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Printed circuit board model updating based on response surface method
XU Fei, LI Chuanri , JIANG Tongmin
School of Reliability and Systems Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:Optimization analysis based on response surface is used in finite element (FE) model updating in recent years. Printed circuit board (PCB) model updating process based on response surface was presented. First six modal frequencies of PCB were calculated using ANSYS and correlated with modal test results. Three objective functions were formed using the first three orders of FE analytical and measured modal frequencies. The fourth objective function was formed by the square of residuals between first six resonant frequencies. Each resonant frequency was given the same weight. A multi-objective genetic algorithm (MOGA) was used in optimized analysis to minimize the four objective functions. A case was presented to illustrate the proposed updating procedure. The results show that the model updating technique based on response surface can be used to improve the accuracy of PCB FE model, and can be analyzed directly by using commercial FE softwares now available, which is conducive to engineering application.
Key words: response surface     printed circuit board     finite element analysis     model updating     optimization analysis

1 基于响应面方法的模型修正技术

1) 初步选择优化参数,确定其上下限,利用参数敏感性分析和重要度分析确定重要参数,从而确定最终的优化参数;

2) 利用DOE构造采样点并调用有限元模型计算输出参数;

3) 利用输入参数和输出参数构造响应面并对响应面进行回归误差分析;

4) 测试结构的振动响应特性,并利用有限元分析和试验结果构造目标函数;

5) 在响应面模型的范围内进行迭代分析,从而对目标函数进行优化,得到优化后的输入参数和优化后的有限元模型.

 图 1 基于响应面方法的模型修正技术流程图Fig. 1 Procedure of model updating technique based on response surface method
1.1 参数选择

1.1.1 敏感性分析

1.1.2 重要度分析

R2越接近1,表明选定的输入参数越重要,越接近0,表明该参数可从修正参数集中去掉,在有限元计算时作为常数.

1.2 试验设计方法

1.3 响应面回归分析

1.4 目标函数构造与优化分析

 图 2 MOGA流程图Fig. 2 Procedure of MOGA

1) 创建初始群体;

2) 利用交叉运算和变异运算创建新群体;

3) 利用新群体更新设计点,计算目标函数并检验其收敛性,若满足则修正完成,否则进入下一步;

4) 检查停止准则,若不满足则回到第2)步继续迭代,若满足则停止,迭代失败,重新定义目标函数,重新进行模型修正.

2 案例分析

 参数 初始值 参数 初始值 Ex/GPa 18.7 μxz 0.18 Ey/GPa 16.7 Gxy/GPa 3.28 Ez/GPa 7.4 Gyz/GPa 2.4 μxy 0.13 Gxz/GPa 2.4 μyz 0.42 ρ/(kg·m-3) 1 836
2.1 模态试验

 图 3 试验设置Fig. 3 Test setup

 图 4 速度频响函数Fig. 4 Mobility

 模态阶数 1 2 3 4 5 6 1 1 0 0.03 0 0 0.01 2 0 1 0 0 0 0.01 3 0.03 0 1 0 0 0.02 4 0 0 0 1 0 0 5 0 0 0 0 1 0.01 6 0.01 0.01 0.02 0 0.01 1

 模态阶数 试验模态频率/Hz 仿真结果 修正前 修正后 频率/Hz 误差/% 频率/Hz 误差/% 1 85.4 79.663 -6.7 85.208 -0.2 2 125.4 120.73 -3.7 125.52 0.0 3 214.9 202.58 -5.7 214.54 -0.2 4 291.6 273.25 -6.3 301.07 3.2 5 338.8 312.58 -7.7 337.85 -0.3 6 353.4 341.88 -3.3 359.4 1.7
2.2 有限元分析

 模态 1 2 3 4 5 6 FEA1 0.94 0.04 0.05 0 0 0.01 FEA2 0.01 0.93 0.02 0 0 0.03 FEA3 0 0 0.86 0.05 0 0.06 FEA4 0 0 0 0.91 0.02 0.05 FEA5 0 0 0 0 0.94 0 FEA6 0 0 0 0 0.03 0.8

PCB的密度和几何尺寸比较容易测量,且相对变化很小,因此本文将PCB的正交各向异性材料属性(3个弹性模量,3个泊松比,3个剪切模量)作为初始输入(修正)变量,前6阶共振频率和一个自定义变量作为输出变量,自定义变量为前6阶模态频率的残差平方和:

 图 5 参数敏感性分析Fig. 5 Parameter sensitivity analysis

 图 6 参数重要度分析Fig. 6 Parameter importance analysis

 图 7 响应面示例Fig. 7 Example of response surface
 图 8 响应面拟合度检验Fig. 8 Response surface fitness check

PCB的前3阶模态频率常常是最重要的,利用前3阶共振频率构造3个目标函数:

 图 9 前3阶共振频率迭代过程Fig. 9 Iteration processes of first three resonant frequencies
 图 10 第4个目标函数迭代过程Fig. 10 Iteration process of fourth objective function
 图 11 两种迭代判据下的收敛过程Fig. 11 Convergence process with two different iterative criteria

 修正状态 Ex/GPa Ey/GPa Gxy/GPa 修正前 18.7 16.7 3.28 修正后 22.7 18 3.76<
3 结 论

1) 在自由边界条件下,Ex,Ey和Gxy对PCB的模态频率影响最大.

2) 修正后PCB前6阶模态频率的误差最大为3.2%,最小为0,达到了模型修正的目的.

3) 多目标函数修正过程收敛速度较快,可利用已有商业有限元软件直接进行分析,易于工程应用.

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#### 文章信息

XU Fei, LI Chuanri, JIANG Tongmin

Printed circuit board model updating based on response surface method

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(3): 449-455.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0219