三维泰勒展开边界元方法及其数值验证

Three-dimension Taylor expansion boundary element method and it's validation

边界元方法广泛应用于势流问题的求解,其分为常值面元法和高阶面元法两类。研究表明:常值面元法在计算光滑物面处的切向速度时精度较好,而在物面拐角处的切向速度精度则很差;而采用高阶面元法可以改进其精度,但高阶面元法在处理非光滑边界处主值时比较困难。为此该文提出了泰勒展开边界元方法(TEBEM方法)。该方法主要通过对边界积分方程中的偶极强度进行泰勒展开并保留至一阶导数项,同时在格林第三公式中关于场点沿边界取切向导数封闭方程组,然后联立方程组直接求解出速度势及其沿物面的切向速度。通过与传统分布源方法的数值计算比较发现,采用TEBEM方法计算非光滑边界处的切向诱导速度的精度有相当大的提高。

The boundary element method is widely used in solving potential flow problems, which is divided into constant panel method and high order panel method. However, numerous literatures show that the accuracy of constant panel method （CPM） is good for the induced velocity when the points locate in the smooth boundary surface, while it＇s quite poor on the non-smooth part. Although using high order boundary clement method can greatly improve its accuracy, the treatment of the chief value at non smoothed surface is quite complicated. So the Taylor expansion boundary element method （TEBEM） is proposed, which mainly applies the Taylor expansion to the dipole strength of the BIE, and then keeps the first-order derivative, finally takes the two tangential derivatives with respect to the field points on the boundary to form the closed equations. Combine the equations then solve the potential and its tangential derivative. Compared to traditional CPM method, the TEBEM method can extremely improve the accuracy of the tangential derivative on the non-smooth body surface.