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Article Excerpt
1. Introduction
Dimensional variation control is one of the most challenging problems in mechanical assemblies, in particular, in multi-station assembly systems, such as in automotive and aerospace industries. A typical automotive body assembly system involves several hundreds of parts, about 50 stations, thousands of locators and hundreds of measuring points. The models for assemblies in automotive and aerospace industries may have thousands of design variables or Key Control Characteristics (KCCs), and hundreds of design responses (Key Process Characteristics (KPCs)). It incorporates both product and process factors. The latest research advances have provided models for controlling variation in complex assembly systems (Chase et al., 1990; Jin and Shi, 1999; Ding et al., 2002; Ceglarek et al., 2004; Ding et al., 2005; Shi, 2006; Huang, Lin, Bezdecny, Kong and Ceglarek, 2007; Huang, Lin, Kong and Ceglarek, 2007). The so-called Stream of Variation (SOVA) model for an assembly system can be mathematically expressed as a Multi-Input-Multi-Output (MIMO) system (Shi, 2006; Huang, Lin, Bezdecny, Kong and Ceglarek, 2007; Huang, Lin, Kong and Ceglarek, 2007). However, neither well established nor recently emerging techniques are capable of effectively dealing with the tolerance synthesis problems of such complex systems.
The tolerance design synthesis for an assembly was formulated as a probabilistic optimization problem by Lee and Woo (1989). The objective is designed to optimally meet functional and economic requirements through properly assigning tolerances to component dimensions. The functional requirements are defined as the process capability to make quality products or yield which defines an implicit constraint function. In multivariate cases the yield assessment is difficult and relies exclusively on Monte Carlo simulations. When embedded in internal iteration loops of optimization algorithms, it can be extremely computationally intensive. Simplifications have been introduced in the literature to convert the probabilistic optimizations into deterministic problems. Despite the simplicity, the major concerns in the simplification of multivariate problems are threefold: (i) difficulty achieving a specified system process capability through individual tolerance assignment; (ii) creation of conservative solutions (Shiu et al., 2003); and (iii) lack of generality (e.g., for non-normal KPCs and asymmetric or irregular tolerance regions).
Targeting these challenges, a new technique is proposed in this paper. The uniqueness of the approach is due to its ability in: (i) the formulation of the synthesis problem in a general framework in terms of process capability; (ii) the construction of a yield surrogate model using candidate subspace searching and a novel sampling technique (computer experiment); and (iii) surrogate-model-based optimization.
The outline of this paper is as follows. Section 2 presents a literature review, followed by a briefing on three-dimensional assembly models in Section 3. Section 4 formulates the tolerance synthesis. The proposed yield Surrogate Model (SM) technique is presented in Section 5. A validation case and comparative studies are presented in Section 6. Conclusions are drawn in Section 7.
2. Related works
Tolerance synthesis represents the computer-aided design technique of assigning tolerances to individual components to minimize the manufacturing cost and ensure that all the function requirements of an assembly are met. The function requirements are represented as conformity to specifications of key dimensions of an assembly. Various cost functions, such as reciprocal, exponential, etc., are available (Wu et al, 1988). These functions are designed to characterize the trend that the cost is inversely affected by tolerances.
2.1. Tolerance synthesis for manufacturing and design
There are two types of synthesis problems: synthesis for manufacturing (tolerance transfer) in process planning and synthesis for assembly in design.
In order to reduce the manufacturing cost, tolerance allocation based on sequences, set-ups and multiple process alternatives have been extensively investigated by various authors. Tolerance charting, developed in the 1950s and popularized in the 1960s as a simple manual tool, receives considerable interest from industrial practitioners. It involves converting design tolerances into tolerances for the manufacturing processes and is a part of process design. Extensive efforts have been devoted to automating the manual tolerance charting and create a computer-aided charting procedure. A comprehensive review on tolerance charting can be found in Ngoi and Kuan (1995). Considerable effort has been devoted to relating tolerance allocation with process planning (Zhang and Wang, 1993a, 1993b; Roy and Feng, 1997). The purpose of process planning in terms of tolerance allocation is to minimize cost through the selection of set-ups and operation sequences. Related process planning topics, such as set-ups in NC machining, fixture planning and sequence of operations were extensively investigated by Zhang and Wang (1993a). The simultaneous synthesis of design and process tolerances has also been investigated in the last decade. Zhang and Wang (1993b) developed a general mathematical model of optimal tolerancing supporting concurrent engineering to determine optimal machining tolerances in product design. Comprehensive reviews on synthesis for manufacturing can be found in Ngoi and Ong (1998) and Hong and Chang (2002).
One of the primary concerns in tolerance synthesis in design is the assembly model which relates component tolerances to assembly key dimension tolerances. An assembly model characterizes the dimensional variation flow in an assembly system. Several variation propagation (stack-up) models have been developed and are summarized in Hong and Chang (2002).
