Transient thermal boundary value problems in the half-space with mixed convective boundary conditions

Abstract

Numerous problems in thermal engineering give rise to problems with abrupt changes in boundary conditions particularly as regards structural and construction applications. There is a continual need for improved analytical and numerical techniques for the study and analysis of such canonical problems. The present work considers transient heat propagation in a two-dimensional half-space with mixed boundary conditions of the Dirichlet and the Robin (convective) type. The exact temperature field is obtained in double integral form by applying the Wiener–Hopf technique, and this integral is then efficiently evaluated numerically with the help of numerical algorithms developed for the present problem and by exploiting the properties of the integrand. The results are then compared to the numerical solutions using finite element methods, and excellent agreement is shown. Some potential application areas of the theory are also discussed, specifically in the regime where edge effects can influence thermal convection processes.

Appendices

Appendix A Comparison of the factorisation algorithm to the conventional approach

The proposed algorithm involved in the function factorisation is explained in Sect. 4. This algorithm allows one to evaluate the Hilbert transform of an arbitrary function $\phi (p)$ which is defined as follows:

$$\begin{aligned} \mathcal{H}[\phi ](p)=\frac{1}{\pi }\int _{-\infty }^{\infty }\frac{\phi (\xi )}{(\xi -p)}\,\text {d}\xi , \end{aligned}$$

(A.1)

where $\mathcal{H}[.]$ stands for the Hilbert transform. There exists a connection between the Fourier transform of the Hilbert transform of $\phi (p)$ and the Fourier transform of $\phi (p)$ itself. This relations is expressed as

$$\begin{aligned} \mathcal {F}[\mathcal {H}[\phi ]](p)=(-\text {i}\,\text {sgn}(p))\mathcal {F}[\phi ](p), \end{aligned}$$

(A.2)

where $\mathcal {F}[.]$ is the Fourier transform and $\text {sgn}(p)$ is the signum function. According to the last relation one can utilise the FFT which is implemented in many software. Specifically, one should apply the FFT to $\phi (p)$ and then apply the inverse FFT to obtain (A.1).

One of the main advantages of the FFT is that it can be performed extremely fast and usually this algorithm is already optimised in computational software. Nevertheless, it possesses several drawbacks. Firstly, it is well suited for fast decaying functions at infinity. However, in many cases, the functions decay reasonably slowly, and a large integration interval should be taken in order to achieve accurate results. For instance, in the problems presented in the main body of the text, the functions behave as $O(|p|^{-1})$ as $p\rightarrow \infty .$ Moreover, if the interpolation points where the function $\phi (\xi )$ is computed are $\xi _j=(j-N/2)\Delta \xi ,\,j=0,1,\ldots , (N-1)$ with the step size $\Delta \xi$ and the total number of points N,  then evaluation points of the FFT are $p_j=(j-N/2)\Delta p,\,j=0,1,\ldots , (N-1)$ with the step size $\Delta p=1/(N \Delta \xi ).$ Thus, the scales are linked, and they depend on the number of evaluation points.

Fig. 8

Relative error of the Hilbert transform through different algorithms (the FFT-based algorithm and the algorithm discussed in this work) and the exact results in (A.3)

On the contrary, the proposed algorithm is well adapted for functions with slow decay. Furthermore, the scales of interpolation points and evaluation points are not connected and one can use a small number of interpolation points of $\phi (\xi )$ adapted to the particular function, e.g. to intensify the mesh in regions of rapid change of function curvature, and obtain a large number of the evaluation points of $\mathcal {H}[\phi ](p)$ (Fig. 8).

