等几何分析中采用Nitsche法施加位移边界条件

等几何分析使用NURBS基函数统一表示几何和分析模型,消除了传统有限元的网格离散误差,容易构造高阶连续的协调单元.对于结构分析,选择合适的几何参数可以得到光滑的应力解,避免了后置处理的应力磨平.但是由于NURBS基函数不具备插值性,难以直接施加位移边界条件.针对这一问题,提出一种基于Nitsche变分原理的边界位移条件"弱"处理方法,它具有一致稳定的弱形式,不增加自由度,方程组对称正定和不会产生病态矩阵等优点.同时给出方法的稳定性条件,并通过求解广义特征值问题计算稳定性系数.最后,数值算例表明Nitsche方法在h细化策略下能获得最优收敛率,其结果要明显优于在控制顶点处直接施加位移约束.

IMPOSING DISPLACEMENT BOUNDARY CONDITIONS WITH NITSCHE’S METHOD IN ISOGEOMETRIC ANALYSIS

Isogeometric analysis uses the exact geometric representations for the modeling and numerical simulations by NURBS basis functions.It eliminates the geometric approximation errors during the mesh discretization, and the high-order conforming NURBS elements can be conveniently constructed.In the structural analysis,smooth stress fields can be directly computed without the stress recovery procedure as in the finite element method.Due to the lack of the interpolation properties for the NURBS basis functions,it is difficult to enforce the displacement boundary conditions in isogeometric analysis.The imposition of prescribed values is not as straightforward as the conventional approaches.In order to solve this issue,a weak imposition method was proposed basing on the Nitsche’s variational principle.It has some attractive advantages:(i) the consistent and stabilized weak form,(ii) the degree-of-freedoms are not increased,(iii) the resulting system is symmetric and positive,(iv) the matrix condition number is not very large in order to ensure convergence.Meantime,the stability conditions were derived for the structural analysis.The stability parameters were evaluated by solving a generalized eigenvalue problem.Through several numerical examples,the optimal rates of convergence were observed under the h-refinement of the NURBS meshes.Contrasting with directly imposing into the control points,the better results were obtained by the proposed method.