QUANTIFYING DESIGN AND MANUFACTURING ROBUSTNESS [PDF]

Proceedings of The 1996 ASME Design Engineering Technical Conferences and Computers in Engineering Conference August 18-22, 1996, Irvine, California

96-DETC/DAC-1600 QUANTIFYING DESIGN AND MANUFACTURING ROBUSTNESS THROUGH STOCHASTIC OPTIMIZATION TECHNIQUES David Kazmer Assistant Professor University of Mass. Amherst

Philip Barkan Professor Emeritus Stanford University

ABSTRACT Critical design decisions are often made during the detailed design stage assuming known material and process behavior. However, in net shape manufacturing processes such as stamping, injection molding, and metals casting, the final part properties depend upon the specific tool geometry, material properties, and process dynamics encountered during production. As such, the end-use performance can not be accurately known in the detailed design stage. Moreover, slight random variations during manufacture can inadvertently result in inferior or unacceptable product performance and reduced production yields. These characteristics make it difficult for the designer to select the tooling, material, and processing details which will deliver the desired functional properties, let alone achieve a robust design which is tolerant to process variation. This paper describes a methodology for assessing the design/manufacturing robustness of candidate designs at the detailed design stage. In the design evaluation, the fundamental sources of variation are explicitly modeled and the effects conveyed through the manufacturing process to predict the distribution of end-use part properties. This is accomplished by utilizing optimization of manufacturing process variables within Monte Carlo simulation of stochastic process variation, which effectively parallels the industry practice of tuning and optimizing the process once the tool reaches the production floor. The resulting estimates can be used to evaluate the robustness of the candidate design relative to the product requirements and provide guidance for design and process modifications before tool steel is cut, as demonstrated by the application of the methodology for dimensional control of injection molded parts.

Kosuke Ishii Associate Professor Stanford University

use of the design that ultimately creates wealth for society. While design synthesis utilizes deterministic data, the environment surrounding the manufacture and end-use of the design is largely uncontrolled and stochastic. As such, the design robustness largely determines the product’s efficiency, reliability, and perceived quality (Ford, 1995). Figure 1 illustrates a segment of a typical product development process – needs assessment, conceptual design, and many other tasks have been omitted for simplicity. In this process, the design specification acts as a contract between the customers and the product development team. During the detail design stage, every effort is spent to ensure that the physical manifestation of the design will meet the required design specifications. Multiple design iterations are commonly evaluated before a ‘robust’ solution is accepted (Dixon, 1986). Design Specification

Synthesize Design

Design Evaluation

Are Specifications Met?

Production and Testing

INTRODUCTION The synthesis of new concepts is the primary added value activity of design. However, it is the implementation and end-

Stochastic Variation

Figure 1: A Common Development Process

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Unfortunately, evaluation techniques available in the early stages of the product development process do not consider the effect of systematic and stochastic variation during production or end-use of the design. This is an important consideration when net shape manufacturing processes are utilized as the design evaluation occurs before production or testing, so the downstream production data are not known and can not be used to influence the design. As such, the resulting product may differ considerably from the idealized design, failing to meet the required design specifications. One or more external design iterations may then be necessary to bring the product to acceptable quality or performance levels as indicated by the dashed line in Figure 1. As (Dacey, 1990) has indicated, these late design iterations often require costly tooling changes and delay the product launch. This article describes one method for explicitly considering the stochastic variation in production processes within existing detailed design and design evaluation methodologies. The goal is to enable the designer to understand and account for the effects of manufacturing and end-use variation. This knowledge could then be applied in the configuration and detailed stages of design to select and tune final design parameters, thereby delivering robust engineering designs. PROBLEM FORMULATION A net-shaped part is ordinarily one component in an assembled product, designed and manufactured to deliver its share of functionality to the integrated product. As such, the net-shaped part must be designed with a set of functional requirements and specifications, some implicitly understood, others explicitly stated. Designs to be produced using net shape manufacturing processes pose two significant challenges for the designer which this paper will address. First, the final part properties and performance are determined by processing as well as geometric design and material properties. Second, there is significant coupling between design, material properties, and processing. For instance, small changes in the specification of a wall thickness for a molded part may result in large swings in the cavity pressure distribution which, in turn, may inadvertently affect the material shrinkage and part dimensions thereby rendering the product unacceptable. These two phenomena make it very difficult to 1) predict the end-use properties, and 2) understand the effect of manufacturing variation on product performance. Figure 2 demonstrates the unknown inputs to this system which interfere with prediction of end-use properties of parts manufactured via net-shape processes – the mystery function within the black box is both unknown and intrinsically non-linear. These form-fabrication-function relationships are compounded in technical applications with multiple requirements, subject to process dynamics and limitations which are unknown to the part designer. To overcome these difficulties, improved analysis techniques have been developed to better predict part performance for candidate designs. In

theory, more accurate analysis techniques could eliminate the need for costly mold tooling and evaluation iterations. In reality, even the most advanced analyses remain incapable of providing accurate estimates of performance for candidate designs given the effects of uncertain material properties and stochastic process variation. As such, the product development process for net-shape manufacturing processes is forced to utilize iterative evaluations in which steel must be cut with no guarantee that the mold alterations will deliver the desired product performance. n ies ics tio e rt am ria n op a y r D lP sV ss ria es ce ate o roc r P M P d n tic ne as ow efi ch kn D o n t l Il S U

