通过 (α, β) 距离变换计算瞬态热传导问题中边界元法的二维域积分数值计算

Axioms Pub Date : 2024-07-22 DOI:10.3390/axioms13070490
Yunqiao Dong, Zhengxu Tan, Hengbo Sun
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引用次数: 0

摘要

当采用时变边界元法(也称为伪初始条件法)求解瞬态热传导问题时,必须对域积分进行数值评估。因此,精确计算域积分对分析瞬态热传导至关重要。然而,随着时间步长逐渐减小并趋近于零,域积分的积分项接近奇异,直接使用标准高斯正交会产生较大误差。为了解决这个问题并进一步提高域积分的计算精度,提出了一种 (α, β) 距离变换。距离变换是一种简单有效的消除近奇异性的方法,通常用于近奇异积分。首先,介绍了 (α, β) 坐标变换。然后,通过用时间步长代替最短距离,为域积分建立了一种新的距离变换。采用新方法后,域积分的整数得到了大幅平滑,有效消除了域积分中因小时间步长而产生的奇异性。因此,通过 (α, β) 距离变换可以获得更精确的结果。在数值示例中考虑了不同的时间步长、源点位置和积分元素形状。使用各种方法对域积分的数值结果进行比较研究表明,所提出的方法具有更高的精度和效率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems
When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method.
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