Generalized Chessboard Structures Whose Effective Conductivities Are Integer Valued

Let h be a large positive integer and consider a periodic composite material with period equal to 1/h. The stationary heat conduction problem can then be formulated by the following minimum principle: where Here, u is the temperature; the conductivity matrix C_h(x) is given by where C(·) is periodic relative to the unit cube of ℝ^m, g is the source field, Ω is a bounded-open subset of ℝ^m, and the minimization is taken over some Sobolev space depending on the boundary conditions. It is possible to prove that the “energy” E_h converges to the so-called homogenized “energy” E_hom as h → ∞, defined by where The matrix σ^* is often called the homogenized or effective conductivity matrix and is found by solving a number of boundary value problems on the cell of periodicity. For an elementary introduction to the theory of homogenization, see, for example, the book Persson et al. [1].

There are very few microstructures where all elements of the effective conductivity matrix are known in terms of closed form explicit formulae. Laminates and chessboards are the most classical. Mortola and Steffé [2] studied in 1985 a chessboard structure consisting of four equally sized squares in each period with (isotropic) conductivities a, b, c, and d, respectively (see Figure 1). They conjectured that the corresponding effective conductivity matrix of this composite structure is given by where Here, A, G, and H denote the arithmetic mean, the geometric mean and the harmonic mean, respectively, given by the formulae: Mortola and Steffés conjecture was ultimately proved by Craster and Obnosov [3] and Milton [4] in 2001 (see also [5]).

For the connected case, it is easily verified that σ₁₁ = σ₂₂ = σ, where σ is given by the formula: Assume that a and b are positive integers. In order to obtain an integer value of σ, there must exist integers r₁, r₂, k, and l such that where We note that (2.2) is equivalent with Hence, It is clear that $$ (otherwise, $$ and r₂ = dt for some integers ts and d > 1, which would mean that $$, and since $$, we obtain that $$, which contradicts (2.3)). According to (2.5), this gives that for some integer s₂. By checking the four combinations when r₁ and r₂ are odd/even, we can easily verify that Hence, using (2.2), we obtain that where and s is some positive integer. The corresponding integer value of σ is then given by In order to have positive values of a and b, (2.8) shows that we only can choose values of r₁ and r₂ such that

Summing up, all integers a and b making σ integer valued are of the form (2.8) for some positive integers s, r₁ and r₂, satisfying (2.11), where d is given by (2.9). The corresponding value of σ is Conversely, all a and b of the form (2.8), satisfying (2.11), are positive integers and make σ integer valued.

For the chessboard case, σ₁₁ = σ₂₂ = σ, where This case is very simple. We just use the representation: where r₁, r₂, and k are integers (giving σ = kr₁r₂).

For the laminate case, we find that σ₁₁ = A(a, b) and σ₂₂ = H(a, b). It is possible to show that the integers a and b making the harmonic mean integer valued are precisely those on the form: (in this case, H(a, b) = 2tqp) and the form: (in this case, H(a, b) = t(2q + 1)(2p + 1)) where p, q, and t are positive integers. For a proof of this fact, see [6]. Therefore, if σ₁₁ = A(a, b) and σ₂₂ = H(a, b) are integer valued, (a, b) must belong to the class (3.3) or (3.4). The latter is directly seen to generate integer values also for A. However, (3.3) gives integer values of A only if both p and q are odd (for which (3.3) may be written on the form (3.4) with t replaced by 2t) or both even, or t is even. In any case, if both H and A are integers, we end up with the form: (in this case σ₁₁ = t(p+q)² and σ₂₂ = 4tqp) and the form: (in this case σ₁₁ = t(p+q+1)² and σ₂₂ = t(2q + 1)(2p + 1)).

As mentioned in the introduction, the effective conductivity matrix is found by solving a number of boundary value problems on the cell of periodicity. For the connected case, the effective conductivity σ can be found by solving the following boundary value problem: Here, Y is the unit cell Y = [0,1]² and the conductivity λ is defined by In addition, we must assume continuity of normal component of λ(x)  grad u through the four surfaces where λ(x) changes its value from a to b. The effective conductivity σ is then found by calculating the integral: The above boundary value problem can be solved numerically by using the finite-element method with relatively good accuracy. The hardest case is assumed to be the one when b = 0, but even in this case, we obtain a numerical value close to the exact one. Using a couple of thousand elements (built up by second-order polynomials), the numerical value of σ for the case a = 1 and b = 0 turns out to be 0.5773, which is close to the exact value: Using the class of integer values for a and b giving integer valued effective conductivity, we may relatively easily make internet-based student projects where the students themselves are supposed to train their skills in using FEM programs, simply by randomly generating a and b from (2.8) and ask the students to find the integer which is the closest to their numerical estimate. Without knowing the exact formula, there are no way that they can guess the correct value without doing the numerical FEM calculation. The actual evaluation can be done by a simple Java-script or other programs which can register the students progress.

For the chessboard structure, we can use the same method with the only difference that λ(x) is defined by However, the numerical estimation of the effective conductivity turns out to be significantly more difficult. In order to illustrate, we have made a numerical estimation for the case when a = 1 and b = 10000. Even with about 10000 quadrilateral 8-node elements (with increasing number of elements close to the midpoint of the unit-cell), our numerical value turned out to be as high as 557, which is more than 5 times higher than the actual value which is σ = 100. There exists a numerical method which is much more efficient than the finite-element method in such problems. Concerning this, we refer to the paper [7].