Decomposition-based Assembly Synthesis of a 3d Body-in-white Model for Structural Stiffness

This paper presents an extension of our previous work on decomposition-based assembly synthesis for structural stiffness [1], where the 3D finite element model of a vehicle body-inwhite (BIW) is optimally decomposed into a set of components considering the stiffness of the assembled structure under given loading conditions, as well as the manufacturability and assembleability of components. Two case studies, each focusing on the decomposition of a different portion of a BIW, are discussed. In the first case study, the side frame is decomposed for the minimum distortion of front door frame geometry under global bending. In the second case study, the side/floor frame and floor panels are decomposed for the minimum floor deflections under global bending. In each case study, multi-objective genetic algorithm [2,3] with graph-based crossover [4,5], combined with FEM analyses, is used to obtain Pareto optimal solutions. Representative designs are selected from the Pareto front and trade-offs among stiffness, manufacturability, and assembleability are discussed. INTRODUCTION Complex structural products such as automotive bodies are made of hundreds of components joined together. While a monolithic design is ideal from a structural viewpoint, it is virtually impossible to economically manufacture complex structures as one piece, requiring them to be assemblies of smaller sized components with simpler geometry. Therefore, during the conceptual design stage designers need to decide a set of components by decomposing the overall product geometry of the whole structure. In industry, a handful of basic decomposition schemes considering geometry, functionality, and manufacturing issues are used. However, these decomposition schemes are usually non-systematic and depend mainly on the designers’ experience, which may cause the following problems during design and the production phases: Problems of the insufficient assembled structure stiffness: Components and joining methods specified by designers may not meet the desired stiffness of the assembled structure. Problems of manufacturability and assembleability: Components decided by designers can not be produced or assembled in an economical way. Since these problems are directly related to the component and joint configurations and therefore usually found in the production phase, solving them requires costly and timeconsuming iteration from an early design stage. Hence introducing more systematic method of finding components set considering overall structural characteristics, manufacturability and assembleability will have a significant impact on industry. Assembly synthesis [6] refers to such a systematic method where entire product geometry is decomposed to components and joints. Since joints are often structurally inferior to components, it is important the decomposition and joint allocation are done in an optimal fashion, such that the reduction in structural performances (eg., stiffness) is maximized while achieving economical manufacture and assembly. As an extension of our previous work on decompositionbased assembly synthesis for structural stiffness [1], the present method optimally decomposes the 3D finite element model of a vehicle body-in-white (BIW) into a set of components considering the stiffness of the assembled structure under given loading conditions, as well as the manufacturability and assembleability of components. The stiffness of the assembled structure is evaluated by FEM analyses, where joints are 1 Copyright © 2003 by ASME modeled as linear torsional springs. Manufacturability of a component is evaluated as a estimated manufacturing cost based on the size and geometric complexity of components. Assuming assembly efforts are proportional to the total number of weld spots, assembleability is simply accounted for as the total rate of torsional springs. In order to allow close examination of the trade-off among stiffness, manufacturability, and assembleability, the optimization problem is solved by a multi-objective genetic algorithm, which can efficiently generate a well-spread Pareto front over multiple objectives. A graph-based crossover scheme is adopted for the improved convergence of the algorithm. RELATED WORK Design for Assembly/Manufacturing Design for assembly (DFA) and design for manufacturing (DFM) refers to design methodologies to improve product and process during the design phase of a product, thereby ensuring the ease of assembly and manufacturing. Boothroyd and Dewhurst [7] are widely regarded as major contributors in the establishment of DFA/DFM theories. In their work [8], assembly costs are first reduced by the reduction of part count, followed by the local design changes of the remaining parts to enhance their assembleability and manufacturability. One of the main functions of DFA/DFM is manufacturability analysis of the product design, eg., by evaluating the capability of production within the specified requirements such as low production costs and short production time. In general, to manufacturability analysis requires a product to be decomposed into features containing a manufacturing meaning, such as, surfaces, dimensions, tolerances and their correlations [9]. While existing DFA/DFM methods share the idea of simultaneous engineering with the present approach, they analyze or improve existing designs from the viewpoint of assembly and manufacturing by modifying geometry of given (i.e., already decomposed) components. On the other hand, the decomposed-based assembly synthesis method presented in this paper starts with no prescribed components and generates optimized components set considering assembleability, manufacturability and structural characteristic of the assembled structure. Automotive Body Structure Modeling In automotive body design, high stiffness is one of the most important design factors, since it is directly related the improved ride and NVH (Noise, Vibration, and Harshness) qualities and crashworthiness [10]. Therefore evaluating the structural characteristics of a vehicle, including stiffness, became a crucial factor in designing a vehicle. Before mathematical modeling techniques were not available, structural analysis was usually carried out only for the stresses in specific hardware items, such as door hinges, drive train and suspension components. Overall structural behavior could not be predicted until a vehicle prototype was built and tested. Therefore, any changes recommended from the test results were bound to be costly to implement [11]. Prior to the use of Finite Element Methods(FEM) in the automotive body analysis in the middle of 1960s, preliminary structural analysis was performed by Simple Structural Surface method (SSS method) [12,13], where the