Comprehensive research has been conducted in the last two decades on assembly modeling for tolerancing and variation analysis (Chase et al., 1990; Jin and Shi. 1999; Ding et al., 2002; Ceglarek et al., 2004; Ding et al, 2005; Shi, 2006; Huang, Lin, Bezdecny, Kong and Ceglarek, 2007; Huang, Lin, Kong and Ceglarek, 2007). Efforts have also been made in the last decade to characterize variation propagation in multi-station assembly processes. Jin and Shi (1999) and Ding et al. (2000) initiated a State Space Model (SSM) for variation modeling, wherein a spatial indexed model and observation equation were established. Ding et al (2002), Ceglarek et al. (2004), Ding et al (2005), Chen et al. (2006), and Shi (2006) developed a two dimensional SSM for automotive body assemblies and applied it to process-oriented tolerancing problems. More recently, a SSM-based three-dimensional assembly model was developed, which integrates both part errors and fixture errors and thus enables integration of product and process factors in a tolerance synthesis (Huang, Lin, Bezdecny, Kong and Ceglarek, 2007; Huang, Lin, Kong and Ceglarek, 2007; Loose et al., 2007). Efforts have also been made to model compliant assemblies (Liu et al., 1995; Chang and Gossard, 1997; Rong et al., 2000; Shiu et al., 2003; Camelio et al., 2003). Elastic deformation and locked-in stresses created by closing the gaps between parts were analyzed using mechanics principles and finite element analysis. Shiu et al. (2003) introduced a beam compliant assembly model for tolerance allocation. For complex assemblies the quality characteristics are usually composed of multiple responses, e.g., an automotive body uses hundreds of measurement points to ensure dimensional integrity. Therefore, these methods, when applied to industrial problems, will result in high-dimensional MIMO models.
The complexity and high dimension of these models pose tremendous challenges in optimization modeling and algorithms in tolerance synthesis in both manufacturing and design.
2.2. Optimization models and algorithms
Tolerance synthesis in manufacturing and design is primarily an optimization problem which involves the following strategies.
1. Optimization formulation: deterministic models or probabilistic optimization models.
2. Optimization algorithms.
One of the key issues in problem formulation is how to define a constraint model. The primary goals in a tolerance synthesis are to achieve: (i) a minimum cost; (ii) a specified capability of producing quality product, which is represented by process capability (e.g., yield (Lee and Woo, 1989)) for given specifications. The constraint function should represent this capability requirement. The yield was introduced in tolerance synthesis (Lee and Woo, 1989):
yield = Prob([intersection]{[y.sub.i][member of][-[T.sub.si], [T.sub.si]]}), i = 1, 2, ..., n.
where [y.sub.i] and [T.sub.si] are ith quality and its specification limit, respectively.
In more general settings with an irregular tolerance region (Spence and Soin, 1997; Kotz and Johnson, 2002):
yield = Prob([intersection]{[y.sub.i][member of] [R.sub.n]]})
For a univariate normal KPC, yield is easy to assess and can be directly translated into a constraint on the tolerance or variance. In multivariate cases, for given KPC distributions and tolerance region [R.sub.n] the yield can be exclusively assessed by numerical sampling methods, e.g., Monte Carlo. Although the algorithm is straightforward the computation can be extremely intensive, which motivated the simplifications and led to various deterministic models, for example the reliability index model (Lee and Woo, 1989) and multivariate capability index models (Kotz and Johnson, 2002).
The deterministic constraint model was formulated by Shi et al. (2003) and Ding et al. (2005). Using a linear model approximation, the root sum of squares (RSS) and worst case represent two simplified deterministic and widely used constraints to a design response:
RSS: [n.summation over (i = 1)] [[absolute value of ([[partial derivative]f]/[[partial derivative][x.sub.i]])].sup.2][t.sub.id.sup.2][less than or equal to][T.sub.f.sup.2]; worst case: [n.summation over (i = 1)][([[partial derivative]f]/[[partial derivative][x.sub.i]])][t.sub.id][less than or equal to][T.sub.f], (1)
where f, [x.sub.i], [t.sub.id] and [T.sub.f] are the assembly function, ith component dimension, [x.sub.i]'s tolerance and the tolerance of the design response (KPC), respectively. The implication of Equation (1), under normality assumption, is yield = 99.73% for a 3[sigma] tolerance on each individual KPC with the hope of achieving the same total yield. When using RSS in Equation (1) for a multivariate case, it is equivalent to the Bon-ferroni method and may result in a much lower yield than expected. For example, suppose there are 100 KPCs and constraints are set individually on the variance of KPCs as [T.sub.si] [greater than or equal 3[[sigma].sub.i], i = 1, ..., 100, if...
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