Design Parameters Material Properties

Net Shape Process

End-Use Properties

Process Conditions

Figure 2: Interference with Prediction of End-Use Properties Let us consider an example: what wall thickness should be used to minimize the cost of a die-cast part while ensuring adequate manufacturability and structural performance? The product development team must specify geometric design parameters, material properties, and process conditions. These design decisions influence certain output characteristics such as cooling time, part weight, flow length, and moment of inertia which are of concern to the development team. However, it is the exact state of the net-shape process during a part’s manufacture which the development team can not know apriori (let alone measure in-situ!) which will ultimately determine the actual end-use properties of the manufactured product. As such, several design-build-test iterations may be required to achieve the desired performance. Unexpected variation can result in unsatisfactory part performance, low production yields, and increased product cost. The objective of this design methodology is to enable the creation of robust designs whose manufactured part properties are within desired specifications, even in the presence of uncertain material properties and stochastic process variations. Robustness has been defined in terms similar to the process capability index (Cp) which is used in characterized manufacturing process (Boyles, 1991) for “the nominal the better” situations (Ford, 1995):

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ℜ=

((USL − LSL) − 2 µ − τ ) , where:

(1)

6σ USL ≡ upper functional limit of product requirement LSL ≡ lower functional limit of product requirement τ ≡ target of product requirement µ ≡ mean of product performance σ ≡ standard deviation of product performance

Other types of targets, including one-sided “the smaller/larger the better” specifications have been defined elsewhere, and are not considered in this study due to space considerations. This definition of design robustness indicates the predicted yield of manufactured parts that satisfy the specified product requirement – a robustness equal to one represents product performance at the target level with three standard deviations to the closest specification limit. If a 12σ level of quality is specified (Denton, 1991), the design robustness is required to be 2 or higher. The robustness of a design with multiple requirements may be evaluated as (Ford, 1995): −1 −1 ⎛ 1 1 n ⎞ Φ ⎜ − Π ( 1 − 2 Φ ( −3 ℜ i ) ) ⎟ , where: ⎝ 2 2 i =1 ⎠ 3 ℜ i ≡ Robustnessof i - th performance parameter, eq. (1) ℜ=

(2)

Φ ≡ Normal probability density function Φ −1 ≡ Inverse normal probability density function n ≡ Number of performance parameters

Thus, design robustness is an aggregate performance measure which includes the consequences of product and tolerance design, process capability, and stochastic variation. There are several beneficial properties of this definition for robustness: • directly extensible to multiple objectives; • allows for direct inclusion of different kinds of specifications; • consistent with Taguchi’s concept of tolerance design since it promotes central tendencies with small deviations in part properties, rather than a goal post mentality (DeVor, 1992); and, • consistent with many design axioms to minimize information content since the production yield will tend to decline geometrically as the number of requirements rise (Suh, 1990). RELATION TO PREVIOUS RESEARCH Deterministic optimization techniques have been traditionally applied in the detailed design stage of product development to enhance product performance or reduce unit cost. Examples include shape optimization, wall thickness minimization, and cycle time reduction (Ali, 1994; Burns, 1994; Santoro, 1992). The application of optimization techniques in these instances was possible because well defined relationships existed between the independent design variables and their performance attributes. These deterministic methods, however, do not consider or predict

the impact of stochastic variation in actual material properties, manufacturing processes, or end-use operation. To address these issues, two different approaches have been developed, each eventually leading to the establishment of a novel research area. Knowing that input variation is unavoidable, Taguchi (1993) developed methods of parameter and tolerance design utilizing direct experimental techniques to minimize product variation by maximizing the signal-to-noise ratio. Since the 1970’s, Taguchi has shown that robustness can be enhanced in a wide range of applications through use of his Parameter Design Methodology. These methods have now become commonplace in modern engineering design and manufacturing practice. Wilde (1992) and Sundaresan (1989) have developed other efficient means for maximizing design robustness when computer models exist of the manufacturing process. Stochastic and probabilistic optimization (Charnes and Cooper, 1963) is a separate approach which considers the effect of random variation in the assessment and optimization of a design’s performance. As with all optimization problems, the approach and formulation are critical components in developing a useful model relating input variation to end use properties. In stochastic optimization, variables are described by distribution functions instead of deterministic constants. The goal is to determine an optimal design which satisfies the required specifications with the highest reliability. Eggert and Mayne (1991) and Lewis and Parkinson (1994) have provided overviews of this research area. Recent work has focused on utilizing stochastic optimization techniques to enhance performance and robustness of candidate designs. Eggert and Mayne (1993) developed and investigated a recursive approach for successively estimating probability distribution functions with moment matching methods to compare probabilistic optimization with deterministic optimization. Lewis and Parkinson (1994) have utilized a second order tolerance model with moment matching methods in a nonlinear optimization process to assess the effect of parameter variation in developing robust designs – these methods can be validated using Monte Carlo simulation (Siddall, 1983). There are two fundamental differences between the proposed methodology and previous research. First, previous research requires the design parameter distributions be known apriori to estimate the effect of variation on system robustness. The proposed method takes one step back, examining the underlying sources of variation in the manufacturing process to account for the intrinsically non-linear nature between the geometric design, manufacturing process, and end-use performance. Moreover, the proposed methodology also incorporates an estimate of the manufacturing response to improve the part properties during production when faced with instances of significant variation. In this paper, the core sources of stochastic variation are explicitly modeled and the effects conveyed through the manufacturing process to predict the distribution of end-use

