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A refined analytical model of spatially resolved diffuse reflectance with small source-detector separations (SDSs) for the in vivo skin studies is proposed. Compared to the conventional model developed by Farrell et al., it accounts for the limited acceptance angle of the detector fiber. The refined model is validated in the wide range of optical parameters by Monte Carlo simulations of skin diffuse reflectance at SDSs of units of mm. Cases of uniform dermis and two-layered epidermis-dermis structures are studied. Higher accuracy of the refined model compared to the conventional one is demonstrated in the separate, constraint-free reconstruction of absorption and reduced scattering spectra of uniform dermis from the Monte Carlo simulated data. In the case of epidermis-dermis geometry, the recovered values of reduced scattering in dermis are overestimated and the recovered values of absorption are underestimated for both analytical models. Presumably, in the presence of a thin mismatched topical layer, only the effective attenuation coefficient of the bottom layer can be accurately recovered using a diffusion theory-based analytical model while separate reconstruction of absorption and reduced scattering fails due to the inapplicability of the method of images. These findings require implementation of more sophisticated models of light transfer in inhomogeneous media in the recovery algorithms. [ABSTRACT FROM AUTHOR]

The capability of the in-house transient wave-multibody computational tool, ITU-WAVE, is extended to predict the wave power absorption with Wave Energy Converters (WECs) arrays placed in front of a vertical breakwater. The hydrodynamic forces are approximated by solving boundary integral equation at each time interval. The reflection of incoming waves due to a vertical wall is predicted with method of images. The constructive or destructive performance of WECs arrays with different array configurations is measured with mean interaction factor. The behaviour of the hydrodynamic forces of each WEC due to a vertical wall effect shows considerable differences than those of WECs arrays without a vertical wall. When the wave power absorption with WECs arrays with and without a vertical wall effect are compared, the numerical results show that WECs placed in front of a vertical wall have much greater effects on wave power absorption. This can be attributed to the hydrodynamic interaction, standing waves, and nearly trapped waves in the gap between a vertical wall and WECs arrays. The analytical and other numerical results are used for the validation of present ITU-WAVE computational results for exciting and radiation forces, and mean interaction factor of WECs arrays which show satisfactory agreements. [ABSTRACT FROM AUTHOR]

A refined analytical model of spatially resolved diffuse reflectance with small source-detector separations (SDSs) for the in vivo skin studies is proposed. Compared to the conventional model developed by Farrell et al., it accounts for the limited acceptance angle of the detector fiber. The refined model is validated in the wide range of optical parameters by Monte Carlo simulations of skin diffuse reflectance at SDSs of units of mm. Cases of uniform dermis and two-layered epidermis-dermis structures are studied. Higher accuracy of the refined model compared to the conventional one is demonstrated in the separate, constraint-free reconstruction of absorption and reduced scattering spectra of uniform dermis from the Monte Carlo simulated data. In the case of epidermis-dermis geometry, the recovered values of reduced scattering in dermis are overestimated and the recovered values of absorption are underestimated for both analytical models. Presumably, in the presence of a thin mismatched topical layer, only the effective attenuation coefficient of the bottom layer can be accurately recovered using a diffusion theory-based analytical model while separate reconstruction of absorption and reduced scattering fails due to the inapplicability of the method of images. These findings require implementation of more sophisticated models of light transfer in inhomogeneous media in the recovery algorithms.

This paper focuses on deriving new approximate analytical solutions in wedge‐shaped aquifers. The proposed methodology is applicable to any type of aquifer namely, leaky, confined and unconfined, under both steady state and transient flow conditions. By applying the method of images and separating the flow field into sections using physical arguments, approximate analytical expressions are obtained for the drawdown function, which in contrast to the conventional theory, are applicable to any arbitrary wedge angle. Moreover, the solutions fully observe the boundary conditions, while they preserve the continuity of the drawdown, which can be calculated directly at any point of the flow field. Nevertheless, comparison of the results of the new approximate analytical solutions to numerical ones, has been considered necessary to check their validity. MODFLOW, a well‐known numerical tool is used to calculate the numerical results. The discrepancies between the numerical results and those of the approximate analytical solution are negligible. The main advantages of the proposed methodology are the following: (a) The model needs only finite number of terms compared to conventional analytical and numerical solutions that involve infinite series, (b) The computational load is low, so it can be easily used in conjunction with meta‐heuristic algorithms to solve groundwater resources optimization problems, (c) Stream depletion rate can be calculated rather accurately and (d) The method is applicable to related flow problems. Key Points: New functions for drawdown calculation in wedge‐shaped aquifers are presentedThe proposed functions are suitable to be used in conjuction with meta‐heuristic algorithms to solve groundwater optimization problemsConvergence of the results obtained from the set of functions and numerical methods (MODFLOW) point out the validity of the proposed solutions [ABSTRACT FROM AUTHOR]

