Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity.

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Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity.

Marc Bonnet

To cite this version:

Marc Bonnet. Regularized direct and indirect symmetric variational BIE formulations for three- dimensional elasticity.. Engineering Analysis with Boundary Elements, Elsevier, 1995, 15, pp.93-102.

�10.1016/0955-7997(95)00022-G�. �hal-00092368v2�

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Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity.¹

Marc Bonnet

Laboratoire de M´ecanique des Solides (URA CNRS 317) Ecole Polytechnique, 91128 Palaiseau Cedex, France

[email protected], (1)-69333026 (fax)

1Eng. Anal with Bound. Elem.,15, 93–102 (1995)

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1 Introduction and motivation.

Although variational Galerkin-type boundary integral equation (GBIE) formulations have received a lot of attention in the past, mostly in the fielf of applied mathematics (see e.g. Bendali [2], Cortey-Dymont [8], Nedelec [15], Wendland [22]), developments and applications in the field of solid mechanics (see Sirtori et al. [21], Parreira and Guiggiani [18], Becache [1], Nishimura and Kobayashi [17], Kane and Balakrishna [11]) have been few compared to the very popular collocation boundary integral equation (CBIE) methods.

Variational approaches do have attractive conceptual features compared to usual collocation methods: known convergence results are more extensive [15] and actually better than those of CBEM using the same discretization. They lead to symmetric stiffness-like BEM matrices, so that the solution CPU time, which dominates for large DOF numbers, is asymptotically half that necessary for solving the full nonsymmetric system produced by collocation methods. Indeed the numerical examples in Balakrishna et al. [12] show clearly that, for a given discretization, GBEA becomes computationally more efficient than CBIE for sufficiently large numbers of DOFs.

GBIE formulations can be viewed as weighted residual formulations of CBIEs, and thus usually involve double surface integrals (see however Bui [7], where a symmetric BIE formulation with a single surface integration is discussed). In order to get a symmetric GBIE formulation, weighted residual forms of displacement and traction CBIEs are used respectively on the Dirichlet and Neumann parts of the boundary. Although the collocation traction BIE is known to require aC^1,α continuity for the boundary displacement u at the collocation point, which causes some technical difficulties especially when using curved elements for 3-D situations, its Galerkin counterpart requires only aC^0,αcontinu- ity foru. The usualC⁰ conformal shape functions can then be used without difficulty in a GBEA context.

On the other hand, the implementation of GBIE formulations for general engineering problems is somewhat more involved than that of displacement CBIE, which explains perhaps its relative lack of popularity. Also, investigation of GBIE formulations for mixed boundary value problems began only recently. Another important issue is the handling of singular integrals: the double boundary integrals that appear in Galerkin-type BIE

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formulations are convergent but some of the successive simple integrals are not. One possibility is a proper direct handling of the strongly singular and hypersingular simple surface integrals that appear, using either a numerical method (Guiggiani et al. [9]) or analytical integration of the singularity (Sirtori et al. [21], Balakrishna et al. [12]; the latter used symbolic computation). Another possibility is to perform, as is done here, an analytical regularization before any discretization.

In the present paper, we investigate the statement of a GBIE formulation for the mixed boundary value problem of linear elastostatics, for either interior or exterior problems, obtained from the stationarity conditions associated to the minimization principle of an augmented potential energy. ThisvariationalBIE formulation appears to be precisely that obtained from a weighted residual viewpoint. Also, the present formulation is regularized in the spirit of previous works on indirect regularization of displacement CBIEs [4], [5], [6] and of traction BIEs ([17]). The resulting variational BIE formulation is expressed in terms of weakly singular inner surface integrals followed by outer regular integrals. Also, it is a direct formulation, whose unknowns are the physical displacements and tractions on the boundary. As a by-product, the indirect variational BIE formulation for mixed problems, whose unknowns are fictitious densities, is also established from the direct one.

Finally, some implementation features are discussed.

2 Elastic problem, potential energy and stationarity conditions

Let Ω denote a three-dimensional elastic body, either bounded or unbounded;n denotes the unit normal vector outwards from Ω. The elastic displacement u(y) (y ∈ Ω) is governed by the homogeneous Navier equation:

C_ijabu_a,bj = 0 (1)

whereCdenotes the Hooke tensor of elastic constants. The comma indicates partial differ- entiation; Einstein summation convention is used throughout. Besides, mixed boundary

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conditions are considered:







u = u^D on S_u Tⁿ(u) = t^D on ST

(2) whereS_u, S_T =∂Ω−S_u are complementary portions of ∂Ω and

Tⁿ(u) = [C :ε(u)].n

is the elastic traction vector associated withu. If Ω is unbounded, suitable decay conditions at infinity must be added to (2).

