Adaptive finite element analysis of free vibration problems with frequency error control alone as the objective
SUN Haohan1,2，YUAN Si3
1.CAEP Software Center for High Performance Numerical Simulation, Beijing 100088, China;
2.Institute of Applied Physics and Computational Mathematics, Beijing 100088, China;
3.Department of Civil Engineering, Tsinghua University, Beijing 100084, China
摘要对于自由振动问题，基于单元能量投影（element energy projection ,EEP）技术，对频率和模态同时进行误差控制的自适应有限元分析已建立，并被证明可靠且高效。在实际应用中，也存在另一类需求，即只需保证频率的精度，而并不关心模态误差大小。本文提出了频率超收敛计算方案，继而建立了整体频率误差和局部模态误差的转换关系，从而在整体上以频率误差估计控制算法停机，在局部上以模态误差估计驱动网格更新，最终建立了以频率误差控制为目标的自由振动问题自适应有限元分析策略。本方法的有效性在二阶Sturm-Liouville问题及弹性薄膜自由振动问题上得到了应用验证。
Abstract：Adaptive finite element analysis of free vibration problems based on element energy projection (EEP) method has been proved to be reliable and efficient, where the errors of frequencies and modes are simultaneously controlled. In practical applications, the error evaluation of frequencies can sometimes be more required than the error distribution of modes. The paper proposes an approach to calculate the super-convergent frequencies, and then sets up the transformation relation between the global frequency error and the local mode error, so that the former is used to control the termination of the adaptive algorithm on the whole and the latter is used to drive mesh refinement locally. As a result, an adaptive finite element analysis of free vibration problem with frequency error control alone as the objective is established. The method is verified by its application to second order Sturm-Liouville problem and free vibration problem of elastic membrane.
孙浩涵1,2，袁驷3. 以频率误差控制为目标的自由振动问题自适应有限元分析[J]. 振动与冲击, 2023, 42(4): 106-115.
SUN Haohan1,2，YUAN Si3. Adaptive finite element analysis of free vibration problems with frequency error control alone as the objective. JOURNAL OF VIBRATION AND SHOCK, 2023, 42(4): 106-115.
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