Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, ItalyTriangulation Method for Structure Functions ofMulti-Directional Heat-FlowsLorenzo Codecasa, Dario D’Amore, Paolo MaffezzoniPolitecnico di Milano, Milan, Italye-mail: {codecasa, damore, pmaffezz}@elet.polimi.itAbstract— In this paper previous results proposed by theauthors for localizing defects in components and packages bymeans of structure functions in three-dimensinal heat diffusionproblems have been generalized. To this aim a novel traingulationapproach is presented based on the use of structure functionscorresponding to dinstinct heat sources.I. INTRODUCTIONStructure functions have been originally used by V. Székeyet. al. as means for inferring on the spatial distributions ofthermal properties in one-directional heat flows [1]. Precisely,a 1-D heat diffusion problem in which power is injected atone boundary and temperature rise is measured at the sameboundary, referred to as one-directional heat flow, definesa short-circuited RC transmission line. This short-circuitedRC transmission line is characterized by a structure function,relating the cumulative thermal resistance and capacitancealong the line. Such structure function can be determined fromthe port response of the short-circuited RC transmission line,by solving an inverse problem. In this way information on thespatial distribution of thermal properties are recovered fromthe port response of a one-port dynamic thermal network.For the general case of a one-port passive dynamic thermalnetwork modeling a 3-D heat diffusion problem, referred toas multi-directional heat flow, the authors have shown in [2]that a structure function can still be defined and in [3] thata relation exists between structure function and the spatialdistribution of thermal properties.In this paper such results on structure functions are exploitedand a novel method, based on triangulation, for spatiallylocalizing defects in components and packages is provided. Asshown by the authors in [3], for a given heat source, and forthe corresponding one-port passive dynamic thermal network,the values of the structure function up to a given value of thecumulative thermal resistance R and capacitance C dependsby all and only the values of the thermal conductivity andvolumetric heat capacity within a given spatial region Ω. Theboundary of Ω is solution of an eikonal equation [3]. Thisfact suggests the following procedure for localizing defects bymeans of structure functions. Let C be a reference componentor package whose spatial distribution of thermal propertiesis known and let C 1 be a component or package presentingdefects in the spatial distribution of thermal properties withrespect to C.LetC(R) and C 1 (R) be respectively the structurefunctions of C and C 1 , with respect to a given heat source.By comparing the structure functions C(R) and C 1 (R) andby evaluating the smallest values of C and R at which theydiffer, the region Ω in which no defects are present can bedetermined by solving an eikonal equation.However in this way only a raw localization of a defects canbe in general achieved. Accurate localizations are shown to bein general feasable by repeating this procedure with respect todifferent heat sources. This approach has been numericallyimplemented. To this aim the well known Fast MarchingMethod [4] has been used for solving the eikonal equation,in order to determine the boundaries of the regions withoutdefects. Using the implemented numerical method, accuratelocalizations of defects have been achieved in a referenceexample.The remaining of this paper is organized as follows. Insection II structure functions for generic three-dimensionalheat diffusion problems are recalled. In section III the relationbetween spatial distributions of thermal properties and structurefunctions in described. The novel triangulation approachis presented in section IV. Examples of localization of defects,both analytical and numerical are presented in sections V andVI respectively. .II. STRUCTURE FUNCTIONS OF Multi-Directional HEATFLOWSAs it has been shown by the authors [2], structure functionscan be introduced for one-port passive dynamic thermalnetworks modelling generic three-dimensional heat diffusionproblems. Precisely, let us consider a three-dimensional heatdiffusion problem in a bounded spatial region Ω, referred to asmulti-directional heat flow. The relation between the generatedpower density G(r,t), the temperature rise distribution u(r,t)and the heat flux density q(r,t) is ruled by the First Principleof Thermodynamics and by Fourier’s law as follows∇·q(r,t)+c(r) ∂u (r,t)=G(r,t), (1)∂tq(r,t)=−k(r)∇u(r,t), (2)in which c(r) is the volumetric heat capacity and k(r) isthe thermal conductivity. Conditions on the boundary ∂Ω,assumed of Robin’s type, areq ν (r,t)=h(r)u(r,t), (3)in which h(r) is the heat transfer coefficient, ν(r) is the unitvector outward normal to ∂Ω and q ν (r,t)=q(r,t) · ν(r).Initial condition is assumed to be zerou(r, 0) = 0. (4)©EDA Publishing/THERMINIC 2008 8ISBN: 978-2-35500-008-9