二维稳态热传导问题改进的EFG-SBM法

Improved Element-Free Galerkin Scaled Boundary Method for Two-Dimensional Steady Heat Transfer Problems

EFG-SBM法作为一种新型的边界型无网格法,兼顾了无单元Galerkin法和比例边界有限元法的优点,在环向用无单元Galerkin法进行离散简化了前处理和后处理工作量,径向解析可以直接求得物理场函数值,形函数高阶连续可以获得更加准确的计算结果。然而基于移动最小二乘法构造的形函数缺乏Kronecker Delt,a函数性质,因此在本质边界条件的施加上存在困难。本文将滑动Kriging插值法与EFG—SBM法相结合提出了改进的EFG-SBM法,由于滑动Kriging插值法构造的形函数满足Kronecker Delta函数性质,因此这方便了本质边界条件的施加。进一步将这种方法用于二维稳态热传导问题的求解,通过裂纹体和无限域传热等五个算例表明,改进的EFG-SBM法比传统的比例边界有限元法具有更高的计算精度和更快的收敛速率,同时在热流密度的处理上避免了传统的比例边界有限元法需采用的磨平技术。

Element-free Galerkin scaled boundary method （EFG-SBM） possesses the advantages of EFG and scaled boundary finite element method （SBFEM） as a newly boundary-type meshless method. Unlike the conventional SBFEM, EFG-SBM weakens the governing differential equations along the circumferential coordinate direction by the EFG approach so that the preprocessing and postprocessing processes are simplified. The physical fields can be solved analytically in the radial direction similarly to SBFEM. The more accuracy can be obtained because of higher continuity of the shape functions. The moving least squares （MLS） method is the mostly used in the shape flmction construction, but it is difficult to enforce the essential boundary conditions since shape functions constructed by the MLS approximation lack the Kronecker Delta property. This paper presents an improved EFG-SBM which combines the moving Kriging （MK） interpolation method and EFG-SBM. Because the MK interpolation satisfies the Kronecker Delta property, it is more convenient in enforcing the essential boundary conditions than the MLS-based EFG-SBM. This paper applies the improved EFG-SBM to solve two-dimensionM steady heat transfer problems. The proposed method is verified via five numerical examples including problems with thermal crack and unbounded domain. The numerical solutions show that EFG-SBM has higher accuracy and better convergence than the traditional SBFEM. An accurate smooth heat flux can be obtained directly without the necessity of using the recovery procedure.