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Author: Yeh-Liang Hsu, Ming-Sho Hsu (2000-02-18); last updated: Yeh-Liang Hsu (2001-02-16); recommended: Yeh-Liang Hsu (2000-04-18).
Note: This paper is published in Computers in Industry, Vol. 46/2, October 2001, p. 61~73.

Weight reduction of aluminum disc wheels under fatigue constraints using a sequential neural network approximation method

Abstract

This paper describes a weight reduction problem of aluminum disc wheels under cornering fatigue constraints. It is a special structural optimization problem because of the existence of the implicit fatigue constraint. A sequential neural network approximation method is presented to solve this type of discrete-variable engineering optimization problems. First a back-propagation neural network is trained to simulate the feasible domain formed by the implicit constraints using just a few training data. A search algorithm then searches for the “optimal point” in the feasible domain simulated by the neural network. This new design point is checked against the true implicit constraints to see whether it is feasible, and the new training data is then added to the training set. This process continues in an iterative manner until we get the same design point repeatedly and no new training point is generated. In each iteration, only one evaluation of the implicit constraints is needed to see whether the current design point is feasible. No precise function value or sensitivity calculation is required.

Keywords: structural optimization, neural network, implicit constraint, fatigue.

Introduction

Disc wheels intended for normal use on passenger cars have to pass three types of tests before going into production: the dynamic cornering fatigue test, the dynamic radial fatigue test and the impact test. From the statistics of local wheel manufacturers, the dynamic cornering fatigue test has the highest failure rate among the three tests.

On the other hand, to provide better riding comfort, weight reduction is always an important concern to the design of disc wheels for passenger cars. For wheel manufacturers, reduction in weight also means reduction in material cost. Therefore, wheel manufacturers often try to reduce the weight of wheels as much as possible, while the wheels are still able to pass the required structural performance tests.

To the authors’ knowledge, few researchers paid special attention on fatigue prediction and weight reduction of aluminum disc wheels. Karandikar and Fuchs [1990] developed a computer-based system for predicting the fatigue life of wheels. This system is comprised of a CAD package, a finite element analysis program, and a fatigue life computation program. It can successfully simulate the dynamic corner fatigue tests and the development time of new wheels can be shorten using this system.

Kaumle and Schnell [1998] also developed a technique for fatigue testing using a suitable rapid-prototyping system. A light metal wheel is used to demonstrate this technique. On the rapid-prototyping wheel the stresses will be measured, areas with high stress can be detected early, and it is possible to eliminate these areas. The fatigue behaviour will be tested with the optimized rapid-prototyping wheel.

Marron and Teracher’s work [1996] emphasized on reducing the weight of wheel discs. They examined different high-strength sheet steel grades and showed that considerable weight reduction can be obtained by designing the stamping sequence in order to use the steel grade at its maximum without damage.

This paper describes a weight reduction problem of aluminum disc wheels under cornering fatigue constraints. In this paper, finite element models of aluminum wheels are constructed to simulate the dynamic cornering fatigue test. Based on Goodman and Gerber’s fatigue criterions, the analysis results are used to predict whether a wheel can pass the cornering fatigue test. The possible failure positions are also identified. This procedure proves to be a practical and reliable tool for predicting whether the wheels can pass the dynamic cornering fatigue test. Matching with physical test results of a local wheel manufacturer in a sample of 26 wheels, this procedure have a 96% success rate in correctly predicting whether the wheels will pass the dynamic cornering fatigue test. Therefore design engineers can decide whether and how to redesign the wheel based on the finite element analysis result.

The design issue for the wheels that are predicted to pass the fatigue test is, “Is there any room to further reduce the weight of the wheel?” A common approach is to put weight reduction holes on the back of the spokes of the wheels. The position, shape, and size of the weight reduction holes become the design variables that the engineers have to decide.

