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 哈尔滨工业大学学报  2019, Vol. 51 Issue (7): 178-183  DOI: 10.11918/j.issn.0367-6234.201810147 0

### 引用本文

ZHANG Zebin, ZHANG Pengfei, GUO Hong, LI Yong. Implementation of Kriging model based sequential design on the optimization of sliding bearing[J]. Journal of Harbin Institute of Technology, 2019, 51(7): 178-183. DOI: 10.11918/j.issn.0367-6234.201810147.

### 文章历史

Kriging序贯设计方法在滑动轴承优化中的应用

Implementation of Kriging model based sequential design on the optimization of sliding bearing
ZHANG Zebin, ZHANG Pengfei, GUO Hong, LI Yong
School of Mechanical Engineering, Zhengzhou University, Zhengzhou 450001, China
Abstract: A built-in prime benefit of Kriging surrogate model resides in its unbiased prediction and the associated confidence intervals, comparisons have been done between the classical DoE methods and Kriging model based sequential design. This latter considers both the design space global exploration and optimum-neighborhood exploitation. A parallel point-adding training strategy and its corresponding convergence criterion were introduced to improve the model precision. The approach had been illustrated with two classical optimization test functions. Results show not only a more accurate model but also a possibility to reduce the number of sampling points. Finally, the training strategy was experimented for the optimal design of a sliding bearing. Friction power loss per unit load capacity was taken as the objective function to be modeled. Two surrogate model based optimizations had been per-formed based on classical DoE such as Orthogonal Latin Hypercube and Kriging sequential strategy respectively. These two optimization results are compared with a previously optimization using Complex-optimization method. Within a limited number of iterations, the Kriging model based training strategy showed the best convergence to the global optimum among those 3 methods.
Keywords: Kriging     Surrogate model     Sequential Design     Hybrid Bearing     Optimization

1 Kriging代理模型介绍

Kriging方法是一种基于统计学理论的插值方法，其模型包含均值项和随机项.前者为回归函数，后者为符合一定预设统计学规律的随机变量，即

 $\hat{y}(x)=\boldsymbol{F}^{\mathrm{T}}(x) \beta+z(x).$

 $E[z(\boldsymbol{w}) z(\boldsymbol{x})]=\sigma^{2} \boldsymbol{R}\left(\theta, \boldsymbol{w}_{j}, \boldsymbol{x}_{j}\right).$

 $\boldsymbol{R}\left(\theta, \boldsymbol{w}_{j}-\boldsymbol{x}_{j}\right)=\prod\limits_{j=1}^{n} \exp \left(-\theta\left(\boldsymbol{w}_{j}-\boldsymbol{x}_{j}\right)^{2}\right).$

 $\hat{y}(\boldsymbol{x})=\boldsymbol{c}^{\mathrm{T}} \boldsymbol{Y}=\boldsymbol{F}^{\mathrm{T}}(\boldsymbol{x}) \hat{\boldsymbol{\beta}}+\boldsymbol{r}^{\mathrm{T}}(\boldsymbol{x}) \boldsymbol{R}^{-1}(\boldsymbol{Y}-\boldsymbol{F} \hat{\boldsymbol{\beta}}).$

 $\hat{\beta}=\left(\boldsymbol{F}^{\mathrm{T}} \boldsymbol{R}^{-1} \boldsymbol{F}\right)^{-1} \boldsymbol{F}^{\mathrm{T}} \boldsymbol{R}^{-1} \boldsymbol{Y},$
 $\sigma^{2}=\frac{1}{m}(\boldsymbol{Y}-\boldsymbol{F} \hat{\boldsymbol{\beta}})^{\mathrm{T}} \boldsymbol{R}^{-1}(\boldsymbol{Y}-\boldsymbol{F} \hat{\boldsymbol{\beta}}).$

 $\boldsymbol{R}=\left[\begin{array}{ccc}{\boldsymbol{R}\left(\boldsymbol{w}_{1}, \boldsymbol{x}_{1}\right)} & {\cdots} & {\boldsymbol{R}\left(\boldsymbol{w}_{1}, \boldsymbol{x}_{n}\right)} \\ {\vdots} & {\ddots} & {\vdots} \\ {\boldsymbol{R}\left(\boldsymbol{w}_{n}, \boldsymbol{x}_{1}\right)} & {\cdots} & {\boldsymbol{R}\left(\boldsymbol{w}_{n}, \boldsymbol{x}_{n}\right)}\end{array}\right].$

rT(x)是随机模型相关系数矩阵，表示样本点与预测点之间的相关性，可以表示为

 $\boldsymbol{r}^{\mathrm{T}}(\boldsymbol{x})=\left(\boldsymbol{R}\left(\boldsymbol{w}_{1}, \boldsymbol{x}_{1}\right), \boldsymbol{R}\left(\boldsymbol{w}_{2}, \boldsymbol{x}_{1}\right), \ldots, \boldsymbol{R}\left(\boldsymbol{w}_{n}, \boldsymbol{x}_{1}\right),\right)^{\mathrm{T}}.$

Kriging模型预估计值的均方差为

 ${\rm{MSE}}\left\{ {\hat y\left( \mathit{\boldsymbol{x}} \right)} \right\} = {\sigma ^2}\left\{ {1 - {\mathit{\boldsymbol{r}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{r}} + {{\left( {1 - {\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{r}}} \right)}^2}/{\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{F}}} \right\}.$ (1)

 $\psi \left( \theta \right) = - \frac{{m\ln \left( {{{\hat \sigma }^2}} \right)}}{2} - \frac{{\ln \left| \mathit{\boldsymbol{R}} \right|}}{2}.$

2 基于序贯设计的模型训练

1) 根据工程优化设计要求，确定研究对象并建立相应数学模型、约束条件，并确认设计变量及取值范围.