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•

part properties. Once the model has been developed, the robustness of different candidate designs and processing strategies may be evaluated. The contribution of the methodology is the development of a method from a more fundamental basis, relating variability in the critical process and operating parameters to the resulting variability in the enduse properties of the product. METHODOLOGY In detailed design synthesis, a set of design variables, represented by vector x, are selected to produce the desired product performance. For instance, component functionality, geometric form, and dimensional tolerances are selected to deliver the desired product performance – these are all controllable design variables. There are also design parameters, represented by vector p, whose values are fixed by the design specification and are not controllable by the designer. Examples of design parameters may include material properties, component functionality, and tooling characteristics. For a set of nominal values {x, p} there will be natural fluctuations {δx, δp} about the norm. To evaluate the robustness of a design, the impact of the manufacturing dynamics must also be considered. Similar to the design synthesis, the production process can be ‘designed’ to optimize the manufactured product properties. There are process variables, represented by vector y, which are selected to produce the desired process performance. For instance, die/material temperatures, injection/punch velocities, and pressures/forces are routinely chosen to positively influence part properties – these are controllable process variables. Moreover, there are also process parameters, represented by the vector q, whose values are fixed by the process specification and are not controllable by the process engineer. Examples of process parameters include material properties and machine/controller dynamics. For a set of nominal values {y, q} there will be natural fluctuations {δy, δq} about the norm. The described methodology, presented in Figure 3, explicitly considers stochastic variation in both the design and manufacturing processes – the technique utilizes optimization of the manufacturing process conditions within a Monte Carlo simulation to evaluate the robustness of a candidate design for a stochastic manufacturing process. The output of the methodology is a robust design which does not require iterations during tool commissioning. The methodology begins with the product specifications and design synthesis, identical to the design process described in Figure 1. However, the design evaluation has been replaced with a stochastic optimization which explicitly considers the probabilistic nature of the design and manufacturing processes. To enable evaluation of the design and manufacturing robustness, the following items are required: • a set of product specifications, indicated by the vector τ;

a candidate design represented by the design variables, x, and design parameters, p, as well as initial estimates of the manufacturing process described by process variables, y, and process parameters, q; • an estimate of the sources and levels of variation within the design and manufacturing processes (variables and parameters): δx, δp, δy, δq; and, • a set of design to manufacturing relationships, usually implemented as a process simulation, to predict the properties of manufactured parts from design, material properties, and process dynamics. Design Specifications Iterative Design Synthesize Design Monte-Carlo Simulation Sources & Levels of Variation

Generate Random Set of Properties

Design & Manufacture Relations

Perform Simulation

Process Optimization

Improve Process Conditions Optimal? Optimal Manufacturing Performance

Convergence? Evaluation of Design Robustness

Acceptable? High Yield, Robust Design

Production and Testing

Figure 3: Robust Product & Mfg. Design Methodology Once a candidate design is synthesized, a random set of design and process conditions is instantiated as consistent with {δx, δp, δy, δq}. Given this instance of the design, ~ ~ ~ } , a simulation is performed to estimate a set of the y, q { ~x, p, manufactured product’s end-use properties, represented by the vector µ. The expected value of each part property, µi, will be compared to its specification, τi. The yield is then predicted for this specific instance using [1] and [2]. Since the initial process conditions will not likely result in near-optimal product properties, an optimization of the process variables, ~y , is performed to maximize the yield of acceptable ~ q ~} . In industry practice, this parts given the specific { ~x, p, process optimization is performed by a process engineer on the production floor through ‘trial and error.’ In this methodology,

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the process variables may be improved through Simplex and other optimization techniques. Once the process conditions have been optimized to maximize the yield of good parts for a given set of design and process conditions, another set of stochastic conditions is randomly instantiated. The simulations and optimization are then again repeated to estimate the predicted yield for this set of conditions. With many iterations of the Monte Carlo simulation, a distribution of the end-use properties develops which leads to convergence of the design and manufacturing robustness, ℜ . As previously defined, a robustness equal to one represents a candidate design which is tolerant of design and manufacturing variations and should produce the desired product performance at the target quality levels. Low values of robustness indicate that the stochastic variations in design and processing will result in either undesirable and/or inconsistent product performance. These candidate designs are incapable of satisfying the product specifications with acceptable levels of yield and quality. As in a conventional development process, the designer may synthesize a new set of design variables, x, to re-evaluate. It should be stressed that once the evaluation methodology is in place, there is no additional burden on the design engineer. However, the development of this methodology is non-trivial – the details of development will now be presented. Design and Manufacturing Relations The first step in the design methodology is to identify critical properties in the final product. Process relations are then necessary to link the design variables/parameters, {x, p}, and process variables/parameters, {y, q}, to the end-use part properties, µ: µ = f ( x, p, y, q), where:

(3)

µ ≡ set of end use properties x ≡ set of design variables p ≡ set of design parameters y ≡ set of process variables q ≡ set of process parameters

There are many reliable methods for developing functional models, including empirical, analytical, and numerical techniques. Unfortunately, net-shape manufacturing processes are notoriously complex, with highly non-linear interactions between design, material properties, process conditions, and end-use properties. This is a big step – the number of factors and complex interaction between factors make it difficult to predict the resulting part properties. In the absence of available models, one might well profit from a Taguchi-style design of experiments to identify the critical variables and their effects/interactions.

Sources and Levels of Variation The second step in the design methodology is to identify the root sources of variation and understand the mechanisms of variation in production. Sources and levels of stochastic variation must be assessed to evaluate the robustness of the product design and process capability in the presence of unknown material properties, random process variation, and other factors. Some of the real-life sources of variation which could be represented by the stochastic model include: • inconsistencies in material properties, such as batchto-batch variation; • effect of unmodeled or unknown material properties; • systematic errors in melt temperature and other process conditions; • random, time-varying process noise; and • inaccurate steel dimensions. All of these factors will vary stochastically across an application’s production. The evaluation methodology requires probabilistic ranges to be applied to each of the root cause variables. In this method, each of the many design/process variables/parameters, {x, p, y, q}, are assumed to be stochastic and normally distributed with standard deviations, {δx, δp, δy, δq}. The methodology, however, is not restricted to any unique probability function and may easily be extended to consider arbitrary sources and distributions of variation. Monte Carlo Simulation A Monte Carlo technique was implemented as the physics of net-shape manufacturing processes are complex and intrinsically non-linear. Compared to moment matching methods, Monte Carlo methods are easy to implement, highly accurate, and enable consideration of arbitrary, complex, and mixed probability distributions. The one predominant disadvantage, of course, is computation time with thousands of function calls being required to estimate robustness. If the function call is a complex numerical simulation, evaluation time can exceed hours or even days. During each iteration of the Monte Carlo simulation, instances of random variables are generated for the design and ~~ ~} . y, q For manufacturing variables and parameters, { ~x,p, instance, a randomized set of stochastic design variables, ~ x, may be generated as: ~ x = N ( x , δx) = Φ −1 ( random(1)) ⋅ δx + x , where: ~ x ≡ set of design variables with Gaussian distribution

(4)

Φ −1 ≡ Inverse normal probability density function δx ≡ set of design variable standard deviations x ≡ set of design variable expected mean

After the specific instances of the design and manufacturing conditions have been assessed, optimization of the manufacturing process is performed. This mirrors the actual

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industry practice of tuning the manufacturing process to obtain improved part properties. The result of one iteration, as shown in Figure 3, is an estimate of the production yield for one given set of parameter instantiation. With many additional iterations of stochastic design instances, the overall robustness of the candidate design and manufacturing process will converge. Optimization With the process relations defined, the methodology progresses by optimizing the manufacturing variables, ~ y , to maximize the global production yield. If two design goals are conflicting, τi and τj, then a set of process conditions will be selected that makes a compromise between the two to maximize the overall utility: ⎧ −1 ⎛1 1 n ℜ = maximize ⎨ Φ −1 ⎜ − Π ( 1 − 2 Φ ( −3 ℜ i ~ ⎝ 2 2 i =1 ⎩ 3 y where: ℜi =

( (UFL

i

− LFL i ) − 2 µ i − τ i

~,~ ~) x ,p y,q µ = f (~

⎫

) ) ⎞⎟⎠ ⎬ ,

(5)

⎭

)

6σ i

There are two types of constraints to be applied in the optimization: design constraints and process constraints. Design constraints are imposed by the designer on the allowable range of adjustable design variables, x. For instance, constraints are typically used to guarantee that a length must be maintained within tolerances or a wall thickness must be less than 0.5 cm. Process constraints on y stem from the physical limitations of the manufacturing processes – that the injection pressure must fall between 0 and 150 MPa, for instance. Output Robustness Through the described methodology, the design/manufacturing robustness is evaluated. A robustness of 1.0 corresponds to the predicted molded part property being centered between the upper and lower specification limits, with a predicted standard deviation of one sixth of the tolerance band – this corresponds to a production yield of 99.3%. For higher values of robustness, the production yield approaches 100%, indicating that the manufacturing process can always be adjusted to compensate for random sources of variation and produce parts within the required specifications. If a candidate design is not feasible or a manufacturing process is overly inflexible, then the robustness will be considerably less than 1. In the manufacture of complex net-shape parts, such as injection molding of automotive instrument panels, production yields of ~95% are often considered acceptable (Choineire, 1994). By utilizing process relationships with wide probabilistic spreads in the evaluation, a predicted robustness near 1.0 indicates that the process flexibility exists to meet the required product specifications and that re-tooling or additional design iterations should not be necessary in production.