This is a typical structural optimization problem: to minimize the weight of the wheel, subject to the fatigue constraint. This is also a typical engineering optimization problem that cannot be solved by simply applying existing numerical optimization algorithms. In this optimization problem, the fatigue constraint is the so-called “implicit constraint.” It cannot be expressed as an analytical function in terms of the design variables. Evaluating the fatigue constraint is quite expensive. Moreover, the constraint is in the form of “pass-or-fail,” “feasible-or-infeasible.” The numeric function value of the constraint does not exist, let alone its first derivative.

To handle this type of structural optimization problems, a “Sequential Neural-Network Approximation Method (the SNA method)” is presented in this paper. One important category of numerical optimization algorithms is the sequential approximation methods. The basic idea of the sequential approximation methods is to use a simple sub-problem to approximate the hard, exact problem. By “simple” sub-problems, we mean the type of problems that can be readily solved by existing numerical algorithms. For example, linear programming sub-problems are commonly used in the sequential approximation methods. The solution point of the simple sub-problem is then used to form a better approximation to the hard, exact problem for the next iteration. In an iterative manner, it is expected that the solution point of the simple approximate problem will approach the optimum point of the hard exact problem.

The SNA method is also a sequential approximation method. In this method, first a back-propagation neural network is trained to simulate the feasible domain formed by the implicit constraints using just a few training data. A training data consists of two pieces of information: a design point (i.e., a set of values for the design variables) and whether this design point is feasible or infeasible. A search algorithm then searches for the “optimal point” in the feasible domain simulated by the neural network. This new design point is checked against the true implicit constraints to see whether it is feasible, and the new training data is then added to the training set. The neural network is trained again with this added training data, in the hope that the network will better approximate the boundary of the feasible domain of the exact optimization problem. Then we search for the “optimal point” in this approximated feasible domain again. This process continues in an iterative manner until we get the same design point repeatedly and no new training point is generated.

This paper starts by describing the modeling of the cornering fatigue test of disc wheels and the prediction of fatigue failure using the finite element model. Then the SNA method is presented to solve this structural optimization problem.

Modeling the Dynamic Cornering Fatigue Test

The dynamic cornering fatigue test simulates the loading condition of the wheels in normal driving. Figure 1 shows a typical setup of the 90-degree loading method of cornering fatigue, according to SAE J32 [SAE, 1992]. In the Figure, the downside outboard flange of rim of the wheel is clamped securely to the test device, and a rigid load arm shaft is attached to the mounting surface of the wheel. A test load applies on the arm shaft to provide a constant cyclical rotation bending moment. After being subjected to the required number of test cycles, there shall be no evidence of failure of the wheel, as indicated by propagation of a crack existing prior to test, new visible cracks penetrating through a section, or the inability of the wheel sustain load.

Figure 1. Typical setup of 90 degree loading method of cornering fatigue [SAE, 1992].

The finite element method is used to simulate the dynamic cornering fatigue test. Figure 2 shows the finite element model of an aluminum wheel. All degrees of freedom of the nodes on the downside outboard flange of the rim are fixed. The dynamic cyclical load is represented by 24 discrete loads, 15 degrees apart. For each node, the maximum and minimum von mises stresses during the load cycle are extracted to obtain the mean stress  and the stress amplitude  of the node, as defined below:

                                                                                   (1)

                                                                                   (2)

The  of all 14,000 nodes of the wheel model are plotted in Figure 3.

 

(a) Top view                                           (b) Front view

(c) Cyclical load

Figure 2. The finite element model of a wheel

Figure 3. Mean stress and stress amplitude of all nodes

Several different criterions are commonly used in predicting fatigue failure. In this study, the Goodman and Gerber’s criterions are used:

Goodman’s criterion:                                                                    (3)

Gerber’s criterion:                                                          (4)

where  is the endurance limit,  is the ultimate strength of the material, and n is the design factor [Bannantine et. al., 1990].

The Goodman and Gerber’s lines of the material of the wheel are plotted in Figure 4 (design factor n = 1). The top 1% of the nodes that are closest to the Goodman’s line are also left in the Figure. These nodes have the greatest possibility to fail in the fatigue test. The positions of these nodes on the wheel are also recorded. The average of  and  of these top 1% nodes are calculated, as shown by the solid circle  in Figure 4. This point  is used to represent the wheel when checking with Goodman and Gerber’s lines to predict whether the wheel will pass the cornering fatigue test.

Figure 4. The Goodman and Gerber’s lines for design factor n = 1.

Predicting the Fatigue Failure of Wheels

Finite element models of 28 aluminum wheels from a local manufacturer are constructed to simulate the cornering fatigue tests. The  for each wheel is calculated. Figure 5 shows the  of all 28 wheels. These wheels are already physically tested. In Figure 5, a circle represents the wheels that actually passed the cornering fatigue test, while a cross represents the wheels that did not pass the cornering fatigue test.

Figure 5. The Goodman and Gerber’s lines for design factor n = 2.6.

These historical test data provide a reference for choosing the design factor n in Eq. (4) and (5) for this specific case. It is found that for n = 2.6, the Goodman and Gerber’s lines fit the 28 test data well, as shown in Figure 5. The region inside the Goodman’s line is the “safe zone.” Five out of the 28 wheels fall in region, and 4 of them did pass the cornering fatigue test. The region between the Goodman’s line and the Gerber’s line is the “dangerous zone.” Fourteen wheels fall in this region, 8 of them did not pass the cornering fatigue test. Finally the region outside of the Gerber’s line is the “failure zone.” Nine wheels fall in this region, and 7 of them did not pass the cornering fatigue test. Note that the two wheels in this zone that passed the test are very close to the Gerber’s line.

This procedure proves to be a practical and reliable tool for predicting whether the wheels can pass the dynamic cornering fatigue test. Following this study, another batch of 26 new aluminum wheels are predicted using the same procedure and the same design factor. The prediction results are then matched with the physical test results. As shown in Figure 6, all 10 wheels that are in the safe zone did pass the cornering fatigue test. Three wheels fall in the dangerous zone and 1 of them did not pass the cornering fatigue test. Thirteen wheels fall in the failure zone, and 12 of them did not pass the cornering fatigue test. In this sample of 26 wheels, this procedure has a 96% success rate in correctly predicting whether the wheels will pass the dynamic cornering fatigue test.

Figure 6. Prediction results of 26 new aluminum wheels

For the wheels that are predicted to pass the cornering fatigue test, design engineers may consider to put weight reduction holes on the back of the spokes to further reduce the weight of the wheel. It becomes a typical structural optimization problem: to minimize the weight of the wheel, subject to geometry and fatigue constraints. This problem can be formulated as follows:

min. 

s.t.   

                                                                                                (5)

where x is the vector of design variables, such as the size, shape, depth, and position of the weight reduction holes;  are the explicit constraints, that is, the constraints that can be written explicitly in terms of design variables, such as the geometry constraints on the size of the hole.

Constraint  is a binary constraint. If the current design x (with the weight reduction holes) falls in the safe zone inside the Goodman’s line, then , otherwise . The fatigue constraint is the so-called “implicit constraint.” It cannot be expressed as an analytical function in terms of design variables. Evaluating the constraint is quite expensive. Moreover, the constraint is in the form of “pass-or-fail,” “feasible-or-infeasible.” The numeric function value of the constraint does not exist, let alone its first derivative.

This type of implicit constraints is quite common in engineering optimization problems involving complicated computer simulation or experiments. This type of optimization problems cannot be solved by simply applying certain numerical optimization algorithms. In the next section, a “Sequential Neural-Network Approximation Method (the SNA method)” is presented to handle this type of structural optimization problem.

