Introduction

Currently, automotive industries are dedicating a lot of attention to improve product quality and reliability already in a virtual simulation environment. A major issue that designers and manufacturers in this field have to face is to improve vehicle lifetime. Since product life is highly determined by the fatigue life of its components, the presence of variability in the material parameters (Young's modulus, tensile strength, ...), as well as in geometrical properties, may have a strong effect on the overall fatigue life. Current deterministic approaches used so far are unable to take into account all the variabilities without oversizing structures or assuming a too pessimistic view on the actual material properties and on the operating conditions. Traditionally, safety factors have been applied to the product specifications and dimensions in order to make sure that the structure complies with the internal and external requirements, while being also robust to variability in the design parameter values. Unfortunately, these safety factors are often rather arbitrary and typically overestimate the actual variability. This might result in a design with a much higher level of robustness than what is actually required, at a cost of added weight, which is quite undesirable.

In order to achieve better performances together with improved safety, a new design process is needed to build automotive structures. This new process requires a shift from the traditional design approach to a new approach that incorporates all the variabilities and uncertainties in the analysis phase and in the design flow and to use computer simulation methods to guarantee the design reliability. This paper gives an outline of the methodology that has been used to analyze and optimize the reliability of a vehicle knuckle in terms of fatigue life, together with a description of the most commonly used methods to improve the reliability of automotive structures. Particularly, a Reliability-based Design Optimization (RBDO) procedure will be illustrated, that uses reliability methods not only to assess the reliability of a given design, but sets a step further: it allows computing a more reliable design for the given product variability.

Test Case Description

The vehicle lifetime is highly determined by the fatigue life of its components; in turn, variability in material properties may affect the fatigue life. For demonstration purposes, an automotive analysis project has been created, to improve the reliability of a vehicle knuckle fatigue life. A Finite Element (FE) model is used to represent this structure in a simulation environment and realistic variability has been assigned to the structure's material properties. The RBDO procedure has then been used to improve its design.

The structure considered (shown in Figure 1) is made of steel and has properties (with inherent variabilities) as specified in Table 1. A fatigue analysis is performed, based on a validated FE model, to increase the fatigue life in the presence of variability.

Figure 1: Knuckle finite element model

In this paper, all the input variables are considered to be statistically independent. Unfortunately this approximation is not always true in reality because the realistic characterization of the correlation coefficients is often very difficult and therefore considered impractical for most engineering applications.
The loads applied on the structure have been obtained from a multibody simulation of an assembly consisting of the structure and the wheels, the steering mechanism and the chassis [1]. All these contributions have been simulated over a time period of 630s and then applied on the different interface points of the knuckle. Then the fatigue effect of the cyclic application of the load histories has been evaluated.
The types of loads applied are summarized here:

• Wheels load components along the X, Y and Z axes + momentums around axes X and Y
• Steering load components along the X, Y and Z axes
• Upper chassis connection load components along the X, Y and Z axes
• Lower chassis connection load components along the X, Y and Z axes

Examples of some of the load time histories are shown in Figure 2.

(a)

(b)
Figure 2: Wheel load conditions in Newton for X and Y directions (a) and for the Z direction (b) over a time interval of 630s.

In general, fatigue is the degradation of a material due to repeated cyclic loading. For metals, this typically means that there is an initiation of small cracks from active slip bands in grains on the free surface of a specimen, component or structure, which eventually link to form large cracks that either break or severely degrade the performance of a component.

Two basic approaches have been developed to estimate the crack initiation life of components and to address fatigue analysis and design for durability: the stress-life approach and the strain-life approach. The goal of both approaches is to estimate the crack initiation life of structures, by combining a mechanics-based analysis of the stresses or strains in the structure with the results of basic material property tests. The type of fatigue analysis approach to use will depend on the type of failure mode that is expected for the structure.

When repeated fatigue life tests are performed, one does not find exactly repeated results, but rather a scatter of results. This scatter comes from several sources. There are natural variations in material properties, component dimensions, customer service loads and manufacturing tolerances. The largest variation is typically found in the customer usage. One can use probabilistic distributions to characterize the variation of these variables, and include this variability in the design process to better explain and predict the scatter obtained in fatigue test results. Probabilistic methods offer a powerful tool to assess the inherent risk of designs for durability through the characterization of the inherent variabilities, by allowing the design engineer to minimize the weight and costs requirements, while guaranteeing with sufficient reliability that targets are met.

