﻿ 基于逆有限元方法的结构变形重构研究
 舰船科学技术  2024, Vol. 46 Issue (5): 133-140    DOI: 10.3404/j.issn.1672-7649.2024.05.024 PDF

1. 华中科技大学 船舶与海洋工程学院，湖北 武汉 430074;
2. 中国舰船研究设计中心，湖北 武汉 430064

Research on structural deformation reconstruction based on inverse finite element method
XU Geng-hui1, CHEN Wen2, HUANG Hui1, HU Gao-bo1
1. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
2. China Ship Development and Design Center, Wuhan 430064, China
Abstract: An inverse finite element method for reconstructing the displacement field using strain information is proposed to address the problem that the structural health status of an offshore platform is challenging to monitor in real-time. This paper is based on Timoshenko beams and least squares variational theory to calculate the neutral axis strain in a section by extracting the strain from the surface of the structure. The strain field and displacement field functions are also derived and the strain information is used as an input condition for the inverse finite element method to reconstruct the structural deformation. To improve the tolerance of the proposed method, the effects of strain gauge position deviation and signal noise on the accuracy are analyzed. The finite element analysis results and the measured structural strain information are used to reconstruct the displacement field using the marine platform column as the test object, respectively. The test results show that the proposed inverse finite element method has a high reconstruction accuracy under different load forms.
Key words: inverse finite element     offshore platform column     deformation reconstruction     structural health monitoring
0 引　言

1 逆有限梁单元理论 1.1 Timoshenko 梁理论

 图 1 悬臂圆柱梁结构示意 Fig. 1 Cantilever cylindrical beam structure

 $\begin{gathered} {u_x}(x,y,z) = u(x) + z{\theta _y}(x) - y{\theta _z}(x) ，\\ {u_y}(x,y,z) = v(x) - z{\theta _x}(x) ，\\ {u_z}(x,y,z) = w(x) + y{\theta _x}(x)。\end{gathered}$ (1)

 $\begin{gathered} {e_1}(x) = \frac{{\partial u(x)}}{{\partial x}}{\text{ ,}}{e_4}(x) = \frac{{\partial w(x)}}{{\partial x}} + {\theta _y}(x) ，\\ {e_2}(x) = \frac{{\partial {\theta _y}(x)}}{{\partial x}},{e_5}(x) = \frac{{\partial v(x)}}{{\partial x}} - {\theta _z}(x) ，\\ {e_3}(x) = - \frac{{\partial {\theta _z}(x)}}{{\partial x}},{e_6}(x) = \frac{{\partial {\theta _x}(x)}}{{\partial x}}。\\ \end{gathered}$ (2)

 $\begin{gathered} {\varepsilon _x}(x,y,z) = {e_1}(x) + z{e_2}(x) + y{e_3}(x) ，\\ {\gamma _{xz}}(x,y) = {e_4}(x) + y{e_6}(x) ，\\ {\gamma _{xy}}(x,z) = {e_5}(x) - z{e_6}(x)。\\ \end{gathered}$ (3)

 图 2 IFEM梁结构模拟流程 Fig. 2 IFEM beam structure simulation process

 $\begin{gathered} N = {A_x}{e_1}{\text{ }}{{{M}}_x} = {J_x}{e_6} ，\\ {Q_y} = {G_y}{e_5}{\text{ }}{{{M}}_y} = {D_x}{e_2}，{\text{ }} \\ {Q_z} = {G_y}{e_4}{\text{ }}{{{M}}_z} = {D_x}{e_3} 。\\ \end{gathered}$ (4)

 ${{\boldsymbol{k}}^e}{{\boldsymbol{u}}^e} = {{\boldsymbol{f}}^e}，$ (8)

 ${\boldsymbol{KU}} = {\boldsymbol{F}} ，$ (9)
 ${\boldsymbol{U}} = {{\boldsymbol{K}}^{ - 1}}{\boldsymbol{F}}。$ (10)
1.3 表面与中心轴应变转换关系

