本論文用非奇異之形式展現外部問題的格林方程及其法向導數在三維之Laplace式上。對於所描述之物理公式，採用直接積分的方式去求解，因此一般處理物理問題之邊界元素近似法(此時需使每個切割元素的奇異點規律化)將不採用。對於弱奇異點我們同時採用高斯流量(Guass Flux)定理和均勢體(associated equipotential)來處理；對於強奇異點則以共同內部問題之邊界公式處理之(弱、強奇異點於內容中有定義)。
修正後之積分方程將被假設在無限領域之球體上先作測試，附帶全軟(Dirichlet)或全硬(Neumann)之邊界條件。在自由液面、波動和物體間之相互干擾問題上，使用邊界積分公式須在某些離散之頻率下作分解，且用標準積分式及其法向導數之線性組成來去除不規則之頻率。計算之結果將列成附加質量、阻尼係數、激盪力分別和波數所呈現之關係，並與其他可行之近似處理法作相關結果之比較。

The paper presents the non-singular forms of Green's formula and its normal derivative of exterior problems for three-dimensional Laplace's equation. The main advantage of these modified formulations is that they're amenable to solution by directly using quadrature formulas. Thus, the conventional boundary element approximation, which locally regularizes the singularities in each element, is not required. The weak singularities are treated by both the Gauss flux theorem and the property of the associated equipotential body. The hypersingularities are treated by further using the boundary formula for the associated interior problem. The efficacy of the modified formulations is examined by a sphere, in an infinite domain , subject to Neumann and Dirichlet conditions , respectively.
The modified integral formulations are further applied to a practical problem, ie. surface-wave -body interaction. Using the conventional boundary integral equation formulation is known to break down at certain discrete frequencies for such a problem.Removing the ‘irregular’ frequencies is performed by linearly combiming the standard integral equation with its normal derivative. Computations are presented of the add-mass and damping coefficients and wave exciting force on a floating hemisphere .Comparing the numerical results with that by other approaches demonstrates the effectiveness of the method.

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