A State-of-the-Art Review on Integral Transform Technique in Laser–Material Interaction: Fourier and Non-Fourier Heat Equations

Real-time heat distribution and phase transformation based on operating conditions and material properties can be estimated using heat equations. The corresponding characteristic functions are used to analyze heat conduction processes in various fields, including laser and electron beam processing. A powerful universal analytical and numerical method that transforms partial differential equations into a coupled system of ordinary differential equations is the wavelet transform method. Fourier and non-Fourier heat equations can be implemented for both equilibrium and non-equilibrium thermodynamic processes, including a wide range of processes such as the two-temperature model, ultrafast laser irradiation, and biological processes. The ultrafast laser heating process of nanofilms is characterized by ultrashort duration and ultrasmall spatial size, in which the classical Fourier law based on the local equilibrium hypothesis is no longer applicable. Based on the Cattaneo-Vernotte model and the double phase delay model, two-dimensional analytical solutions of thermal conductivity in two-dimensional structures under the action of ultrafast laser are obtained using the integral transform method. The results show that there is a thermal wave phenomenon inside the film, which becomes increasingly obvious as the temperature gradient delay time elapses. In this paper, non-Fourier heat conduction problems with temperature and heat flux relaxations are studied based on the wavelet finite element method and solved by the central difference scheme for one-dimensional and two-dimensional media. The heat wave model and the double phase delay model are used to formulate the finite elements, and a new formulation of the wavelet finite element solution is proposed to solve the computational optimization problem. Compared with the current methodologies for the heat wave model and the dual phase delay model, the present model is a direct model that describes the thermal behavior with a single equation with respect to temperature. The developed method can be used for arbitrary shapes. A new iteration update methodology is also proposed for the dual phase delay model to solve the computationally efficient problems. The time iteration algorithms do not use the global stiffness matrix. This allows for optimized calculations. Numerical calculations were performed in comparison with the classical finite element method and the spectral finite element method. The comparisons in accuracy, efficiency, flexibility and applicability confirm that the developed method is an effective and alternative tool for thermal analysis of local volumes of two-dimensional materials.

This work studied the optothermal response of plasmonic nanofocusing structures under picosecond pulsed laser irradiation. The surface plasmon polariton is simulated to calculate the optical energy dissipation as the Joule heating source and the thermal transport process is studied using a two temperature model (TTM). At the picosecond time scale that we are interested in, the Fourier heat equation is used to study the electron thermal transport and the hyperbolic heat equation is used to study the lattice thermal transport. For comparison, the single temperature model (STM) is also studied. The difference between TTM and STM indicates that TTM provides more accurate estimates in the picosecond time scale and the STM results are only reliable when the local electron and lattice temperature difference is negligible.

A century ago, Edgar Buckingham presented data and a theoretical conceptualization of soil moisture movement. His work constitutes a milestone in the history of soil physics and more generally, of movement of multiple fluid phases in porous media. Starting from first principles, Buckingham formulated a conceptual model to make rational sense of long‐term observations of evaporation from soil columns. Central to his model were the notion of a capillary potential, soil moisture retention curve, and potential‐dependent hydraulic conductivity. Buckingham recognized that whereas heat capacity and thermal conductivity were independent of temperature in Fourier's heat equation, in the case of soil moisture, the slope of the soil‐moisture retention curve (analogous to specific heat) and capillary conductivity were both strong functions of capillary potential. Noting that available solutions of Fourier's linear differential equation did not apply to moisture movement in soils, Buckingham was skeptical that the nonlinear problem could be solved mathematically. This is perhaps why he did not present a partial differential equation for soil‐moisture movement. Such an equation would be gvien in 1931 by Richards. Despite considerable efforts, analytical solutions to Richards' equation can be obtained only under simplifying assumptions. While these solutions give valuable insights into patterns of soil moisture movement, they cannot adequately address problems of the natural soil environment. Although Buckingham's model remains the only workable physical‐mathematical conceptualization for studying moisture movement in soils, his own skepticism of its ability to reliably describe moisture movement in soils is still valid. More profound, his skepticism captures the limitations inherent in precisely describing the behavior of earth systems. This paper examines Buckingham's central ideas in light of developments in groundwater hydrology and soil mechanics and reflects on the limits of our ability to quantitatively understand moisture movement in unsaturated soils.