The capability of the in-house transient wave-multibody computational tool, ITU-WAVE, is extended to predict the wave power absorption with Wave Energy Converters (WECs) arrays placed in front of a vertical breakwater. The hydrodynamic forces are approximated by solving boundary integral equation at each time interval. The reflection of incoming waves due to a vertical wall is predicted with method of images. The constructive or destructive performance of WECs arrays with different array configurations is measured with mean interaction factor. The behaviour of the hydrodynamic forces of each WEC due to a vertical wall effect shows considerable differences than those of WECs arrays without a vertical wall. When the wave power absorption with WECs arrays with and without a vertical wall effect are compared, the numerical results show that WECs placed in front of a vertical wall have much greater effects on wave power absorption. This can be attributed to the hydrodynamic interaction, standing waves, and nearly trapped waves in the gap between a vertical wall and WECs arrays. The analytical and other numerical results are used for the validation of present ITU-WAVE computational results for exciting and radiation forces, and mean interaction factor of WECs arrays which show satisfactory agreements.

A new technique for two-dimensional vortex methods is presented. The vorticity field is discretized and represented by vortex blobs. Viscosity and roughness are incorporated into vortex simulations by means of the corrected core-spreading method with LES theory. A deterministic and efficient grid-free method simulates viscous effects by maintaining small vortex core sizes through a splitting algorithm that controls the consistency error. The LES theory also enables the implementation of the roughness model. The effectiveness of this method is shown in calculating vortex interactions and decay in aircraft wakes with crosswind near a rough ground plane. The numerical results of the trajectory of primary vortical structures are compared with experimental data (when possible), suggesting the validity of the method. In general, the control of the roughness height size appears as an important factor to interfere on the trajectory of primary vortical structures in the ground effect with crosswind. The effect of the relative roughness height of ε/Δs = 0.001 shows that the primary vortical structures survive the interaction with the ground plane and can attain a maximum height in the order of 0.95 h (h is the release height of the primary vortical structures) during the loop for crosswind velocity of U∞ = 0.02 at Re = 7650. On the other hand, the combined effects of roughness ε/Δs = 0.001 and of crosswind U∞ = 0.04 at Re = 75,000 indicate that the primary vortical structures attain a maximum height about 0.83 h during the loop, tending to leave the runway faster, with sufficient intensity to disturb a smaller aircraft operating on a parallel runway. [ABSTRACT FROM AUTHOR]

External barrier options are financial securities that have two assets for stochastic variables, where the payoff depends on one underlying asset and the barrier depends on another state variable such that it determines whether the option is knocked in or out. In this study, considering the financial derivatives subject to the default risks of the option writer in over-the-counter markets since the global financial crisis of 2007–2008, we study vulnerable external barrier option prices by utilizing multivariate Mellin transforms and the method of images and then examine the behaviors and sensitivities of the vulnerable external barrier option prices in terms of the model parameters. Based on the results obtained, our study has two main contributions. First, by using multivariate Mellin transform approaches, we can find an explicit-form pricing formula for the option prices more effectively and easily, resolving the complexity of calculation of the option prices by using probabilistic or other methods. Second, we verify that our closed-form solution has been accurately and efficiently obtained by comparing the closed-form solution with the Monte Carlo simulation solution. [ABSTRACT FROM AUTHOR]

A new technique for two-dimensional vortex methods is presented. The vorticity field is discretized and represented by vortex blobs. Viscosity and roughness are incorporated into vortex simulations by means of the corrected core-spreading method with LES theory. A deterministic and efficient grid-free method simulates viscous effects by maintaining small vortex core sizes through a splitting algorithm that controls the consistency error. The LES theory also enables the implementation of the roughness model. The effectiveness of this method is shown in calculating vortex interactions and decay in aircraft wakes with crosswind near a rough ground plane. The numerical results of the trajectory of primary vortical structures are compared with experimental data (when possible), suggesting the validity of the method. In general, the control of the roughness height size appears as an important factor to interfere on the trajectory of primary vortical structures in the ground effect with crosswind. The effect of the relative roughness height of ε/Δs = 0.001 shows that the primary vortical structures survive the interaction with the ground plane and can attain a maximum height in the order of 0.95 h (h is the release height of the primary vortical structures) during the loop for crosswind velocity of U∞ = 0.02 at Re = 7650. On the other hand, the combined effects of roughness ε/Δs = 0.001 and of crosswind U∞ = 0.04 at Re = 75,000 indicate that the primary vortical structures attain a maximum height about 0.83 h during the loop, tending to leave the runway faster, with sufficient intensity to disturb a smaller aircraft operating on a parallel runway.

