This study presents a comprehensive theoretical and experimental investigation into the fabrication of three-dimensional gold nanostructures with tailored optical properties using a laser-induced method, aimed at enhancing applications based on localized surface plasmon resonance (LSPR) and Surface-Enhanced Raman Spectroscopy (SERS). The effects of the second harmonic (without filtering the fundamental harmonic) of Nd:YAG laser pulses, along with key parameters such as laser fluence and pulse count, are analyzed to achieve precise control over nanoparticle size, shape, and arrangement. Theoretical modeling using Fourier heat equations and the novel Generalized Finite Difference Time Domain (G-FDTD) method provides initial insights into the optical fluences necessary for the formation of surface gold nanoparticles. Experimentally, 100 nm gold thin films are irradiated with different fluences (0.48, 0.12, and 0.053 J/cm2) and pulse numbers (ranging from 1 to 50 pulses) of Nd:YAG laser with a time width of 60 ns. Results demonstrate that manipulating laser parameters significantly influences the localized surface plasmon resonance, offering a scalable and reproducible platform for designing stable, biocompatible, and high-performance nanostructures for applications based on LSPR, like SERS and enhanced fluorescence. Also, the formed gold elliptical NPs are simulated to understand the obtained results and the efficiency of the samples in SERS applications.

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duration and ultrasmall spatial size, in which the classical Fourier law based on the local equilibrium hypothesis is no longer applicable.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.

In a recent work, assuming a Beer–Lambert optical absorption and a Gaussian laser time profile, it was shown that the exact solutions for a 1D photoacoustic (PA) boundary-value-problem predict a null pressure for optically strong absorbent materials. In order to overcome this inconsistency, a heuristic correction was introduced by assuming that heat flux travels a characteristic length during the duration of the laser pulse [M. Ruiz-Veloz et al., J. Appl. Phys. 130, 025104 (2021)] τp. In this work, we obtained exact analytical solutions in the frequency domain for a 1D boundary-value-problem for the Dual-Phase-Lag (DPL) heat equation coupled with a 1D PA-boundary-value-problem via the acoustic wave equation. Temperature and pressure solutions were studied by assuming that the sample and its surroundings have a similar characteristic thermal lag response time τT; therefore, the whole system is assumed to have a similar thermal relaxation. A second assumption for τT is that it is considered as a free parameter that can be adjusted to reproduce experimental results. Solutions for temperature and pressure were obtained for a one-layer 1D system. It was found that for τT<τp, the DPL temperature has a similar thermal profile of the Fourier heat equation; however, when τT≥τp, this profile is very different from the Fourier case. Additionally, via a numerical Fourier transform, the wave-like behavior of DPL temperature is explored, and it was found that as τT increases, thermal wave amplitude is increasingly attenuated. Exact solutions for pressure were compared with experimental PA signals, showing a close resemblance between both data sets, particularly in time domain, for an appropriated value of τT; the transference function was also calculated, which allowed us to find the maximum response in frequency for the considered experimental setup.

In this paper, we consider the Moore–Gibson–Thompson–Fourier system made by coupling the Moore–Gibson–Thompson (MGT) equation with the classical Fourier heat equation known as the MGT–Fourier model. For σ=αβ−γ>0, the authors used the semi-group method to prove the existence and uniqueness of global solutions and the exponential stability of total energy. Our contribution will consist in studying numerical method based on finite element discretization in the spacial variable x and finite difference schema in time of the MGT–Fourier model. A discrete stability property and a priori error estimates are proved. Finally, the numerical simulation agrees well with theoretical results.

This article presents a new approach for calculating the heat distribution of a cylindrical wire carrying an alternating current. It is an approximation method that uses the skin-depth factor to distribute the heat flow into two different directions. The main objective of this method is to develop a relatively simple heat equation to calculate the temperature in cylindrical wire without using Basel functions. First, a Fourier heat equation for direct current is shown and compared with 2D FEM simulation results. Then the approximation formula will be derived from the Fourier heat equation for the case of alternating current (AC). A 2D FEM simulation is also performed for this case to validate the results of the approximation formula. The results show that the approximation formula is very suitable for most applications.