Using Neural Networks for Optimization

Hopfield and Tank [1985] extended neural network applications to optimization problems. In the Hopfield model, the optimization problem is converted into a problem of minimizing an energy function formulated by the design restraints. Lee and Chen [1991] further explored the idea of applying neural network technology in design optimization. In their approach, neural networks were used to simulate the constraint functions. Oda and Mizukami [1993] developed an application of a “mutual combination-type” neural network to the optimal design problems of truss structure. The Hopfield model and the quadratic energy function are used to solve the design problems, and the method is able to obtain feasible solutions for several optimal design problems.

In the SNA method presented in this paper, the neural networks are used to simulate the feasible domain formed by the implicit constraints. In this design case, Eq.(5) can be thought of as a hard, exact problem because of the existence of the fatigue constraint. It is approximated by a “simple” approximate problem below

min. 

s.t.   

                                                                                                      (6)

where the binary constraint  approximates the feasible domain of the fatigue constraint. If , the design point  is feasible; if , the design point  is infeasible. For convenience, the optimization model in Eq.(5) will be denoted  throughout the paper, and the approximate model in Eq.(6) will be denoted .  is a “simple” model because once the training of the neural networks is completed, the computational cost of evaluating  is much less than that of evaluating the implicit constraint .

To illustrate the SNA method, the wheel design case with circular weight reduction holes is first presented, as shown in Figure 7. There are two design variables h (depth of the hole) and r (radius of the hole) in this case, and each variable has 11 possible discrete values:

,

.

Note that these discrete values are not equally spaced. Higher resolution is used when the radius and the depth of the hole are large. The position of the center of the hole is fixed. The explicit geometry constraint  is already reflected in the range of the discrete values. Note that the objective function  is also an explicit function. In this case,

                                                                                    (7)

where W is the weight of the wheel without the holes.

Figure 7. Weight reduction holes of the wheel with only two variables

Figure 8 is a flow chart of the SNA method. A back-propagation neural network is trained to simulate the feasible domain of the implicit constraints using just a few training data (the set of initial training data X0). The initial optimization model  is formed. After the initial training is completed, a search algorithm is started to search for the solution point  of . This solution point is then evaluated to see whether it is feasible in , and whether  generates the same solution point as in the previous iteration. If the feasible solution point  is the same as , no new training data is generated. The feasible domain simulated by the neural networks will not change in the next iteration, so this procedure has to stop. If a new solution point  is generated, it is added to the set of training points X, and the neural network is trained again. With the increasing number of training points, it is expected that the feasible domain simulated by the neural network in  will approach to that of the exact model , and the solution point of  will approach the true optimum point.

Figure 8. Flow chart of the SNA method

Representation of a Design Point

There are three layers in the neural network, the input layer, the hidden layer, and the output layer. The size of the input layer depends on the number of variables and the number of discrete values of each variable. In this example, there are 2 design variables and each variable has 11 possible discrete values. Thus a total of 2x11 neurons are used in the input layer. Each neuron in the input layer has value 0 or 1 to represent a discrete value the corresponding variable takes. There is only a single neuron in the output layer to represent whether this design point is feasible (the output neuron has value 0) or infeasible (the output neuron has value 1).

In the wheel design example, the center point of the discrete search domain  is used as one of the initial training points. This training point is represented as shown in Figure 9, where a blank circle represents a “0” in the node, a solid circle represents a “1” in the node. For illustrative purpose, the 22 input nodes are arranged into two rows in Figure 9 to represent two design variables. The first 6 nodes of the first row have value 1 to represent that the first variable  has the discrete value 11. The first 6 nodes of the second row also have value 1 to represent that the second variable  also has the discrete value 20. Checking with , we can determine that  is a feasible design point. So the single node in the output layer has the value of 0 to represent that this point is feasible.

Input layer

 

 

Output layer

(Feasible)

Figure 9. Representation of a feasible training point .