The interaction between spherical cavitation bubble and flat wall is transformed into that between the real bubble and imaging bubble by the method of images. Firstly, we investigate the dynamics of real bubble and matched, inversed or mis-matched imaging bubble driven by a small amplitude ultrasound, revealing the characteristics of the interaction between cavitation bubble and rigid, soft and impedance walls. Then, we emphatically study the dynamics of real bubble and mis-matched imaging bubble driven by a finite amplitude ultrasound, and the interaction characteristics between cavitation bubble and real impedance wall are revealed. The results show that the cavitation bubble is always close to the rigid wall and far away from the soft wall; For the impedance wall, whether the cavitation bubble is far away or close depends on the specific wall parameters. Moreover, the direction and magnitude of bubble's translation velocity can be changed by adjusting the driving parameters. Understanding the interaction between cavitation bubble and impedance wall is of great significance for efficient application of ultrasonic cavitation.

Since the initial isolation of few-layer graphene, a plethora of two-dimensional atomic crystals has become available, covering almost all known materials types including metals, semiconductors, superconductors, ferro- and antiferromagnets. These advances have augmented the already existing variety of two-dimensional materials that are routinely realized by quantum confinement in bulk-semiconductor heterostructures. This review focuses on the type of material for which two-dimensional realizations are still being actively sought: magnetoelectrics. We present an overview of current theoretical expectation and experimental progress towards fabricating low-dimensional versions of such materials that can be magnetized by electric charges and polarized electrically by an applied magnetic field – unusual electromagnetic properties that could be the basis for various useful applications. The interplay between spatial confinement and magnetoelectricity is illustrated using the paradigmatic example of magnetic-monopole fields generated by electric charges in or near magnetoelectric media. For the purpose of this discussion, the image-charge method familiar from electrostatics is extended to solve the boundary-value problem for a magnetoelectric medium in the finite-width slab geometry using image dyons, i.e. point objects having both electric and magnetic charges. We discuss salient features of the magnetoelectrically induced fields arising in the thin-width limit.

Adequate knowledge of physical phenomena has been shown to be essential for the development of new technologies. Such knowledge allows several engineering problems to be studied and solutions to be found. More and more complex problems have been studied and solved, especially in recent centuries. Infinite and semi-infinite spaces presented great discretization difficulty due to the scarcity of resources for this. With the emergence of numerical methods, these problems had their closest possible solution to being found. The present work presents the formulation and mathematical principles that comprise the Boundary Element Method model to approach twodimensional problems with infinite and semi-infinite domains. The particulars regarding the definition of the relevant variables, the characteristics of the fundamental solutions in these cases, and the behavior of primal variables at infinitely distant points are discussed. Aspects related to the discretization and numerical issues are also addressed by solving two particular semi-plane problems. The numerical results obtained showed satisfactory accuracy, confirming that the Boundary Element Method is the most adequate technique even today to solve such problems. [ABSTRACT FROM AUTHOR]

Method of images is used in many fields like electrostatics, fluid dynamics, etc. It requires many different formulae. This article aims to help the students comprehend these formulae and highlight the relation between electrostatic images with the results of optics. [ABSTRACT FROM AUTHOR]

Planar mirrored-coil structures are often used in the field of biomedicine as a sensor to measure physiological signals wirelessly. Although there are many studies on interrogation methodologies, research on the sensor itself remains understudied. In this article, we report an analytical formulation and a data-fitted formula to calculate the resonant frequency for a planar mirrored-coil sensor. Compared to the measured results, our analytical formula and data-fitted formula deviate by 13% and 9% median errors, respectively, which is much more accurate than the conventional one, 23%. It shows that both methods provide a way to quickly evaluate and design a planar mirrored-coil structure with high precision. [ABSTRACT FROM AUTHOR]

The evaluation of the tunnel correction remains an actual problem, especially for the effect of tunnel walls. Even if the experimental campaign meets the basic similitude criteria (Mach, Reynolds etc.), the wall effect on the measured data is always present. Consequently, the flow correction due the limited by walls must be evaluated. Solid wall corrections refer to the aerodynamic interference between the experimental model and the walls of the wind tunnel. This interaction affects the measured forces and implicitly the angle of attack. Usually, these effects are introduced through semi-empirical correction factors which change the global measured forces. The present paper refers to the mathematical and numerical modeling of aerodynamic interferences between the experimental model and the solid walls based on the potential flow model. The main goal is to asses a method allowing an estimate of the corrections for each configuration with a minimum computational resource.

