伽辽金最小二乘无网格法在几何非线性问题中的应用

Application of the meshless Galerkin least square method in geometrically nonlinear problems

目的在不需要划分单元的情况下求解几何非线性问题。方法伽辽金最小二乘无网格法(MGLS)采用移动最小二乘近似函数作为试函数,并用罚函数法施加本质边界条件,内部区域用最小二乘域,边界区域用伽辽金域,是一种与单元划分无关的无网格方法。在求解几何非线性问题时,采用了增量和修正的Newton-Raphson迭代分析的方法,并在整个分析过程中所有变量的表达格式都采用更新的拉格朗日格式。结果通过对受均布载荷作用的悬臂梁用MGLS法进行内力分析,由于考虑大变形的影响,结构呈现出比线性分析结果刚硬的性质,结果与解析解符合的很好。结论算例表明:MGLS法在求解几何非线性问题时具有可行性,而且计算精度也较好。

Aim Solving geometrically nonlinear problems in the instance of no cell be compartmentalized. Methods The meshless Galerkin least square （MGLS） method uses the moving least square approximation as a trial function, and the essential boundary conditions are imposed by the penalty factor method. MLS method applied to the interior region and Galerkin method applied to the exterior region. Only nodal data are necessary and there is no need to make elements with nodes in this method. An incremental and iterative solution procedure using modified Newton-Raphson iterations is used to solve the geometrically nonlinear problem, and measurements of strain and stress are related back to the original configuration, namely, the updated Lagrangian method is used. Results Through carrying on the analysis of internal force to the cantilever beam under uniformly distributed loads function with MGLS method, the structure demonstrates the properties just harder than the linear analysis result because of considering the great influence out of shape. What the result accords with analyzing solving is very good. Conclusion Examples show that in solving the geometrically nonlinear problem the meshless Galerkin least square method achieves results of not only feasibility but also good accuracy.