2) 初始样本选取.通过OLH获取能较好反映设计空间的空间特性的初始样本，并数值求解得到对应的响应，从而得到Kriging模型的启动样本.

3) 建立代理模型.选择二次回归函数和高斯相关函数，并根据步骤2)获取的初始样本集，运用DACE工具箱[15]构建初始Kriging代理模型.

4) 加点准则.通过选择合理的加点法则循环选择新增训练样本点进行迭代求解，用以提高Kriging模型精度.

5) 收敛判定.根据代理模型精度(最优点附近的相对误差)作为收敛准则.如果模型满足收敛条件，则代理模型满足精度要求，建模过程即终止，可以基于模型进入优化阶段.否则进入下一次循环返回步骤4).

2.1 加点准则

MSE准则是直接运用Kriging代理模型提供的均方差估计(式(1))最大处作为新样本点的位置.该方法对提高模型全局精度，避免优化结果收敛于局部最优有较好的效果，但收敛的速度偏慢.

EI准则是综合考虑预测值以及方差加权的一种高效全局优化(EGO)方法.设当前最优点响应值为Ymin，且Kriging代理模型预测值满足均值${\hat y}$(x)和均方差σ2的正态分布，其概率密度为

 $P\left( \mathit{\boldsymbol{x}} \right) = \frac{1}{{\sqrt {2\pi } \sigma \left( \mathit{\boldsymbol{x}} \right)}}\exp \left[ { - \frac{{{{\left( {{Y_{\min }} - I - \hat y\left( \mathit{\boldsymbol{x}} \right)} \right)}^2}}}{{2{\sigma ^2}\left( \mathit{\boldsymbol{x}} \right)}}} \right].$

 $I\left( \mathit{\boldsymbol{x}} \right) = {Y_{\min }} - y\left( \mathit{\boldsymbol{x}} \right).$

 $\left[ {I\left( \mathit{\boldsymbol{x}} \right)} \right] = \sigma \left( \mathit{\boldsymbol{x}} \right)\left[ {u\mathit{\Phi }\left( u \right) + \varphi \left( u \right)} \right],$
 $u = \frac{{{Y_{\min }} - \hat y\left( \mathit{\boldsymbol{x}} \right)}}{{\sigma \left( \mathit{\boldsymbol{x}} \right)}}.$

2.2 收敛判定

 $\left| {\hat f\left( {{x_k}} \right) - f\left( {{x_k}} \right)} \right| \le {\varepsilon ^ * }f\left( {{x_k}} \right).$ (2)

3 序贯设计方案的理论测试

3.1 测试函数

 $f\left( {{x_1},{x_2}} \right) = x_1^2\left( {4 - 2.1x_1^2 + \frac{{x_1^4}}{3}} \right) + {x_1}{x_2} + x_2^2\left( { - 4 + 4x_2^2} \right).$

 图 1 测试函数等值线图 Fig. 1 Test function contour map

Rosenbrock函数：

 $f\left( {{x_1},{x_2}} \right) = {\left( {1 - {x_1}} \right)^2} + 100{\left( {{x_2} - x_1^2} \right)^2}.$

3.2 方案评价标准

 ${\rm{RMSE}} = \sqrt {\sum\limits_{i = 1}^{{n_t}} {{{\left( {{e^{\left( i \right)}}} \right)}^2}/{n_i}} } ,{e^{\left( i \right)}} = \left\| {\left( {\hat y_{\rm{t}}^{\left( i \right)} - y_{\rm{t}}^{\left( i \right)}} \right)} \right\|.$ (3)

3.3 不同方案结果分析

 图 2 六峰值驼背函数序贯设计样本点分布情况 Fig. 2 Sample distribution of sequential design for Six-Hump Camel Back function
 图 3 Rosenbrock函数序贯设计样本点分布情况 Fig. 3 Sample distribution of sequential design for Rosenbrock function

2个测试函数均采用OLH设计构建初始点集，采用并行加点准则增加新样本点，直到代理模型收敛.利用式(3)评价各测试函数下模型精度，结果如表 12所示.

4 动静压滑动轴承优化设计

 $\min F\left( x \right) = \left( {\frac{{{{\bar H}_f} + {{\bar H}_p}}}{{{{\bar F}_r}}}} \right).$
 图 4 带深浅腔的圆柱动静压滑动轴承 Fig. 4 Cylindrical hybrid bearing with deep and shallow pocket

 ${\rm{s}}.\;{\rm{t}}.\left\{ \begin{array}{l} l/d \in \left[ {0.5,0.8} \right]\\ {z_1},{z_2} \in \left[ {1,7} \right]\\ {\theta _q} \in \left[ {44,53} \right] \end{array} \right..$

5 结论

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