Three dimensional dielectric systems are simulated to obtain the relaxational polarization responses. The dielectric system consists of randomly distributed permanent dipoles that fluctuate due to thermal activation in double well potentials between two plane-parallel electrodes. The barrier height between the wells is assumed to be proportional to the dipole length. The electrodes' effect is considered with the method of images. The relaxational phenomena of two dielectric systems, one with fixed and the other with distributed dipole lengths having single and distributed relaxation times, respectively, are calculated with a Monte-Carlo simulation method. The local field at every dipole and consequently the transition probabilities are computed to determine the transient polarization response iteratively. Both dielectric systems are simulated over a wide range of temperatures for interacting and non-interacting cases. The mean relaxation times exhibit a Vogel-Fulcher law for interacting systems and an Arrhenius law for non-interacting systems. [ABSTRACT FROM AUTHOR]

In the present study, the problems of diffraction and radiation of water waves by a cylindrical heaving wave energy converter (WEC) placed in front of a reflecting V-shaped vertical breakwater are formulated. The idea conceived is based on the possible exploitation of amplified scattered and reflected wave potentials originating from the presence of V-shaped breakwater, towards increasing the WEC’s wave power absorption due to the wave reflections. An analytical solution based on the method of images is developed in the context of linear water wave theory, taking into account the hydrodynamic interaction phenomena between the converter and the vertical wall. Numerical results are presented and discussed concerning the hydrodynamic forces on the absorber and its wave power efficiency for various examined parameters, namely, the breakwaters’ forming angle, the distance between the converter and the vertical walls and the wave heading angle. The results show that the amount of the harvested wave power by the WEC in front of the walls is amplified compared to the wave power absorbed by the same WEC in the open sea.

Highlights • Method of images is successfully adopted to predict the LVG vortex dynamics. • A mixed LVG configuration is suggested to enhance the heat transfer performance. • LVG aspect ratio and channel height are very critical to the LVG effectiveness. Abstract This paper reports a numerical investigation of fluid flow and heat transfer in a rectangular channel with delta-shaped winglet longitudinal vortex generators (LVGs) under different configurations. In addition to the conventional common-flow-down configuration and the common-flow-up configuration, a unique mixed configuration is suggested. The "method of images" is successfully adopted to analyze the dynamics of the longitudinal vortices due to wall interference. The Nusselt number, friction factor and overall performance coefficient for the three configurations are compared at various Reynolds numbers (all less than 2200), LVG row numbers, channel heights, and LVG aspect ratios. It is found that the channel height and LVG aspect ratio are the two most critical factors influencing the effectiveness of the different LVG configurations, which may explain why inconsistent conclusions have been presented by previous studies comparing the performance of the common-flow-down and common-flow-up configurations. When the channel height is relatively small, the unique mixed configuration is found to be the most effective at enhancing fluid mixing (and therefore improving heat transfer) at streamwise cross sections. When the LVG aspect ratio becomes large, the common-flow-down configuration outperforms the common-flow-up configuration. This paper sheds some insight on how to design the optimal LVG configuration for enhancing heat transfer performance. [ABSTRACT FROM AUTHOR]

The evaluation of the tunnel correction remains an actual problem, especially for the effect of tunnel walls. Even if the experimental campaign meets the basic similitude criteria (Mach, Reynolds etc.), the wall effect on the measured data is always present. Consequently, the flow correction due the limited by walls must be evaluated. Solid wall corrections refer to the aerodynamic interference between the experimental model and the walls of the wind tunnel. This interaction affects the measured forces and implicitly the angle of attack. Usually, these effects are introduced through semi-empirical correction factors which change the global measured forces. The present paper refers to the mathematical and numerical modeling of aerodynamic interferences between the experimental model and the solid walls based on the potential flow model. The main goal is to asses a method allowing an estimate of the corrections for each configuration with a minimum computational resource. [ABSTRACT FROM AUTHOR]

A mathematical model of cylindrical shaped plane-ended scanning Kelvin probe output signal is proposed considering the case of an infinite plane sample's surface with local defects represented by dot charges. Modeling results were obtained for the case of two closely situated dot charges anddifferent combinations of scanning Kelvin probe tip's diameter and sample-to-probe gap. It was found that the most effective way to improve the lateral resolution of a scanning Kelvin probe is to reduce the sample-to-probe gap in line with the reduction of sensor's vibration amplitude.

