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A Rigorous Framework for Optimization of Expensive Functions by Surrogates
, 1998
"... The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approxima ..."
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Cited by 204 (15 self)
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The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approximations to the objective function and managing the use of these approximations as surrogates for optimization. The result is to obtain convergence to a minimizer of an expensive objective function subject to simple constraints. The approach is widely applicable because it does not require, or even explicitly approximate, derivatives of the objective. Numerical results are presented for a 31variable helicopter rotor blade design example and for a standard optimization test example.
Trust Region Augmented Lagrangian Methods for Sequential Response . . .
 Journal of Mechanical Design
, 1997
"... A common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques. The use of approximation strategies is designed to reduce the number of detailed, costly computer simulat ..."
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Cited by 71 (21 self)
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A common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques. The use of approximation strategies is designed to reduce the number of detailed, costly computer simulations required during optimization while maintaining the pertinent features of the design problem. To date the primary focus of most approximate optimization strategies is that application of the method should lead to improved designs. This is a laudable attribute and certainly relevant for practicing designers. However to date few researchers have focused on the development of approximate optimization strategies that are assured of converging to a solution of the original problem. Recent works based on trust region model management strategies have shown promise in managing convergence in unconstrained approximate minimization. In this research we extend these well established notions from the literature on trustregion methods to manage the convergence of the more general approximate optimization problem where equality, inequality and variable bound constraints are present.The primary concern addressed in this study is how to manage the interaction between the optimization and the fidelity of the approximation models to ensure that the process converges to a solution of the original constrained design problem. Using a trustregion model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, one can show that the optimization process converges to a solution of the original problem. In this research an approx1 Graduate Research Assistant.
Filter Pattern Search Algorithms for Mixed Variable Constrained Optimization Problems
 SIAM Journal on Optimization
, 2004
"... A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for gene ..."
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Cited by 55 (6 self)
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A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for general nonlinear constraints. In generalizing existing algorithms, new theoretical convergence results are presented that reduce seamlessly to existing results for more specific classes of problems. While no local continuity or smoothness assumptions are required to apply the algorithm, a hierarchy of theoretical convergence results based on the Clarke calculus is given, in which local smoothness dictate what can be proved about certain limit points generated by the algorithm. To demonstrate the usefulness of the algorithm, the algorithm is applied to the design of a loadbearing thermal insulation system. We believe this is the first algorithm with provable convergence results to directly target this class of problems.
On Managing The Use Of Surrogates In General Nonlinear Optimization And Mdo
 Proceedings of the 7th AIAA/USAF/NASA/ISSMO Multidisciplinary Analysis & Optimization Symposium, AIAA 984798
, 1998
"... This paper is concerned with a trustregion approximation management framework (AMF) for solving the nonlinear programming problem, in general, and multidisciplinary optimization problems, in particular. The intent of the AMF methodology is to facilitate the solution of optimization problems with hi ..."
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Cited by 13 (1 self)
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This paper is concerned with a trustregion approximation management framework (AMF) for solving the nonlinear programming problem, in general, and multidisciplinary optimization problems, in particular. The intent of the AMF methodology is to facilitate the solution of optimization problems with highfidelity models. While such models are designed to approximate the physical phenomena they describe to a high degree of accuracy, their use in a repetitive procedure, for example, iterations of an optimization or a search algorithm, make such use prohibitively expensive. An improvement in design with lowerfidelity, cheaper models, however, does not guarantee a corresponding improvement for the higherfidelity problem. The AMF methodology proposed here is based on a class of multilevel methods for constrained optimization and is designed to manage the use of variablefidelity approximations or models in a systematic way that assures convergence to critical points of the original, highfidel...
VizCraft: A Multidimensional Visualization Tool for Aircraft Configuration Design
 IN PROCEEDINGS OF IEEE VISUALIZATION'99
, 1999
"... We describe a visualization tool to aid aircraft designers during the conceptual design stage. The conceptual design for an aircraft is defined by a vector of 1030 parameters. The goal is to find a vector that minimizes an objective function while meeting a series of constraints. VizCraft integrat ..."
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Cited by 11 (0 self)
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We describe a visualization tool to aid aircraft designers during the conceptual design stage. The conceptual design for an aircraft is defined by a vector of 1030 parameters. The goal is to find a vector that minimizes an objective function while meeting a series of constraints. VizCraft integrates the simulation code that evaluates the design with visualizations for analyzing the design individually or in contrast to other designs. VizCraft allows the designer to easily switch between the view of a design in the form of a parameter set, and a visualization of the corresponding aircraft. The user can easily see which, if any, constraints are violated. VizCraft also allows the user to view a database of designs using parallel coordinates.