This chapter deals with the validity/limits of the integral transform technique on finite domains. The integral transform technique based upon eigenvalues and eigenfunctions can serve as an appropriate tool for solving the Fourier heat equation, in the case of both laser and electron beam processing. The crux of the method consists in the fact that the solutions by mentioned technique demonstrate strong convergence after the 10 eigenvalues iterations, only. Nevertheless, the method meets with difficulties to extend to the case of non-Fourier equations. A solution is however possible, but it is bulky with a weak convergence and requires the use of extra-boundary conditions. To surpass this difficulty, a new mix approach is proposed with this chapter resorting to experimental data, in order to support a more appropriate solution. The proposed method opens in our opinion a beneficial prospective for either laser or electron beam processing.

Stress induced birefringence due to asymmetry in axial and radial directions that is generated because of the interaction of ultrashort laser pulses with a transparent material is numerically studied. The coupled equations of nonlinear Schrodinger and plasma density evolution are solved to calculate the deposited energy density and initial temperature profile. Fourier's heat equation and the equations related to the thermo-elasto plastic model are solved to calculate the temperature evolution and distribution of induced displacement inside the material, respectively. Finally, by numerically calculating the distribution of the induced refractive index changes experienced by both axially and radially probe beams, induced stress birefringence is calculated for different characteristics of writing pulses. Furthermore, the induced stress birefringence is experimentally realized, and the effect of the energy of the writing pulse is investigated. To know how the induced refractive index changes and birefringence distributions depend on parameters of the writing pulse is crucial to obtain high performance guiding structures and polarization-sensitive as well as polarization-independent components.

At present, adjusting mill velocity is usually applied to guarantee the control precision of finishing temperature FDT for many hot plants, so the forecast of strip velocity progression is frequently not realized as a result of current production conditions. It is difficult to eliminate the strip velocity fluctuation effect on cooling temperature for run-out table cooling control system. In order to improve control precision of cooling temperature on run-out table, the finite-difference model for hot strip cooling process is established basing on Fourier's heat equation. Then taking values easily measured, such as strip finishing temperatures, strip velocity, acceleration rate, value actual on/off status, etc. as auxiliary variables, online monitor function of cooling temperature on run-out table of hot strip mills is developed using soft-sensing technique. With help of this online monitor, twice compensation for strip velocity disturbances is added to run-out table cooling control system. The work in this paper as the matching technology of ultra-fast cooling equipment of Northeastern University of China has been applied to a local 2150 mm hot strip plant successfully.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Extensive applications of YBCO tapes claim for a comprehensive knowledge of their behavior in every possible operating condition. In particular, one of the main topics to explore is the tape stability against heat and current disturbances, in terms of minimum quench energy and normal zone propagation velocity (NZPV). In this work we present the results of NZPV measured in a 35 cm long YBCO coated conductors manufactured by SuperPower Inc. The tape is cooled by means of a cryocooler and kept under vacuum in order to achieve near-adiabatic conditions. Two independent heaters are positioned on the tape: one on the YBCO side and the other on the substrate side. The quench is monitored by recording the voltage and the temperature evolution along the tape. The NZPV, at 77 K, reaches a value larger than 2.5 cm/s for a bias current <formula formulatype="inline"><tex Notation="TeX">${\rm I}=50\ {\rm A}$</tex></formula> which represents the 56% of the critical current value. The quench dynamics of the tape are discussed with the help of a simulation program based on the finite difference discretization of the tape with the Fourier's heat equation: the model is able to reproduce the experimental behavior. </para>

Permanent metallic implants, such as dental fillings and cardiac devices, generate streaks-like artefacts in computed tomography (CT) images. In this article, we propose a strategy to perform metal artefact reduction (MAR) that relies on the total variation-H− 1 inpainting, a variational approach based on a fourth-order total variation (TV) flow. This approach has never been used to perform MAR, although it has been profitably employed in other branches of image processing. A systematic evaluation of the performance is carried out. Comparisons are made with the results obtained using classical linear interpolation and two other partial differential equation-based approaches relying, respectively, on the Fourier's heat equation and on a second order TV flow. Visual inspection of both synthetic and real CT images, as well as computation of similarity indexes, suggests that our strategy for MAR outperforms the others considered here, as it provides best image restoration, highest similarity indexes and for being the only one able to recover hidden structures, a task of primary importance in the medical field.