actual vehicle geometry was replaced with an equivalent boxlike structure composed of shear panels and reinforcing beams. With SSS methods, designers can identify the type of loading condition that is applied to each of the main structural members of a vehicle and also the nominal magnitudes of the loads to be determined based on the static conditions with load path in the structure. However, this method can be used only to the simplified conceptual design and it can not be used to solve for loads on redundant structures with more than one load path [13]. The availability of high-powered computers, user-oriented FEM codes and economical solution methods enabled full-scale finite element vehicle models in the early 70’s. To predict the stiffness of a body structure with finite element model more accurately, Chang [14] modeled joints as torsional springs, and demonstrated that the model can accurately predict the global deformation of automotive body substructures. Garro and Vullo [15] analyzed the dynamic behavior of typical body joints under two typical actual loading conditions. They addressed that the plates along spot welds tend to detach from each other when joint deformations occur. Lee and Nikolaidis [16] proposed a 2-D joint model to consider joint flexibility, the offset of rotation centers and coupling effects between the movements of joint branches. Recently, correlation between torsional spring properties of joints and the length of structural member was studied to assess the accuracy of joint model [17]. Long [18] studied the method of correlating the performance targets for a design of individual joint in the automotive to design variables that specify the geometry of the joint design. Kim, et al. [19] employed an 8-DOF beam theory for modeling joints to consider the warping and distortion in vibration analysis. These works, however, focus on the accurate prediction of the structural behavior of a given (i.e., already “decomposed”) assembly and individual joint design and do not concern the selection of optimal joint locations and properties, which is addressed in the present method. Multi-objective Optimization Algorithm Engineering problems generally involves multiple objectives. Among the techniques to solve multi-objective optimization problems, evolutionary algorithms that simulate natural evolution process have shown to be effective in many engineering problems [20]. The major advantages of evolutionary algorithms in solving multiobjective optimization problems are 1) they can obtain Pareto optimal solutions in a single run, and 2) they do not require derivatives of objective functions. Many evolutionary multiobjective optimization algorithms (MOGA [21], NSGA [22], NSGA-II [23], and NPGA [24]) were developed based on the two ideas suggested by Goldberg [25]: Pareto dominance and niching. Pareto dominance is used to exploit the search space in the direction of the Pareto front. Niching technique explores the search space along the front to keep diversity. Another important operator that has been shown 2 Copyright © 2003 by ASME to improve the performance of multiobjective algorithm is elitism, which maintains the knowledge of the previous generations by conserving the individuals with best fitness in the population or in an auxiliary population (SPEA [26] and PAES [27]). Considering a proven efficiency and simplicity of NSGAII, the present work utilizes an implementation based on NSGA-II with Pareto ranking selection. DECOMPOSITION-BASED ASSEMBLY SYNTHESIS FOR STRUCTURAL STIFFNESS Overview The decomposition-based assembly synthesis method simultaneously identifies the optimal components set and joint attributes considering the stiffness of the assembled structure. It consists of the following two major steps: 1. A 3D finite element model is transformed to a structural topology graph representing the liaisons between basic members, the smallest decomposable components of the given structure, specified by the designer. 2. The product topology graph is automatically decomposed, through an optimization process, to a set of subgraphs representing components connected together by edges representing joints. Detailed procedure covered throughout this section uses a simple structural model composed of a plate with reinforcing beam frame shown in Figure 1. This type of structure is widely used in automotive and airspace industries. Step 1: Construction of structural topology graph An entire structure is divided into substructures each of which can be manufactured by a single process (Figure 2 (b) and (c)). This prevents the synthesis of the components that cannot be manufactured with a single process. Then, basic members are defined in each substructure (Figure 2 (d) and (e)) by the designer. In this example, 4 basic members (B0~B3) are defined in the beam substructure and 6 basic members (P0~P5) are defined in the plate substructure. Since components are represented as a group of basic members, the definition of basic member determines the diversity and resolution of the resulting components. Figure 1. (a) simple structure with a plate reinforced by a beam, and (b) decomposition with 2 beam and 3 plate components. (a) (b) Figure 2. (a) Overall structure, (b) beam substructure and (c) plate substructure separated from (a), (d) 4 basic members (B0~B3) defined in (b), and (e) 6 basic members (P0~P5) defined in (c). Figure 3. Constructing structural topology graph for eachsub structure. (a) basic members of beam substructure, (b) structural topology graph GB of (a), (c) basic members of plate substructure, and (d) topology graph GP of (c). In (b) and (d), JD* represents the joint design at each potential joint position defined for each edge. Then, structural topology graph G = (V, E) is constructed such that: 1. A basic member mi is represented as a node ni in set V. 2. The connections (liaisons) between two basic members mi and mj are represented as edge e = {ni, nj} in set E. As illustrated in Figure 3, structural topology graph GB (Figure 3 (b)) of the beam substructure with 4 nodes (nB0~nB3) and 4 edges (eB0~eB3) is constructed based on the basic members of Figure 3 (a). Similarly, structural topology graph GP (Figure 3 (d)) of the plate substructure with 6 nodes (nP0~nP5) and 7 (eP0~eP6) edge is constructed from the basic members in Figure 3 (c). Joints can occur at each connection between basic members. Hence, joint designs (JD), attributes of joints, are assigned to every edge in GB and GP (tables in Figure 3 (b) and (d)). In addition, the entire structural topology graph GE is defined to represent the joints between substructures. In Figure 4, joint designs between the beam and plate components (Figure 4 (c)) are assigned to 10 edges between the beam and plate basic members (eBP0~eBP9) shown as thick edges in Figure 4 (b). (d) (a) (b) B0