If the predicted robustness is significantly lower than 1, rework of the design or consideration of a different manufacturing process may be necessary to increase the robustness of the product – the results of the evaluation will indicate which constraint or objective is causing the loss in the product robustness, suggesting a starting point for the redesign. This may involve changing the gating scheme, varying the thickness, increasing allowable tolerances, or other numerous actions. When corrective actions have been completed, the relative success of the new design may be evaluated. As with all optimization techniques, the designer’s experience plays a crucial role in the evaluation and acceptance of a candidate design. If the ‘optimal’ design is not acceptable, the designer must re-formulate the optimization problem, adjust the relationships between design variables and part performance, and guide the design to a more satisfactory design space. APPLICATION TO DIMENSIONAL DESIGN FOR THE INJECTION MOLDING PROCESS Injection molding of thermoplastics has emerged as the premier vehicle for delivering high quality, value-added commercial products. Continued global competitiveness has increased standards for product capability and quality while requiring reduced product development time and unit cost. It is becoming apparent, however, that the injection molding process is neither flexible nor robust enough to ensure the production of acceptable molded part’s on time and under budget. In fact, (Lassor, 1995) has testified that “we are starting to see the migration of customers to other manufacturing processes for time-critical applications.” A methodology is next described which evaluates the robustness of candidate designs and molding practices, with respect to dimensional design and tolerancing. This application will consider variation in tooling, material properties, and processing to estimate the production yield for three sets of design and manufacturing methods: • a center-gated part with common molding practice; • a center gated part with industry best practices; and, • a three-gated part with an advanced molding process. With this quality information, the development team can determine the correct implementation strategy. While this application was developed to provide a concrete example of the methodology, it should be clear the methodology described herein is readily extensible to other types of design specifications and manufacturing processes. Overview of Application Mold design involves the conversion of the part’s geometric form to an image in steel from which the part is molded. Mold design includes completion of the detailed part design as well as tool design issues which include, in part, layout of ejector pins, specification of draft to allow ejection of the molded part, design and sequencing of cores and slides, and structural tool design to

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withstand large, cyclic forces. These activities are not the focus of this design methodology, since they are well understood and do not commonly induce lengthy delays or excessive costs during product development. Some aspects of mold design which are crucial in product development include: • selection of material and grade of thermoplastic resin; • layout and sizing of feed system; • selection and adjustment of wall thickness; and, • compensation for part shrinkage and warpage. These mold design details are crucial to product development since they largely determine the quality of molded parts. A mold design can be considered robust if defects do not arise during production or, if defects do arise, the mold can be easily modified to obtain acceptable quality levels.

L1 L2 L3

15.00 ± 0.03 25.00 ± 0.05 15.00 ± 0.03

Problem Formulation The molded part dimensions are the primary measures of performance, µ . To deliver the desired product performance, the development team can adjust the tool dimensions represented by x, the design variables. The material shrinkage behavior is a design parameter represented by the vector p, which is not a controlled variable. The only primary control variable during manufacturing is the injection pressure, yinj, which determines the cavity pressures, y, which affect the molded part shrinkage. This nomenclature is summarized in Table 2. The following sub-sections describe the values and relationships of these variables. Table 2: Nomenclature for Application Variable

Description

ℜ µ

Measure of production yield, [1] Vector of molded part dimensions Vector of tool steel dimensions Stochastic relationship between { y,q} and µ

x p y yinj

L1

Design:Manufacturing Relations The molded part dimensions can be calculated directly from the tool dimensions and the material shrinkage: µ~i = ~ xi ⋅ (1 − ~ pi )

L2

L3

Figure 4: Typical molded part and specified dimensions In this particular example, the robustness of different gating schemes, dimensional designs, and manufacturing methods will be evaluated via the stochastic optimization methods previously described. Figure 4 shows a typical molded housing with four bosses used to attach a cover to the base. In this application, three critical dimensions (as shown in the right hand figure) have been specified to provide a proper fit with a mating part. The required tolerances are listed below in Table 1 – note that these tolerances are not stringent and represent a typical industry standard of ~0.2% (Beris, 1991). Table 1: Specified dimensions for molded part Dimension

Stochastic vector of cavity pressures Controlled value of injection pressure

Required (cm)

(6)

The use of the twiddle indicates that the molded part’s dimensions are dependent upon the stochastic tool dimensions and material properties which vary stochastically during production. In particular, the material shrinkage can be modeled as: ~ pi = ~ s0 + ~ s1 ⋅ ~ yi

(7)

This relationship implies that there is a base level of shrinkage, s0, which is adjusted by changes in cavity pressure. The critical aspect of this approach is that the design parameter, p, is dependent upon 1) stochastic parameters representative of the material properties, {s0, s1}, and 2) the controllable cavity pressures represented by the stochastic manufacturing variable, ~ y . This linear relationship was used since 1) industry practitioners utilize a very similar ‘rule of thumb’ and 2) this behavior was observed in physical molding trials (Kazmer, 1996). As stochastic parameters, the shrinkage behavior must

