Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions

This paper presents a set of fully analytical solutions, together with explicit expressions, in the time and frequency domain for the heat conduction response of homogeneous unbounded and of bounded rectangular spaces (three-, two-, and one-dimensional spaces) subjected to point, line, and plane heat diffusion sources. Particular attention is given to the case of spatially sinusoidal, harmonic line sources. In the literature this problem is often referred to as the two-and-a-half-dimensional fundamental solution or 2.5D Green’s functions. These equations are very useful for formulating three-dimensional thermodynamic problems by means of integral transforms methods and/or boundary elements. The image source technique is used to build up different geometries such as half-spaces, corners, rectangular pipes, and parallelepiped boxes. The final expressions are verified here by applying the equations to problems for which the solution is known analytically in the time domain.


Introduction
Problems in thermodynamics can often be solved with the aid of formulas or expressions known as Green's functions. These functions, or fundamental solutions, relate the field variables (heat fluxes and temperatures) at some location in a solid body caused by thermodynamic sources placed elsewhere in the medium.
The fundamental solutions most often used are point sources in a three-dimensional (3D), infinite homogeneous space; line sources acting within two-dimensional (2D) spaces; and plane sources heating one-dimensional (1D) spaces. The reason for these choices is that these three fundamental solutions are known in closed-form in time domain and have a relatively simple structure [1].
They are frequently combined to simulate heat conduction in the time domain or in a transform space defined by the Laplace transform, in half-spaces, infinite plates, rectangular 2D spaces, wedges, and rectangular 3D spaces [1][2][3]. Solutions have also been proposed to deal with multilayer systems, and they include the matrix method [1], the thermal quadrupole method [3], the thin layer method [4], and methods based on the definition of potentials [5][6][7]. Chen et al. have described the use of image method to solve 2D and 3D problems in unbounded and half-space domains containing circular or spherical shaped boundaries [8][9][10].
This paper compiles alternative fundamental solutions in explicit form, specifically Green's functions for harmonic 2D and 3D and harmonic (steady state) line sources whose amplitude varies sinusoidally in the third dimension. This last solution, which is often referred to in the literature as the 2.5D problem, can be of significant value when formulating 3D thermodynamics problems via boundary elements together with integral transforms. In addition, the proposed Green's functions are combined using an image source technique to model a half-space, a corner, a layer system, a laterally confined layer system, a solid rectangular column, a solid rectangular column with an end cross section, and a 3D parallelepiped inclusion. To the best of our knowledge, this is the first such derivation that promises to be efficient for formulating 3D thermodynamics problems using boundary elements and integral transforms.

Fundamental Solution
The transient heat transfer by conduction in an infinite, homogeneous space can be described by the diffusion equation in Cartesian coordinates: in which is time, ( , , , ) is the temperature at a point ( , , ) in the domain, and is the thermal diffusivity defined by /( ), where is the thermal conductivity, is the density, and is the specific heat of medium.
The solution of (1) can be obtained in the frequency domain after the application of a Fourier transform in the time domain, which leads to the following equation: where = √ −1, 1 = √− / and is the frequency. Consider first an infinite, homogeneous space subjected at ( 0 , 0 , 0 ) to a harmonic point heat source of the form where = √( − 0 ) 2 + ( − 0 ) 2 . Consider next an infinite, homogeneous space subjected to a spatially varying line heat source of the form ( − 0 ) ( − 0 ) ( − ) , with being the wavenumber in . This source acts in one of the three coordinate directions, passes through ( 0 , 0 ), and varies sinusoidally in the (i.e., third) dimension. This type of source is often referred to in the literature as a 2.5D source. The response to this source can be obtained by applying a spatial Fourier transform in the direction to the equations for a point heat load.
Applying a Fourier transformation in the direction leads to the solutioñ(  inverse Fourier transformation can be expressed as a discrete  summation if we assume the existence of virtual sources,  equally spaced at  along , which enables the solution to  be obtained by solving a limited number of 2D problems, ( , , , ) with being the axial wavenumber given by = (2 / ) . The distance chosen must be big enough to prevent spatial contamination from the virtual sources.
Equation (4) can be further manipulated and written as a continuous superposition of heat plane phenomena, where 1 = √− / − 2 − 2 and Im( 1 ) ≤ 0, and the integration is performed with respect to the horizontal wave number ( ) in the direction. Assuming the existence of an infinite number of virtual sources, we can discretize these continuous integrals. The integral in the above equation can be transformed into a summation if an infinite number of such sources are distributed along the direction, spaced at equal intervals . The above equation can then be written as where 0 = − /(2 ), = − 1 | | , = − ( ) , 1 = √− / − 2 − 2 and Im( 1 ) ≤ 0, and = (2 / ) , which can in turn be approximated by a finite sum of equations ( ). Note that = 0 is the 2D case,̃( , , ) = 0 ∑ =+∞ =−∞ ( / 1 ) with 1 = √− / − 2 . Next, the above Green's functions are combined so as to define Green's functions for a half-space, a corner, a single layer system, a U system, a solid rectangular pipe, a solid open box, and a 3D parallelepiped box. Expressions in frequency and time solutions are provided. The time solutions obtained after the application of inverse spatial and frequency Fourier transforms are compared with those given by Green's functions defined directly in the time domain.
Green's functions for the different spaces are determined using the image source method. By this method a distribution of virtual sources and sinks are combined so as to give null temperatures (Dirichlet boundary conditions) or heat fluxes on the required boundaries (Neumann boundary conditions). Other boundary conditions, such as Robin, are not studied in this paper. In the case of solid bodies bounded by two parallel surfaces the number of sources, placed perpendicular to the surfaces, is theoretically infinite.
The superscripts , , and identify the position of the virtual sources along the , , and directions, respectively. The upper value of , , and is defined by the convergence criteria. Each value of , , and is associated with four possible source positions, which are identified by the subscripts , , and for the , , and directions, respectively. Thus, , , and may take the values of 1, 2, 3, and 4.
The use of complex frequencies allows the contribution of the sources placed at greater distances to vanish and so to limit the number of the virtual sources. The use of complex frequencies with a small imaginary part, taking the form = − (where = 0.7Δ and Δ is the frequency increment), has the additional effect of avoiding the aliasing phenomena. This shift in the frequency domain is subsequently taken into account in the time domain by means of an exponential window, , applied to the response.
Green's functions are validated assuming that the medium is subject to a Dirac delta source. This type of source would require the solution to be computed in the frequency domain [0.0, ∞] Hz. However, the response does not need to be computed for a very large number of frequencies since it decays very quickly as the frequency decays. Note that the static response for the frequency 0.0 Hz can be calculated thanks to the use of complex frequencies.
The number of virtual sources used depends directly on the predefined convergence criterion. As we move from one dimension to two dimensions and then to three dimensions, the number of sources grows significantly. Thus, although the method converges rapidly, the cost of computation grows significantly as we move from a one-dimensional to a threedimensional problem.

