Comments about the paper entitled “A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity” by A. J. Kassab and E. Divo
Introduction
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 (: field point), i.e. for temperature distributions T such thatIn order to do so, the variant kind of fundamental solution, denoted , associated with a forcing term containing a singular source at a given point , is introduced:
Section snippets
Examination of the sampling property
The following, crucial, requirement on the forcing term (eqs. (5c) and (5d) of Ref. [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 ):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 Ref. [1]
Limiting process of vanishing neighbourhood
Let be a fixed point on the boundary ∂Ω of a two-dimensional domain Ω. We consider an exclusion neighbourhood of , of radius ≤ε (Fig. 1). For any ε>0, is always an external point for the domain whose boundary ∂Ωε is given bywhere sε=Ω⋂∂vε, , and Γε=∂Ω−eε ( is the closure of vε).
The singular integral equation for heat conduction problems (T: unknown temperature field) in non-homogeneous media can then be sought as the limiting case, for ε
Discussion
From the analysis conducted in Section 3, the integral representation in Eq. (13)and the integral equation in Eq. (15)involve, in general, a domain term. It is thus not correct to state, as was done in Ref. [1], that recourse to the kernel E leads to boundary-only integral formulations for heat conduction problems in non-homogeneous media. However, since E is in some sense an “averaged” version associated with the “averaged” conductivity k̃, of the (unknown) fundamental solution associated with
Illustrative numerical examples
In order to illustrate the above considerations and have some quantitative idea of the approximate character of Eq. (3), two 2D examples are now considered.
The first example is analytical. We take k=(1+x)(1+y) and Ω={0≤x, y≤2}. A solution to Eq. (1)is T=(1+x)2−(1+y)2. One then findsTo keep analytical calculations simple, we consider only source points on the horizontal median of Ω, i.e. such that yi=1. After some
Conclusion
From the present analysis, our contention is that the approach proposed in Ref. [1], contrary to the claim made in that paper, does not lead to an integral equation without domain integrals. In particular, the integral equation, eq. (10) of Ref. [1], is found to be only approximately, not exactly, satisfied. The arguments detailed here for two-dimensional situations extend to three dimensions as well.
One may notice that the equations in Eq. (3)are not discussed or verified a posteriori in the
References (1)
- et al.
A generalized boundary integral solution for heat conduction problems in non-homogeneous media
Engineering Analysis with Boundary Elements
(1996)
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