The solutionuto the elastic problem (1-2) minimizes the following augmented potential energy functionalE(v):

E(v) = Z

Ω

ε(v) :C :ε(v) dV − Z

ST

t^D.vdS− Z

Su

t.(v−u^D) dS (3) The last integral term in (3) has the effect thatuis an unconstrainedminimizer ofE(v):

no kinematical constraints are imposed on v. Thus u can be sought such that the first variation ofE vanishes atv =u, i.e., letting v=u+δu, ve have:

δE(u).δu= Z

Ω

ε(u) :C :ε(δu) dV − Z

S_T

t^D.δudS− Z

Su

t.δudS = 0 (4) with no kinematical constraint onδu. Indeed, from

Z

Ω

ε(u) :C :ε(δu) dV = Z

∂Ω

Tⁿ(u).δudS− Z

Ω

div [C :ε(u)].δudV

it is readily shown that u solves (1-2) and that the Lagrange multiplier t equals the traction vector Tⁿ(u) on S_u.

Now let us suppose that the minimization ofE(v) is attempted only for thosevwhich satisfy the local equilibrium equation (1). Since this is also satisfied by the solution u, one is then led to restrict (4) to the trial functions δu =v−u which themselves satisfy (1). Upon integration by parts of the domain integral, the stationarity condition (4) thus takes the form

δE(u).δu= Z

Su

u^D.Tⁿ(δu) dS+ Z

ST

u.Tⁿ(δu) dS

− Z

ST

t^D.δudS− Z

Su

t.δudS = 0 (5)

which constitutes the starting point for the derivation of a direct variational BIE formulation.

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3 Test functions

The next important step is the actual construction of test functions δu that satisfy (1).

In order to do so, recall that any such δu admits the integral representation (x interior to Ω):

δu_k(x) =− Z

∂Ω

δu_i(y)n_j(y)Σ^k_ij(x,y) dS_y + Z

∂Ω

δt_i(y)U_i^k(x,y) dS_y (6) whereδtis the traction vectorTⁿ(δu). Similarly, anyδu⁺which solves (1) in theexterior domain Ω⁺ =R³−Ω satisfies at any x interior to Ω the complementary (with reference to Ω⁺) integral representation:

0 =− Z

∂Ω

δu⁺_i (y)n_j(y)Σ^k_ij(x,y) dS_y+ Z

∂Ω

δt⁺_i (y)U_i^k(x,y) dS_y (7) where δt⁺ stands for Tⁿ(δu⁺). The normal n outwards from Ω is used in both (6) and (7). The symbolsU_i^k(x,y),Σ^k_ij(x,y) denote the i−and ij-components of the elastic displacement and stress fields created at y by a unit point force applied at x along the k-direction (elastostatic fundamental solution). The full space (Kelvin) solution, a half space solution with a free-surface condition (Mindlin), or any other fundamental solution defined on a subset ofR³ that includes Ω may be used for this purpose. Any of them has the symmetry property:

U_k^a(x,y) =U_a^k(y,x) (8) which in turn implies:

Cijab

∂

∂x_bU_a^k(x,y) = Σ^a_ij(y,x) Cijab

∂

∂x_bΣ^a_kl(x,y) = Ck`ab

∂

∂y_bΣ^a_ij(y,x) (9) Next, (7) is subtracted from (6), giving:

δuk(x) = Z

∂Ω

˜

ui(y)nj(y)Σ^k_ij(x,y) dSy− Z

∂Ω

t˜i(y)U_i^k(x,y) dSy

using the displacement and traction jumps accross ∂Ω: ˜u =δu⁺−δu and ˜t =δt⁺−δt.

It is indeed legitimate to restrict the above formula by imposingδu⁺=δu, i.e. ˜u=0on S_u and δt⁺ =−δt, i.e. ˜t =0 on S_T, which gives:

δu_k(x) = Z

ST

˜

u_i(y)n_j(y)Σ^k_ij(x,y) dS_y − Z

Su

˜t_i(y)U_i^k(x,y) dS_y (10)

=δu^T_k(x) +δu^U_k(x) (11)

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Such test functions δu satisfy (1) and reflect the boundary condition structure of the initial mixed elastostatic problem under study. Note that, since ˜u must be continuous all over∂Ω, one has:

˜

u|_∂S˜T=0 (12)

The stress tensorδσ =σ(δu) is then given from (10) by the representation:

δσij(x) = Cklab

Z

ST

˜

uk(y)n`(y) ∂

∂y_bΣ^a_ij(y,x) dSy− Z

Su

˜tk(y)Σ^k_ij(y,x) dSy (13) where use has been made of (8), (9).