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be estimated through the Monte Carlo simulation based upon the estimates of variation described in the next section.

differ both in y-axis intercept and slope, providing significant behavioral differences in the dimensions of molded parts. 1 0.9

Sources and Levels of Variation Variation in tool steel dimensions may occur for many reasons: imprecision during tool manufacture, wear during production, uneven cooling across mold, etc. Considering only errors in tooling and uneven temperature variations during molding, the stochastic steel dimensions can be estimated as: (8)

This implies that each of the tool steel dimensions will be normally distributed with mean, x, and standard deviation of 0.05%. This level of variance represents standard tooling tolerances (Trucks, 1987). More complicated is the prediction of the material shrinkage, which is both material and process dependent. Material suppliers have historically quoted shrinkage as a range of values, for instance 0.5% to 0.7%. Also, molders have long used a rule of thumb that shrinkage can be decreased 0.05% for every 7 MPa (1,000 psi) increase in cavity pressure. These characteristics have been experimentally validated (Kazmer, 1996). The coefficients for [7] can be assigned deterministically as: s0 = 0.8% − 0.05% s1 = 7 MPa

Shrinkage (% cm/cm)

~ x = N ( x,0.05% ) .

0.8

(9)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

10

20

30

40

50

60

Cavity Pressure (MPa)

Figure 5: Stochastic shrinkage behavior of polycarbonate Finally, the relationship between the controlled injection pressure and resulting cavity pressures must be assessed. To accomplish this, numerical flow simulation was utilized as described by (Wang, 1986). This resulted in deterministic relationships between the injection pressure and cavity pressure distributions, as shown in figure 6. The pressures are highest in the center of the part, near the gate, and decrease towards the end of fill. The pressures are higher around the dimension L3 due to stagnation of flow and an over packing of this side of the cavity.

This relationship, however, can not be used with confidence to tune a mold since 1) the true dependency of shrinkage is fantastically complex, dependent upon flow direction, geometric constraints, micro-mechanics, and thermoviscoelastic properties, and 2) stochastic variations in material properties and process conditions will lead to significant deviation in material shrinkage (Shulz, 1992). As such, a stochastic representation of shrinkage may be used to emulate this uncontrollable behavior: ~ s0 = N ( µ = 0.8%, σ = 01% . ) ~ s1 = N ( µ = 0.05%, σ = 0.01% ) 7 MPa

(10)

This level of variation was chosen to mimic the magnitude of variation found in experimental moldings (Kazmer, 1996). By taking a number of normally distributed samples, the stochastic variations in the molding process can be simulated. Figure 5 plots thirty such relationships between cavity pressure and linear shrinkage. The dark wide line represents the deterministic relationship while the region enclosed in the dashed box indicates the shrinkage range commonly assumed by designers and molders. The stochastic shrinkage models

Figure 6: Deterministic Cavity Pressure Distribution Utilizing coefficients of variation as investigated by (Speight, 1996), the cavity pressures may be represented by:

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⎧~ ⎧ N (15,3) ⎫ y1 ⎫ ⎪~ ⎪ ⎪ ⎪ ~ y = ⎨ y 2 ⎬ = yinj − ⎨ N (10,3) ⎬ ⎪~ ⎪ ⎪ N ( 25,3) ⎪ ⎩ y3 ⎭ ⎩ ⎭

(11)

Optimization For each candidate design, the stochastic natures of the design and manufacturing process were estimated by instantiating many sets of steel dimensions, material properties, and cavity pressure distributions. For each instance, the injection pressure was adjusted to optimize the level of cavity pressures and resulting molded part dimensions. This would be equivalent to a process engineer changing the injection time and pack pressure levels to affect the molded part properties. Roughly 1000 iterations were performed to evaluate the design robustness with a high degree of certainty. The optimization problem can be stated as: ⎧ −1 ⎛1 1 n ⎞⎫ ℜ = maximize ⎨ Φ −1 ⎜ − Π ( 1 − 2 Φ ( −3 ℜ i ) ) ⎟ ⎬ , ⎝ ⎠⎭ = 1 i 3 2 2 ⎩ y inj

(12)

where: ℜi =

( (UFL

i

~ −τ − LFLi ) − 2 µ i i 6σ i

),

One Gate, Standard Practice A typical gating scheme for the housing would involve direct sprue-gating into the center of the part as shown in Figure 6. In this example, neither the cavity pressure nor the part dimensions were optimized. Rather, the cavity inlet pressure was maintained at a constant 40 MPa, the pressure for which the deterministic shrinkage would correspond exactly to that of the assumed linear shrinkage, 0.5% at 30 Mpa cavity pressure. Since this is typical of molding practice, this candidate design provides an industry benchmark against which to compare more advanced designs and process control strategies. Table 3 lists the molded part dimensions resulting from the stochastic process relationships previously described. For this design, L2 is the most well-behaved dimension, with a centered mean and the lowest deviation (0.015 cm with a ± 0.05 cm tolerance). The other dimensions generally exhibit more shrinkage since they are farther from the gate and experience lower cavity pressures. Note that L1 exhibits a low predicted yield of 76.5% due to undersized molded part dimensions which are approaching the lower specification limit of 14.97 cm. Table 3: Performance of center gate at constant pressure

