A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials

Lisin

2022

EEJP

The paper presents the simulation results of heat transfer in single-crystal lithium niobate (LiNbO3) in the form of cylinder of diameter mm and height mm in interaction with continuous-wave laser radiation with the output power of W and the wavelength of nm. The density of the LiNbO3 crystal is kg/m3; the thermal conductivity along the [001] direction is W/(m×K); the thermal conductivity in the (001) plane is W/(m×K); the specific heat at constant pressure is J/(kg×K); the absorption coefficient is %/cm @ 1064 nm. The laser beam propagates along the optical axis of the crystal. The laser beam intensity profile is represented as a Gaussian function, and the absorption of laser radiation of the single-crystal lithium niobate is described by Beer-Lambert’s law. The numerical solution of the non-stationary heat conduction problem is obtained by meshless scheme using anisotropic radial basis functions. The time interval of the non-stationary boundary-value problem is 2 h 30 min. The results of numerical calculations of the temperature distribution inside and on the surface of the single-crystal lithium niobate at times s are presented. The time required to achieve the steady-state heating mode of the LiNbO3 crystal, as well as its temperature range over the entire time interval, have been determined. The accuracy of the approximate solution of the boundary-value problem at the n-th iteration is estimated by the value of the norm of relative residual . The results of the numerical solution of the non-stationary heat conduction problem obtained by meshless method show its high efficiency even at a small number of interpolation nodes.

Section: Numerical Resultsmentioning

confidence: 99%

Simulation of Heat Transfer in Single-Crystal Lithium Niobate in Interaction with Continuous-Wave Laser Radiation

Lisin

2022

EEJP

“…k AHorp x x x as basis functions in the implementation of the meshless method for solving three-dimensional non-stationary heat conduction problems in materials with anisotropy, described in [1]. In this approach, the combination of the dual reciprocity method [22] using anisotropic radial basis function and the method of fundamental solutions [23] is used for solving boundary-value problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Currently, meshless methods for the numerical solution of boundary-value problems are being actively developed [1][2][3][4][5]. In particular, methods that implement the approximation of a differential equation in the strong form (collocation methods) using compactly supported radial functions as basis [6][7][8][9][10].…”

mentioning

confidence: 99%

Family of the Atomic Radial Basis Functions of Three Independent Variables Generated by Helmholtz-Type Operator

2021

EEJP

The paper presents an algorithm for constructing the family of the atomic radial basis functions of three independent variables generated by Helmholtz-type operator, which may be used as basis functions for the implementation of meshless methods for solving boundary-value problems in anisotropic solids. Helmholtz-type equations play a significant role in mathematical physics because of the applications in which they arise. In particular, the heat equation in anisotropic solids in the process of numerical solution is reduced to the equation that contains the differential operator of the special form (Helmholtz-type operator), which includes components of the tensor of the second rank, which determines the anisotropy of the material. The family of functions is infinitely differentiable and finite (compactly supported) solutions of the functional-differential equation of the special form. The choice of compactly supported functions as basis functions makes it possible to consider boundary-value problems on domains with complex geometric shapes. Functions include the shape parameter , which allows varying the size of the support and may be adjusted in the process of solving the boundary-value problem. Explicit formulas for calculating the considered functions and their Fourier transform are obtained. Visualizations of the atomic functions and their first derivatives with respect to the variables and at the fixed value of the variable for isotropic and anisotropic cases are presented. The efficiency of using atomic functions as basis functions is demonstrated by the solution of the non-stationary heat conduction problem with the moving heat source. This work contains the results of the numerical solution of the considered boundary-value problem, as well as average relative error, average absolute error and maximum error are calculated using atomic radial basis functions and multiquadric radial basis functions.

“…The software «AnisotropicHeatTransfer3D» for the numerical solution of threedimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4].…”

mentioning

confidence: 99%

Software for Simulation of Non-Stationary Heat Transfer in Anisotropic Solids

Hariachevska

2022

GoS

The software «AnisotropicHeatTransfer3D» for the numerical solution of three-dimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4]. The dual reciprocity method with anisotropic radial basis functions is used to obtain particular solution, and the method of fundamental solutions is used to obtain a homogeneous solution of boundary-value problem.