 图 3 梁单元截面示意 Fig. 3 Cross section of beam element

 $\left\{ \begin{gathered} {\varepsilon _x} = \frac{{{\sigma _x}}}{E} ，\\ {\varepsilon _\theta } = - \frac{v}{E}{\sigma _x} = - v{\varepsilon _x}，\\ {\gamma _{x\theta }} = \frac{{{\tau _{x\theta }}}}{G} 。\\ \end{gathered} \right.$ (11)

 ${\varepsilon _\beta } = {\varepsilon _x}{\cos ^2}\beta + {\gamma _{x\theta }}\sin \beta \cos \beta + {\varepsilon _\theta }{\sin ^2}\beta，$ (12)

 ${\varepsilon _\beta } = {\varepsilon _x}({\cos ^2}\beta - v{\sin ^2}\beta ) + {\gamma _{x\theta }}\sin \beta \cos \beta ，$ (13)

r=R时，结合切应变几何关系，可得：

 ${\gamma _{x\theta }} = {e_4}\cos \theta - {e_5}\sin \theta + {e_6}R ，$ (14)

 \begin{aligned}[b] & {{\varepsilon }_{{{x}^{*}}}}\left( {{x_i},\theta ,\beta } \right) =\\ & {e_1}({x_i})({\cos ^2}\beta - v{\sin ^2}\beta ) + {e_2}({x_i})({\cos ^2}\beta - v{\sin ^2}\beta )\sin \theta {R_{ext}} + \\ & {e_3}({x_i})({\cos ^2}\beta - v{\sin ^2}\beta )\cos \theta {R_{ext}} + {e_4}({x_i})\cos \beta \sin \beta \cos \theta - \\ & {e_5}({x_i})\cos \beta \sin \beta \sin \theta + {e_6}({x_i})\cos \beta \sin \beta {R_{ext}} ，\end{aligned} (15)

 ${Q}_{y}=\frac{\partial {M}_{z}}{\partial x}，{Q}_{z}=\frac{\partial {M}_{y}}{\partial x}，$ (16)

 $\begin{gathered} {e_2} = {m_1}{e_4}x + {c_1}，\\ {e_3} = {m_2}{e_5}x + {c_2}。\\ \end{gathered}$ (17)

 $\begin{split} & {e_1}\left( x \right) = {a_1},{e_2}\left( x \right) = {m_1}{a_4}x + {a_2},{e_3}\left( x \right) = {m_2}{a_5}x + {a_3}，\\ & {e_4}\left( x \right) = {a_4},{e_5}\left( x \right) = {a_5},{e_6}\left( x \right) = {a_6} ，\\[-1pt] \end{split}$ (18)

 ${{{\boldsymbol{e}}}^{\boldsymbol{\varepsilon}} }\left( {\boldsymbol{x}} \right) = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0 \\ 0&1&0&{{m_1}x}&0&0 \\ 0&0&1&0&{{m_2}x}&0 \\ 0&0&0&1&0&0 \\ 0&0&0&0&1&0 \\ 0&0&0&0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{a_1}} \\ {{a_2}} \\ {{a_3}} \\ {{a_4}} \\ {{a_5}} \\ {{a_6}} \end{array}} \right] = {{{T_1}}}\left( x \right){{{p_1}}}$ (19)

 $\begin{split}{{{\boldsymbol{e}}}^{\boldsymbol{\varepsilon}} }\left( {{{\boldsymbol{x}}_{\boldsymbol{j}}}} \right) = {T_1}\left( {{x_j}} \right){\left( {{T}\left( {{x_i},{\theta _i},{\beta _i}} \right) * {{e}^\varepsilon }\left( {{x_i}} \right)} \right)^{ - 1}}{{\varepsilon }_{{{x}^{*}}}}\left( {{x_i},{\theta _i},{\beta _i}} \right)。\;\;\;\;\\[-2pt] \end{split}$ (20)