UFL − LFL = { 0.06 , 0.10 , 0.06} , T

Ave. Dimension Std. Deviation Max. Dimension Min. Dimension Predicted Yield

τ = x = {15, 25,15} , T

dy = {15,10 , 25} T , ~ = N ( x , 0.05% ) ⋅ I ⋅ 1 − ~ µ s0 − ~ s1 ⋅ y inj − N ( dy , 3)

( (

(

~ s0 = N ( 0.8%,0.1% ) , and ~ s1 = N ( 0.05%,0.01% ) 7 MPa .

))) ,

The only constraints on the problem are: yinj < 100 MPa yinj > 0 MPa

,

(13)

meaning that injection pressure must remain within the limits of the machine. It should be noted that this optimization problem is non-linear: the objective function is a product of multiple yields which are, in turn, a non-linear function of the molded part dimension. However, the problem is convex and was easily solved by a quasi-Newton search method with a central differencing technique (Davidon, 1975; Fletcher, 1963). Evaluation of Candidate Designs Three different design candidates are reviewed to demonstrate the application of the methodology. The first case considers standard industry practice for a molded part. The second case considers the increase in design robustness using design and processing best practices, without additional capitol investment. The third case considers the robustness for an alternative, highly flexible manufacturing process technology.

L1

L2

L3

14.98

25.01

15.00

0.01 15.02

0.02 25.05

0.02 15.03

14.95

24.97

14.97

76.5

94.8

87.3

As previously defined, the robustness is the expected yield of parts which meet the required dimensional tolerances. According to our definition, this design has a robustness of 0.22. This corresponds to a process yield of approximately 63% – roughly 37% of all molded parts will be rejected due to poor tool design and stochastic variation in the manufacturing process. This result is consistent with industry yields during the manufacturing startup of complex molded parts with multiple requirements which requires tool modifications to increase the production yield. One Gate, Best Practices The previous design utilized a constant pack pressure across all the runs. A more capable approach, becoming more common in industry, is to qualify the process for a given mold geometry on a specific machine with a specific lot of material – to optimize the process to achieve higher manufacturing yields. While there is stochastic variation between molding machines

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and material lots, the variation within a batch of parts is greatly reduced. Another best practice in industry, moreover, is the utilization of differential shrinkage estimates in cutting tool dimensions. In this way, the lower pressures at dimensions L1 and L3 can be offset by revising tool dimensions to allow higher shrinkages. Knowing the deterministic pressure distribution across the cavity [11], this example utilized differential shrinkage estimates of 0.55% for L1, 0.5% for L2, and 0.60% for L3. For each set of stochastic relationships, the input cavity pressure was re-optimized to maximize the total yield – Table 4 summarizes the results. The robustness of this process is 0.74 with a yield of approximately 93%. This is a considerable improvement in design robustness, especially considering that no additional investment in technology or process capability is required, just a change in the operation of the molding machine.

pressures during the injection molding process (Kazmer, 1996). In this application, this process technology can be used to directly control the cavity pressure at multiple locations in the cavity and adjust for variation, rather than assuming the cavity pressure to be an uncontrolled variable resulting from the mold design, material properties, and machine inputs. An intuitive design might utilize three gates in the center of each of the critical dimensions as shown in Figure 7.

L1

L2

Table 4: Performance of center gate with estimated shrinkage and optimal injection pressures

Ave. Dimension Std. Deviation Max. Dimension Min. Dimension Predicted Yield

L1

L2

L3

15.00

25.00

15.00

0.01 15.02

0.02 25.03

0.01 15.02

14.94

24.91

14.95

97.5

96.7

97.6

Time Adjustable Valves

L3

Figure 7: Multi-Pressure Control of Three Gates

Note that all three dimensions are now centered even though the cavity pressure distribution remains non-uniform. Moreover, the standard deviation of the dimensions has been reduced along with the range of dimensions.

The pressure at each gate can now be optimized for each iteration of the Monte Carlo simulation to adjust for the varying levels of shrinkage. The optimization problem has additional degrees of freedom: ⎧ −1 ⎛1 1 n ⎞⎫ ℜ = maximize ⎨ Φ −1 ⎜ − Π ( 1 − 2 Φ ( −3 ℜ i ) ) ⎟ ⎬ , ⎝ ⎠⎭ 1 i = 3 2 2 y1 , y 2 , y 3 ⎩

(14)

where: ℜi =

( (UFL

i

~ −τ − LFLi ) − 2 µ i i 6σ i

),

.