The number of neurons in the hidden layer depends on the nature of the problem, and on the number of neurons in the input layer. We decided to put 16 neurons in the hidden layer in this case after a few trial runs. The transfer functions used in the hidden and output layer of the network are both log-sigmoid functions. The neuron in the output layer has a value range [0, 1]. After a training is completed, a threshold value is applied to the output layer when simulating the boundary of the feasible domain. In other words, given a discrete design point in the search domain, the network always output 0 (if output neuron’s value is less than the given threshold) or 1 (otherwise) to indicate whether this discrete design point is feasible or infeasible.

Note that the artificial neural network can be a continuous input model, and could be used with discrete values if needed. However, the representation of a training point in Figure 9 is designed to have 2x11 input element instead of just two (r and h). In this representation, all training data are in a crisp  0-1 pattern, which makes the training process relatively fast. A conjugate gradient algorithm (Powell-Beale Restarts) is used for the training. In this example, the error goal of 1e-5 is usually met within 100 epochs, even for later cases with many training points.

Searching in the Feasible Domain

Using three initial training points , , and , Figure 10 is a rough map of the feasible domain of  after the training is completed. As shown in the Figure, these three initial training points are the lower extreme, the center point, and the upper extreme of the design domain. The mathematical optimization model  is now transformed into , in which the feasible domain of the optimization model is simulated by .

Figure 10. The feasible domain (represented by blan circles) of  by 3 initial training points.

A discrete search algorithm is designed to search for the solution point  in . This algorithm has to start from a feasible design point. If the new design point generated from the previous iteration is a feasible design point in , it is used as the starting point in the current search. If the new design point is infeasible in , then use the same starting point of the previous iteration in the current search.

From the feasible starting point, the search algorithm moves one variable at a time to the neighboring discrete value. At the beginning of the search all design variables form a “usable set”, which is denoted U. The gradient of the objective function Eq.(7) with respect to the design variables is calculated at the starting point. Then one variable is picked to move to the neighboring discrete value. From all variables , the variable that has the largest  is picked. Since we are minimizing the objective function, if , then  is moved to the higher neighboring discrete value; on the contrary, if , then  is moved to the lower neighboring discrete value.

This new design point is checked against  to see if it is feasible in . If the new design point is feasible, this move is accepted and the search algorithm is started again from this new design point. If the new design point is infeasible in , this move is discarded, the variable  is not usable, so we update the usable set, . Another variable with the largest gradient value is picked from the usable set, and we proceed with the same procedure. This search process terminates when , since all moves from the current design point for lower objective function values are infeasible in . Therefore the current design point is the optimum design point (at least locally) in .

Back to our wheel design example, the search algorithm described above is then started to search for the optimum point in the feasible domain in Figure 10. The feasible design point  is used as the starting point, and the search algorithm terminates at a new design point . This new design point is then evaluated using the true constraints of  in Eq.(5), and is determined infeasible. Thus a new training point  is added to the set of training points, and the neural-network is trained again. After the training is completed,  is updated, the search algorithm is started again to search for the optimum point in the feasible domain simulated by  with four training points.

The iteration continues, with one more training point added into the set of training points in each iteration. The whole process terminates when we get the same design point repeatedly and no new training point is generated. In our wheel design example, the SNA method terminates after 6 iterations. The final design point obtained by the SNA method is . The weight of the wheel decreases from 4683g to 4446g, a 5.1% reduction. Figure 11 shows the iteration history of this example; the points represented by circles are the feasible design points. Note that only 8 training points are used, including 3 in the initial training, and 5 during the iterations. That is, only 8 evaluations (out of the 11x11=221 possible combinatorial combinations) of the implicit constraints are needed in this case. No gradient calculation is required.

Figure 11. Iteration history of the wheel design example with 2 variables.

To check the quality of the design point obtained by the SNA method, Figure 12 compares the feasible domain simulated by  at the end of the process and the true feasible domain. The design point obtained by the SNA method is indeed the global minimum of . Note that in the SNA method, as well as in most numerical optimization algorithm, global optimality is not guaranteed. At the end of the process, the feasible domain of  in the neighborhood of the final design point is close to that of the , because most training points (nodes enclosed by square brackets in the Figures) fall in this region. The final design point is a local optimum in this neighborhood.

