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Comparing various multi-disciplinary optimization approaches for performance enhancement and weight reduction of a vehicle chassis frame.

ABSTRACT

Designing a vehicle chassis involves meeting numerous performance requirements related to various domains such as Durability, Crashworthiness and Noise-Vibration-Harshness (NVH) as well as reducing the overall weight of chassis. In conventional Computer Aided Engineering (CAE) process, experts from each domain work independently to improve the design based on their own domain knowledge which may result in sub-optimal or even non-acceptable designs for other domains. In addition, this may lead to increase in weight of chassis and also result in stretching the overall product development time and cost. Use of Multi-Disciplinary Optimization (MDO) approach to tackle these kind of problems is well documented in industry. However, how to effectively formulate an MDO study and how different MDO formulations affect results has not been touched upon in depth.

This study implements various MDO formulations on an established SUV chassis frame for further weight reduction and performance improvement. Results from different MDO formulations are compared and insights are provided into ways of formulating an MDO problem in order to achieve desired results. How the choice of optimization algorithm, number of objectives and constraints etc. affect MDO results is also discussed. 18 component thicknesses are selected based on domain understanding as design variables (DVs) for optimization. Various attributes/load-cases considered during optimization are -bending and torsional stiffness for Durability, global mode shapes from modal analysis for NVH, energy absorption and displacements at specific locations during frontal offset crash for crashworthiness. As a first step, a Design of Experiment (DoE) study is performed for each domain individually and results are analysed to identify conflicting targets. Response Surface (RSM) based optimization method is selected for optimization studies considering time and resource availability. Commercial software package, modeFRONTIER, is used as a process integration, design optimization and data analysis tool. Optimal designs from all MDO studies are compared with each other to evaluate effectiveness of formulations.

CITATION: Bhosale, S., Malladi, A., and Londhe, A., "Comparing Various Multi-Disciplinary Optimization Approaches for Performance Enhancement and Weight Reduction of a Vehicle Chassis Frame," SAE Int. J. Mater. Manf. 9(2):2016,

INTRODUCTION

With vast increase in computing power, design of complex engineering systems such as automobiles is becoming increasingly computer centric. Designers are now able to capture system behaviors at greater depth and solve complex problems in a time bound manner which may not have been possible previously. Most of these systems have contributing factors from multiple disciplines or subsystems, often resulting in multiple conflicting objectives. Designers/Analysts always aim to optimize designs using formal/informal methods. Use of formal optimization techniques across various disciplines is also well known. As the complexity of systems increase, the need for system level collaborative optimization increases even more.

In the recent years Multi-Disciplinary Optimization (MDO) has emerged as a well-established optimization method as the computer speed increased, the simulation methodologies matured, and specialized MDO software tools have been developed. In MDO, designers aim to optimize all disciplines simultaneously as shown in figure 1 rather than optimizing each discipline sequentially. It enables designers to find a solution which is optimum for all disciplines at the same time, hence reducing the computational efforts and time required during the entire design process. Various MDO techniques, such as, Collaborative Optimization (CO) [1], Analytical Target Cascading (ATC) [2], Concurrent Subspace Optimization (CSSO) [3], Bi-Level Integrated System Synthesis (BLISS) [4] etc. have been developed over the years. While these approaches have differences in the way they formulate and handle an MDO problem, all of them essentially target to capture interactions between various disciplines and produce optimum designs acceptable to all disciplines. While Gradient-Based optimization methods can be used for MDO studies, the evolutionary optimization techniques are more popular due to their ability to better handle complex problems and find global optimum solutions. Although MDO has a lot of advantages, there are still few challenges to make the use of the method widespread. Including all disciplines simultaneously, significantly increases the complexity of the problem. In an organization, getting experts from all disciplines to work together and compromise on individual disciplines to achieve greater system level optimum may prove to be a challenge in itself. Techniques such as surrogate based optimization have been developed to tackle such challenges. MDO is now a widespread method in automobile industry. Its application in fields such as crashworthiness, NVH and Durability is well documented in [5, 6, 7, 8, and 9].

PROBLEM DEFINITION

SUV chassis is taken for optimization study. Figure 2 shows the chassis frame with selected members for optimization. Gauges of 18 structural members are selected and used as design variables in this study. Upper and lower limits have been decided by varying base thicknesses by [+ or -]20%. Bending and torsional stiffness are selected from durability domain as these stiffness define the global performance of chassis. Frontal offset crash at chassis level is selected from crash domain. Untrimmed modal analysis is selected from NVH domain. Vertical bending, lateral bending, torsional and match box modes are selected as targets. Objective of this study is to compare MDO formulations with two objectives 1) Weight reduction while maintaining all performance targets at base level and 2) Performance improvement while maintaining mass at base level.

