, terres de colonisation. Essai sur les Marches sino-tibétaines du Yunnan à travers l'implantation de la Mission du Tibet[END_REF]Gros, , 2001. 5 . 5 The anthropological research I undertook about the Drung (Dulong) people, and the several fieldwork assignments I conducted between 1998 and 2003 for a total of eighteen months, was largely funded by a Franco-Chinese Bilateral Grant(1997)(1998), a Lavoisier Grant of the French Foreign Affairs (1999), the financial support of the France Fundation (1999), and that of the Louis Dumont's Fund for SocialAnthropology (2003).

See Corlin (1994) who suggests that the "ritual language" of ethnic relations is patterned after a general conceptual model regarding the universe and man's place in the world. The all-pervading and integrative marriage system also contributes to the blurriness of ethnic categories; see Robinne, this volume.

The total Drung (Dulong) population in China is estimated as 7,426 (National Bureau of Statistics of China 2002: 97), making up one of the smallest official minority nationalities in China. More than 5000 Drung live in the remote Dulong valley bordering Burma.

See Galey (1980: 145) on the creditor-debtor paradigm. For further details, see[START_REF] Gros | Terres de confins, terres de colonisation. Essai sur les Marches sino-tibétaines du Yunnan à travers l'implantation de la Mission du Tibet[END_REF][START_REF] Gros | Ritual and Politics: Missionaries Encounters with Local Culture in Northwest Yunnan[END_REF]Gros , 2005: chap. 3 and 4. : chap. 3 and 4.

See Goré, 1923: 388-390; Ward, 1934: 52.

This point was highlighted by Lazcano Nebreda,

1998: 229-231. 19 About the pawning of persons, seeTestart, 2001: chap. 3. 20 From a theoretical point of view, the discussion inTestart (2001: 115, 151-152) is to me the most rigorous.

About the Kachin,Leach (1968: 161) mentioned that chiefs gained their reputation by having many slaves, but there were many economic advantages to the servile condition. Contractual slavery was a means to obtain economic credit and political protection from the chief.

There are historical documents that refer to the period of the seventeenth century, and later on to the period when the French missionaries were present (end of the nineteenth century).

On that point see alsoSagant, 1990: 163.

Most probably Huobajie, the Chinese name for the Festival of the torches.

See Bouchery, this volume about the Feasts of Merit among the Naga. These feasts could also accurately be refered to as "cults of fertility."

The problem of religious conversion among the Drung is the topic of a work in progress.

This point was already made clear in a comparative perspective in Russell (ed.), 1989.