Heat diffusion in layered media via the thin‐layer method

Abstract

SUMMARYThe thin-layer method is a semi-analytical, numerical tool that has been successfully used in the past for wave motion in layered media, and differs from finite elements in that the medium is discretized only along a subset of the problem's dimensions, i.e. it combines partial discretization with analytical solutions. This paper shows that it can be applied just as well to heat diffusion, and results are given for impulsive as well as distributed point and dipole thermal sources with formulations in Cartesian and cylindrical coordinates. In essence, the method involves the discretization of the governing equations in the direction of layering together with analytical solutions for the remaining directions and for the variation in time. The solutions in response to spatially and temporally impulsive sources can also be used as fundamental solutions in the context of a boundary element formulation. This is illustrated herein by means of an example of a concrete slab overlain by two layers of carbon fiber reinforced polymers (CFRP), where a penny-shaped crack is meant to simulate an area where the CFRP has debonded from the concrete.

A method of recursive images extended to solve diffusion in multilayer materials under convective boundary conditions

“…The temperature of the convective medium is 1 K above the initial temperature of the slab, the convection constant is h = 1 W/m 2 Á K, while the density is 1 kg/m 3 , the specific heat is 1 K/kg and the thermal conductivity is 1 W/m Á K. It is apparent from this figure that the analytical solution obtained from the separation of variables Eqs. (13) and (14) and that obtained using this method of recursive images are virtually identical justifying thus the validity of the latter method also in the case where convection is present. Indeed, in both cases the temperature in the slab starts to rise gradually until it reaches an uniform temperature of 1 K after 5 s.…”

“…This diffusion problem raises continuing and considerable interest and various approaches have been proposed for its solution [6][7][8][9][10][11][12][13][14]. Recently, we have proposed an extension to the method of images which could give exact solutions for the diffusion equation in a material with any number of layers, in a way that is conceptually simple [15].…”

A local high-order doubly asymptotic open boundary for diffusion in a semi-infinite layer

“…For such real positive values of a, the modal impedance coefficient S(a) given in Eq. (25) is always complex, the cut-off phenomenon known from wave propagation does not exist for diffusion. Nevertheless, it is obvious from Fig.…”

Perfectly matched layers in the thin layer method

“…Since its inception in the early 1970's [27,28,41], the TLM has found widespread use in soil dynamics and soil-structure interaction [37,38], non-destructive evaluation methods, seismic source simulations, wave propagation in waveguides of complex cross-section, wave propagation in laminated, anisotropic materials [20], waves in piezoelectric materials [8], heat diffusion in layered composites [15], consolidation in poroelastic media, solid-fluid interaction [39], and in many more areas of application. Although the origin and early development of the TLM technique hark back to the early 1970's, the designation TLM became common only since the beginning of the 1990's.…”