4 Derivation of the direct symmetric bie formulation

Basically the symmetric BIE formulation comes from the substitution of (10), (13) into the stationarity equation (5). This leads to a formulation in terms of double surface integrals. However, integrability problems, related to the presence of kernels behaving like r⁻¹, r⁻², r⁻³ where r=|y−x|(especially the last two), require a limiting process of some kind.

4.1 Introduction of an auxiliary surface

Let ˜Sbe a closed, regular surface exterior to Ω (hence exterior to∂Ω for interior problems, but interior to ∂Ω for exterior problems), defined by means of a one-to-one mapping F onto∂Ω:

y∈∂Ω→z=F(y)∈S˜

which is left unspecified. This surface ˜S is substituted to ∂Ω in (10), (13), following an approach used by Sirtori et al. [21]. The idea is to perform some analytical manipulations, for regularization purposes, on the double surface integrals over ∂Ω×S˜ (in which case the elastic kernels are nonsingular) and then consider the limiting process ˜S →∂Ω.

The test functions δu, δσ are thus given by:

δu_k(x) = Z

S˜T

˜

u_i(y)n_j(y)Σ^k_ij(x,y) dS_y− Z

S˜u

˜t_i(y)U_i^k(x,y) dS_y (14) δσij(x) = Cklab

Z

S˜T

˜

uk(y)n`(y) ∂

∂y_bΣ^a_ij(y,x) dSy− Z

S˜u

˜tk(y)Σ^k_ij(y,x) dSy (15)

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x

y z S~

n _

Ω _

_ _

Figure 1: Boundary ∂Ω and auxiliary surface ˜S

(with ˜S_u =F(S_u),S˜_T =F(S_T)). Recall that eqns. (14), (15) apply for Ω either bounded or unbounded, nbeing the unit normal outwards from Ω.

Upon substitution of (14), (15), the stationarity condition (5) for E(u) can be split into two independent parts:







∀u˜ δE^T = 0

∀˜t δE^U = 0

(16) whereδE^T andδE^U collect the terms containing ˜u and ˜t respectively that arise from the substitution of (14), (15) into (5).

4.2 Evaluation of δE^U

From eqns (5), (14), (15), one has:

δE^U =I1+I2 (17)

I₁ = Z

Su

t_k(x) Z

S˜u

˜t_i(z)U_i^k(x,z) dS_zdS_x +

Z

ST

t^D_k(x) Z

S˜u

˜t_i(z)U_i^k(x,z) dS_zdS_x (18) I₂ =−

Z

Su

u^D_i(x)n_j(x) Z

S˜u

˜t_k(z)Σ^k_ij(z,x) dS_zdS_x

− Z

S_T

u_i(x)n_j(x) Z

S˜u

˜t_k(z)Σ^k_ij(z,x) dS_zdS_x (19) First, the termI₁ is made of a (potentially) weakly singular surface integral followed by a regular one and thus needs no regularization; it is left unchanged.

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Then, I₂, eqn. (19), is rewritten as:

I₂ =− Z

S˜u

˜t_k(z) Z

Su

[u^D_i(x)−u^D_i(y)]n_j(x)Σ^k_ij(z,x) dS_xdS_z

− Z

S˜u

t˜_k(z) Z

ST

[u_i(x)−u^D_i(y)]n_j(x)Σ^k_ij(z,x) dS_xdS_z

− Z

S˜u

t˜_k(z)u^D_i(y) Z

∂Ω

n_j(x)Σ^k_ij(z,x) dS_xdS_z (20) wherey ∈Su is the point such that F(y) =z ∈S˜u (hence u(y) =u^D(y)). Integrations over∂Ω and ˜S have been interchanged, this operation being valid because of the current nonsingular character of the integrals.

Besides, one notes that the fundamental stress tensor satisfies, by virtue of equilibrium, the following identity (the source pointz being exterior to Ω):

Z

∂Ω

n_j(x)Σ^k_ij(z,x) dS_x =κδ_ik (21) whereκ= 0 (Ω bounded,nexterior to∂Ω) orκ= 1 (R³−Ω bounded,ninterior to∂Ω).

