基于自然单元法的动力学问题分析

Dynamics Problems Based on the Natural Element Method

针对有限元法模拟动力学问题受单元尺寸限制的问题,本文采用自然单元法来消除单元尺寸的限制。在二维自然单元法的理论基础上,把自然单元法的理论推广到三维空间,采用Voronoi结构的边元素构造三维自然单元的插值函数,并利用该函数推导动力学问题的离散格式,对于离散格式中时间域求解采用本中心差分和Newmark常平均加速度法相结合的第一种积分格式进行,空间域采用高斯积分。采用大型数值软件ANSYS模拟悬臂梁动力响应的计算结果作为基准,通过分片试验和悬臂梁等算例分别验证本文推导插值函数和动力学问题的离散格式的正确性。通过对比可以发现大变形条件下,有限元方法将出现网格畸变使计算无法进行,而本文方法则不会出现网格畸变,计算仍然可以进行。

The finite element method is limited by the problem of the element size in simulating dynamics. Natural element method was used in this paper to eliminate the limitation from the element size. Based on the theory of two-dimensional natural element method, the natural element method was developed to three-dimensional space, and the boundary element of Voronoi structure was used to construct the three-dimensional natural element interpolation function, which was used to deduce the discrete format of dynamics problems, For the time domain solving of discrete format, this problem was solved by the combination of the center differential solution and Newmark normal average acceleration solution of the first kind of integral format, and space domain was solved by gauss integral. The cantilever beam dynamic response was simulated using the large-scale numerical software ANSYS, and the calculation results were taken as the benchmark, the subdivision test, cantilever beam as well as other examples were used respectively to verify the correctness of the discrete format of interpolation function and dynamics problems. Based on the comparison, it can be found that the distortion of the grid of the finite element method will cause the calculation discontinue under the large deformation condition, which is not the case described in the method in this paper.