1 Introduction
2 Optimization Objectives of Fixture Layout Optimization in Thin-Walled Part Assembly
2.1 Considering In-Plane Variations
2.2 Considering Out-of-Plane Deformations
2.3 Comparative Analysis of Different Optimization Objectives
Objectives | Physical meaning | Engineering significance | References | |
---|---|---|---|---|
Considering in-plane variations | Minimize \(F\left(X\right)=\frac{1}{\sqrt{2{N}_{KPC}}}\sqrt{\sum_{i=1}^{{N}_{KPC}}[{{\sigma }^{2}\left(\delta {x}_{0}\right)}_{i}+{{\sigma }^{2}\left(\delta {y}_{0}\right)}_{i}]}\) \({{\sigma }^{ }\left(\delta {x}_{0}\right)}_{i}, {{\sigma }^{ }\left(\delta {y}_{0}\right)}_{i}:\) The \(i\)th KPC’s variations in the x and y directions \({N}_{KPC}\): Number of KPCs | Minimizing the pooled standard deviation of resultant errors at all KPCs | Minimizing variations caused by source variations and improving robustness | Cai [5] |
Minimize \(F\left(X\right)=\sum_{i=1}^{m}ST {D}_{x}^{2}\left(i\right)+ST {D}_{z}^{2}\left(i\right)\) \(ST {D}_{x}\), \(ST {D}_{y}^{ }:\) Standard deviations in the X and Z directions | Minimizing the summation of squared standard in-plane deviations | Masoumi et al. [10] | ||
Minimize \({S}_{{\text{max}}}={\lambda }_{{\text{max}}}({D}^{{\text{T}}}D)\) \(D:\) a sensitivity index reflects the relationship between the final variation and source variations | Minimizing the square of the 2-norm of sensitivity matrix | Minimizing the sensitivity of the part to source variations and improving robustness | ||
Minimize \(F\left(X\right)=\sqrt{{\text {cond}}({S}^{{\text{T}}}S)}\) \(S:\) a sensitivity index reflects the relationship between the final variation and source variations | Minimizing the square root of the condition number of the sensitivity matrix | Li et al. [12] | ||
Considering out-of-plane deformations | Minimize \(F\left(X\right)=\sum_{i=1}^{m}{{w}_{i}(X)}^{2}\) \({w}_{i}\left(X\right):\) the deformation perpendicular to the part surface at the \(i\) th node | Minimizing the sum of squares of nodal deformations | Minimizing deformations caused by forces and reducing assembly errors caused by deformations | Cai et al. [6] |
Minimize \(F=\sum_{i=1}^{n}{u}_{i}\) \({u}_{i}\): strain energy of the \(i\)th finite element | Minimizing the sum of strain energy of finite elements | Ahmad et al. [16−18], Bi et al. [19] | ||
Minimize \(F={\text{max}}\left\{\frac{{e}^{i}}{{T}^{i}} \space \,{\text {for}}\space \,i=1,\cdots ,M\right\}\) \({e}^{i}:\) profile error of the \(i\)th point \({T}^{i}\): profile tolerance of the \(i\)th point | Minimizing the maximum ratio of error to tolerance | De Meter [20] | ||
Minimize \(F={(\sum_{i=1}^{N}{\Delta }_{i})}^{{\text{T}}}(\sum_{i=1}^{N}{\Delta }_{i})\) \({\Delta }_{i}:\) the rigid body motion at the \(i\)th fixturing point | Minimizing the total rigid body motion | Minimizing positioning errors caused by elastic deformations and improving positioning accuracy | ||
Minimize \(H\left(x\right)=\sum_{i=1}^{{m}_{0}}{\varphi }_{i}(x)/{m}_{0}\) \({\varphi }_{i}\left(x\right):\) the dimensional gap at node \(i\) along the interface between the compliant parts to be assembled \({m}_{0}:\) the number of the nodes along the assembly interface between two parts | Minimizing the average dimensional gap along the interface between the compliant parts to be assembled | Reducing the assembly gap between two parts and improving the weld quality | Du et al. [23] | |
Minimize \(f=\frac{C}{{C}_{\text {max}}}+\sum_{k=1}^{K}p({C}_{puk})\) \(C\): total cost of production \({C}_{pu}:\) upper process capability index \(p\left({C}_{puk}\right):\) a penalty function relative to \({C}_{pu}\) of the \(k\)th KPC | Minimizing the expense of production | Reducing the expense while meeting quality requirement | Aderiani et al. [24] |
2.4 Epilog
3 Modeling Methods of Assembly Variation or Deformation for Fixture Layout Optimization
3.1 Mechanism-Based Modeling Methods
3.1.1 Jacobian Matrix Method
3.1.2 State Space Method
3.1.3 Finite Element Method
3.2 Data-Based Modeling Methods
