热传导方程的无网格Galerkin方法数值模拟研究

Element free Galerkin method to solve heat equation

为了更好地数值模拟热传导方程,将无网格Galerkin(EFG)方法引入热传导问题的求解中,时间导数采用θ加权方法离散,同时与有限元(FE)方法的数值结果进行了比较,并研究了EFG方法中若干参数的选取对数值结果的影响。计算结果表明:相对于有限元方法,EFG方法能更好地吻合微分方程的解析解,EFG方法在节点布置较稀疏时,也可以获得很高的计算精度;θ≥1/2,EFG方法无条件稳定,且θ=1时数值解精度最高;算例中影响半径取为1.2h≤r<2.8h,EFG方法可获得较为理想的计算结果。

In order to simulate the heat equation well,the Element Free Galerkin（EFG） method was adopted to numerically simulate the heat equation,and the temporal derivative was discretized by the θ-weighted method.The results were compared with the results obtained from the Finite Element（FE） method.Furthermore,this paper discussed the efficiency of EFG method under different parameter setting.Simulation results show that compared with FE method,EFG method can better match the analytical result of the differential equation.In addition,EFG method can get high precision even the nodes are arranged sparsely.In particular,EFG method is unconditionally stable when θ≥1/2,and it has highest numerical precision when θ=1.When the value of influence radius satisfies 1.2h≤r2.8h,EFG method can get perfect simulation result.