An efficient boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials under heat source load

Abstract

This paper presents a boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials (FGMs) with heat source load. In the proposed scheme, Laplace transformation and the numerical inverse Laplace transformation (NILT) are implemented to avoid the troublesome time-stepping effect on numerical efficiency. The collocation Trefftz method (CTM) coupled with composite multiple reciprocity method is used to obtain the high accurate results in the solution of nonhomogeneous problems in Laplace-space domain. The extended precision arithmetic is introduced to overcome the ill-posed issues generated from the CTM simulation, the NILT process and the large-size-ratio FGM. Heuristic error analysis and numerical investigation are presented to demonstrate the effectiveness of the proposed scheme for transient thermal analysis. Several benchmark examples are considered under large-size-ratio FGMs with some specific spatial variations (quadratic, exponential and trigonometric functions). The proposed scheme is validated in comparison with known analytical solutions and COMSOL simulation.

Appendix

Consider a two-dimensional problem with the domain

$$ \Omega = \left\{ {\left( {r,\theta } \right)\left| {0 \le r} \right. < R,0 \le \theta \le 2\pi } \right\} $$

(A1)

where $r,\theta$ denote the polar coordinates in 2D problems, it can be generated from the Cartesian coordinates $\left( {x,y} \right)$ with the origin at the center of 2D computational domain ${\mathbf{x}}_{c} = \left( {x_{c} ,y_{c} } \right)$, namely, $r = \sqrt {\left( {x - x_{c} } \right)^{2} + \left( {y - y_{c} } \right)^{2} }$ and $\theta = \arctan \left( {\frac{{y - y_{c} }}{{x - x_{c} }}} \right)$. The high-order Trefftz functions of common-used operators are listed as follows:

• (i) High-order Trefftz functions of Laplacian operator $\Delta^{n + 1}$

For Laplace equation $\Delta u = 0$, when n = 0, its Trefftz function are given in the literature as

$$ 1,r^{m} \cos \left( {m\theta } \right),r^{m} \sin \left( {m\theta } \right)\quad m = 1,2, \ldots ,\quad \left( {r,\theta } \right) \in \Omega $$

(A2)

which are known as zero-order Trefftz function $u^{T0}$. The corresponding nth order Trefftz function $u^{Tn}$ is presented as follows

$$ A_{n} r^{2n} ,A_{n} r^{m + 2n} \cos \left( {m\theta } \right),A_{n} r^{m + 2n} \sin \left( {m\theta } \right)\quad m = 1,2, \ldots ,\quad \left( {r,\theta } \right) \in \Omega $$

(A3)

where $A_{n} = \frac{{A_{n - 1} }}{{4n\left( {m + n} \right)}},A_{0} = 1$.

• (ii) High-order Trefftz functions of Helmholtz operator $\left( {\Delta + \lambda^{2} } \right)^{n + 1}$

In the Helmholtz operator, $\lambda > 0$ is a real number and assume that $\lambda^{2}$ is not an eigenvalue of Laplace operator. Then the zero-order Trefftz function $u^{T0}$ can be written as

$$ J_{0} \left( {\lambda r} \right),J_{0} \left( {\lambda r} \right)\cos \left( {m\theta } \right),J_{0} \left( {\lambda r} \right)\sin \left( {m\theta } \right)\quad m = 1,2, \ldots ,\quad \left( {r,\theta } \right) \in \Omega $$

(A4)

where $J_{0}$ is the Bessel function of the first kind. And the following functions are the corresponding nth order Trefftz functions $u^{Tn}$.

$$ \begin{aligned} & A_{n} \left( {\lambda r} \right)^{n} J_{n} \left( {\lambda r} \right),A_{n} \left( {\lambda r} \right)^{n} J_{m + n} \left( {\lambda r} \right)\cos \left( {m\theta } \right), \\ & A_{n} \left( {\lambda r} \right)^{n} J_{m + n} \left( {\lambda r} \right)\sin \left( {m\theta } \right),\quad m = 1,2, \ldots ,\quad \left( {r,\theta } \right) \in \Omega \\ \end{aligned} $$

(A5)

where $A_{n} = \frac{{A_{n - 1} }}{{2n\lambda^{2} }},A_{0} = 1$.

• (iii) High-order Trefftz functions of modified Helmholtz operator $\left( {\Delta - \lambda^{2} } \right)^{n + 1}$

In the modified Helmholtz operator, $\lambda$ is again a real number and $\lambda > 0$. Then the zero-order Trefftz function $u^{T0}$ can be given in the form

$$ I_{0} \left( {\lambda r} \right),I_{0} \left( {\lambda r} \right)\cos \left( {m\theta } \right),I_{0} \left( {\lambda r} \right)\sin \left( {m\theta } \right)\quad m = 1,2, \ldots ,\quad \left( {r,\theta } \right) \in \Omega $$

(A6)

where I₀ is the Bessel and Hankel functions with a purely imaginary argument. The corresponding nth order Trefftz function $u^{Tn}$ is presented as follows

$$ \begin{aligned} & A_{n} \left( {\lambda r} \right)^{n} I_{n} \left( {\lambda r} \right),A_{n} \left( {\lambda r} \right)^{n} I_{m + n} \left( {\lambda r} \right)\cos \left( {m\theta } \right), \\ & A_{n} \left( {\lambda r} \right)^{n} I_{m + n} \left( {\lambda r} \right)\sin \left( {m\theta } \right),\quad m = 1,2, \ldots ,\quad \left( {r,\theta } \right) \in \Omega \\ \end{aligned} $$

(A7)

where $A_{n} = \frac{{A_{n - 1} }}{{2n\lambda^{2} }},A_{0} = 1$.