1. Introduction
Rubber bumpers built into air spring structures perform several critical tasks, such as working together with the air spring as a secondary spring, thus modifying the original characteristics of the air spring when pressed together. In the product design and development cycle, engineers are faced with several predefined requirements that are difficult to fulfill and time consuming, and thus remain a challenging task. The product investigated is applied in the air springs of lorries, where the force–displacement characteristic for the compression load is one of the most challenging technical requirements. Design engineers manage to achieve the required working characteristics by modifying the shape of the product, which leads to an iterative design process. This process is termed shape optimization whose simplest solution is to determine the optimal geometry through a series of trials with a study called “what if,” based on design engineers’ experiences. Owing to the continuum mechanics background and hyperelastic material model, available trials can be carried out by applying a finite element analysis. If there is an opportunity to parameterize the process from creating a geometry to obtaining the results, then conversion that meets the technical requirements can be automated; furthermore, there might also be an opportunity to use optimization algorithms in the design process.
Design optimization is an engineering design methodology that uses a mathematical formulation of a design problem to support the selection of the optimal design among many alternatives [
1]. Several researchers have investigated shape optimization of rubber products, out of which [
2,
3] are the least efficient “trial and error” procedures. Many articles combined finite element analysis with a variety of optimum search methods, such as one presented by Kaya [
4] involving a differential evolution algorithm-based shape optimization of the 2D rubber bushing. Interaction of a genetic algorithm and finite element code was used in the shape optimization of a rubber bumper for a new pickup vehicle [
5]. A bush-type engine mount using a parameter optimization method was designed by Kim [
6], and Fletcher’s method using the concept of quadratic convergence and was applied as an optimization algorithm.
Several papers can be found where metamodel-based design optimization was used for rubber product design. The problem of a rubber component design for automotive application by changing the shape with five design variables was discussed by Previati et al. [
7]. The optimization of the bushing was performed concerning two objectives, mass reduction and the fatigue life of the component. The finite element model of the bushing was used to simulate about 200 different combinations of parameters for four different material models. These simulations were used to calibrate the parameters of a series of interpolating functions. A parameter optimization methodology for a rubber mount, based on finite element analysis and the genetic neural network model, was proposed by Li [
8]. The orthogonal experiment table was adopted to design the geometric parameters of the samples on which the numerical analyses were run. The results of finite element analyses were used as samples to train the error backpropagation neural network model which defines the nonlinear global mapping relationship between the geometric parameters of the rubber mount and its primary stiffness in the three principal directions.. Multiobjective shape optimization of a rubber isolator of an automotive cooling module was performed to maximize fatigue life and vibration isolation. The response values of interest were evaluated with the integration of various computer-aided engineering tools. The analysis procedure was automated using a commercial process integration and design optimization tool, therefore the cycle time of the complex analysis was reduced. A regression-based sequential approximate optimizer was used successfully to obtain the optimal shapes of the rubber isolators for two different cooling module types [
9].
Optimization methods can be divided into local and global optimum search procedures. Most local search procedures require the calculation of gradient while most optimum search ones are direct and belong to gradient-free optimization algorithms. Applying the latter kind of algorithms needs to meet few requirements, as they only use function value and implementing them is generally simpler. The direct optimal algorithm includes localized stochastic search, the Nelder–Mead method (simplex method), simulated annealing, tunneling, and several other methods [
10]. The random search methods for optimization are based on randomly exploring the design area to find a point that minimizes the objective function. These techniques have all or some of the following advantages relative to most other search methods. The advantages, like ease of programming, inexpensive realization, flexibility, or reasonable computational efficiency, were mentioned by Karnopp [
11]. In direct random search, several methods are known, depending on the field of application for noise-free or noise problems. For noise-free object function, three algorithms were presented in [
12]. One of the presented methods is a simple random or blind search. In this case, uniformly distributed random numbers were created above the range of each design variable. This is a unique algorithm from the point of view that it does not contain any adjustable algorithm coefficients that need to be selected by the user. Localized and enhanced localized random search methods are slightly more sophisticated in that random sampling depends on the position of the current best-estimate object function.
The most primitive version of the localized random search was applied by Mátyás [
13]. This utilizes selection of the next point from a normally distributed hyperplane or hypersphere and then moving to a better position. This technique can be improved in several ways based on observations. One of them is that if the newly selected point results in a higher function value, the opposite direction can often lead to a lower objective function. The other observation is that successive success or failure searches in a certain direction should bias or discourage subsequent searching toward that direction. It could be handled with the use of a bias term as the mean value of the random vector [
14]. In [
15], a modified algorithm such as enhanced localized random search was introduced with these guidelines. Luus-Jaakola [
16] utilized a local search algorithm with the use of pseudorandom numbers over an exponentially decreasing search region. To test its effectiveness, mathematical, chemical, and engineering examples were chosen.
