Peridynamics for Heat Conduction

2020 ◽  
Author(s):  
Yozo Mikata
Keyword(s):  
Heat Conduction ◽  
Constitutive Equation ◽  
Heat Equation ◽  
Governing Equation ◽  
Heat Conduction Equation ◽  
Transient Heat Conduction ◽  
Anisotropic Materials ◽  
Time Dependent ◽  
Heat Sources ◽  
Exact Analytical Solutions

Abstract Peridynamics for transient heat conduction problems in general anisotropic materials is developed. In order to develop a new peridynamic governing equation for heat conduction problems, the microconductivity (or microdiffusivity), which contains equivalent information as the constitutive equation for classical heat conduction, is determined by directly requiring the resulting peridynamic equation to converge to a classical heat conduction equation for anisotropic materials as the generalized material horizon approaches 0. Therefore, the convergence proof is built into the theory from the perspective of the governing equation. For the application of the newly obtained peridynamic governing equation, a time-dependent 3D peridynamic heat equation is analytically solved with two types of heat sources, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic heat conduction.

scholarly journals Analytic transient solutions of a cylindrical heat equation

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2617-2628
Author(s):  
K.Y. Kung ◽  
Man-Feng Gong ◽  
H.M. Srivastava ◽  
Shy-Der Lin
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Transient Temperature ◽  
Transient Solutions

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

scholarly journals Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

2022 ◽  
Vol 2022 ◽  
pp. 1-8
Author(s):  
Endalew Getnet Tsega
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Numerical Solution ◽  
Numerical Method ◽  
Heat Conduction Equation ◽  
Three Dimensional ◽  
Transient Heat Conduction ◽  
Cylindrical Coordinates ◽  
Central Difference ◽  
Central Difference Method

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

scholarly journals Construction and investigation of new numerical algorithms for the heat equation : Part 2

2020 ◽  
Vol 10 (4) ◽  
pp. 339-348
Author(s):  
Mahmoud Saleh ◽  
Ádám Nagy ◽  
Endre Kovács
Keyword(s):  
Heat Conduction ◽  
Numerical Methods ◽  
Heat Conduction Equation ◽  
Space Dimension ◽  
Numerical Test ◽  
Numerical Algorithms ◽  
Heat Conducting ◽  
Heat Sources ◽  
Test Results ◽  
New Algorithms

This paper is the second part of a paper-series in which we create and examine new numerical methods for solving the heat conduction equation. Now we present numerical test results of the new algorithms which have been constructed using the known, but non-conventional UPFD and odd-even hopscotch methods in Part 1. Here all studied systems have one space dimension and the physical properties of the heat conducting media are uniform. We also examine different possibilities of treating heat sources.

Linear peridynamics for triclinic materials

2020 ◽  
pp. 108128652096988
Author(s):  
Yozo Mikata
Keyword(s):  
Constitutive Equation ◽  
Plane Wave ◽  
Governing Equation ◽  
Wave Solution ◽  
Anisotropic Materials ◽  
Dispersion Curves ◽  
Plane Wave Solution ◽  
Peridynamic Theory

The governing equation of linear peridynamics is developed for the most general anisotropic materials (triclinic materials). As a departure from the standard peridynamic theory, the linear constitutive equation in the form of a micromodulus is determined by directly requiring the resulting peridynamic equation to converge to a comparable classical elastodynamic equation for a triclinic material as the generalized material horizon approaches zero. As a result, a new peridynamic governing equation is obtained for triclinic peridynamic materials. As an application of the newly obtained peridynamic equation, a plane wave solution is analytically obtained and discussed, and dispersion curves are plotted for triclinic peridynamic materials.

scholarly journals Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
R. T. Al-Khairy ◽  
Z. M. AL-Ofey
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Closed Form Solution ◽  
Infinite Medium ◽  
Form Solution ◽  
Time Dependent ◽  
Hyperbolic Heat Conduction ◽  
Laser Heat Source

This paper presents an analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by while the semi-infinite body has insulated boundary. The solution is obtained by Laplace transforms method, and the discussion of solutions for different time characteristics of heat sources capacity (constant, instantaneous, and exponential) is presented. The effect of absorption coefficients on the temperature profiles is examined in detail. It is found that the closed form solution derived from the present study reduces to the previously obtained analytical solution when the medium velocity is set to zero in the closed form solution.

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