The generalized finite element method (GFEM) without extra degrees of freedom (dof) is extended to solve incompressible Navier-Stokes (N-S) equations. Unlike the existing extra-dof-free GFEM, we propose a new approach to construct the nodal enrichments based on the weighted least-squares. As a result, the essential boundary conditions can be imposed more accurately. The Characteristic-Based Split (CBS) scheme is used to suppress oscillations due to the standard Galerkin discretization of the convective terms, and the pressure is further stabilized by the finite increment calculus (FIC) formulation. Hence, equal velocity-pressure interpolation and the incremental version of the split scheme can be used without inducing spurious oscillations. The developed extra-dof-free GFEM is very flexible and can achieve high-order spatial accuracy and convergence rates by adopting high-order polynomial enrichments. In particular, better accuracy could be obtained with special enrichments reflecting a-priori knowledge about the solution. This is demonstrated by numerical results. Benchmark examples such as the Lid-Driven Cavity flow and the flow past a circular cylinder are also presented to further verify the effectiveness of the proposed extra-dof-free GFEM for incompressible flow.

中文翻译：

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一种用于不可压缩 Navier-Stokes 方程的无额外自由度广义有限元方法


扩展了没有额外自由度 （dof） 的广义有限元法 （GFEM） 以求解不可压缩的纳维-斯托克斯 （N-S） 方程。与现有的无 extra-of-free GFEM 不同，我们提出了一种基于加权最小二乘法构建节点富集的新方法。因此，可以更准确地施加基本的边界条件。基于特征的分裂 （CBS） 方案用于抑制对流项的标准 Galerkin 离散引起的振荡，并通过有限增量微积分 （FIC） 公式进一步稳定压力。因此，可以使用等速-压力插值和拆分方案的增量版本，而不会引起杂散振荡。开发的无自由度 GFEM 非常灵活，可以通过采用高阶多项式富集来实现高阶空间精度和收敛率。特别是，通过反映有关解的先验知识的特殊富集可以获得更好的准确性。数值结果证明了这一点。还提出了基准示例，例如盖子驱动腔流和流经圆形圆柱体的流，以进一步验证所提出的无额外自由度 GFEM 对不可压缩流的有效性。