SUPERCONVERGENT PATCH RECOVERY SOLUTIONS AND ADAPTIVE MESH REFINEMENT ANALYSIS OF FINITE ELEMENT METHOD FOR THE VIBRATION MODES OF NON-UNIFORM AND VARIABLE CURVATURE BEAMS
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摘要: 该文提出变截面变曲率梁振型的有限元后处理超收敛拼片恢复方法，建立各阶振型的超收敛解，并基于振型超收敛解进行变截面曲梁面内和面外自由振动的自适应分析。在位移型有限元后处理阶段，引入超收敛拼片恢复方法和高阶形函数插值技术，得到振型(位移)的超收敛解。利用振型超收敛解估计当前网格下振型有限元解的能量模形式下的误差，并指导网格进行自适应细分加密分析，获得优化的网格和满足预设误差限的高精度解答。数值算例表明该算法适于求解不同曲线形态、多类边界条件、变截面、变曲率形式的曲梁面内和面外自由振动连续阶频率和振型，解答精确、分析过程高效可靠。Abstract: It presents a superconvergent patch recovery method for the superconvergent solutions of modes in the finite element (FE) post-processing stage of non-uniform and variable curvature curved beams. An adaptive method for the in-plane and out-of-plane free vibration of curved beams with variable cross-section is also proposed. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and the high-order shape function interpolation technique are introduced to obtain the superconvergent solution of mode (displacement). Using the superconvergent solution of mode to estimate the error of the FE solution of mode in the energy form under the current mesh, an adaptive mesh refinement is proposed by mesh subdivision to derive the optimized mesh and accurate FE solution to meet the preset error tolerance. Numerical examples show that the proposed algorithm is suitable for solving the continuous orders for frequencies and modes in the in-plane and out-of-plane free vibration of different kinds of curve shapes, boundary conditions, non-uniform cross-section, and variable curvature forms of the non-uniform curved beams. The computation procedure can provide accurate solutions. The analysis process is efficient and reliable.
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图 1 平面变截面变曲率梁坐标系和符号
Figure 1. Coordinate systems and symbols of planar non-uniform and variable curvature curved beam
图 3 本文方法解答与密集网格高精度解的误差
Figure 3. Errors of results of proposed method and high-precision solutions under dense mesh
图 4 第1阶振型
$(u_1^h,\;v_1^h,\;\psi _{{\textit{z}}1}^h)$ Figure 4. First order vibration mode
$(u_1^h,\;v_1^h,\;\psi _{{\textit{z}}1}^h)$ 
图 5 第2阶振型
$(u_2^h,\;v_2^h,\;\psi _{{\textit{z}}2}^h)$ Figure 5. Second order vibration mode
$(u_2^h,\;v_2^h,\;\psi _{{\textit{z}}2}^h)$ 
图 6 第3阶振型
$(u_3^h,\;v_3^h,\;\psi _{{\textit{z}}3}^h)$ Figure 6. Third order vibration mode
$(u_3^h,\;v_3^h,\;\psi _{{\textit{z}}3}^h)$ 
图 7 变截面变曲率梁几何模型
Figure 7. Geometric model of non-uniform and variable curvature curved beam
图 9 第7阶振型
$(w_7^h,\;\psi _{y7}^h,\;\varphi _{x7}^h)$ Figure 9. Seventh order vibration mode
$(w_7^h,\;\psi _{y7}^h,\;\varphi _{x7}^h)$ 表 1 抛物线曲梁面内自由振动无量纲频率值
Table 1. Non-dimensional frequencies of in-plane vibration of parabolic curved beam
高跨比${h / L}$ 阶次k 频率值$\bar \omega $ 两端固支 一端固支一端简支 两端简支 $\bar \omega _k^h$ $\bar \omega _k^h$[25] $\bar \omega _k^h$ $\bar \omega _k^h$[25] $\bar \omega _k^h$ $\bar \omega _k^h$[25] 0.2 1 46.0245 46.0252 36.5733 36.6037 28.7587 28.7644 2 87.1265 87.2729 78.0592 78.1589 68.2433 68.3075 3 126.287 126.280 123.343 123.3630 123.282 123.2460 0.8 1 10.9353 10.9359 8.34175 8.34340 6.37925 6.38010 2 25.7756 25.7958 21.7083 21.7199 17.9637 17.9707 3 45.6961 45.7441 40.3252 40.3514 35.3552 35.3749 
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表 2 变截面变曲率梁面内自由振动频率值
Table 2. Frequencies of in-plane vibration of non-uniform and variable curvature curved beam
阶次k 频率值$\omega /{\rm{Hz}}$ $\omega _k^h$ $\omega _k^h$[26] 1 72.1035 72.05 2 150.842 150.78 3 267.482 267.34 4 407.912 407.77
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表 3 椭圆弧曲梁面内自由振动无量纲频率值
Table 3. Non-dimensional frequencies of in-plane vibration of elliptic curved beam
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表 4 抛物线曲梁面外自由振动无量纲频率值
Table 4. Non-dimensional frequencies of out-of-plane vibration of parabolic curved beam
阶次k 频率值$\bar \omega $ 两端固支 一端固支一端简支 两端简支 $\bar \omega _k^h$ $\bar \omega _k^h$[27] $\bar \omega _k^h$[28] $\bar \omega _k^h$ $\bar \omega _k^h$[27] $\bar \omega _k^h$[28] $\bar \omega _k^h$ $\bar \omega _k^h$[27] $\bar \omega _k^h$[28] 1 17.0442 17.03 17.12 11.1276 11.12 11.15 6.08274 6.079 6.090 2 48.3965 48.37 48.77 38.9657 38.94 39.10 30.4037 30.38 30.40 3 95.0213 94.97 96.06 82.1935 82.14 82.61 70.0338 69.99 70.03 4 109.942 109.9 109.9 109.828 109.8 109.8 109.839 109.7 109.8 5 156.526 156.4 158.7 140.486 140.4 141.4 125.037 125.0 125.0 6 203.793 203.7 203.8 203.788 203.7 203.8 193.982 193.8 194.0 7 230.935 230.8 234.7 212.172 212.1 213.8 203.764 203.7 203.8
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