Optimization using Surrogates for Engineering Design
, 2002
"... The goal of these lectures is to acquaint the audience with some approaches to a class of nasty optimization problems involving nonconvex nonlinear extendedvalued functions. Such functions arise often in multidisciplinary optimization (MDO). The first three lectures are meant to set the context for ..."
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Cited by 2 (0 self)
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The goal of these lectures is to acquaint the audience with some approaches to a class of nasty optimization problems involving nonconvex nonlinear extendedvalued functions. Such functions arise often in multidisciplinary optimization (MDO). The first three lectures are meant to set the context for applying our algorithms. The context determines the form of the algorithms, and to present this context requires a bit more than just a short list of assumptions. Briefly though, the objective function and constraints depend not only on the optimization variables, but also on some ancillary variables such as the solutions of some coupled systems by standalone solvers for partial differential equations, table lookups, and other nonsmooth simulation codes. This has important algorithmic implications. First, the function and constraint values may be very expensive. Second, the functions may be nondifferentiable and discontinuous. In fact, they are often treated as extended valued since a function call may not return a value even if all the specified constraints are satisfied. The approach we treat in these lectures has been successful for some real problems in engineering design. We hope to convince engineers and mathematicians alike that not only are the algorithms given here useful, but the mathematics involved is interesting and relevant. We hope to convince mathematicians that good applied problems produce good mathematics, and that contrary to what they may have heard, they will suffer no loss of virtue as a direct result of considering them.
Twolevel OPTIMIZATION of Composite Wing Structures Based on PANEL GENETIC OPTIMIZATION
, 2001
"... ..."
The Cost of Numerical Integration in Statistical Decisiontheoretic Methods for Robust Design Optimization
"... Abstract: The Bayes principle from statistical decision theory provides a conceptual framework for quantifying uncertainties that arise in robust design optimization. The difficulty with exploiting this framework is computational, as it leads to objective and constraint functions that must be evalua ..."
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Cited by 1 (1 self)
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Abstract: The Bayes principle from statistical decision theory provides a conceptual framework for quantifying uncertainties that arise in robust design optimization. The difficulty with exploiting this framework is computational, as it leads to objective and constraint functions that must be evaluated by numerical integration. Using a prototypical robust design optimization problem, this study explores the computational cost of multidimensional integration (computing expectation) and its interplay with optimization algorithms. It concludes that straightforward application of standard offtheshelf optimization software to robust design is prohibitively expensive, necessitating adaptive strategies and the use of surrogates. Engineers increasingly rely on computer simulation to develop new products and to understand emerging technologies. In practice, this process is permeated with uncertainty: manufactured products deviate from designed products; actual products must perform under a variety of operating conditions. Most of the computational tools developed for design optimization
SobieszczanskiSobieski and Haftka
"... . Abstract Design for manufacturing is often difficult for mechanical parts since significant manufacturing knowledge is required to adjust part designs for manufacturability. The traditional trial and error approach usually leads to expensive iterations and compromises the quality of the final des ..."
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. Abstract Design for manufacturing is often difficult for mechanical parts since significant manufacturing knowledge is required to adjust part designs for manufacturability. The traditional trial and error approach usually leads to expensive iterations and compromises the quality of the final design. The authors believe the appropriate way to handle product design for manufacturing problems is not to formulate a large design problem that exhaustively incorporates design and manufacturing issues, but to separate the design and manufacturing activities and provide support for collaboration between engineering teams. In this paper, the Collaborative Multidisciplinary Decisionmaking Methodology (CMDM) is used to solve a product design and manufacturing problem. First, the compromise Decision Support Problem is used as a mathematical model of each engineering teams' design decisions and as a medium for information exchange. Second, game theoretic principles are employed to resolve couplings or interactions between the teams' decisions. Third, design capability indices are used to maintain design freedom at the early stages of product realization in order to accommodate unexpected downstream design changes. A plastic robot arm design and manufacturing scenario is presented to demonstrate the application of this methodology and its effectiveness for solving a complex design for manufacturing problem in a streamlined manner, with minimal expensive iterations. Keywords: Collaborative Design, Design for Manufacturing, Game Theory, and Multidisciplinary Decision Making 2 Design for manufacturing Concurrent engineering involves separating product realization activities so that design activities can be executed independently while simultaneously incorporating relevant information from downstream domains such as manufacturing, assembly, or recycling Alternatively, a manufacturing team that understands the purpose of a design and its functional requirements may be more capable of adjusting it to facilitate manufacturing. The authors believe the appropriate