Thus, the last integral in (20) becomes:

Z

S˜u

t˜_k(z)u^D_i(y) Z

∂Ω

n_j(x)Σ^k_ij(z,x) dS_xdS_z =κ Z

S˜u

˜t_k(z)u^D_k(y) dS_z (22) These manipulations result in a regularization of the initially strongly singular integral w.r.t. x in (19). Indeed one notes that, since

U(z,x)∼ |z−x|⁻¹ Σ(z,x)∼ |z−x|⁻²

the expressions (18), (20) involve weakly singular integrals w.r.t. x followed by nonsingular integrals w.r.t. z, provided the regularity requirementu ∈C^0,α(∂Ω), i.e.:

∃C >0, ∃α∈]0,1] |u(y)−u(x)| ≤C|y−x|^α (23) is met. Then, the limiting expression of I₂, I₁ for ˜S → ∂Ω is obtained by a mere substi-

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tution of ( ˜S_u,z) by (S_u,y) in eqns. (18), (20). The result is thus:

I₁ = Z

Su

t_k(x) Z

Su

t˜_i(y)U_i^k(x,y) dS_ydS_x +

Z

ST

t^D_k(x) Z

Su

˜t_i(y)U_i^k(x,y) dS_ydS_x (24) I₂ =−

Z

Su

˜t_k(y) Z

Su

[u^D_i(x)−u^D_i(y)]nj(x)Σ^k_ij(y,x) dSxdSy

− Z

Su

˜t_k(y) Z

ST

[u_i(x)−u^D_i(y)]n_j(x)Σ^k_ij(y,x) dS_xdS_y

−κ Z

S˜u

˜t_k(z)u^D_k(y) dS_y (25)

For eqn. (25) above, use has been made of the substitution of (21) into (19).

4.3 Evaluation of δE^T

From eqns (5), (14), (15), one has:

δE^T =J₁+J₂ (26)

J₁ =− Z

Su

t_k(x) Z

S˜T

˜

u_i(z)n_j(z)Σ^k_ij(x,z) dS_zdS_x

− Z

ST

t^D_k(x) Z

S˜T

˜

u_i(z)n_j(z)Σ^k_ij(x,z) dS_zdS_x (27) J₂ =

Z

Su

u^D_i(x)n_j(x)C_klab Z

S˜T

˜

u_k(z)n_`(z) ∂

∂z_bΣ^a_ij(z,x) dS_zdS_x +

Z

ST

u_i(x)n_j(x)C_klab Z

S˜T

˜

u_k(z)n_`(z) ∂

∂z_bΣ^a_ij(z,x) dS_zdS_x (28) First, J₁ is subjected to a treatment similar to that made to I₂. For this purpose, eqn.

(27) is rewritten, upon addition and subtraction of ˜u(x) to ˜u(z), as:

J₁ =− Z

Su

t_k(x) Z

S˜_T

˜

u_i(z)n_j(z)Σ^k_ij(x,z) dS_zdS_x

− Z

ST

t^D_k(x) Z

S˜T

[˜u_i(z)−u˜_i(x)]n_j(z)Σ^k_ij(x,z) dS_zdS_x

− Z

ST

t^D_k(x)˜u_i(x) Z

S˜T

n_j(z)Σ^k_ij(x,z) dS_zdS_x (29) where the fact that ˜u = 0 for x ∈ Su has been used. Besides, the fundamental stress tensor satisfies the following identity, similar to (21) but with the source point x now being interior to Ω:

Z

S˜

nj(z)Σ^k_ij(x,z) dSz = (κ−1)δik (30)

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where againκ = 0 (Ω bounded, nexterior to ∂Ω) or κ = 1 (R³−Ω bounded, n interior to∂Ω). The last integral in (29) then becomes:

Z

ST

t^D_k(x)˜u_i(x) Z

S˜u

n_j(z)Σ^k_ij(x,z) dS_zdS_x

= Z

ST

t^D_k(x)˜u_i(x)

(κ−1)δ_ik− Z

S˜u

n_j(z)Σ^k_ij(x,z) dS_z

dS_x

= (κ−1) Z

ST

t^D_k(x)˜u_k(x) dS_x− Z

ST

t^D_k(x)˜u_i(x) Z

S˜u

n_j(z)Σ^k_ij(x,z) dS_zdS_x (31) Substitution of (31) into (29) allows for an expression ofJ₁ which is made, in the limiting case ˜S →∂Ω, of weakly singular integrals over S_T followed by nonsingular integrals over S_u, S_T, as follows:

J₁ =− Z

Su

t_k(x) Z

ST

˜

u_i(y)n_j(y)Σ^k_ij(x,y) dS_ydS_x

− Z

ST

t^D_k(x) Z

ST

[˜u_i(y)−u˜_i(x)]n_j(y)Σ^k_ij(x,y) dS_ydS_x + (1−κ)

Z

ST

t^D_k(x)˜u_k(x) dS_x +

Z

ST

t_k(x)˜u_i(x) Z

Su

n_j(y)Σ^k_ij(x,y) dS_ydS_x (32) The weakly singular character of the inner integral in the above equation relies upon ˜u being C^0,α-continuous over ∂Ω, eqn. (23). The property (12) of ˜u, together with (23), implies:

∃C > 0, ∃α∈]0,1] |u(y)| ≤C|y−x| x∈S_u,y∈S_T and thus also contributes to the weakly singular character of (32):

Next, let us consider the second integral J₂ in (26), which involves the hypersingular kernel C_klab_∂z^∂

bΣ^a_ij(z,x). Following an argument given by Nishimura and Kobayashi [16]

and using the symmetry property (9), one notes that, for x6=y:

∂

∂x_j

C_klab ∂

∂x_bΣ^a_ij(z,x)

=C_klab ∂

∂x_b ∂

∂x_jΣ^a_ij(z,x)

= 0

∂

∂z_l

C_klab ∂

∂z_bΣ^a_ij(z,x)

=C_ijab ∂

∂x_b ∂

∂z_lΣ^a_kl(x,z)

= 0 This implies the existence of a fourth-order tensor Asuch that:

C_klab ∂

∂z_bΣ^a_ij(z,x) =e_iepe_{jf q}e_kgre_lhs ∂

∂x_e

∂

∂x_f

∂

∂z_g

∂

∂z_hA_pqrs(z,x) (z6=x) (33)

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wheree_ijkare the components of the permutation tensor. For instance, the tensorA(z,x) associated with the three-dimensional isotropic Kelvin fundamental solution is given (Ned- elec [15]) by:

A_pqrs(z,x) = µ 8π

δ_prδ_qs+δ_psδ_qr+ 2ν

1−νδ_pqδ_rs

|z−x| (34) The decomposition (33) allows for the use of the following variant of Stokes’ formula:

Z

S

ge_abcn_af_,bdS = − Z

S

f e_abcn_ag_,bdS

where S is any regular surface; moreover either S is closed or f = 0 on ∂S. Here it is applied twice (once w.r.t. z and once w.r.t. x) to J₂, eqn. (28); this gives:

J₂ = Z

Su

(Ru^D)_iq(x) Z

S˜T

(R˜u)_ks(z)B_ikqs(z,x) dS_zdS_x +

Z

ST

(Ru)_iq(x) Z

S˜T

(Ru)˜ _ks(z)B_ikqs(z,x) dS_zdS_x (35) using the notations

(Ru)_iq =e_{jf q}n_ju_i,f (36)

B_ikqs(z,x) =e_iepe_kgr ∂

∂x_e

∂

∂z_gA_pqrs(z,x) (37) The result (35) makes use of the property (12) of ˜u. One notes that

B(z,x)∼ |z−x|⁻¹

which stems from (33), (37). This implies that eqn. (35) involves a weakly singular inner (w.r.t. x) integral followed by a nonsingular outer (w.r.t. z) integral. The limiting expression of J₂ for ˜S → ∂Ω is thus obtained by a simple replacement of ( ˜S_T,z) with (ST,y) in (35):

J2 = Z

Su

(Ru^D)iq(x) Z

ST

(Ru)˜ ks(y)Bikqs(y,x) dSydSx

+ Z

S_T

(Ru)_iq(x) Z

S_T

(R˜u)_ks(y)B_ikqs(y,x) dS_ydS_x (38)

4.4 Statement of the direct symmetric BIE formulation

The previous results are now put together. From (16), the definitions (17), (26) and the results (24), (25), (32), (38), the direct symmetric BIE formulation for the mixed elastic

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boundary value problem reads:

Find u∈ V_u,t ∈ V_T







Buu(u,u) +˜ Btu(t,u) =˜ Lu( ˜u) B_ut(u,˜t) +B_tt(t,˜t) = L_t(˜t)