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

The presence of sub-continuum effects in nano-scale systems, including size and boundary effects, causes the continuum-level relations (e.g., Fourier heat equation) to break down at such scales. The thermal sub-continuum effects are manifested as a temperature jump at the system boundaries and a reduced heat flux across the system. In this work, we reproduce transient and steady-state results of Gray lattice Boltzmann simulations by developing a one-dimensional, transient, modified Fourier-based approach. The proposed methodology introduces the following two modifications into the Fourier heat equation: (i) an increase in the sample length by a fixed length at the two ends, in order to capture the steady-state temperature jumps at the system boundaries, and (ii) a size-dependent effective thermal diffusivity, to recover the transient temperature profiles and heat flux values. The predicted temperature and heat flux values from the proposed modified Fourier approach are in good agreement with those predicted by the Gray lattice Boltzmann simulations.

Numerical treatment of nonlinear Fourier and Maxwell-Cattaneo-Vernotte heat transport equations

X-ray synchrotron sources, possessing high power density, nanometric spot size and short pulse duration, are extending their application frontiers up to the exploration of direct matter modification. In this field, the use of atomistic and continuum models is now becoming fundamental in the simulation of the photoinduced excitation states and eventually in the phase transition triggered by intense X-rays. In this work, the X-ray heating phenomenon is studied by coupling the Monte Carlo method (MC) with the Fourier heat equation, to first calculate the distribution of the energy absorbed by the systems and finally to predict the heating distribution and evolution. The results of the proposed model are also compared with those obtained removing the explicit definition of the energy distribution, as calculated by the MC. A good approximation of experimental thermal measurements produced irradiating a millimetric glass bead is found for both of the proposed models. A further step towards more complex systems is carried out, including in the models the different time patterns of the source, as determined by the filling modes of the synchrotron storage ring. The two models are applied in three prediction cases, in which the heating produced in Bi2Sr2CaCu2O8+δ microcrystals by means of nanopatterning experiments with intense hard X-ray nanobeams is calculated. It is demonstrated that the temperature evolution is strictly connected to the filling mode of the storage ring. By coupling the MC with the heat equation, X-ray pulses that are 48 ps long, possessing an instantaneous photon flux of ∼44 × 1013 photons s-1, were found to be able to induce a maximum temperature increase of 42 K, after a time of 350 ps. Inversely, by ignoring the energy redistribution calculated with the MC, peaks temperatures up to hundreds of degrees higher were found. These results highlight the importance of the energy redistribution operated by primary and secondary electrons in the theoretical simulation of the X-ray heating effects.

The paradox of fourier heat equation: A theoretical refutation

For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation. Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier&ndash;Plancherel formula for non-commutative harmonic analysis, applied to locally-compact, non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences. The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties. One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory. The content of this Special Issue will highlight papers exploring non-commutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.

A semi-analytical-numerical solution is theorized to describe the laser additive manufacturing via laser-bulk ceramic interaction modeling. The Fourier heat equation was used to infer the thermal distribution within the ceramic sample. Appropriate boundary conditions, including convection and radiation, were applied to the bulk sample. It was irradiated with a Gaussian spatial continuous mode fiber laser (λ = 1.075 µm) while a Lambert-Beer law was assumed to describe the laser beam absorption. A close correlation between computational predictions versus experimental results was validated in the case of laser additive manufacturing of silicon nitride bulk ceramics. The thermal field value rises but stays confined within the irradiated zone due to heat propagation with an infinite speed, a characteristic of the Fourier heat equation. An inverse correlation was observed between the laser beam scanning speed and thermal distribution intensity. Whenever the laser scanning speed increases, photons interact with and transfer less energy to the sample, resulting in a lower thermal distribution intensity. This model could prove useful for the description and monitoring of low-intensity laser beam-ceramic processing.