Patent ID: 8204690

Claim:
A computer-implemented method for modeling an average flow of a substance or heat through a periodic and a non-periodic material, said method comprising: said computer defining an effective conductivity tensor, K å , comprising main diagonal values and off-diagonal values, wherein said main diagonal values are computed using K i ⁢ ⁢ i a . = 0.5 ⁢ ( K i ⁢ ⁢ i a + K i ⁢ ⁢ i g ) where K i ⁢ ⁢ i a ⁢ ⁢ and ⁢ ⁢ K i ⁢ ⁢ i g are arithmetic and geometric averages of Cardwell and Parsons bounds, and wherein said off-diagonal values are computed using K j ⁢ ⁢ i a . = K i ⁢ ⁢ j a . = - a 2 ⁢ ⁢ sin ⁡ ( 2 ⁢ θ ) ⁢ ⁢ if ⁢ ⁢ K i ⁢ ⁢ i a . ≠ K j ⁢ ⁢ j a . ⁢ ⁢ wherein a 2 = K i ⁢ ⁢ i a . - K j ⁢ ⁢ j a . 2 ⁢ cos ⁡ ( 2 ⁢ θ ) , or K j ⁢ ⁢ i a . = K i ⁢ ⁢ j a . = - b 2 ⁢ ⁢ sin ⁡ ( 2 ⁢ θ ) ⁢ ⁢ if ⁢ ⁢ K i ⁢ ⁢ i a . = K j ⁢ ⁢ j a . ⁢ ⁢ wherein b 2 = K ^ i ⁢ ⁢ i + K ^ j ⁢ ⁢ j 4 - K ~ i ⁢ ⁢ i + K ~ j ⁢ ⁢ j 4 ; said computer identifying a pertinent angle of rotation, θ, that defines said off-diagonal values; and said computer determining a sign of said off-diagonal values to model said flow of said substance or heat through said periodic and said non-periodic material.