     

(a)                                                                  (b)

Figure 12. The feasible domains (represented by blan circles) of  and .

Weight Reduction Holes Described by Five Design Variables

To demonstrate the capability of the SNA method in handling multiple variables, the same weight reduction problem is solved again in this section. As shown in Figure 13, this time the weight reduction hole is described by 5 design variables, and the shape of the hole is much more flexible. Each variable has 11 possible discrete values:

Figure 13. Weight reduction holes described by 5 design variables.

Again, the lower extreme , the center point , and the upper extreme  of the design domain are used as the initial training points. The SNA method terminates after 7 iterations. The final design is . The weight of the wheel decreases from 4683g to 4376g, a 6.5% reduction, which is better than the 5.1% of the previous case. Figure 14 shows the iteration history. Note that only 9 evaluations (out of the  possible combinatorial combinations) of the implicit constraints are needed to obtain this design.

Figure 14. Iteration history of the wheel design example with 5 variables.

Discussions and Conclusions

This paper describes a weight reduction problem of aluminum disc wheels under cornering fatigue constraints. This is a special structural optimization problem because of the existence of the implicit fatigue constraint. Numerical optimization algorithms are usually derivative-based methods that require at least the first derivatives of the implicit constraints with respect to the design variables. Typically, only the local numerical information from the solution point of the previous iteration is used in the current iteration. Knowledge about the whole search domain is not accumulated for formulating better approximation, or for better understanding of the optimization model. Precious function evaluations and sensitivity calculations of the implicit constraints are often not most efficiently utilized.

The SNA method presented in this paper is attractive in two ways. First, in each iteration, only one evaluation of the implicit constraints is needed to see whether the current design point is feasible. No precise function value or sensitivity calculation is required. Therefore it is especially suitable for engineering optimization problems with expensive implicit constraints. Secondly, in the SNA method, precious function evaluations of the implicit constraints are accumulated and utilized repeatedly to form better approximation to the feasible domain. In the wheel design example, the SNA method demonstrated the potential of being a more practical and efficient approach than the derivative-based numerical approximation algorithms.

Acknowledgements

This research is partially supported by the National Science Council, Taiwan, Project No. NSC 89-2212-E-155-004, and by Ensure Co., Ltd. This support is gratefully acknowledged.

References

Bannantine, J.A., Comer, J.J., and Handrock, J.L., 1990. Fundamentals of Metal Fatigue Analysis, Prentice-Hall, Inc.

SAE Handbook, 1992. Society of Automotive Engineers, Inc, pp. 4:31.01~4:31.16.

Hopfield, J.J., and Tank, D.W., 1985. “Neural Computation of Decisions in Optimization Problems,” Biological Cybernetics, v 52, p 141-152.

Karandikar, H.M., and Fuchs, W., 1990. “Fatigue life prediction for wheels by simulation of the roating bending test,” SAE Transactions, v 99, p 180-190.

Kaumle, F., and Schnell, R., 1998. “Entwicklungszeit von Leichtmetallradern verkurzen,” Materialpruefung/Materials Testing, v 40, p 394-398.

Marron, G., and Teracher, P., 1996. “Application of high-strength, hot-rolled steels in auto wheels,” Minerals, Metals & Materials Soc., JOM 48, p 16-20.

Lee, S.-J., and Chen, H., 1991. “Design optimization with back-propagation neural networks,” Journal of Intelligent Manufacturing, v 2, p 293-303.

Oda, J., and Mizukami, T., 1993. “Technique for optimum design of truss structures by neural network,” Nippon Kikai Gakkai Ronbunshu, A Hen/Transactions of the Japan Society of Mechanical Engineers, Part A, v 59, p 273-278.