way to handle complex product design problems such as DfM is not to formulate large design problems but to support cooperation and collaboration between multidisciplinary engineering teams. Towards this end, the Collaborative Multidisciplinary Decisionmaking Methodology (CMDM) is established (Xiao et al. 2005) to enable collaborative decision making between design and manufacturing teams through 'collaboration by separation'. Separation signifies that the responsibility for DfM is transferred from the design team to the downstream manufacturing team; whereas, collaboration signifies that satisfactory systemslevel solutions are coordinated with minimal information exchange and iteration. Unlike many mathematical multidisciplinary optimization (MDO) approaches (Balling and SobieszczanskiSobieski 1996 1. Exchanging Information. The information required for decisionmaking in an activity must be transferred completely from one team to another, and the recipient teams should be able to understand the team's intentions without requiring additional information flows or causing iterations. The compromise Decision Support Problem (DSP) 2. Accommodating interactions between activities. Some activities in a DfM process may be coupled, such that each design team makes decisions that affect the decisions of other teams. Game theory is used in the CMDM to model different degrees of collaboration and manage interactions between engineering teams, with little or no expensive, systemslevel iterations. 3. Maintaining feasible and satisfactory overall designs. When design activities are separated, design teams must make decisions without full knowledge of their impact on downstream activities. If single point solutions are exchanged, downstream designers are prevented from adjusting designs for feasibility or satisfactory local performance, and iterations often ensue. With setbased approaches, however, ranges or sets of solutions are shared and gradually narrowed during the design process, thereby reducing or eliminating the need for global, systemslevel iterations In a product realization problem, the dependencies between any two activities, such as designmanufacturing, designdesign, or manufacturingmanufacturing, may be interactive or sequential. Game theory is used to resolve the interactive couplings and design capability indices are used to handle the sequential relationships. The authors believe the sequential relationships are more significant in DfM problems, mostly due to the upstream/downstream nature of the designmanufacturing relationship. As shown in <Figure 1 goes about here> Collaborative multidisciplinary decision making methodology The CMDM is implemented in three steps: Step 1 Representing decision making information in a compromise DSP which serves as an information medium to eliminate iterations caused by information exchange and communication; Step 2 Representing cooperation styles among engineering teams with game theoretic protocols to eliminate iterations caused by interdisciplinary interactions; and Step 3 Reformulating the compromise DSPs using design capability indices for finding superior ranged set of solutions that eliminate or reduce costly iterations caused by unexpected downstream requirements and constraints. Step 1, modeling product realization activities using compromise DSPs In order to resolve the first challenge, that of exchanging information, Section 1, a compromise Decision Support Problem, DSP, is used. A compromise DSP is a multiobjective decision modela hybrid formulation based on mathematical programming and goal programming that is used to find the values of design variables that satisfy a set of constraints and achieve a set of conflicting goals as closely as possible For a given product realization activity, a compromise DSP is capable of representing a team's decisionmaking knowledge, as well as the design rationale underlying its decision. A team's decision is represented with a feasible design space, a set of design objectives, and a tradeoff strategy between these design objectives. As shown in The compromise DSP resolves the first challenge of exchanging information, but it does not address the second challenge of enabling the separation of activities. There are three possible relationships between any two compromise DSPs; they may be solved concurrently, sequentially, or as coupled problems. Given the disk brakes in a passenger vehicle as an example, there is no direct information exchange between the brake design and exhaust system design. From a decisionmaking perspective, the two compromise DSPs (brake and exhaust system) do not share any unknown variables. They can be solved concurrently, and the solution remains the same regardless of the teams' cooperation styles. Meanwhile, the brake pad cannot be designed without knowledge of the rotor design team's results, whereas the rotor has to be designed with knowledge of the geometric shape, surface finish, and other details of the brake pad. This situation is reflected as shared variables between the rotor and brake pad design compromise DSPs. Neither compromise 6 DSP can be solved independently, and the result is always affected by the two design teams' cooperation styles, namely, which team solves its compromise DSP first. Game theoretic protocols are used to address the second challenge of separating the coupled activities. In addition, a manufacturing team must design the fixtures, determining the processing parameters based on the final rotor and brake pad designs. The manufacturing compromise DSP includes variables that are determined only by solving the design compromise DSPs. This is a sequential process and the teams' cooperation styles do not affect the solution. However, the downstream manufacturing team may need to modify the design, causing potential iterations. Design capability indices are used to address this third challenge. Since design and manufacturing activities are separated and the responsibility for DfM is transferred from the upstream design team to the downstream manufacturing team, the third challenge becomes more significant in this study. Step 2, representing cooperation styles using game theory The second challenge is resolving couplings