This article compares machine learning methods and a numerical method of determination of the doped lubricating layer with experimental data. Based on the sweep method, the one-dimensional Fourier heat equation with boundary and initial conditions is solved. As a result of comparing numerical and predictive data with experiments, it can be concluded that machine learning models are better at predicting results compared to numerical data

Symmetry can play an important role in the study of boundary value problems, which are a type of problem in mathematics that involves finding the solutions to differential equations subject to given boundary conditions. Integral transforms play a crucial role in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and integral equations. This article focuses on extending a single-valued Sawi transform to a double-valued ST, which we call the double Sawi (DS) transform. We derive some fundamental features and theorems for the proposed transform. Finally, we study the applications of the proposed transform by solving some boundary value problems such as the Fourier heat equation and the D’Alembert wave equation.

Convective thermal metamaterials are artificial structures where convection dominates in the thermal process. Due to the field coupling between velocity and temperature, convection provides a new knob for controlling heat transfer beyond pure conduction, thus allowing active and robust thermal modulations. With the introduced convective effects, the original parabolic Fourier heat equation for pure conduction can be transformed to hyperbolic. Therefore, the hybrid diffusive system can be interpreted in a wave-like fashion, reviving many wave phenomena in dissipative diffusion. Here, recent advancements in convective thermal metamaterials are reviewed and the state-of-the-art discoveries are classified into the following four aspects, enhancing heat transfer, porous-media-based thermal effects, nonreciprocal heat transfer, and non-Hermitian phenomena. Finally, a prospect is cast on convective thermal metamaterials from two aspects. One is to utilize the convective parameter space to explore topological thermal effects. The other is to further broaden the convective parameter space with spatiotemporal modulation and multi-physical effects.

Time-dependent dielectric breakdown (TDDB) is a crucial issue for the dielectric reliability. In this work, we present a full three-dimensional mechanistic model for calculation of the TDDB process in polycrystalline thin films. The model is based on the multiphonon trap-assisted tunneling theory and takes into account the intrinsic three-dimensional discreteness of traps at the dielectric grain boundaries. The leakage current density is calculated by solving coupled three-dimensional master equation and Poisson equation. The net phonon emission associated with each charge trapping and release event is treated as a local point heat source, which then enters the Fourier heat equation for three-dimensional temperature distribution calculation. The generated trap is determined by local temperature and electric field, which is subsequently included in the next round of calculation of electric and thermal properties. A positive feedback loop gradually leads to an increase of trap density, temperature, and leakage current density, and finally the dielectric breakdown. Our model can, to a good approximation, reproduce the experimental leakage current density-voltage characteristics and the Weibull distribution of time to breakdown at different dielectric thicknesses, stress voltages, and environmental temperatures. We find that in realistic devices, the three-dimensional trap-to-trap transport of electrons contributes a non-negligible part to the leakage current when the dielectric approaches breakdown. Our approach of three-dimensional mechanistic simulation is computationally efficient such that evolution of ${10}^{3}$ traps during the TDDB process can be easily performed on a standard desktop computer.

The Purpose of the paper. The study of thermal physics of the process of drilling polymer composite materials based on the capabilities of the engineering package Comsol Multiphysics. Determination of the values of temperatures arising in the cutting zone when drilling with a cutting tool in polymer composite materials, namely, in carbon fiber. Establishment of the depth of heat propagation inside the workpiece from the edge of the formed hole of the polymer composite material. In this work, the Fourier heat equations are used. Numerical simulation of the drilling process wascarried out in the engineering package COMSOL Multiphysics. A technique for modeling thermal effects during drilling of carbon fiber reinforced plastics in the Comsol Multiphysics environment has been developed. As a result of computer calculation, temperature fields were obtained at the edge of the carbon fiber reinforced plastic hole. The study of the thermal state of carbon plastics has been carried out. On the basis of the conducted research, it was revealed that the temperature during drilling of carbon fiber reaches 650 K. The distance over which heat spreads from the edges of the hole into the workpiece for carbon fiber is 3 mm. The developed technique for modeling the thermal impact of cutting polymer composite materials in the COMSOL Multiphysics environment can significantly simplify complex analytical calculations, help to avoid overheating of the part during drilling, which improves the quality of processing.

Mathematical analysis and numerical simulation of the Guyer–Krumhansl heat equation

Thermal diffusivity from Fourier’s inverse problem supervised by an optimization model: Theoretical analysis and experimental validation

Large amplitude vibration of annular and circular functionally graded composite plates under cooling thermal shocks

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.

