A Finite Difference Algorithm Applied to the Averaged Equations of the Heat Conduction Issue in Biperiodic Composites—Robin Boundary Conditions

Abstract

This note deals with the heat conduction issue in biperiodic composites made of two different materials. To consider such a nonuniform structure, the equations describing the behavior of the composite under thermal (Robin) boundary conditions were averaged by using tolerance modelling. In this note, the process of creating an algorithm that uses the finite difference method to deal with averaged model equations is shown. This algorithm can be used to solve these equations and find out the temperature field distribution of a biperiodic composite.

Influence of Composite Structure on Temperature Distribution—An Analysis Using the Finite Difference Method

“…Using the created algorithm of the Finite Difference Method [ 23 , 25 , 26 , 28 ], calculations were conducted for the three structures described above. The two-dimensional, non-stationary heat-conduction problem of composites with base dimensions equal to L 1 = L 2 = 1 [m] was analyzed.…”

“…To create an algorithm of the Finite Difference Method, it is necessary to divide an analyzed structure into ranges, redefine the tolerance model equations to formulate them in index notation, replace the derivatives appearing in these equations by appropriate differential quotients, and define the boundary conditions, because this affects the number of equations to solve [ 26 ]. All of the structures, whether periodic, biperiodic, or with a functional gradation of properties, were discretized along the x 1 -direction into m ranges, so the number of the nodes along the x 1 -direction equals m + 1, and analogously along the x 2 -direction into n ranges, so the number of the nodes along the x 2 -direction equals n + 1; refer to Figure 5 .…”

“…The technique, used to average the discontinuous coefficients in Equation (1), is constantly developed and used in the analysis of periodic [ 18 , 19 , 20 , 21 , 22 , 23 ], biperiodic [ 24 , 25 , 26 ] and functionally graded structures [ 27 , 28 ]. The most relevant assumption of the Tolerance Averaging Technique, from the point of view of the carried-out considerations, is the micro–macro decomposition assumption, according to Equation (2): where the total temperature field θ is divided into an averaged part ϑ and an oscillating part in the form of product g a ·ψ a .…”

“…Each cell is composed of two (periodicity) or four (biperiodicity) sub-cells, and the above-mentioned material properties within a given sub-cell are constant. The technique, used to average the discontinuous coefficients in Equation ( 1), is constantly developed and used in the analysis of periodic [18][19][20][21][22][23], biperiodic [24][25][26] and functionally graded structures [27,28]. The most relevant assumption of the Tolerance Averaging Technique, from the point of view of the carried-out considerations, is the micro-macro decomposition assumption, according to Equation (2):…”

Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core

“…[ 25 ]. Within the literature, one can find multiple applications of this technique in various mechanical issues, such as stability analysis [ 26 , 27 , 28 , 29 ], dynamics [ 30 , 31 , 32 , 33 ] or even heat conduction issues [ 34 , 35 , 36 , 37 ].…”