UFL − LFL = { 0.06 , 0.10 , 0.06} , T

Three Gates, Advanced Molding Process A 93% product yield may or may not be acceptable in high volume production, however. There are three possible ways to possibly improve the yield: • increase allowable dimensional tolerances through product design; • reduce stochastic variability by qualifying materials and machines; or, • add degrees of freedom to compensate for variability. Presumably, the specifications have originated from a design which has considered these issues; the tolerances are fairly loose. Also, stochastic variation can not be completely removed; the model presented in equation [10] is a good representation of stochastic process variation. A new process technology, multi-cavity pressure control, has been developed to provide dynamic control of cavity

τ = x = {15, 25,15} , ~ = N ( x , 0.05% ) ⋅ I ⋅ 1 − ( ~ µ ( s0 − ~s1 ⋅ y ) ) , ~ s = N ( 0.8%,0.1% ) , and T

0

~ s1 = N ( 0.05%,0.01% ) 7 MPa .

Table 5 lists the predicted dimensional properties. The resulting robustness was 1.47, corresponding to a process yield of roughly 99.5% for the same set of stochastic shrinkages assumed in evaluation of the other designs. Note that the maximum and minimum part dimensions are completely within the tolerance limits. In fact, the dimensional deviations are such that the tolerances can be tightened significantly without incurring extra cost, possibly enabling enhanced product quality or new product capabilities. Table 5: Performance of three gates at optimal pressures

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Ave. Dimension Std. Deviation Max. Dimension Min. Dimension Predicted Yield

L1

L2

L3

15.00

25.00

15.00

0.01 15.01

0.01 25.02

0.01 15.01

14.99

24.99

14.99

99.7

99.8

99.7

CONCLUSIONS The described methodology extends the current research regarding robust design by explicitly considering the fundamental sources of variation for evaluation of design/manufacturing robustness. This approach enables the optimization of the simulated manufacturing process to counter these sources of variation, just as process conditions are tweaked during actual production to improve the quality of the manufactured products. Using this a-priori manufacturing optimization with Monte Carlo simulation and the estimates of variation, the design engineer can investigate not only the effect of variation on product performance but also the capability of the manufacturing process to maintain the performance specifications in the face of unexpected variation. In this way, the robustness of the design and manufacturing pair are evaluated simultaneously. This methodology was applied to a typical dimensional control problem for an injection molded application. Results of the analysis indicated that standard industry practice (of utilizing standard molding conditions and constant shrinkage estimates across the part) results in inferior product quality and low production yields. The methodology led to synthesis of two improved design/manufacturing candidates: 1) using differential shrinkage estimates to adjust for complex cavity pressure distributions, and 2) using an advanced dynamic feed system to add degrees of freedom to the manufacturing process to simultaneously deliver multiple product requirements. While this example focused on dimensional control for injection molded parts, the methodology can be directly extended to more complex designs, other production processes, and other types of performance specifications. ACKNOWLEDGMENTS This work would not have been possible without the involvement of the Stanford Integrated Manufacturing Association, GE Plastics, Kona Corporation, and Dynisco Instruments. The authors gratefully acknowledge their support. Also, parts of this research were funded through the Department of Energy Innovative Industrial Concepts Program.

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Parts, Doctoral Dissertation submitted to the Stanford University ME Design Division. Lassor, R., 1995, Personal communication with Dick Lassor General Manager, GE Plastics Polymer Processing Development Center, Pittsfield, MA, September 29. Lewis, L., Parkinson, A., 1994, “Robust Optimal Design Using a Second Order Tolerance Model,” Research in Engineering Design, v. 6, pp. 25. Santoro, E., 1992, “Global Methods in Multi-Objective Optimization and Their Application to a Mechanical Design Problem,” Computers in Industry, v. 18, n 2, pp. 169. Schulz, M., Alig, I., 1992, “Influence of Stochastic Environments of Gaussian Chains on Dynamic Shear and Bulk Properties,” Journal of Chemical Physics, v. 97, n. 4, pp. 2772. Siddall, J. N., 1983, Probabilistic Engineering Design: Principles and Applications, Marcel Dekker, New York. Speight, R. G., Yazbak, E. P., Coates, P., 1996, “In-Line Process Measurements for Integrated Injection Molding,” Proceedings of the 54th Annual Tecnical Meeting of the Society of Plastics Engineers. Suh, N. P., 1990, The Principles of Design, Oxford University Press, New York. Sundaresan, S., Ishii, K., Houser, D., 1989, “A Procedure Using Manufacturing Variance to Design Gears with Minimum Transmission Error,” Proceedings of the 15th Annual ASME Design Automation Conference. Taguchi, G., Tsai, S.C., 1993, Taguchi on Robust Technology Development: Bringing Quality Engineering Upstream, ASME Press, New York. Trucks, H. E., 1987, Designing for Economical Production, 2nd Edition, Society of Manufacturing Engineers, Dearborn, Michigan. Wang, V.W., Hieber, C. A., Wang, K. K., 1986, “Dynamic Simulation and Graphics for the Injection Molding of ThreeDimensional Thin Parts,” Journal of Polymer Engineering, v. 7, n. 1, pp. 21. Wilde, D. J., 1992, “Product Quality in Optimization Models,” Proceedings of the ASME Design Theory and Methodology Conference.

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