∀u˜ ∈ V_u⁰, ∀˜t∈ V_T (39) using the bilinear forms

B_uu(u,u) =˜ Z

ST

(Ru)_iq(x) Z

ST

B_iqks(y,x)(Ru)˜ _ks(y) dS_ydS_x (40) B_tu(t,u) =˜ −

Z

Su

Z

ST

t_k(x)˜u_i(y)n_j(y)Σ^k_ij(x,y) dS_ydS_x (41) B_ut(u,˜t) =−

Z

Su

Z

ST

˜t_k(y)[u_i(x)−u^D_i(y)]n_j(x)Σ^k_ij(y,x) dS_xdS_y (42) B_tt(u,˜t) =

Z

Su

Z

Su

t_k(x)˜t_i(y)U_i^k(x,y) dS_ydS_x (43) and the linear forms

L_u( ˜u) = (κ−1) Z

ST

t^D_k(x)˜u_k(x) dS_x +

Z

ST

t^D_k(x) Z

ST

[˜u_i(y)−u˜_i(x)]n_j(y)Σ^k_ij(x,y) dS_ydS_x

− Z

ST

t^D_k(x)˜ui(x) Z

Su

nj(y)Σ^k_ij(x,y) dSydSx

− Z

ST

(Ru^D)_iq(x) Z

Su

B_iqks(y,x)(R˜u)_ks(y) dS_ydS_x (44) L_t(˜t) =κ

Z

Su

˜t_k(x)u^D_k(x) dS_x +

Z

Su

t˜_k(y) Z

Su

[u^D_i(x)−u^D_i(y)]n_j(x)Σ^k_ij(y,x) dS_xdS_y

− Z

Su

Z

ST

t^D_k(x)˜t_i(y)U_i^k(x,y) dS_ydS_x (45) and where the spaces of admissible functionsV_u,V_u⁰,V_T are defined as:

V_u =

u|u∈C^0,α(S_T) and u=u^D on∂S_T V_u⁰ =

˜

u|u˜ ∈C^0,α(S_T) and ˜u=0 on∂S_T VT =˜t|˜t piecewise continuous onST

Note that, contrary to (u,u), no restriction is made on the values of˜ t,˜t along∂S_T. Eqns. (39) to (45) constitute the main result of this paper. Being the stationarity condition (5) applied to an appropriate family of test functions δu, they define a variational BIE formulation for the mixed elastostatic boundary-value problem. In addition,

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the formulation is symmetric by virtue of the relations

∀(u,u)˜ ∀(t,˜t)











B_uu(u,u) =˜ B_uu( ˜u,u) Btt(t,˜t) = Btt(˜t,t) B_ut(u,˜t) = B_tu(˜t,u)

(46)

which are visible consequences of formulae (40), (41), (42), (43). Those symmetry properties may be viewed as consequences of the symmetry properties (8), (9), which are true for any elastostatic fundamental solution.

Moreover, the present variational symmetric BIE formulation is direct, i.e. expressed in terms of the physical elastic boundary unknownsu|_S_T,t |_S_u. The prescribed displacements u^Dand tractions t^D appear in the linear forms L_u and L_t.

Also, the formulation (39 to 45) is valid whatever fundamental solution is used. How- ever, the kernelB(y,x) depends on the actual fundamental solution used and is currently known analytically only in a few cases like the Kelvin solution (see eqn. (34) for the isotropic case or Becache [1] for the anisotropic case).

The present formulation is established for elastic domains either bounded or unbounded. They slightly differ from each other by the presence of the coefficient κ in the expressions (44), (45) for the linear forms Lu,Lt.

Finally, the regularization of theJ₂ term, subsection 4.3, was first proposed by Nedelec [15], for Neumann problems. It is valid providedu,u˜ areC^0,α-continuous over∂Ω andS_T respectively. On the other hand, the weighted residual formulation of the traction CBIE requires a C^1,α continuity for u at the collocation point no matter which integration technique is used. This is of practical importance since C^0,α interpolations are easier to handle and more flexible, especially for 3-D situations.

4.5 Relation to weighted residual statements

The displacement and traction exterior representation formulas are respectively:

0 = Z

∂Ω

ui(x)nj(x)Σ^k_ij(z,x) dSx− Z

∂Ω

ti(y)U_i^k(z,x) dSx (47) 0 =n_j(z)

C_ijab

Z

∂Ω

u_k(x)n_`(x) ∂

∂z_bΣ^a_k`(z,x) dS_x− Z

∂Ω

t_k(y)Σ^k_ij(z,x) dS_x

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