METHODOLOGY

Response surface based optimization method is used in current work. Direct optimization based on solver results lead to large number of simulations and there is no control over the number of simulations required to converge to the global optima. Generally, for direct optimization, one needs to analyze results for optima after certain number of simulations and stop the optimization run if the satisfactory design is found. But this may lead to getting stuck at local optima most of the times. In RSM based optimization method, meta-models based on DOE matrices are created and used for optimization. Considering time, resources and non-linearity of problem, number of simulations for specific loadcase were decided. Commercial optimization software modeFRONTIER is used as a front end for process integration and optimization. Nastran is used for performing stiffness and modal analysis. LS Dyna explicit is used to evaluate crash loadcase.

Crash Loadcase Set-Up

Performing the current MDO study on SUV chassis with full vehicle crash model requires lot of simulations involving huge run time and is very resource intensive with high server occupancy. Hence, to simplify and solve the problem within available resources of time and computational abilities, a representative set up for frontal offset deformable barrier (ODB) impact load case was considered. The chassis is set to impact against a stationary rigid wall with overlap matching with that during an offset crash at the full vehicle level. Thus loading conditions (figure 4) are replicated in the closest possible way ensuring chassis deformation (figure 5) to be similar to that observed during an offset impact.

Impact speed of the chassis is estimated on the basis of chassis energy absorption during full vehicle crash. Critical regions and components on chassis are tracked to measure deformations and energy absorption during the event which, in combination and independently, are set as objectives and constraints based on the MDO problem formulation.

DESIGN OF EXPERIMENTS

Design of experiment matrix for each response is created separately. Number of simulations for each response is decided based on linearity or non-linearity of the response. The algorithm used was uniform latin hypercube (ULH) method. Uniform Latin Hypercube is useful when a random sample is needed which is a case for RSM generation. But at the same time, it ensures to be relatively uniformly distributed over each dimension. Uniform latin hypercube type is well proven for creating accurate response surfaces. Initial DOE size chosen is 150 for stiffness and modal responses and 220 for frontal offset crash considering the non-linearity of the problem.

This whole process of DOE set-up, integration with high performance cluster (HPC) and post-processing of results is established in modeFRONTIER.

As shown in figure 6, DOE matrix is generated according to ULH, input LS-dyna file was modified according to each design point in the table, supporting files are fied along with the main file. All these files were transferred to HPC for solving and results are pulled back and post-processed for extracting responses. Similar workflows are created for stiffness and modal loadcases. LS-PrePost is used as a post-processor for extracting crash responses. Modal frequency responses and displacements for stiffness were extracted from ASCII (American Standard Code for Information Interchange) output files.

DOE spread is checked using variable level scatter matrix. Figure 7 shows example of scatter matrix chart for one of the DOEs. As shown in the figure, the spread is said to be better if probability distribution function chart is fat in nature and correlation between two variables is close to zero. This was ensured for DOEs created for all responses.

DOE ANALYSIS AND VARIABLE SCREENING

Data mining can be done on DOE data and lot of information like trend between inputs and outputs, main effects, interaction effects and non-linearity of the problem can be obtained. Half-normal plot is a useful graphical tool to judge effect significance and helps to know which factors (including main effects and interactions) are important and which are insignificant. All the large estimated effects appear in the upper right corner and fall above the line through the small estimated effects. Effects ranking table quantifes the effect significance.

Figure 8 shows DOE analysis charts for bending stiffness. These charts are plotted for each response. Figure 8 shows middle long members as most infuential DVs for bending stiffness of Chassis frame. Redundant DVs are identified and removed for response surface generation. This step is important for reducing dimensionality of the response surface, which will result in improving accuracy of response surface. Factors contributing up to 90 % cumulative effect are screened and used for RSM creation. Care was taken to ensure that there is no co-linearity between design variables as it leads to wrong interpretation of effects. This helped in 7increasing accuracy of RSM in terms of R squared values and enabled efficient exploration of response surface by optimization algorithm as dimensionality is reduced.