between activities. Traditionally, a trial and error approach is used to solve coupled compromise DSPs. Since a team has to make assumptions about another team's decisions to initiate the trial and error process, this traditional approach may not guarantee consensus (convergence) and usually fails to achieve superior results. Game theory facilitates interaction among multiple engineers without integrating a product realization process into a single large optimization problem or causing iterations. There are three game protocols representing different types of interactions between teams (or players in game theory terminology): cooperative, noncooperative, and leader/follower. Rao and colleagues In a leader/follower game, the leader makes a set of rational decisions by predicting the A is a subset of X A which must be determined using information from team B, and x B B must be determined using The feasibility of the solution is ensured by using a BRC to predict the follower's behavior. Generally, a leaderfollower game protocol facilitates collaborative decision making without requiring iteration, hence the coupled activities can be accomplished separately. This solves the second challenge. Step 3, maintaining design freedom using design capability indices The third challenge is to eliminate, or at least reduce, costly iterations between upstream and downstream activities by having the upstream team identify ranges of design variables, rather than single point values, that are as broad as possible without deviating from a desired range of 8 performance, as shown in <Figure 3 goes about here> As presented in In the design variable set, X, if any design variable is discrete, say x j , the location and deviation of the performance measures have to be conservatively estimated using: In many cases, calculating the min/max values of a performance measure requires exhaustive search, but the performance range estimated in this manner will cover all the possible values even though they are not continuous. If a performance variable is discrete, the design capability indices are not applicable. Given that all the performance variables are continuous, design capability indices are embedded into the compromise DSP by formulating the design goals using C dk , adding constraints C dk ≥1, and formulating the deviation function to maximize the overachievements of C dk . Moreover, the constraints are reformulated using Equation Clearly, constraint g k (X) must be differentiable. If any design variable is discrete, the constraints can be calculated using Equation (4). The bounds of design variables are still formulated using constant values. The resulting compromise DSP is shown in <Figure 4 goes about here> In Step 1, the challenge is to provide a method for exchanging information, and it is solved by representing the decision making information in compromise DSPs. In Step 2, the challenge is accommodating interactions between activities, and it is addressed with game theory. In Step 3, the challenge is to maintain feasible and satisfactory overall designs, and it is addressed by reformulating the compromise DSPs using design capability indices. The CMDM provides a normative framework that facilitates collaborative product realization by separating the decision making activities. A robot arm design and manufacturing scenario The authors have developed a distributed product realization environment called the Rapid Tooling <Figure 5 goes about here> A design team, a rapid tooling team, and an injection molding team are assigned to this task. Correspondingly, the product realization process is partitioned into three activities, as shown in In this scenario, DfM includes not only adjusting the geometric shape, but also the entire rapid tooling activity. Designing the mold pairs requires knowledge about rapid tooling; hence, it is very difficult for the design team. Since the robot arm is designed without knowledge of the downstream manufacturing process, the design team will have to modify its design based on feedback from manufacturing experts. For a simple product realization process like this one, it is possible to collect all the manufacturing related information and formulate a large design problem to solve all the design variables, i.e., robot arm geometry shape, mold geometry shape, and some manufacturing parameters, like that in Concurrent Engineering. For complex realworld problems, this implies a design problem containing large numbers of design variables and complex analyses; therefore it is not practical to solve as a single problem. The traditional approach to solving a DfM problem is a trial and error approach, which will cause extensive information exchange and iteration. In this example, all three of the challenges addressed by the CMDM exist. Information exchange is required between these activities (challenge 1). Rapid tooling and injection molding are coupled activities; thus iterations exist between them (challenge 2). The geometric shape of the 12 robot arm may have to be modified in order to fabricate the batch with given time and cost; this forces the upstream design team to redesign the geometry (challenge 3). Engineering teams' compromise DSPs The first step of the CMDM is modeling each activity as a standard compromise DSP as shown in Three design goals are determined based on the customer requirements: (i) the maximum deformation under working load should be as close to 0.5mm as possible, (ii) the maximum von Mises stress under working load should be as close to 6MPa as possible, and (iii) the weight of the robot arm should be as close to 3.5g as possible. The design compromise DSP is shown in Please note at this step, no design capability indices or game protocol is yet involved. Mathematical equations for deformation, stress, and weight can be found in (Xiao 2003). <Figure 6 goes about here> The rapid tooling team designs the injection mold halves, On the other hand, the mold life determines the number of mold halves, N m , that must be built in the rapid tooling activity, which thus affects all of the process parameters in this activity. Therefore, injection molding and rapid tooling are coupled activities. As shown in If we combine all compromise DSPs into