船舶與海洋結構物受力分析常以邊界元素法搭配源點分佈技巧處理，其中三維平移脈動源(Three-Dimensional Translating Pulsating Source)可滿足航行船舶運動邊界值問題的線性自由液面、遠場輻射等邊界條件，而別所型格林函數(Bessho Green Function)因屬單積分式，具工程應用優勢。此外，將全船浸水表面離散為有限個平面小板，在各小板皆佈置單極源點，套用船體運動邊界條件，來建構源點強度求解矩陣，係屬慣用作法；不過，各小板僅用單極源點表徵流場，理論嚴謹性較薄弱；因此，本文將探討可反映整體小板源點貢獻的板源函數(Panel Source Function)，以改善嚴謹性。
本研究將建構基於別所型格林函數(Bessho-from Green Function)的板源解析解函數，惟該函數受限於指數相位函數y分量絕對值緣故，衍生小板分割問題，增加計算複雜性，故本研究再以雙重高斯積分(Double Gaussian Quadrature)技巧搭配源點函數的作法來獲得板源近似解，藉兩者結果比較以評估其計算可行性。
在波浪項誘導速度勢與梯度項計算上，雙重高斯積分近似解皆與解析解高度吻合，藉不同高斯點數測試，顯示33配置兼具精度與效率。此外，以板源解析解函數、高斯積分技巧以及單極源點作法，對不同遠近場點的誘導速度勢進行計算與比較，隨小板與場點間距放大，顯示單極源點解與板源解間誤差同步縮小，惟在梯度項上，因梯度項數量級低，誤差未能收斂，而雙重高斯積分近似解誤差極低；綜前，雙重高斯積分法有效可行，能提升船舶受力運動分析嚴謹性，並可迅速落實於工程應用上。

The study of hydrodynamic load analysis using the Boundary Element Method with translating–pulsating sources, aims to improve the conventional practice utilizing a single monopole source on each hull panel. Although commonly used, this point‑source representation is theoretically insufficient. To remedy insufficiency, an analytical panel source function based on the Bessho-from Green function is introduced, but its dependence on the absolute y‑component in the exponential phase term requires panel subdivision and consequently increases computational cost in some cases.
To balance accuracy and efficiency, an approximate panel source formulation using double Gauss quadrature is constructed. Comparisons of wave‑term induced velocity potentials and gradients show that the Gauss‑based approximation closely matches the analytical panel source results. Further tests with different Gauss point configurations indicate that a 3×3 Gauss point framework provides an effective compromise between computational effort and precision.
Further comparisons of induced potentials at various field‑point locations reveal that monopole‑source results gradually approach panel‑source solutions as distance between source and field pints increases. However, for gradient terms, solution based on monopole‑source discrepancies is apparent due to their small magnitude, whereas the Gauss quadrature maintains reliable accuracy. Overall, the double Gauss quadrature method offers an efficient and rigorous approach for evaluating panel sources and is well suited for engineering applications in naval hydrodynamics.

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