In this paper, an iterative boundary element method (IBEM) was proposed to solve for a wing-in-ground (WIG) effect. IBEM is a fast and accurate method used in many different fields of engineering and in this work; it is applied to a fluid flow problem assessing a wing in ground proximity. The theory and the developed code are validated first with other methods and the obtained results with the proposed method are found to be encouraging. Then, time consumptions of the direct and iterative methods were contrasted to evaluate the efficiency of IBEM. It is found out that IBEM dominates direct BEM in terms of time consumption in all trials. The iterative method seems very useful for quick assessment of a wing in ground proximity condition. After all, a NACA6409 wing section in ground vicinity is solved with IBEM to evaluate the WIG effect.

In this paper, an iterative boundary element method (IBEM) was proposed to solve for a wing-in-ground (WIG) effect. IBEM is a fast and accurate method used in many different fields of engineering and in this work; it is applied to a fluid flow problem assessing a wing in ground proximity. The theory and the developed code are validated first with other methods and the obtained results with the proposed method are found to be encouraging. Then, time consumptions of the direct and iterative methods were contrasted to evaluate the efficiency of IBEM. It is found out that IBEM dominates direct BEM in terms of time consumption in all trials. The iterative method seems very useful for quick assessment of a wing in ground proximity condition. After all, a NACA6409 wing section in ground vicinity is solved with IBEM to evaluate the WIG effect.

In this paper, method of current images is used to calculate magnetic field of a rectangular loop in presence of high-permeability material. We provide closed form description for the images of the current loop placed between two semi-infinite blocks of high-permeability material, in the cases when the plane of the loop is parallel or perpendicular to the surfaces of blocks. A case study of using the method of images to calculate magnetic field of rectangular loop inside cube made of high-permeability material is also provided. [Projekat Ministarstva nauke Republike Srbije, br. TP32016]

• Investigating the interaction between bubble and wall by the method of images. • Complete bubble behavior near the wall is obtained, including soft and impedance walls. • Bubbles can still translate toward some walls which are very close to soft wall. • Direction and magnitude of bubble translation velocity can be changed. The interaction between spherical cavitation bubble and flat wall is transformed into that between the real bubble and imaging bubble by the method of images. Firstly, we investigate the dynamics of real bubble and matched, inversed or mis-matched imaging bubble driven by a small amplitude ultrasound, revealing the characteristics of the interaction between cavitation bubble and rigid, soft and impedance walls. Then, we emphatically study the dynamics of real bubble and mis-matched imaging bubble driven by a finite amplitude ultrasound, and the interaction characteristics between cavitation bubble and real impedance wall are revealed. The results show that the cavitation bubble is always close to the rigid wall and far away from the soft wall; For the impedance wall, whether the cavitation bubble is far away or close depends on the specific wall parameters. Moreover, the direction and magnitude of bubble's translation velocity can be changed by adjusting the driving parameters. Understanding the interaction between cavitation bubble and impedance wall is of great significance for efficient application of ultrasonic cavitation. [ABSTRACT FROM AUTHOR]

This paper studies the large- t behavior of the boundary generated by the method of images for the first-passage-time problem. We show that this behavior is characterized by certain properties of the Laplace transform of the input measure. Such properties also determine the asymptotic behavior of the first-passage-time density. Most of the paper assumes a positive input measure, which generates a concave boundary. The last section, however, discusses a non-positive measure. We obtain a sufficient condition for the boundary to be convex. [ABSTRACT FROM AUTHOR]

An electromagnetic model is proposed to compute translational motion eddy currents in a conductive plate. The eddy currents are due to the movement of the plate in a dc magnetic field created by a permanent magnet (PM) inductor. First, the magnetic field, due to the PMs, is computed in three dimensions, where the iron yokes’ influence is considered because of the method of images. Then, the motional eddy currents are computed, such that the edge effects are correctly considered through an iterative procedure that uses magnetic images. The computations are very fast, and the obtained results are close to those issued from the three-dimensional finite-element method and the experiments. [ABSTRACT FROM AUTHOR]