 $\begin{split} & {e_1}\left( x \right) = {a_1},{e_2}\left( x \right) =\\ & {m_1}{b_4}{x^2} + {m_1}{a_4}x + {a_2},{e_3}\left( x \right) = {m_2}{b_5}{x^2} + {m_2}{a_5}x + {a_3}，\\ & {e_4}\left( x \right) = {b_4}x + {a_4},{e_5}\left( x \right) = {b_5}x + {a_5},{e_6}\left( x \right) = {a_6} ，\\[-4pt] \end{split}$ (21)

 \begin{aligned} {{{\boldsymbol{e}}}^{\boldsymbol{\varepsilon}} }\left( {\boldsymbol{x}} \right) =\\& \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0 \\ 0&1&0&{{m_1}x}&0&0&{{m_1}{x^2}}&0 \\ 0&0&1&0&{{m_2}x}&0&0&{{m_2}{x^2}} \\ 0&0&0&1&0&0&x&0 \\ 0&0&0&0&1&0&0&x \\ 0&0&0&0&0&1&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{a_1}} \\[-1pt] {{a_2}} \\[-1pt] {{a_3}} \\[-1pt] {{a_4}} \\[-1pt] {{a_5}} \\[-1pt] {{a_6}} \\[-1pt] {{b_4}} \\[-1pt] {{b_5}} \end{array}} \right] = \qquad\qquad\qquad\qquad\\& {T_2}\left( x \right){p_2}。\end{aligned} (22)

 ${{{\boldsymbol{e}}}^{\boldsymbol{\varepsilon}} }\left( {{{\boldsymbol{x}}_{\boldsymbol{j}}}} \right) = {T_2}\left( {{x_j}} \right){\left( {{T}\left( {{x_i},{\theta _i},{\beta _i}} \right) * {{e}^\varepsilon }\left( {{x_i}} \right)} \right)^{ - 1}}{{\varepsilon }_{{{x}^{*}}}}\left( {{x_i},{\theta _i},{\beta _i}} \right) 。$ (23)

2 IFEM位移重构结果分析 2.1 计算模型及应变测点方案

 图 4 悬臂梁有限元模型加载方式 Fig. 4 Cantilever beam finite element model loading method

 图 5 悬臂梁模型有限元应变信息提取方案 Fig. 5 Extraction scheme of Finite Element strain information of cantilever beam model

2.2 有限元与IFEM对比分析

 图 6 有限元与IFEM位移结果对比 Fig. 6 Comparison of FEM and IFEM displacement results

 ${{RMS}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^n {{{\left( {{\delta ^{iFEM}}(x) - {\delta ^{Ansys}}(x)} \right)}^2}} }}{n}} 。$ (24)

 图 7 悬臂梁结构不同载荷下误差分析 Fig. 7 Error analysis of cantilever beam structure under different loads

2.3 试验与IFEM对比分析

 图 8 海洋平台立柱试验台架 Fig. 8 Offshore platform column test bench

 图 9 试验与仿真重构位移对比 Fig. 9 Comparison of experimental and simulated reconstructed displacements

3 应变片测量对重构精度分析

3.1 位置偏差对重构精度的影响

3.2 噪声信号对重构精度的影响

4 结　语

1）提出的结构变形重构方法在有限元和试验测试中均呈现较高精度。对多种工况及布置方案进行分析，有限元仿真中C3C6方案最大误差为2%，试验测试T3方案最大误差为3%，该方法可为海洋结构物健康监测提供参考。

2）对于不同载荷形式的立柱结构，因β=45°的表面应变需几何关系转化，增加迭算法代误差。可增加0°或90°方向测点，但至少保留1个β=45°应变测点，可提高逆有限元算法精度。

3）立柱结构试验过程中，轴向偏差不是主要误差因素，周向偏差及噪声信号对重构精度影响较大。针对薄壁柱型位移重构研究，周向位置偏差应在±3°范围内，提取位置尽可能靠近中部位置。

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