Purpose

The boundary integral method (BIM) is very attractive to practicing engineers as it reduces the dimensionality of the problem by one, thereby making the procedure computationally inexpensive compared to its peers. The principal feature of this technique is the limitation of all its computations to only the boundaries of the domain. Although the procedure is well developed for the Laplace equation, the Poisson equation offers some computational challenges. Nevertheless, the literature provides a couple of solution methods. This paper revisits an alternate approach that has not gained much traction within the community. The purpose of this paper is to address the main bottleneck of that approach in an effort to popularize it and critically evaluate the errors introduced into the solution by that method.

Design/methodology/approach

The primary intent in the paper is to work on the particular solution of the Poisson equation by representing the source term through a Fourier series. The evaluation of the Fourier coefficients requires a rectangular domain even though the original domain can be of any arbitrary shape. The boundary conditions for the homogeneous solution gets modified by the projection of the particular solution on the original boundaries. The paper also develops a new Gauss quadrature procedure to compute the integrals appearing in the Fourier coefficients in case they cannot be analytically evaluated.

Findings

The current endeavor has developed two different representations of the source terms. A comprehensive set of benchmark exercises has successfully demonstrated the effectiveness of both the methods, especially the second one. A subsequent detailed analysis has identified the errors emanating from an inadequate number of boundary nodes and Fourier modes, a high difference in sizes between the particular solution and the original domains and the used Gauss quadrature integration procedures. Adequate mitigation procedures were successful in suppressing each of the above errors and in improving the solution accuracy to any desired level. A comparative study with the finite difference method revealed that the BIM was as accurate as the FDM but was computationally more efficient for problems of real-life scale. A later exercise minutely analyzed the heat transfer physics for a fin after validating the simulation results with the analytical solution that was separately derived. The final set of simulations demonstrated the applicability of the method to complicated geometries.

Originality/value

First, the newly developed Gauss quadrature integration procedure can efficiently compute the integrals during evaluation of the Fourier coefficients; the current literature lacks such a tool, thereby deterring researchers to adopt this category of methods. Second, to the best of the author’s knowledge, such a comprehensive error analysis of the solution method within the BIM framework for the Poisson equation does not currently exist in the literature. This particular exercise should go a long way in increasing the confidence of the research community to venture into this category of methods for the solution of the Poisson equation.

中文翻译：

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边界积分法求解泊松方程

 目的

边界积分法 (BIM) 对实践工程师非常有吸引力，因为它可以将问题的维度减少一维，从而使该过程的计算成本比同类方法便宜。该技术的主要特征是将其所有计算仅限于域的边界。尽管拉普拉斯方程的程序已经开发得很好，但泊松方程带来了一些计算挑战。尽管如此，文献还是提供了一些解决方法。本文重新审视了一种在社区中尚未获得太多关注的替代方法。本文的目的是解决该方法的主要瓶颈，努力推广该方法，并批判性地评估该方法在解决方案中引入的错误。

设计/方法论/途径

本文的主要目的是通过傅里叶级数表示源项来研究泊松方程的特定解。即使原始域可以是任意形状，傅里叶系数的评估也需要矩形域。齐次解的边界条件通过特定解在原始边界上的投影来修改。该论文还开发了一种新的高斯求积过程来计算傅里叶系数中出现的积分，以防无法进行分析评估。

 发现

当前的努力已经开发了源术语的两种不同表示形式。一套全面的基准测试成功地证明了这两种方法的有效性，尤其是第二种方法。随后的详细分析发现了由于边界节点和傅里叶模式数量不足、特定解与原始域之间的大小差异以及所使用的高斯求积积分程序而产生的误差。适当的缓解程序成功地抑制了上述每个错误，并将解决方案的准确性提高到任何所需的水平。与有限差分法的比较研究表明，BIM 与 FDM 一样准确，但对于现实规模的问题计算效率更高。在使用单独导出的解析解验证模拟结果后，后来的练习详细分析了翅片的传热物理。最后一组模拟证明了该方法对复杂几何形状的适用性。

 原创性/价值

首先，新开发的高斯求积分程序可以在傅里叶系数评估过程中有效地计算积分；目前的文献缺乏这样的工具，从而阻碍了研究人员采用此类方法。其次，据笔者所知，目前文献中并不存在对BIM框架内的泊松方程求解方法进行如此全面的误差分析。这项特殊的练习应该会大大增强研究界冒险采用此类方法来求解泊松方程的信心。