A full-wave integral formulation has recently been proposed for the simulation of magnetically coupled resonant wireless power transfer (WPT) arrangements in homogeneous medium. In this paper, the formulation is extended to the case where two different types of dielectrics are separated by a planar interface. This configuration has extensive practical interest nowadays, especially for modeling biomedical WPT applications. The new formulation is based on a stationary approximation, which is valid at the typical operating frequencies. The proposed scheme requires much less computational resources compared with standard finite-element simulations. The results are validated against alternative simulations and measured data as well. [ABSTRACT FROM AUTHOR]

Multielectrode array (MEA) measurements of neural activity in brain slices are becoming more and more ubiquitous. To properly interpret the results gathered with their help, it is important to understand the measurement physics - the link between the electrical potential recorded at the electrodes and the underlying neural activity. In this work, we modeled the generation of transmembrane currents in a slice preparation and the extracellular potentials they generate. We addressed this forward modeling problem using Finite Element Methods (FEM) and used these results as ground truth for comparison with simplified modeling schemes in particular, forward modeling by means of the analytical point- and line-source formulas amended by means of the “method of images” (MoI) . To compare the different schemes we tested them on current sources and tissue models of increasing complexity. We calculated the extracellular potential generated at the MEA plane while varying the following: 1) We placed a point current source inside the slice, and varied its position with respect to the MEA plane. We also varied the conductivity profile of the slice to include inhomogeneity, anisotropy, and a thin layer of saline beneath the slice. 2) We used a layer 5 pyramidal cell model (Hay et al., 2011) and tracked the transmembrane currents through its compartments during a spontaneous spike. We placed these transmembrane currents as line current sources on a reconstructed cell within the slice. We then varied the conductivity of the saline bath surrounding the slice. 3) For testing a population of cells, we used a thalamo-cortical column model (Traub et al., 2005), and tracked the transmembrane currents through all its cortical neurons during an evoked response. The MEA potentials were then calculated assuming point-like current sources positioned on the mid points of the compartments in the multicompartment neuron models. We varied the conductivity of the slice to include inhomogeneity, anisotropy, and a thin layer of saline beneath the slice . In all cases we found that the MoI gave quantitatively accurate results compared to FEM. We further investigate how the changes in the extracellular potential due to the variations in the conductivity profile of the slice affect the estimates of current source density (CSD) in a MEA setup. We used MoI to extend the kernel current source density method (Potworowski et al., 2012) to include the geometry of the slice and saline slab. We found that including the saline layer does not increase the accuracy of the estimated CSD.

A method of recursive images applicable to diffusion in multilayer materials, is extended to cases where there is convection at the outer faces. Therefore this old/new method is now applicable to a wider, more diverse and real life diffusion problems. This objective is reached using a result by Bryan [1] who established that the Green function at a convection boundary involves adding a mirror reflecting thermal wave while subtracting a one-sided convolution, of a decreasing exponential with that same reflected wave. This is verified in the case of convection of heat, for a slab with its back face insulated. This method was then applied to find temperature solution for the diffusion in a slab having convection both at the front and at the back face of the slab. Finally the same rationale, of the recently proposed method of generalized method of images, was used to find the temperature solution for a three layered material with radiation convection at its front face. An alternative method is suggested which minimizes computation errors when the convection coefficient becomes very large. Finally a relationship is found between the semi-infinite solutions with and without convection radiation. [ABSTRACT FROM AUTHOR]

An innovative method of recursive images is presented to obtain solutions to the transient diffusion equation in a N -layered material based on the superposition of Green functions for a semi-infinite material. Through a sequential sum of image reflected functions a temperature solution is initially built for a structure of one layer over a substrate. These functions are chosen in order to satisfy in sequence the boundary conditions, first at the front interface then at the back interface then again at the front interface and so on until the magnitude of the added functions becomes negligible. Based on this so-called 1-layer algorithm, a 2-layer algorithm is obtained. This is accomplished through a sequential application of the 1-layer algorithm first to layer 1 then to layer 2 then again to layer 1 and so on. After that it is suggested how the sequential application of the N − 1 algorithm leads to the N -layer algorithm. This present scheme is valid for boundary conditions of the first and second kind but it will not applicable neither to the case where there is a contact resistance between layers or to the case of convective heat transfer at the end interfaces. [ABSTRACT FROM AUTHOR]