The determination of the key dimensions of gear blank preforms with complicated geometries is a highly nonlinear optimization task. To determine critical design dimensions, we propose a novel and efficient dimensionality reduction (DR) model that adapts Gaussian process regression (GPR) to construct a topological constraint between the design latent variables (LVs) and the regression space. This procedure is termed the regression-constrained Gaussian process latent variables model (R-GPLVM), which overcomes GPLVM’s drawback of ignoring the regression constrains. To determine the appropriate sub-manifolds of the high-dimensional sample space, we combine the maximum a posteriori method with the scaled conjugate gradient (SCG) algorithm. This procedure can estimate the coordinates of preform samples in the space of LVs. Numerical experiments reveal that the R-GPLVM outperforms the pure GPR in various dimensional spaces, when the proper hyper-parameters and kernel functions are solved for. Results using an extreme learning model (ELM) obtain a better prediction precision than the back propagation method (BP), when the dimensions are reduced to seven and a Gaussian kernel function is adopted. After the seven key variables are screened out, the ELM model will be constructed with realistic inputs and obtains improved prediction accuracy. However, since the ELM has a problem with validity of the prediction, a genetic algorithm (GA) is exploited to optimize the connection parameters between each network layer to improve the reliability and generalization. In terms of prediction accuracy for testing datasets, GA has a better performance compared to the differential evolution (DE) approach, which motivates the choice to use the genetic algorithm-extreme learning model (GA-ELM). Moreover, GA-ELM is employed to measure the aforementioned DR using engineering criteria. In the end, to obtain the optimal geometry, a parallel selection method of multi-objective optimization is proposed to obtain the Pareto-optimal solution, while the maximum finisher forming force (MFFF) and the maximum finisher die stress (MFDS) are both minimized. Comparative analysis with other numerical models including finite element model (FEM) simulation is conducted using the GA optimized preform. Results show that the values of MFFF and MFDS predicted by GA-ELM and R-GPLVM agree well with the experimental results, which validates the feasibility of our proposed methods. 
Optimization of gear blank preforms based on a new R-GPLVM model utilizing GA-ELM