about the paper entitled “ A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity

An integral formulation for heat conduction problems in non-homogeneous media has recently been proposed in [1]. The goal of this communication is to revisit and clarify two key features of the formulation of [1]. First, the contention that the integral equation formulation proposed in [1] does not possess the desired boundary-only character is made and substantiated; it is shown in particular that Eq. (10) therein does not hold due to the fact that a crucial requirement for the fundamental solution, Eq. (5c-d) of [1] is actually not met. Second, the limiting process associated with a vanishing neighbourhood in connexion with the particular kernel function used in [1] is revisited. The purpose of the paper under discussion [1] was to establish a boundary-only integral formulation for heat conduction with isotropic but spatially varying conductivity k(x) (x = (x, y): field point), i.e. for temperature distributions T such that div [k(x)∇T (x)] = 0 (1) In order to do so, a variant kind of fundamental solution, denoted E(x, ξ), associated to a forcing term D(x, ξ) containing a singular source at a given point ξ = (xi, yi), is introduced: div [k(x)∇E(x, ξ)] = −D(x, ξ) (2) 1 Examination of the sampling property The following, crucial, requirement on the forcing term D(x, ξ) (Eqs. (5c-d) of [1]) must be met in order to get rid of the domain integral that arises in the reciprocal theorem (Ω denotes an arbitrary domain required only to enclose the source point ξ): ∫ Ω T (x)D(x, ξ) dΩ(x) = T (ξ)A(ξ) with A(ξ) = ∫ Ω D(x, ξ) dΩ(x) (3) The above relation is the analogue, for the particular fundamental solution at hand, of the sampling property of the Dirac distribution (the term sifting property was used in [1]). However, the kernel E(x, ξ) obtained by the authors, i.e. (Eq. (23) of [1]): E(x, ξ) = − 1 2π ∫ dr rk̃(r;xi, yi) (4) ∗Engineering Analysis with Boundary Elements, 22:235–240 (1998) †Laboratoire de Mécanique des Solides (UMR CNRS 7649), Ecole Polytechnique, 91128 Palaiseau cedex, France ‡Dipartimento di Matematica, Università degli Studi di Siena, Via del Capitano 15, 53100 Siena, Italy 1 where (in 2D) (x, y) = (xi + r cos θ, yi + r sin θ) and having introduced for convenience the ‘averaged’ conductivity k̃: k̃(r;xi, yi) = 1 2π ∫ 2π 0 k(r, θ;xi, yi)dθ (5) happens to violate the crucial requirement (3). Indeed, the kernel given by Eq. (4) is readily seen to solve the equation div [k̃∇E] = −δ(x− ξ) = 1 r ∂ ∂r [ rk̃ ∂E ∂r ] (6) (the second equality holds because ∂E/∂θ = 0). Hence, by subtracting the above equation from Eq. (2), one gets the following expression for the forcing term D: D(x, ξ) = δ(x− ξ) + 1 r ∂ ∂r [ r(k̃ − k) ∂r ] = δ(x− ξ) + 1 2πr ∂ ∂r [ 1− k k̃ ] ≡ δ(x− ξ) + ∆D (7) Except when the material is homogeneous, the complementary term ∆D is therefore in general nonzero at x 6= ξ, ie outside the source point ξ; this is indeed obvious in some of the examples of the paper under discussion (see Eqs. (40,46) and Figs. 3,5,12). In fact, for sufficiently smooth conductivities, k− k̃ = O(r) so that ∆D is even nonsingular at x = ξ. But then, it is impossible for properties (3) to hold. In fact, using the definition of A(ξ) for a given domain Ω, one has: ∫ Ω T (x)D(x, ξ) dΩ = A(ξ)T (ξ) + ∫ Ω [T (x)− T (ξ)]∆D(x, ξ) dΩ ≡ A(ξ)T (ξ) + J(ξ) (8) and property (3) would mean that the integral J(ξ) in the above formula, which measures the absolute error in the realization of (3), vanishes for any domain shape and temperature distribution, which is false by Haar’s lemma since D is not identically zero outside the source point. For example, Eq. (5c-d) of [1] for the same function T (which incidentally is required to have compact support) but two different domains Ω1 ⊂ Ω2 would lead to: ∫ Ω2\Ω1 T (x)D(x, ξ) dΩ(x) = T (ξ)[A2(ξ)−A1(ξ)] and a contradiction arises. From the above equation, the integral in the l.h.s. should, apart from the geometrical arrangement of ξ,Ω1,Ω2, depend only on the value of T at the source point ξ. But since Ω2 \ Ω1 does not contain ξ, the above equation should hold true for any extension of T outside Ω2 \Ω1, which is absurd. In other words, as soon as D(x, ξ) is nonzero over Ω2 \Ω1, the integral ∫ Ω2\Ω1 T (x)D(x, ξ) dΩ(x) cannot depend on T only through the value T (ξ). Thus, Eq. (3) is in general not true. As a consequence, the two members of the integral equation (10) in [1] are in general not equal. 2 Limiting process for vanishing neighbourhood Let ξ be a fixed point on the boundary ∂Ω of a two-dimensional domain Ω. We consider an exclusion neighbourhood vε(ξ) of ξ, of radius ≤ ε (Figure 1). For any ε > 0, ξ is always an external point for the domain Ωε(ξ) = Ω \ vε(ξ) whose boundary ∂Ωε is given by ∂Ωε = (∂Ω− eε) + sε = Γε + sε, where sε = Ω ∩ ∂vε, eε = ∂Ω ∩ v̄ε, and Γε = ∂Ω− eε (v̄ε is the closure of vε). 2