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A continued-fraction approach for transient diffusion in unbounded medium

机译：无界介质中瞬态扩散的连续分数方法

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摘要

A temporally local method for the numerical solution of transient diffusion problems in unbounded domains is proposed by combining the scaled boundary finite element method and a novel solution procedure for fractional differential equations. The scaled boundary finite element method is employed to model the unbounded domain. In the Fourier domain (ω), an equation of the stiffness matrix for diffusion representing the flux-temperature relationship at the discretized near field/far field interface is established. A continued-fraction solution in terms of iω~(1/2) is obtained. By using the continued-fraction solution and introducing auxiliary variables, the flux-temperature relationship is formulated as a system of linear equations in iω~(1/2). In the time-domain, it is interpreted as a system of fractional differential equations of degree α = 1/2. To eliminate the computationally expensive convolution integral, the fractional differential equation is transformed to a system of first-order differential equations. Numerical examples of two- and three-dimensional heat conductions demonstrate the accuracy of the proposed method. The computational cost of both the temporally global and local approach for transient analysis is examined.
机译：通过结合比例边界有限元方法和分数阶微分方程的新颖求解方法，提出了一种无边界域瞬态扩散问题数值解的时间局部方法。采用比例边界有限元方法对无界域进行建模。在傅立叶域（ω）中，建立了扩散刚度矩阵的方程，该方程表示离散的近场/远场界面处的通量-温度关系。得到以iω〜（1/2）表示的连续分数解。通过使用连续分数解并引入辅助变量，将通量-温度关系公式化为iω〜（1/2）中的线性方程组。在时域中，它被解释为度数= 1/2的分数阶微分方程组。为了消除计算量大的卷积积分，分数阶微分方程被转换为一阶微分方程组。二维和三维热传导的数值例子证明了该方法的准确性。研究了瞬态分析的时间全局和局部方法的计算成本。

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