3.2.1 Regression Modeling Methods
3.2.2 Artificial Neural Network Methods
3.3 Comparative Analysis of Different Modeling Methods
Methods | Scope of application | Advantages | Limitations | References | |||
---|---|---|---|---|---|---|---|
Mechanism-based modeling methods | Jacobian matrix method | Single-station process Robust design Rigid assumption | Simple calculation Direct reflect the influence of fixture deviations on part variations | Applicable to rigid assumptions Limited for complex problems | |||
State space method | Multi-station process Variation propagation Considering source variations | Suitable for multi-station assembly Considering various source variations | Cannot be used for complex problems Complicated derivation and calculation | ||||
FEM | Considering deformations caused by force Compliant assumption | Intuitively reflect the deformations at nodes Easy to understand its principle | Large amount of calculation Calculation accuracy depends on FEA software | Du et al. [23], Haynes and Lee [36], DeVries and Menassa [37], Zhong and Hu [38], Chen et al. [39], Liao et al. [40], Vishnupriyan et al. [41], Kumar and Paulraj [42], Wu et al. [43], Hajimiri et al. [44], Xiong et al. [45], Yang et al. [46], Dou et al. [47], Wen et al. [48], Wu et al. [49], Liu and Hu [50], Aderiani et al. [51], Sayeed et al. [52, 53] | |||
Data-based modeling methods | Regression modeling methods | RSM | Large amount of calculation Multiple input variables | Simple calculation | Reduce the amount of calculation Require a small amount of data | Model accuracy depends on data and parameters Requirements for experimental design and data sampling Time for parameter optimization | |
SVR | Good generalization ability | ||||||
Kriging | Stable accuracy | ||||||
PLSR | Suitable for multiple outputs | Bi et al. [19] | |||||
Grey model | Low requirements for data | Yang et al. [69] | |||||
ANN methods | BPNN | Large amount of calculation Multiple input variables | Relatively simple structure | Reflecting the complex relationship between inputs and outputs More accurate | Need a large amount of data Not interpretable Long time to obtain the model | ||
RBFNN | Fast convergence Global approximation |
3.4 Epilog
4 Algorithms for Fixture Layout Optimization
4.1 Traditional Nonlinear Programming Algorithms
4.2 Heuristic Algorithms
4.2.1 Genetic Algorithm
4.2.2 Particle Swarm Optimization
4.2.3 Other Heuristic Algorithms
4.3 Comparative Analysis of Different Optimization Algorithms Used in Fixture Layout
Methods | Application | Advantages | Limitations | References | |
---|---|---|---|---|---|
Traditional nonlinear programming algorithms | Feasible direction method | Small and medium-sized problems | Keep the feasibility of the solution Local optima can usually be found | Only applicable to small and medium-sized problems The result depends on the initial solution | Li and Melkote [21] |
Sequential quadratic programming | Fast convergence High computational efficiency Global convergence | ||||
Lagrange multiplier | Global convergence Simplify the problem | ||||
Heuristic algorithms | GA | Large scale problem | Global search capability Scalable and easy to combine with other algorithms | Slow convergence Complex encoding and decoding Results are affected by the parameters of algorithms | Tian et al. [13], Chen et al. [39], Vishnupriyan et al. [41], Hajimiri et al. [44], Yang et al. [62], Krishnakumar and Melkote [88], Vallapuzha et al. [87, 89], Chen et al. [91], Liu et al. [92], Zhang et al. [93],Wu et al. [94], Zeshan Ahmad et al. [114], Cheng et al. [95], Xing et al. [96, 97], Kulankara et al. [98], Liao [99] |
PSO | Fast convergence Few parameters need to be adjusted | Poor local search ability Easy to fall into local optimization Results are affected by the parameters of algorithms | |||
ACO | Easy to combine with a variety of heuristic algorithms Good robustness | Slow convergence Easy to fall into local optimization Results are affected by the parameters of algorithms |