RESPONSE SURFACE METHOD

Variables are selected for individual responses to generate response surface models. There are various algorithms present for creating response surfaces. These can be divided into three main categories namely classical models, statistical models and advanced models. Different algorithms are tried out and the best one is selected based on specified error criteria. One system for checking the goodness of an interpolant response surface is the leave-one-out methodology. In turn, each point belonging to the training set is excluded from the training procedure. The value predicted in the excluded point by the so created surface is then compared to the known value. The leading idea is that the smaller this value on average, the better the surface trained on the whole dataset. A severe drawback of this technique is its huge computational demand: n different surfaces have to be created, each using n-1 training points, where n is the size of the original training set. Very often this fact prevents this method from being used. There are advanced algorithms like radial basis function model which uses this approach. This method improves quality of response surface in terms of capturing non-linearity in whole design space. "Train and Validate" approach was used for creating RSMs. 10% of design points were randomly separated for validation and RSMs were trained using 90 % of the design points. This step was repeated by choosing different validation points every time to ensure accuracy of all regions in design space. Different response surface models were used and the best one was selected by comparing mean normalized error and R-squared value. It was observed that accuracy of frontal offset ODB RSM was less in terms of R squared value and there was need for additional design points to improve accuracy of RSM. There are intelligent space filler algorithms designed to improve response surface accuracy. Lipschitz sampling method was used to add space filler design points. New points to be evaluated are chosen among a pre-determined set of candidates trying to maximize both the distance from already evaluated points and an approximation of the Lipschitz Constant. 50 additional design points to the original DOE of 150 were added which gave significant improvement in response surface accuracy. Table 1 shows the improvement in R squared values. Even with addition of 50 design points, residual error of tune 7-8 % was observed in some cases which is compensated by tightening the constraints. This decision was taken considering the project timeline and resource availability to run more simulations. The selected RSMs for all responses were exported to new modeFRONTIER project file for bringing all disciplines at one place.

OPTIMIZATION

Optimization problem can be formulated in multiple ways depending on purpose of optimization. In preliminary phase of product development, the purpose of optimization will be to enhance performance and optimize mass. In later stage, weight reduction opportunities needs to be explored keeping performance at base level. In this study, both the approaches are explored and reported. First formulation was focused on performance enhancement. In this formulation, mass was kept as constraint and selected performance responses were taken as objective. Weightages to different responses were set according to importance of the responses. Second formulation was focused on weight reduction. In this formulation, single objective approach of minimizing mass keeping all the responses as constraints is used. This gives us fexibility of putting targets on each response and these targets can be varied according to importance. Drawback of this method is, sometimes problem becomes over constrained when we are dealing with large number of responses. This leads to getting confused at local optima which is not desirable. Figure 9 shows modeFRONTIER workflow for first formulation of optimization. There are different types of optimization algorithm present in the modeFRONTIER. Optimization algorithms are classified into three categories namely gradient based methods, exploratory methods and evolutionary methods. Gradient based methods are fast but are prone to local optima especially for highly non-linear responses. Evolutionary methods like genetic algorithm and simulated annealing are very famous amongst all because of their efficiency in in-depth design space exploration. There are special optimization engines available which combines advantages of different algorithms. For this case study, Non-dominated sorting genetic algorithm was used.

RESULTS AND DISCUSSION

The results of different optimization formulations are presented in this section. Tradeoffs amongst output variables and between input and outputs need to be understood. There are many variables which conflict with each other. For example, it is observed that torsional stiffness is highly positively correlated with mass as shown in figure 10. This high correlation level would result into huge mass addition. Also torsional stiffness was acting as a hard to satisfy constraint during optimization. Torsional stiffness criteria was revisited and reduced to find a balanced optimal solution.

To flter the designs according to targets and to understand the design space, parallel coordinate chart is used. Figure 11 shows parallel coordinate chart for first formulation. Each vertical axis corresponds to performance targets and mass. Criteria is set for all constraints and Pareto Frontier of feasible designs is selected.

This Pareto front of designs is selected for further screening. Multi criteria decision making tool (Figure 12) is used for further shortlisting designs based on user preferences on performance criteria and mass. Final design is selected based on formulation requirement as explained in below sections.

Table 2 compares two formulation results as mentioned above which shows % change in mass and different performance parameters with respect to base design mass and performance parameters. The results presented in Table 2 are from confrmatory simulations. Parameters 8-14 are frontal ODB crash intrusion levels at different points of interests and energy absorbed by specific parts to ensure specific type of deformation.

Formulation 1

In first formulation, purpose was to look for performance enhancement by keeping mass at base level. Multiple algorithms were evaluated and it was found that simplex and Multi-objective genetic algorithm were best suited for this type of formulation. The result showed a marked improvement of 14% in bending stiffness, 3% in torsional stiffness and 1-3% in all frequencies. Crash domain saw 15-25 % improvement in intrusion levels at some locations while maintained at base level at other locations.

Formulation 2

Formulation 2 produced a mass reduction of 4.2 kgs. Torsional stiffness and lateral bending mode frequency were reduced by 2-2.5 %, while bending stiffness was improved by 4% and all other modal frequencies were maintained at their respective baseline values. Intrusion levels were improved at some locations and maintained at base line for some locations. Key finding from this formulation was identification of optimization driver parameters. In such cases, MDO team needs to take decision for trade off depending on domain knowledge and criticality of the particular response.