Analysis Case Implementation

In order to perform the analysis, the methodology reported in the previous section has been implemented. The optimization process is managed by an OPTIMUS [8] thread that determines the necessary steps to reach the optimum design point that satisfies the probabilistic constraint of Eq. (12). The optimization algorithm selected for all the analysis cases is the Sequential Quadratic Programming (SQP) using a tolerance of 10-3 and forward finite difference gradient estimation. Note that the optimization process is a deterministic process with a probabilistic constraint. Thus the result is an optimum point that is also robust with respect to the required failure probability.

For each iteration of the optimization process, the algorithm requires the evaluation of the performance function at specific values of the input parameters, so that a sequence of static analysis (for the present case the Nastran [9] FE solver has been used) and LMS FALANCS [10] fatigue analysis has to be performed for each evaluation. To accomplish this task, another process management thread using OPTIMUS has been created: when the algorithm (for the estimation of the performance measure) requires a performance function evaluation for a given parameter vector, OPTIMUS launches an FE computation and submits the results back to the algorithm, where the LSF criterion is evaluated.

For the vehicle knuckle considered in this paper, a Finite Element model has been created and the external load configuration has been applied. As significant computational effort is required for each fatigue life evaluation, a hybrid meta-model/FE strategy has been used to limit the overall CPU time. In fact, a full factorial, multi-level Design of Experiments (DOE) plan has been designed in order to explore the fatigue life response within the boundaries of interest. Subsequently, a quadratic Response Surface (RS) model has been computed based on the DOE results. The obtained meta-model has then been used for a first RBDO process using a PMA methodology.

The outline of the procedure used for the analysis has been as follows:

1. Compute the response of the structure with 3 DOE plans
1. Compute the 3 levels DOE plan in the range [ 3$sigma$, +3$sigma$ ]
2. Compute the 5 levels DOE plan in the range [ 6$sigma$ , +6$sigma$ ], re-using previous step results
3. Compute the 7 levels DOE plan in the range [ 9$sigma$ , +9$sigma$ ], re-using previous step results
2. Compute the quadratic least squares response surface model (using Taylor Approximation)
3. Perform an RBDO loop using the RS model and obtain a first tentative optimum point
4. Validate (and refine if needed) the results of the previous step with FE computations.

Thus a total of 49 experiments have been run and the maximum absolute error of the meta-model showed to be less than 0.1 when compared with FE simulation results.

The probabilistic constraint set for the RBDO process is:

$beta_{t}=6 rArr P_{f}=9.86*10^{-10}$

(12)

This means that the design optimum will be a 6$sigma$ design w.r.t. the selected performance function.

Choice of Performance Function

Before discussing the choice of the performance function for the present case, some observations are needed. Various failure modes exist in durability cases, and the identification of the failure mode is important in both testing and analysis. Testing at load levels that are too high will change the failure mode and give the incorrect safety factors or may identify incorrect critical locations. The following failure modes have been considered:

A static (non-fatigue) failure mode: this results in large structural deformation and is controlled primarily by the resistance of the net section.

Low cycle fatigue: this is usually the result of a large plastic zone at a notch, or of a stress concentration. Low cycle fatigue behavior depends on the notch severity and the inelastic material response.

High cycle fatigue: this is representative of a situation in which there is little plasticity. Here the notch severity, manufacturing processes and residual stresses play an important role.

For fatigue analysis, the strain-life approach is typically used in situations where yielding of the material may be present at certain locations in the structure. This approach has been developed based on knowledge of more detailed material behavior when subjected to stress levels above the yield strength of the material. On the contrary, the stress-life approach is typically more suitable in situations where the stresses are below (or not much above) the elastic limit of the material.

Following the previous considerations, the choice of the performance function is related to the Maximum Damage sustained by the structure. For this case, a Maximum Damage value of -1 (in log10 scale) has been selected. The performance function can than be written as

$"MaxDamage"=-1 rArr G=-("MaxDamage" + 1) rArr G>0 quad"for MaxDamage"<-1$

(13)

With this choice, the G function is positive where MaxDamage is less than -1, which is in agreement with the definition of the reliability problem reported.

In this paper, only material properties have been considered. Note however that the methodology is not limited to this class of parameters as similar analyses can be performed with variability in geometrical or strength properties, load conditions, etc.

Results

The results of the optimization process for the analysis case are reported in this section. Two steps have been considered for the vehicle knuckle due to the computational effort required for each fatigue analysis. As a first step, the fatigue life response (in terms of Maximum Damage) of the structure has been assessed with a limited number of evaluations using the proposed DOE strategy.

Using the results of the DOE plans a least squares quadratic Taylor Response Surface has been computed to approximate the FE model response. A plot of the response surface is shown in Figure 3 The RS has been used as a meta-model in order to assess the reliability of the knuckle structure.

Figure 3: Response surface of the Maximum damage (log₁₀ scale) as a function of the Elastic modulus and Tensile Strength.