Green's Functions
Green's functions in the time and frequency domain will be grouped for the following three cases: (i) unbounded space, which includes Green's functions for 1D, 2D, and 3D sources; (ii) two-dimensional space, which contains Green's functions for a half-space, a space bounded by two perpendicular planes, a single layer system, a U system, and a solid rectangular pipe, when subjected to 2D and 3D sources; (iii) three-dimensional space, which compiles Green The notation in Table 1 is used.
Equations for an Unbounded Space, Assuming 1D, 2D, and 3D Heat Sources. 1D source is as follows:   2D source is as follows: as the sum of plane sources.
3D source is as follows: as the sum of 2.5D sources.

Two-Dimensional Space
(a) Half-Space Defined by ≥ 0. Boundary conditions prescribed for the half-space (Cases 1 and 2) are shown in Figure 10.
Equations for the half-space (Cases 1 and 2), subjected to a 2D heat source, are as follows.
Case 2. Consider Equations for the half-space (Cases 1 and 2), subjected to a 3D heat source, are as follows.
as the sum of 2.5D sources. (13) as the sum of 2.5D sources. (14) (b) Bounded Space Defined by ≤ 1 and ≥ 0. Boundary conditions prescribed for the bounded space defined by ≤ 1 and ≥ 0 (Cases 1-4) are shown in Figure 11. Equations for the bounded space defined by ≤ 1 and ≥ 0 (Cases 1-4), subjected to a 2D heat source, are as follows. 10

Journal of Applied Mathematics
Equations defined for the bounded space defined by ≤ 1 and ≥ 0 (Cases 1-4) and subjected to a 3D heat source, are as follows.
as the sum of 2.5D sources.
as the sum of 2.5D sources. (20) as the sum of 2.5D sources.
as the sum of 2.5D sources. ] . (25) Equations for the horizontal layer (Cases 1-4), subjected to a 3D heat source, are as follows.
as the sum of 2.5D sources.
as the sum of 2.5D sources. (28) as the sum of 2.5D sources. (29) as the sum of 2.5D sources. (30) (d) U System Bounded by ≥ 0, ≤ 2 , and ≤ 1 . Boundary conditions prescribed for the U system (Cases 1-4) are shown in Figure 13. Equations for the U system (Cases 1-4), subjected to a 2D heat source, are as follows. . (31) . (32) Journal of Applied Mathematics 13 Case 4. Consider Equations for the U system (Cases 1-4), subjected to a 3D heat source, are as follows.
as the sum of 2.5D sources.
as the sum of 2.5D sources. (36) as the sum of 2.5D sources.
as the sum of 2.5D sources.
Equations for the solid rectangular pipe (Cases 1-4), subjected to a 2D heat source, are as follows.
as the sum of 2.5D sources. (43)

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Journal of Applied Mathematics as the sum of 2.5D sources. (44) as the sum of 2.5D sources. (45) as the sum of 2.5D sources. (46)
Equations for the 3D parallelepiped box (Cases 1-4), subjected to a 3D heat source, are as follows.
as the sum of 2.5D sources.

Conclusions
Fully analytical solutions for heat conduction for unbounded and rectangular spaces subjected to point, line, and plane sources have been presented. Two boundary conditions were assumed, namely, the Dirichlet and Neumann boundary conditions. Particular attention was given to the two-anda-half-dimensional fundamental solution or 2.5D Green's functions defined for spatially sinusoidal, harmonic line sources. The final expressions were validated by applying the equations to the problem of a Dirac delta source, for which the solutions in the time domain are known in analytical form. Excellent agreement was found between the numerical solutions given by Fourier synthesis and the exact solutions.