Metamodel methods applied to replace costly simulation engineering tasks are termed “surrogate models” in the English literature [
17]. Generally, the main goal is to replace the original model with one very similar that requires a lower calculational time. There are several metamodels known, such as the simple and adaptive response surface, kriging method, radial basis function, multivariable adaptive spline regression, and neural network or support vector machines (SVM) [
18]. In machine learning, SVM is a supervised learning algorithm that can efficiently perform a nonlinear classification using the so-called kernel trick [
19]. Based on the SVM algorithm, support vector regression (SVR) was introduced in [
20]. This regression technique has advantages in high dimensionality spaces because SVR optimization does not depend on the dimensionality of the input space. Therefore, it is widely used for function approximation, regression estimation, and signal processing [
21,
22].
Three mathematical problems and two engineering design problems were solved with SVR efficiency and accuracy of which were compared with other metamodels such as kriging, radial basis function (RBF), and polynomial regression. Latin hypercube design was used as a design of the experiment technique. The results showed that in all problems the introduced SVR method performed more efficiently than the other methods [
23]. A two-variable aerodynamic problem was introduced in [
24] where the quasi-Monte Carlo algorithm was used to determine eight sample points. Poor space-filling properties of the quasi-Monte Carlo sampling left a noticeable gap in the data. In this area, inaccuracies were found in the prediction of the kriging and RBF model, and only the SVR method mitigated this problem. In [
25,
26], vehicle crashworthiness designs were presented, where SVR was utilized to construct crashworthiness responses. It was demonstrated that SVR models with Gaussian RBF and exponential RBF have good generalization performance compared to other types of kernels. SVR outperforms polynomial response surface, radial basis neural network, and kriging with good accuracy in approximating the crashworthiness responses. A multiobjective crashworthiness design of a tailor-rolled blank thin-walled structure was constructed based on the ε-support vector regression (ε-SVR) technique and nondominated sorting genetic algorithm-II. The SVR parameters were further optimized with a genetic algorithm to improve the predictive accuracy of ε-SVR [
27,
28]. Shape optimization of rubber bumpers, where obtaining data with finite element simulation is independent on the optimization algorithm, was investigated [
29,
30,
31,
32]. Optimal shape was determined by minimizing the objective function by calculating differences in work energy. Design options obtained from design space, such as learning points, were applied. The SVR model was applied to determine the given values of the objective functions of further constructions. Through this metamodel, the optimal shape was determined. The SVR method was investigated by Huri and Mankovits [
33] to predict the optimum shape of a rubber jounce. From the results, the cubic kernel function showed the best match, therefore it was chosen for the prediction of each combination of design variables. From the predicted values, the optimum shape was selected, which was sufficiently accurate if the number of the learning points was at least 22. This observation is used in the current rubber product optimization process.
Based on the professional papers, the local stochastic search method is suitable for finding the global optimum of engineering optimization tasks. However, it is vital to find the right size of the search space to achieve good results. Due to the stochastic procedure, it is to be noted that one cannot conclude whether the adjustment is correct by simply running one instance. From the literature review, it can be seen that none of the research deals with setting the parameters of the applied optimization algorithms. The parameters highly influence the required finite element run and the accuracy of the determined optimum. All the case studies indicated that if enough set of training vectors are supported, the SVR is a promising metamodel technique for highly nonlinear engineering applications. The selection of the appropriate type of kernel function and hyperparameter choice is of vital importance for the right assessment. This article first introduces what considerations are necessary for the finite element analysis of axisymmetric rubber products, especially when exposed to pressure with special attention to the selected material model type and its parameters. The rubber bumper investigated in the paper is a case study based on an industrial design problem. Secondly, the task of two-variable shape optimization of a rubber bumper is described where the objective function values of 22 uniformly distributed samples from the design area were determined from the results of finite element analysis. The SVR surrogate model with cubic kernel function was trained utilizing the sampled experiments, whereby the relationship between the design variables and objective function values was described. The pre- and postprocessing of the model were automated in the Visual Basic for Applications (VBA) language. Owing to this method, the local stochastic search algorithm was implemented and run with basic options for the shape optimization task. This research aims to investigate the parameter selection of local search algorithms for design optimization of an automotive rubber bumper. The novelty of our research is that the trained SVR surrogate model was integrated into the size selecting process of the search space of the algorithm. Finally, using the stochastic search algorithm and the adjusted parameter, the finite element model was directly run to solve the shape optimization of the rubber bumper. Later on, to validate the goodness of the procedure, it is compared with the initial algorithm in terms of precision and efficiency.