有理Bézier单元求解参数曲面上Laplace-Beltrami方程

Solving Laplace-Beltrami Equation on Parametric Surface by Rational Bézier Elements

用有限元法数值求解时,定义在流形曲面上的偏微分方程的数值解精度会因为传统多边形单元的几何逼近误差而严重降低,为此提出基于有理Bernstein多项式的几何精确有限元法.首先插入重复节点从NURBS曲面直接生成有理Bézier单元,这一过程保持原有几何不变;然后通过Galerkin法建立参数曲面上包含Laplace-Beltrami微分算子的二阶椭圆偏微分方程的等效弱形式;针对Bernstein基函数的非插值性,通过配点法施加Dirichlet类型的边界约束,得到最优收敛的离散格式.数值算例结果表明,该方法能有效地减少网格离散误差,提高分析结果精度.

When solving the partial differential equations on manifold surfaces by the finite element method,the accuracy of the numerical results may be seriously reduced by the geometrical errors which are caused by the approximation of the computational domains with the traditional polygonal elements.The geometric accurate finite element method is presented to remedy the issues by the parameterization of the geometrical domains with the rational Bernstein polynomials.Firstly,the new knots are repeatedly inserted into the parametric interval to convert the NURBS surface to the rational Bézier elements without changing it geometrically or parametrically.Then,the Galerkin method is employed to establish the equivalent weak forms for the second-order elliptic partial differential equations which involve the Laplace-Beltrami operator on the parametric surface.It＇s a difficult task to enforce the Dirichlet boundary conditions in the presented method because of the non-interpolation properties of the Bernstein functions.Therefore,the collocation method is employed to solve it,and the optimal convergence is acquired.Finally,the numerical examples show that the method can effectively reduce the discretization errors and improve the accuracy of the numerical results.