Editorial Type:
Article Category: Research Article
 | 
Online Publication Date: 01 Sept 2006

Multi-Objective Design Problem of Tire Wear and Visualization of Its Pareto Solutions2

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Page Range: 170 – 194
DOI: 10.2346/1.2345640
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Abstract

Since tires carry out many functions and many of them have tradeoffs, it is important to find the combination of design variables that satisfy well-balanced performance in conceptual design stage. To find a good design of tires is to solve the multi-objective design problems, i.e., inverse problems. However, due to the lack of suitable solution techniques, such problems are converted into a single-objective optimization problem before being solved. Therefore, it is difficult to find the Pareto solutions of multi-objective design problems of tires. Recently, multi-objective evolutionary algorithms have become popular in many fields to find the Pareto solutions. In this paper, we propose a design procedure to solve multi-objective design problems as the comprehensive solver of inverse problems. At first, a multi-objective genetic algorithm (MOGA) is employed to find the Pareto solutions of tire performance, which are in multi-dimensional space of objective functions. Response surface method is also used to evaluate objective functions in the optimization process and can reduce CPU time dramatically. In addition, a self-organizing map (SOM) proposed by Kohonen is used to map Pareto solutions from high-dimensional objective space onto two-dimensional space. Using SOM, design engineers see easily the Pareto solutions of tire performance and can find suitable design plans. The SOM can be considered as an inverse function that defines the relation between Pareto solutions and design variables. To demonstrate the procedure, tire tread design is conducted. The objective of design is to improve uneven wear and wear life for both the front tire and the rear tire of a passenger car. Wear performance is evaluated by finite element analysis (FEA). Response surface is obtained by the design of experiments and FEA. Using both MOGA and SOM, we obtain a map of Pareto solutions. We can find suitable design plans that satisfy well-balanced performance on the map called “multi-performance map.” It helps tire design engineers to make their decision in conceptual design stage.

Keywords: inverse problem; multi-objective optimization; multi-objective genetic algorithm; self-organizing map; Pareto solution; wear; finite element analysis
Copyright: The Tire Society
FIG. 1.
FIG. 1.

Analysis and design.

FIG. 2.
FIG. 2.

Pareto solutions and Pareto ranking of two-objective maximization problem. Solutions A, B, C, and D are Pareto solutions because their rankings are 1. Pareto ranking of solution E is 2 because solution B is better than solution E on both objectives, respectively. The ranking of solution H is 4 because it is worse than solutions A, B, and E on both objectives.

FIG. 3.
FIG. 3.

Pareto solutions obtained by MOGA and solution by single-objective optimization.

FIG. 4.
FIG. 4.

History of generation in MOGA.

FIG. 5.
FIG. 5.

Three-dimensional Pareto solutions in objective function space.

FIG. 6.
FIG. 6.

Analytical solutions of each objective function in design space.

FIG. 7.
FIG. 7.

Four-dimensional Pareto solutions obtained by MOGA.

FIG. 8.
FIG. 8.

Self-organizing map of four-dimensional Pareto solutions.

FIG. 9.
FIG. 9.

Design variables mapped onto four-dimensional Pareto solution space.

FIG. 10.
FIG. 10.

Half cross section and design variables of 215/60R16 tire.

FIG. 11.
FIG. 11.

Frictional energy distribution (above) at left cornering mode and mean frictional energy in circumferential direction (below). FES and FEC stand for frictional energy at shouder area and that at center area, respectively.

FIG. 12.
FIG. 12.

Frictional energy at each driving mode of a front tire.

FIG. 13.
FIG. 13.

Frictional energy at each driving mode of a rear tire.

FIG. 14.
FIG. 14.

Computed four-dimensional Pareto solutions.

FIG. 15.
FIG. 15.

Cluster of Pareto solutions.

FIG. 16.
FIG. 16.

Self-organizing map of Pareto solutions.

FIG. 17.
FIG. 17.

Mean values of objective functions of each cluster.

FIG. 18.
FIG. 18.

Design variables maped onto self-organizing map.

FIG. 19.
FIG. 19.

Mean values of design variables of each cluster.

FIG. 20.
FIG. 20.

Frictional energy and wear profile of one solution in cluster #8.

FIG. 21.
FIG. 21.

Two single-objective solutions, OPT1 and OPT2, on SOM of objective functions.

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