Abstract The present work has been conducted in order to propose the extension of standard penalty and stabilization techniques to the improved element-free Galerkin (IEFG) method, for the numerical solution of incompressible fluid flow problems. In principle, the numerical procedures to be implemented in this communication have been conceived for finite element method (FEM)-based solutions, and these include the reduced integration penalty method (RIPM), the streamline upwind Petrov–Galerkin (SUPG) scheme, and a penalty-based immersed boundary method (PBIBM) for the imposition of essential boundary conditions along internal fluid–solid interfaces. The linear momentum balance and mass conservation equations have been coupled via the RIPM, in order to obtain a global weak formulation where the IEFG model is entirely developed in terms of improved moving least squares (IMLS) approximations of the velocity field. A detailed explanation concerning the appropriate extension of both the RIPM and SUPG procedures to the context of IEFG formulations, has also been provided. The resulting formulation has been applied to the solution of two well-known benchmark problems: i) Lid-driven square cavity flow, and ii) Flow past a fixed cylinder. Regarding the flow past a fixed cylinder benchmark problem, the fluid–solid interaction has been imposed as an internal immersed boundary condition via the PBIBM. The feasibility and reliability of implementing the RIPM, SUPG and PBIBM procedures in the IEFG formulation, have been proven by comparison with experimental and mesh-based numerical results reported in the literature. The results obtained in this study have revealed that a proper extension of the aforementioned penalty and stabilization techniques to the IEFG formulation, allows the achievement of accurate and stable numerical results during the solution of incompressible fluid-dynamics problems.

中文翻译：

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标准惩罚、流线迎风和浸入边界技术对不可压缩 Navier-Stokes 方程的改进的基于无单元 Galerkin 的解决方案的合理扩展

摘要 目前的工作是为了提出将标准惩罚和稳定技术扩展到改进的无单元伽辽金 (IEFG) 方法，用于不可压缩流体流动问题的数值解。原则上，本次​​交流中要实施的数值程序是针对基于有限元法 (FEM) 的解决方案设计的，其中包括简化积分惩罚法 (RIPM)、流线逆风 Petrov-Galerkin (SUPG) 方案和一种基于惩罚的浸入边界方法 (PBIBM)，用于沿内部流固界面施加基本边界条件。线性动量平衡和质量守恒方程已通过 RIPM 耦合，为了获得全局弱公式，其中 IEFG 模型完全根据速度场的改进移动最小二乘法 (IMLS) 近似而开发。还提供了有关将 RIPM 和 SUPG 程序适当扩展到 IEFG 公式上下文的详细解释。所得公式已应用于两个众所周知的基准问题的解决方案：i) 盖子驱动的方形腔流，以及 ii) 流过固定圆柱体。关于通过固定圆柱基准问题的流动，流固相互作用已通过 PBIBM 作为内部浸入边界条件强加。在 IEFG 制定中实施 RIPM、SUPG 和 PBIBM 程序的可行性和可靠性，已经通过与文献中报道的实验和基于网格的数值结果进行比较来证明。本研究中获得的结果表明，将上述惩罚和稳定技术适当扩展到 IEFG 公式，可以在不可压缩流体动力学问题的求解过程中获得准确和稳定的数值结果。