CONCLUSION

An innovative strategy to optimize automobile systems for multiple domain requirements is presented. Different optimization approaches for performance enhancement and weight reduction considering all CAE domains simultaneously are explored and reported. Design space exploration technique brings out huge potential for performance improvement in the structure. Statistical data analysis performed on DOE results helps understand behavior and relations among responses from different domains. Different response surface fitting algorithms are evaluated which helps in saving resources and time. Weight reduction of 4.2 kg is achieved in formulation 2 while keeping all targets at base line with some improvement. Multi-disciplinary optimization can be implemented at two stages- Initial stage of program for performance improvement and at mid of program for weight reduction. Maximum beneft of MDO can be extracted by implementing it multiple times in design cycle.

REFERENCES

[1.] Braun R., Gage P., Kroo I., and Sobieski I., "Implementation and performance issues in collaborative optimization. Analysis", 10:1, 1996.

[2.] Papalambros P.Y. and Jiang T., "Target Cascading in Optimal System Design" Ann Arbor, 1001:48109.

[3.] Wujek B.A., Renaud J.E., Batill S.M., and Brockman J.B., "Concurrent subspace optimization using design variable sharing in a distributed computing environment. Concurrent Engineering", 4(4):361, 1996.

[4.] Sobieszczanski-Sobieski J. Agte J.S., and Sandusky R.R.. "Bi-Level Integrated System Synthesis (BLISS)". Technical report, Citeseer, 1998.

[5.] Rakowska, J., Chator, A., Barthelemy, B., Lee, M. et al., "An Iterative Application of Multi-Disciplinary Optimization for Vehicle Body Weight Reduction Based on 2015 Mustang Product Development," SAE Int. J. Mater. Manf. 8(3):685-692, 2015, doi:10.4271/2015-01-0470.

[6.] Chuang, C., Li, G., and Yang, R., "An Optimization and Trade-Off Process for Crashworthiness with Multiple Responses," SAE Technical Paper 2007-01-1543, 2007, doi:10.4271/2007-01-1543.

[7.] Donders, S., d'Ippolito, R., Van der Auweraer, H., Hack, M. et al., "Uncertainty-Based Design in Automotive and Aerospace Engineering," SAE Technical Paper 2007-01-0355, 2007, doi:10.4271/2007-01-0355.

[8.] Koppe, A., Kaufmann, M., Lauber, B., "Multi- Disciplinary Optimization considering Crash and NVH Load Cases", ATZ/MTZ Virtual Product Creation Conference, Stuttgart, Germany, June 20-21, 2005.

[9.] Sobieski, J., S., Kodieyalam, S., Yang, R., J., "Optimization of Car Body under Constraints of Noise, Vibration, and Harshness (NVH), and Crash", Structural and Multi-Disciplinary Optimization", Vol. 22, No. 4, p. 295-305, 2001.

CONTACT INFORMATION

Corresponding Author: Subhash Bhosale bhosale.subhash@mahindra.com

Mob no. : +919789088832

ACKNOWLEDGMENTS

I would like to acknowledge modeFRONTIER support team for help and guidance provided.

DEFINITIONS/ABBREVIATIONS

MDO - Multi-Disciplinary optimization

DOE - Design of experiments

RSM - Response surface method

HPC - High performance cluster

DV - Design variable

CAE - Computer aided engineering

NVH - Noise vibration Harshness

Subhash Hanmant Bhosale, Aditya Malladi, and Abhijit Londhe Mahindra & Mahindra, Ltd.

Table 1. Response surface accuracy improvement

Sr.                         Original   Improved
No.  Description            R squared  R squared

1    A Pillar Y intrusion   0.62       0.85
2    A Pillar Z intrusion   0.45       0.76
3    Swan Neck Y intrusion  0.55       0.91
4    Swan Neck Z intrusion  0.53       0.85

Table 2. Optimized results from formulation 1 and 2

Sr.  Description                 Formulation 1  Formulation 2
No                               % Change       % Change

 1   Mass                        -0.15          -2.04
 2   Vertical Bending Mode        3.12          -0.25
     Frequency
 3   Lateral Bending Mode        -0.44          -2.21
     Frequency
 4   First Torsion Mode           1.62          -0.41
     Frequency
 5   Matchbox Mode Frequency      2.39          -0.17
 6   Global Bending stiffness    14.47           4.21
 7   Global Torsional stiffness   3.27          -2.45
 8   A Pillar Y intrusion         2.58           0.74
 9   A Pillar Z intrusion        -3.65           3.19
10   Swan Neck Y intrusion        3.3            6.27
11   Swan Neck Z intrusion       -0.79          13.22
12   Engine A Mount Y intrusion   2.96           0.87
13   Engine A Mount Z intrusion  21.2           20.49
14   Energy absorbed by LM       15.25          13.79
     Inner component
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Author:Bhosale, Subhash Hanmant; Malladi, Aditya; Londhe, Abhijit
Publication:SAE International Journal of Materials and Manufacturing
Article Type:Report
Date:May 1, 2016
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