中文摘要：

为了探究几何非线性问题的数值求解方法,采用理论推导、MATLAB编程计算、有限元模拟相结合的方法,基于S-R和分解定理及更新拖带坐标描述法,运用插值型无单元Galerkin方法对几何非线性问题的增量变分方程进行了推导,并通过四点Gauss积分法和不动点迭代法对其进行求解.最后以平面悬臂梁的大变形问题为例进行求解计算,发现与ANSYS的计算结果拟合相似度很高,说明了所采用的几何非线性力学理论及数值计算方法的正确性和合理性,为求解几何非线性问题提供了一种新的依据.

英文摘要：

To explore the numerical solution method for geometrically nonlinear problems,the theoretical derivation,the MATLAB programming and the finite element simulation were used together.Based on the S-R decomposition theorem,the interpolated element-free Galerkin method was applied to deduce the incremental variational equations through the updated co-moving coordinate formulation,which were solved with the 4-point Gauss integration method and the fixed point iteration method.Finally,the large deformations of exemplary elastic and elastoplastic planar cantilever beams were calculated and the results agreed well with those from the ANSYS simulation. The examples illustrate the correctness and rationality of the proposed geometrically nonlinear mechanics theory and the present numerical calculation method. The work provides a new basis for the solutions to geometrically nonlinear problems.