Apparently, the optimization procedure seems simple, as there's only a single equality constraint to be formally satisfied. A closer look at the problem reveals that the optimization loop should in fact meet two optimization objectives: the first is the equality constraint and the second is that the newly found design point should also minimize the Maximum Damage.

It's easy to figure out that the solution locus for the first constraint is constituted by all the points at a distance of 6$sigma$ from the LSF. But among these points, there is only one point that minimizes the Maximum Damage of the structure. Thus the multi-objective procedure should satisfy two constraints, one equality constraint and one inequality constraint:

${(beta_(t)=6),("minimize"("MaxDamage")):}$

(14)

Figure 4: RBDO procedure using the PMA and the Hybrid Mean Value (HMV) algorithm.

In order to meet both targets, the optimization procedure has been implemented for an objective that is a combination of the two constraints of Eq. (14). The new objective can be expressed as:

$min(|beta_{S}-beta_{t}|-"MaxDamage")$

(15)

Using this new objective, the optimization process has been carried out using the Performance Measure Approach and the HMV algorithm [4], looking for a robust optimum point in the range [ 3 $sigma$ +3$sigma$ ] for each parameter (i.e. E and TS). The results of the optimization are reported in Table 2 and Table 3 and are shown in Figure 4.

Table 2: RBDO results using PMA

Table 3: RBDO performance using PMA

(a)

(b)
Figure 5: Differences between the FE (a) and the DOE/RS Model (b)

The optimized point found with the use of the response surface model has then been used as a starting point for a refinement optimization procedure involving only FE computations. Before stepping into the refinement process results, some considerations on the choice of using a Response Surface approximation are reported, to point out the advantages and disadvantages of this meta-modeling technique.

To better understand the differences between the Response Surface model and the FE computations, some relevant contours of the Maximum Damage values in the stochastic transformed space are presented. These contours have been computed with the FE model and with the DOE/RS model. Figure 5 compares a quadratic RS model with a contour based on a much higher number of FE analyses than have been needed for the DOE. In the region where the behavior is quite linear, the accuracy of the terms with order higher than linear might not be sufficient. This gives rise to slight errors in the estimates with the RS model in this linear region.

It can however be concluded that the RS model is a suitable first approximation model that can yield sufficiently accurate results with a very limited computational effort. Moreover, it saves many FE computations in the optimization process, by locating an initial point which is very close to the actual solution, so that the subsequent refinement process (that usually implies additional FE computations) becomes much faster.

Refinement of the RBDO results

The results of the refinement process using only FE computations are here reported. Even if the number of required optimization loops is less than the number required with the RS model, the time needed for the complete process to finish is some orders of magnitude larger. The results of the refinement process are reported in Table 4 and Table 5. The calculation times for both approaches are compared in Table 6.

Table 4: RBDO refinement results for the vehicle knuckle using PMA

Table 5: FE RBDO refinement loop performance for the vehicle knuckle case using PMA

Table 6: Performance comparison of the refinement process with the RS model first optimization loop.

The results obtained from the refinement process show that the RS model is indeed conservative. Figure 5 shows that the "real" LSF (obtained with the FE refinement analysis) is located slightly further from the origin than the LSF of the RS model. This means that the optimum point found with the RS model is at a distance a little larger than 6 . However, the difference between the Tensile Strength (TS) of the RS model and the TS of the FE refinement is sufficiently small when compared to the standard deviation of the TS property. This means that the FE refinement process can be considered too expensive (in terms of computational time) w.r.t. the advantage it brings.

To conclude, it must be noted that the limited number of significant digits available in FE computations slows down the convergence of both the PMA iterations and the SQP optimizations. In order to limit this effect, a semi-adaptive step for the finite difference gradient estimation has been used.

Probabilistic characterization of the response.

To assess the effect of the introduced variability for the system response, a probabilistic characterization of the original design point has also been carried out. The results, showed in Figure 6, represent the histograms of the Max Damage response of the nominal system subject to a Monte Carlo simulation. The population of 2.302.586 samples obtained has then been analyzed with a maximum likelihood test that showed how the response of the system follows a normal distribution with mean and standard deviation as in Table 7.

(a)

(b)
Figure 6: Histograms of the Probability (a) and Cumulative (b) density of the MCS population (in blue) with the maximum likelihood distribution selected (red).

Table 7: Mean and standard deviation of the generated population.

The likelihood measures reported in Table 8 have been computed by using different distribution models (with the mean and standard deviation given in Table 7). The model with the likelihood closest to the data likelihood has been selected; as mentioned before, for the present application case, this is the Normal distribution.

Table 8: Likelihood results for other distribution models compared to the selected model.