one problem with a variable set that includes all of the variables of design, rapid tooling, and injection molding and an objective set that includes all of the objectives with the same weights, the results are shown in = 9.25mm, t = 3.10mm. In this case, all design goals achieve their target values and the overall deviation is 0. Then, the rapid tooling and injection molding teams make decisions based on this result. Since these two activities are coupled, the rapid tooling team assumes a value of ML, and expects to acquire converged results after several iterations. Unfortunately in this case, rapid tooling and injection molding teams' solutions do not converge. The reason is the design team makes decisions only considering its own design goals; hence the thickness of robot arm t is too large and the mold life becomes so short that the rapid tooling team must build 10 pairs of molds. This violates the constraints of time and cost. Therefore, the product design must be modified. For simplicity, the intermediate results are not listed here. After several rounds of iterations, the converged results from the traditional approach are as shown in In the trial and error process, iterations happen not only between the coupled rapid tooling and injection molding activities, but also with the upstream design activity. Furthermore, the number and styles of iterations are affected by some unpredictable or uncontrollable factors, such as the teams' experience, and the initial values the teams choose to start the iterative process. So far, this case has demonstrated the difficulties of DfM, and why the traditional approach cannot guarantee the superiority of the final result. By using the CMDM to separate the activities, we expect to change the process shown in <Table 1 goes about here> <Table 2 goes about here> Resolving couplings using Leader/Follower protocol Since the design team's decision is not coupled with the decisions of either of the manufacturing teams, as shown in when LT = 2 mils (7) ML=1696. 10135.46d311.63t+423.03Θ +61.17(d8.11 et al. 1996), which is fundamentally different from using them to approximate the BRCs. Beyond predicting a player's behavior using its BRC, game protocols also govern issues such as the sequence of the players' decision making activities and control over specific variables. All of these factors are determined by the players' cooperation styles. If the injection molding team is selected as the leader, the BRC T is: When 150> ML ≥75, two pairs of molds have to be build, thus LT = 8 mils has to be selected in order to meet the time constraint. Here, we do not consider the situation that several pairs of molds can be built simultaneously in an SLA3500 machine. It can also be observed from BRC T that Θ remains 0 because the rapid tooling team does not know how the draft angle will affect mold life, and strives to reduce surface roughness, SF, of the robot arm which is achieved with Θ = 0. Obviously, the injection molding team will be unable to eject the parts with a zero draft angle. This is the main reason why the traditional trial and error approach does not converge between the rapid tooling and injection molding teams. Compromise DSP for a ranged set of decisions As described in Section 1, the third step is reformulating the compromise DSPs using design capability indices. In the design activity, all the design variables are continuous; therefore the 16 locations and deviations of the performance variables are calculated using Equation (2). The design team's compromise DSP for a ranged set of decisions is shown in <Table 5 goes about here> <Figure 14 goes about here> In The CMDM is especially useful in the early stages of product design, when little is known about the product and approximating a set of correct designs is much more efficient than conducting rounds and rounds of guessing and correcting. The advantage of the CMDM in DfM is that design and manufacturing activities are separated systematically; hence the product realization activities are accomplished in a more streamlined process as shown in 18 Design freedom in the process The reason the CMDM results in more superior results than the traditional trial and error approach is the design freedom in the product realization process. A design freedom metric is presented in where n is the number of performance measures of the system. For the i th performance measure, TR i is the target range, PR i is the feasible performance range, and PR i,initial is the initial feasible performance range. In At the second row, the performance ranges and design freedom at the initial state of this product realization process are listed. The initial design freedom is 0.734, which is smaller than 1 due to the natural limitation of this process and the couplings between activities. In the trial and error process, when the design team makes a specific decision, design freedom is quantified as 0.368, shown at the third row of Step 3. The injection molding team finally makes its decision at Step 4. It is observed that when the design team makes a ranged set of decisions, design freedom is 0.504 at Step 2. At this moment of the product realization process, the rapid tooling and injection molding teams can Closure In this paper, the idea of collaboration by separation is tested in product DfM problems. The CMDM is used to enable the separation without causing costly information exchange and iterations. Generally, the compromise DSP is used as an information medium to separate the activities at the information communication level, game theoretical principles separate the coupled activities, and design capability indices separate upstream and downstream activities. The robot arm design and manufacturing process demonstrates that by using the CMDM, a complex product realization process is implemented in a streamlined manner, with each engineering team focusing on its areas of expertise. With the CMDM, final results are obtained with fewer iterations between design teams and significantly less deviation from target performance, relative to using the traditional trial and error approach.
Variablecomplexity optimization applied to airfoil design
, 2006
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