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Diffusion
Published in Gregory N. Haidemenopoulos, Physical Metallurgy, 2018
In most diffusion problems, the solution of the second order linear PDE(5.32) is sought under certain initial and boundary conditions. We distinguish between two types of solutions. The first type refers to solutions, which are valid at short times or to semi-infinite media, where the dimensions are much larger than the net diffusion distance. These solutions involve the error function. The second type refers to solutions, which are valid at long times or to finite media and involve Fourier series. In cylindrical coordinates the Fourier series are substituted by a series of Bessel functions. In order to solve the diffusion equation, several methods are applied, amongst them the method of separation of variables and the Laplace transform. Solutions to simple one-dimensional diffusion problems will be discussed in the next section. For a detailed treatment of the subject, the reader is directed to the suggested reading list at the end of this chapter, especially Krank’s book.
Distributed sparse optimization for source localization over diffusion fields with cooperative spatiotemporal sensing
Published in Advanced Robotics, 2023
Naoki Hayashi, Masaaki Nagahara
In this paper, we propose a distributed sparse modeling to estimate the initial concentration of the diffusion process. The diffusion process can be described by a partial differential equation called the diffusion equation. A group of sensor agents is deployed over a two-dimensional (2D) field and measures time-series data of the concentration at the own position. Then the sensor agents collaboratively estimate the distribution of the initial concentration to determine a hot spot of the diffusion process. The estimation problem can be formulated as a least squares problem with a system matrix that describes a diffusion process at the position of each agent and a measurement vector consisting of time-series data of the concentration observed by each agent [30,31]. In a large-scale sensor network, each agent cannot observe the entire field due to limited sensing and communication capability. Thus, cooperation through local information exchanges is indispensable to estimate the initial distribution in a distributed manner. Besides, a highly concentrated region at the initial time is usually localized in a small area. Thus, the estimation of the initial distribution can be formulated as a sparse regularization problem whose objective function consists of the least squares term for the estimation and the regularization term for the sparsity of the initial distribution.
Continuous and Laplace transformable approximation for the temporal pulse shape of Xe-flash lamps for flash thermography
Published in Quantitative InfraRed Thermography Journal, 2018
Simon J. Altenburg, Rainer Krankenhagen
Here, we present a non-stitched approximation of the temporal shape of a Xe-flash. It has a simple Laplace transform, which is needed for the application in analytical models that solve the heat diffusion equation in the Laplace domain. It approximates the experimentally obtained pulse shape very well and is applicable for different energy settings of the same lamp as well as for different lamps, provided the numerical parameters are adjusted accordingly. In the second part of the paper, we show that using this approximation and an analytical model, it is possible to determine the thickness of very thin polymer samples from experiments in both transmission and reflection configuration. Thus, the limitations of the Parker method can be overcome. It should be emphasised that assuming a (possibly, according to Degiovanni [13], delayed) Dirac shaped flash or a rectangular pulse shape, appropriate results are only obtained when omitting data from a certain period immediately after the heating pulse, the results strongly depend on the length of this period and are associated with a larger error in transmission configuration. The use of a well-fitting approximation of the heating pulse shape is less ambiguous and leads to a well-fitting description of the temperature development at all times. In a previous publication [11] we analysed the same experimental data as analysed here to show the necessity to take into account semitransparency and thermal losses for proper thickness determination. In that work we already used the pulse shape presented here but did not discuss the necessity to do so. Here, we show that the presented good approximation of the actual temporal pulse shape is actually indicated as well for improved thickness determination.
P3 approximation equation of light transport in a slab medium: steady-state and time domains
Published in Waves in Random and Complex Media, 2022
If the Henyey–Greenstein phase function is used, all higher-order moments of the phase function are included in gl [15], and thus, . Some scholars used the Hankel transform to obtain the solution [6–8,21,22] of the diffusion equation. In the diffusion equation, the solutions of the first two terms are obtained, namely, the solutions of Φ0 (ρ, z) and Φ1 (ρ, z), where Φ0 (ρ, z) is the fluence rate, and Φ1 (ρ, z) is the flux of photons. In this study, using the solution method for the diffusion equation and the inverse Fourier transform, we obtained the Green's function solution of the P3 approximation. We obtained the first four terms of the solution, and Φi (ρ, z) was calculated with the following equation: Equation 1 is the Hankel transform, and the equation can be obtained by using the inverse Fourier transform. s is the variable of integration in the diffusion equation [6–8,21,22]. J0 is the zero-order Bessel function, and φi(z,s) is the result of the Fourier transform. φi(z,s) can be expressed in exponential form. When z < z0, the specific equations are as follows: When z > z0, the equations are as follows: These expressions were consistent with the diffusion equation [6–8,21,22]. The diffusion equation solution is in the form of hyperbolic sines or hyperbolic cosines. We used the exponential form, in which ν can be expressed as follows: where a, b, and c are as follows: In Equation 5, The constant terms A, B, C, and E in Equation 2 and can be expressed as follows: