在对梁振动进行有限元分析时,现有的质量矩阵对角化技术忽略了转角自由度所对应的质量,导致生成的集中质量矩阵失去了其正定性,这对时域分析以及频域分析等都造成了相当大的不便。采用数值流形方法,首先在单位分解法框架下,从Hermite插值中取回单位分解和局部近似;其次通过流形上的积分来求得集中质量矩阵;最后再回到单元的集中质量矩阵。相对于一致质量矩阵的计算,集中质量矩阵的计算精度有了一定的提高,特别是对于求解高阶模态时的速度提升尤为显著。该方法有着严格的数学基础和力学基础,为二维乃至三维结构的动力计算提供了新思路。
Abstract
In the vibration analysis of beam bending problems, none of existing mass lumping schemes that are mathematically rigorous (ignored mass of rotational degree of freedom) are available to maintain the lumped mass matrix symmetric and positive definite up to now, which leads to the difficulty in generating symmetric and positive definite matrix. It has caused great inconvenience to analyses in both the time and frequency domain. Within the framework of the partition of unity (PU), the PU functions and local approximations on the patches are retrieved from Hermitian interpolations via numerical manifold method. Next, the variation principle is integrated using the definition of integral of a scalar function on a manifold to retrieve the diagonal block mass matrix. In addition, compared with consistent mass matrix, our method could achieve higher precision and speed, especially for higher order modes.
关键词
有限元法 /
梁动力分析 /
集中质量矩阵 /
流形元 /
单位分解法
Key words
finite element method /
beam vibration analysis /
mass lumping /
numerical manifold method /
partition of unity
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参考文献
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基金
国家自然科学基金重点项目(51621006)
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