新型曲面四边形边界元精细后处理方法研究

为了精确计算三维静电场的电场强度和电位分布，提出了新型曲面四边形边界元方法．在该方法中，对模型边界面进行二阶四边形单元剖分，对二阶单元顶点上的节点号重新编号，以单元的顶点为求解点，根据二阶四边形曲面参数方程，结合面积比值法定义的曲面单元顶点的形状函数，计算曲面单元顶点的函数值．与一阶平面四边形边界元相比，新型曲面边界元法在没有增加计算节点的情况下，由于采用更接近实际边界的曲面积分，计算精度将明显提高．但由于边界面采用二阶单元粗略剖分，单元数量相对较少，剖分后的模型较粗糙．虽然顶点节点上的函数值比较精确，但只能以平面线性单元的形式显示，离实际模型边界差别较大．本文就此提出边界元精细后处理方法．在该方法中，对曲面单元两边按一定步长等分，再根据曲面的参数方程把曲面单元精细显示出来．单元上新建节点的函数值可由曲面单元顶点上的函数值和面积比值法定义的形状函数插值得到．最后形成经精细显示后的新型曲面边界元方法．算例表明，经精细显示后边界面比未处理前更接近实际边界．

Research on Fine Processing for New Curved Surface Quadrilateral Boundary Element Method

The new curved surface quadrilateral boundary element method（BEM） is put forward to calculate the electric field and potential distribution of 3D electrostatic fields accurately. In this method, the boundaries are meshed into the second-order quadrilateral element, renumbering the node number of vertices of second- order element. The curved quadrilateral element vertices are taken as solution node. With the second-order quadrilateral surface parameter equation and four vertices shape functions defined by the area ratio method, the function value of the vertex of surface elements can be calculated. Due to using the curved surface integral that closer to the actual boundary, the precision of the new curved surface quadrilateral BEM is obviously higher than plane quadrilateral BEM without increasing computing nodes. However, due to the model was meshed into second-order element roughly, the number of elements is relatively few; and the model is rough after meshed. Although the function value of vertex is more accurate, boundary surface is in the form of a planar linear element which is away from the actual model boundary. In this paper, fine processing method is put forward because of that. In this method,both sides of surface elements are divided into a certain step,the surface elements can be fine showed with the parametric equations of the surface. By the function value of ver- tex of the surface element and the shape function defined by the area ratio method, the function value of newnode of the surface element can be interpolated. Finally, the fine display of the new curved surface BEM was {ormed. The calculated results show that the boundaries are closer to the actual boundary than before.