Advances in Mechanical Engineering

International SCIENTIFIC Conference on

Advances in Mechanical Engineering

13-15October 2016, Debrecen, Hungary

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SHAPE DESIGN OF AXIALLY SYMMETRIC RUBBER PART USING FINITE ELEMENT METHOD AND SUPPORT VECTOR MACHINES

1MANKOVITS Tamás PhD, 2VÁMOSI Attila, 3KOCSIS Imre PhD, 4HURI Dávid, 5KÁLLAI Imre, 6SZABÓ Tamás PhD

1,4Department of Mechanical Engineering, Faculty of Engineering, University of Debrecen

E-mail: ,

2,3Department of Basic Technical Studies, Faculty of Engineering, University of Debrecen

E-mail: ,

5Institute of Polymer Product Engineering, Johannes Kepler University Linz

E-mail:

6Institute of Machine Tools and Mechatronics, University of Miskolc

E-mail:

Abstract

Non-linear FEM calculations are indispensable when important technical information like operating performance of a rubber component is desired. Rubber parts may undergo large deformations under load, which in itself shows non-linear behaviour. The material characterization of an elastomeric component is also a demanding engineering task. The shape optimization problem of rubber parts led to the study of FEM based calculation processes. In this paper a novel solution for the shape optimization of compressed rubber parts is presented. A special purpose FEM code written in Fortran has been developed for the analysis of nearly incompressible axially symmetric rubber parts.A rubber shape is evaluated via the work difference and the area between the desired and the actual load-displacement curves. The objective of shape optimization is to find the geometry where the work difference is under a specified limit. The tool of optimization is the SVR method, which provides the regression function for the work difference. The minimization process of the workdifference function leads to the optimum design parameters. The efficiency of the method is verified by a numerical example.

Keywords:rubber, finite element method, support vector regression, shape optimization

1. INTRODUCTION

Engineering rubber parts have to meet several kinds of requirements as a result of their function. One of these is that they must have a predefined load-displacement curve under load while the material characteristics remain the same. Achieving this aim is a problem of optimization.

The load-displacement curves of the investigated rubber parts of different shapes were determined with the help of a special purpose FEM code. Number of papers is devoted to the application of h-version FEM for the analysis of hyperelastic materials [1-6]. The p-version of finite elements is proved to be very efficient tool to analyze linear elastic problems [7], its application geometrically and physically nonlinear problems are relatively recent [8-10].

More and more papers are devoted to find the answer how to obtain robust and non-locking finite elements and the majority prefers the mixed formulation [3] and [11].

A finite element code is developed for the investigation of rubber parts, assuming a material that is nearly incompressible also in case of large deformation. The mixed method is based on a three-field functional where the displacement field is continuously approximated, and the change in volume and the hydrostatic pressure are approximated discontinuously, independently of each other. The same order of approximation is selected for the hydrostatic pressure and the volumetric change. The displacement fields are approximated using the quadratic tensor product space. The program is suitable for the quasi-static calculation of axially symmetric rubber components.

The literature does not devote much room to the shape optimization of the rubber parts. The stiffness of rubber mounts in three directions on the basis of parameter examinations is optimized in [12]. Optimization based on sensitivity analysis using a special purpose finite element code is performed in [13] for material properties and shape, where stiffness was also taken into consideration. Determining the shape had the aim of minimizing the volume of the rubber part. For the purpose of minimizing the cross-sectional area and the maximum stress of the rubber mount and for that of maximizing the life cycle, shape optimization using an Ogden-type material model and commercial finite element software is applied in [14]. Several objective functions in a system where the optimization had several stages are handled. A back-propagation neural network (BPN) is used to find the connection between the input and output data and then a micro-genetic algorithm (MGA) is used for global optimization. A large number of finite element running results are used as learning points. The differential evolution (DE) algorithm produced excellent results for different applications in engineering. The optimization process is performed by Fortran routines coupled with finite element analysis code Abaqus in [15]. A formulation of the non-parametric shape optimization problem of a rubber bushing in order to fit the static load-displacement curve with the desired one was presented in [16].

The support vector regression (SVR) proposed by [17] is a widely used application of support vector machines (SVM) for regression problems, e.g. in optimization models. A great number of applications can be found in the fields of materials science, chemistry, economics and data procession, where connections are sought between a numbers of input data (e.g. some mechanical or chemical property). Although there are results in the field of engineering problems based on SVR models [18], this method is not yet particularly widespread for engineering optimization. The application of SVR in non-linear models has the advantage that the transformation function between input space and the so-called feature space (where a linear regression problem is to be solved) can be hidden [19], and machine learning procedures can be applied to find an appropriate regression function.

2. METHODS

The procedure is based on the finite element method and the support vector regression (SVR) model. A finite element code developed by the authors and based on a three-field functional is used for the rapid and appropriately accurate calculation of the characteristics of rubber parts. The geometry and the load-displacement curve of the investigated rubber part calculated by the FEM code can be seen in Figure 1 and Table 1. A rubber shape is evaluated by the work difference and the area between the desired and the actual load-displacement curves.

Table 1 Data for the investigated rubber part

Figure 1 The geometry and the load-displacement curve of the rubber part

The objective of shape optimization was to find the geometry where the work difference is under a specified limit. The tool of optimization was the SVR method, which provides the regression function for the work difference. The minimization process of the work difference function leads to the optimum design parameters. The work difference is the area (filled with grey) between the desired load-displacement curve and the curve obtained by finite element computation for a specific rubber shape, see in Figure 2.

Figure 2 The derivation of the work difference

The aim is to minimize function for determining the optimum design parameter vector , that is,

where is the optimization range given by inequality conditions coming from technology limitations, is the design parameter vector and is the optimum design parameter vector. The SVR model related to the theory of learning machines and kernel methods plays central role in the investigations. The SVR model is used to find the regression function, the calculations are carried out with the program package of R open source code programming environment using SVR. Values of the regression function provided by the software are available for arbitrary design parameter vectors in . The place of the minimum of function , that is the value of the optimum design parameter vector, can be determined numerically.

3. RESULTS

The initial load-displacement curve obtained by the FEM and the prescribed load-displacement curve can be seen in Figure 3, which has to be reached with the change of three geometry parameters of the rubber part, see in Figure 1.

Figure 3 The optimization task

The outer skirt of the rubber part investigated is described by means of cubic spline in five control points. For design parameters the hole diameter, the and control point positions are chosen. In the investigation the design parameters in mm are defined according to the following conditions:

Under the specified accuracy, the number of possible solutions is . The number of learning points is .

Figure 4 The optimization range and learning points

In determining the learning points the objective was that they should properly cover the optimization range. The optimization range and the learning points can be seen in Figure 4. The optimum values of hyperparameters in the SVR program are determined numerically.

On the basis of the calculation, the minimum work difference out of the possible solutions is , for which the optimum design parameters are , and . The curve obtained for the control FEM calculation run for the optimum design variables and the prescribed load-displacement curve and the optimal geometry are shown in Figure 5.

Figure 5 The optimal rubber geometry and the optimal load-displacement curve

CONCLUSIONS

A three dimensional shape design of a rubber part is presented. The characteristics of the investigated rubber part were determined with the help of the finite element method using special purpose FEM code. In the investigations, the SVR was used by means of open-source software to perform the optimization task. Combining the above two methods into one system, a shape optimization problem was solved to prove the efficiency of the presented procedure for axially symmetric rubber parts.

ACKNOWLEDGEMENT

The described work was carried out as part of a project supported by the National Research, Development and Innovation Office – NKFIH, K115701.

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