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solution to the eikonal equation(∇ψ k (r)) 2 = c(r)k(r)(19)such that ψ k (r) =0defines the surface ∂ω k (0).The family of rays orthogonal to the family of wave frontscan also be introduced, as shown in Fig. 1. In particular for aray γ(τ 1 ) going either from ∂ω c (0) to ∂ω c (τ 1 ) or from ∂ω k (0)to ∂ω k (τ 1 ) it results in∫√c(r)τ 1 =dγ. (20)k(r)γ(τ 1)The wave fronts ∂ω c (τ 1 ), ∂ω k (τ 1 ) and rays γ(τ 1 ) of thewave propagation problem are different from the isothermalsurfaces and trajectory lines of the heat diffusion problems.24-26 September 2008, Rome, Italyfact the following result can be proven [3]For each R 1 , the restriction of the structure function C(R)to 0 ≤ R ≤ R 1 , is affected by all and only the values of c(r)in ω c (τ 1 ) and of k(r), h(r) in ω k (τ 1 ),being∫ √R1dC(R)τ 1 =dR. (21)dRBesides∫ R100√dC(R)dR∫γ(τ dR = 1)√c(r)dγ, (22)k(r)γ(τ 1 ) being any ray going either from ∂ω c (0) to ∂ω c (τ 1 ) orfrom ∂ω k (0) to ∂ω k (τ 1 ).ΩΩω c (t)ω k (t)ΣΣ∂Ω∂Ω(t)Fig. 1. Propagation within Ω of the ∂ω c(t) wave front of v(r,t) and of the∂ω k (t) wave front of j(r,t).The ω c (τ 1 ) and ω k (τ 1 ) regions allow to relate the spatialdistributions of thermal properties to structure functions. InIV. TRIANGULATION METHODThe established relation between spatial distributions ofthermal properties c(r), k(r) and h(r) and structure functionC(R) can be usefully exploited in practical applications. Preciselylet C be a multi-directional heat flow and let its structurefunction C(R) be known. Let C 1 be a second multi-directionalheat flow presenting a difference in spatial distributions ofthermal properties with respect to C and let its structurefunction C 1 (R) be known.If, in addition to structure functions C(R) and C 1 (R), thespatial distribution of thermal properties in C is assumed tobe known, then wave-fronts ∂ω c (τ 1 ) and ∂ω k (τ 1 ) can bedetermined by solving Eqs. (18), (19). From Proposition III thespatial difference of thermal properties between C and C 1 canbe recovered to start on either wave-front ∂ω c (τ 1 ) or ∂ω k (τ 1 ),in which∫ √R1∫ √dC(R)R1τ 1 =0 dRdR = dC1 (R)0 dRdR,R 1 being the smallest value at which the structure functionsC(R) and C 1 (R) differ.This strategy for exploiting structure functions allows an tolocalize defects in components and packages not only whenthe heat flow is approximately one-directional as is done in theconventional approach to structure functions but also when itis multi-directional.However in this way only a raw localization of a defects canbe in general achieved, as shown by the conception example inFig. 1a. Here the source common to C and C 1 is S 1 , the defectin C 1 isassumedtobeD and the region in which no defects arepresent, as a consequence of the structure function procedureis Ω 1 . Thus the defect cannot be well localizing outside Ω 1 .Accurate localizations can in general be obtained by repeatingthis procedure with respect to different heat sources. This isshown by the conception example in Fig. 1b, obtained fromthe example in Fig. 1a by adding the heat sources S 2 , S 3 ,S 4 . The regions in which no defects are present, estimated byrepeating the structure function procedure for the heat sourcesS 1 , S 2 , S 3 and S 4 , is the union of Ω 1 , Ω 2 , Ω 3 and Ω 4 . Thusin this case an accurate localization of D is achieved.©EDA Publishing/THERMINIC 2008 10ISBN: 978-2-35500-008-9