A simple model of thermal conduction in human skin: temperature perception and thermal effusivity

The physics behind the cooling process occurring in an elastocaloric Cu–Al–Ni wire during the martensite to austenite transition after stress release is studied. A previous experiment using infrared imaging determined the temperature map evolution of the sample surface and obtained the qualitative evolution of heat sinks by reversing the Fourier heat equation. The results were only qualitative due to the data smoothing algorithm that was used to correct the experimental noise. We analyze the original data by using a more appropriate Gaussian filter to reject the noise without smoothing the sharp martensite-austenite interfaces. Results show a much better spatial resolution that enables us to obtain new physical variables characterizing the discontinuous, avalanche-like, cooling process: the sink width, the sink cooling power, and the sink velocity.

This paper examines thermophysics of the drilling process of polymeric composite materials, such as carbon-fibre-reinforced plastics (CFRP) and fibreglass by tubular diamond drill bits. Features of the COMSOL Multiphysics engineering software package were used. We employed Fourier heat equations, which express the intensity of heat gain by a mobile source in a moving coordinate system. The research was performed using the proprietary method of modelling spatial thermal action upon drilling polymer composite materials (fibreglass and carbon-fibre-reinforced plastics) in the COMSOL Multiphysics software environment. A tubular diamond drill bit with a diameter of 10 mm with two slots was chosen as a model cutting tool. Solid plates with a thickness of 5.5 mm made of layered fibrous polymer composite materials (fibreglass, carbon-fibre-reinforced plastic) were used as a preform. As a result of computer calculations, we obtained temperature fields of fibreglass and carbon-fibre-reinforced plastic during diamond drilling with a tubular tool. When studying the thermal behaviour of fibreglass and carbon-fibre-reinforced plastics, maximum temperature fields were located. The study revealed that the temperature reaches 413.6 and 448.7 K during CFRP and fibreglass drilling, respectively. It was shown that the distance of heat transfer from the edge of the hole into the preform was 6.42 and 6.40 mm for CFRP and fibreglass, respectively. A method of modelling the thermal effects when cutting polymer composite materials developed in the COMSOL Multiphysics environment allows complex analytical calculations of temperatures induced by drilling to be simplified. In addition, its use prevents overheating of a preform during drilling, allows assessing the depth of heat distribution inside the preform from the edge of the formed hole in different polymer composite materials. These measures increase the machining quality of polymer composite materials.

This paper examines the thermophysics of a drilling process of polymeric composite materials such as carbonfibre-reinforced plastics (CFRP) and fibreglass by tubular diamond drill bits. Features of the COMSOL Multiphysics engineering software package were used. We employed Fourier heat equations, which express the intensity of heat gain by a mobile source in a moving coordinate system. The research was performed using the proprietary method of modelling spatial thermal action upon drilling polymer composite materials (fibreglass and carbon-fibre-reinforced plastics) in the COMSOL Multiphysics software environment. A tubular diamond drill bit with a diameter of 10 mm with two slots was chosen as a model cutting tool. Solid plates with a thickness of 5.5 mm made of layered fibrous polymer composite materials (fibreglass, carbon-fibre-reinforced plastic) were used as a preform. As a result of computer calculations, we obtained temperature fields of fibreglass and carbon-fibre-reinforced plastic during diamond drilling with the tubular tool. When studying the thermal behaviour of fibreglass and carbon-fibre-reinforced plastics, maximum temperature fields were located. The study revealed that the temperature reaches 413.6 K and 448.7 K during CFRP and fibreglass drilling, respectively. It was shown that the distance of heat transfer from the edge of the hole into the preform was 6.42 and 6.40 mm for CFRP and fibreglass, respectively. A method of modelling the thermal effects when cutting polymer composite materials developed in the COMSOL Multiphysics environment allows complex analytical calculations of temperatures induced by drilling to be simplified. In addition, it helps avoid overheating of a preform during drilling, allows the depth of heat distribution inside the preform from the edge of the formed hole in different polymer composite materials to be assessed. These measures lead to increasing the machining quality of polymer composite materials.

Determination of position-dependent solid fraction by a new cooling curve analysis method: Semi-Newtonian Fourier thermal analysis

A comparison of predictive control strategies for a highly segmented injection mold tempering