In order to illustrate our argument, we compare the two procedures for the function $x/(1+x^2)$ the Hilbert transform of which is

$$\begin{aligned} \mathcal {H}\left[ \frac{x}{(1+x^2)}\right] (p)=\frac{1}{(1+p^2)}. \end{aligned}$$

(A.3)

The relative errors for the evaluation of this transform with the two different approaches is shown in (8). The evaluation points on this plot are distributed with the step $\Delta p=0.1$ between 0 and 20. The results are chosen in such a way that the $L_2$-norm (mean-square norm) of the difference between the numerical algorithms and the exact results is the same within this range and is equal to $1.7 \times 10^{-4}.$ With this restriction the number, the total number of interpolation points for the FFT-based algorithm is $N=2 \times 10^{6}$ (the evaluation points are then extracted from the whole range) whereas for the spline interpolation only 30 points were required (although the mesh density is higher close to $\xi =0$). Moreover, the relative error for the spline-based algorithm of this work is lower within the considered interval, regardless of its non-monotonic behaviour. Finally, the evaluation time for the FFT-based algorithm appears to be 0.25 s but the time for the developed algorithm is 0.015 s.

Both algorithms have potential for improvements and optimisations, but this lies outside the scope of the present work.

Appendix B Variational formulation used for the FEM simulations

The numerical simulations of the problem defined by Eqs. (3),(4) and (5) defined for the semi-infinite domain $\varOmega$ require the definition of the finite domain $\varOmega _0.$ We choose a rectangular domain $\varOmega _0=\{(x,y):0\le x<a,-0.5b<y<0.5b\}$ where values a and b are big enough in order to avoid the reflections from the boundaries. The boundary of such domain is formed by $\partial \varOmega _0^{\pm }$ which corresponds to $\partial \varOmega ^{\pm },$ respectively, and the remaining part of the boundary $\partial \varOmega _0'=\partial \varOmega _0\setminus (\partial \varOmega _0^+\cup \partial \varOmega _0^-)$, where temperature is fixed to be zero. The problem is formulated as follows:

$$\begin{aligned} \begin{aligned}&\frac{\partial T}{\partial t} - \nabla ^2T=0\quad \text {in }\varOmega _0,\\&T=T_0f(t)\quad \text {on }\partial \varOmega _0^+,\\&T-h\frac{\partial T}{\partial x}=\varTheta _\mathrm{i} F(t)\quad \text {on }\partial \varOmega _0^-,\\&T=0\quad \text {on }\partial \varOmega _0'. \end{aligned} \end{aligned}$$

(B.1)

Let $\delta T$ be a trial function of T,  the weak formulation of this problem becomes

$$\begin{aligned} \begin{aligned}&\int _{\varOmega _0}\delta T\frac{\partial T}{\partial t}\,\text {d}V+\int _{\varOmega _0}\nabla (\delta T)\cdot \nabla T\,\text {d}V+\int _{\partial \varOmega _0^-}\frac{T-\varTheta _\mathrm{i}F(t)}{h}\,\text {d}S=0,\\&T=T_0f(t)\, \text {on }\partial \varOmega _0^+,\quad \text {and}\quad T=0\, \text {on }\partial \varOmega _0', \end{aligned} \end{aligned}$$

(B.2)

where differential elements of surface and volume are denoted by $\text {d}S$ and $\text {d}V$, respectively, and test function $\delta T$ and function T belong to the spaces $\mathcal {S}_1$ and $\mathcal {S}_2$, respectively

$$\begin{aligned} \mathcal {S}_1 = \left\{ f(x,y)\in H^1(\varOmega _0):f(x,y)=0,(x,y)\in {\partial \varOmega _0^+}\cup \partial \varOmega _0'\right\} , \quad \mathcal {S}_2 = \left\{ f(x,y)\in H^1(\varOmega _0)\right\} . \end{aligned}$$

The space $H^1(\varOmega _0)$ designates the Sobolev spaces of scalar functions, having square-integrable first generalised derivatives. For the simulations, we chose $a=5$ and $b=10$ while the simulation time range was $t=[0,1].$ The border lines were discretised with the element sizes $\Delta x=0.05$ and $\Delta y=0.05$ in x and y directions, respectively. The time step is $\Delta t=0.005$, and piecewise continuous elements were taken.