Considerations for Application of Robust Design Formalization

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Prevailing design models and their stages

A generic product development process starts with needs recognition and ends with the marketing of a finished product (Abouel Nasr & Kamrani, 2007).The major phases of this process are shown in figure 2.1.
A rudimentary model of the design may be described as: F → D (2.1) Where F is the function required, D is a design description and → is some transformation (Gero, 1990). Using different methodologies and tools, this model may be explained in a detailed quantitative model. These models can be evaluated by finding the solutions that satisfy the required function. Decision making in design with quantitative models is discussed in detailed in later chapters.
From a systematic point of view by Pahl and Beitz, the aim of design is to have a comprehensive methodology for all the phases of design and development of technical systems. In line with the above aim, the product design life cycle may be broken down into four phases (Pahl & Beitz, 1996) which are:
• Product planning and clarifying the task.
• Conceptual Design.
• Embodiment Design.
• Detail Design.
Scaravetti (2004) provides a concise and illustrative comparison of the dif-ferent popular design processes from an academic and industrial point of view. This comparison presented in figures 2.2 and 2.3 provides a general overview by decomposing where possible, the design processes into sub-stages and comparing them together with each other.
This comparison reveals that all the presented design processes fall well into the distinct stages of the product design process as shown in figure 2.1.
In addition to the processes defined above, there exist, other processes, whose steps are not similar and classifiable as the models presented earlier. These models include: the Axiomatic Design (Suh, 2001) (Benhabib, 2003), Design of experiments and Taguchi’s Methods (Taguchi, 1978) (Taguchi, 1987) (Benhabib, 2003) (Phadke, 1989) and Set based design (Malak et al., 2009). Axiomatic design is based on two axioms and their corollaries. The design of experiments and Taguchi’s method emphasize the use of experiments and statistical techniques for the search of the parameter values. All of these design phases employ downstream validation and verification steps. In order to integrate variation management early on in the design phase (Maropoulos & Ceglarek, 2010), the verification and validation phase should be expressed early on.

Set based concurrent engineering

The Set based concurrent engineering is an approach popularized by a Japanese automobile manufacturer. In this approach, instead of taking a Point based design approach, the designer takes a set based approach towards design and treats sets of design alternatives at both the conceptual and parametric design levels. These sets are gradually refined and narrowed through the process of elimination of ill suited alternatives until the emergence of the final design. In contrast to the Point based design, where one design is refined, the Set based concurrent engineering (SBCE) maintains the alternatives until the emergence of the final design. Malak and Paredis define the Set based design as an approach in which different design alternatives are evaluated by reasoning and comparing different solutions based on possibilities offered by alternative possible configurations of ”SETS” of design parameters. Set based design aims to delaying commitments to a particular design in favor of gathering information about the problem and reduce imprecision to levels at which indeterminacy is resolved (Malak et al., 2009).

Tolerance accumulation

The aim of tolerance accumulation is to simulate the composition of tolerances i.e. linear tolerance accumulation, 3D tolerance accumulation. Based on the displacement models, several vector space models map all possible manufacturing variations (geometrical displacements between manufacturing surfaces or between a manufacturing surface and the corresponding nominal surface) into a region of hypothetical parametric space. The geometrical tolerances or the dimensioning tolerances are represented by deviation domain (Giordano, 1993; Giordano et al., 2005; Teissandier et al., 1999), T-Map R (Bhide et al., 2007; Davidson & Shah, 2003), or specification hull (Dantan et al., 2003a; Dantan & Ballu, 2002). These three concepts are a hypothetical Euclidean volume which represents all possible deviations in size, orientation and position of features.
For tolerance analysis, this mathematical representation of tolerances allows calculation of accumulation of the tolerances by Minkowsky sum of deviation and clearance domains (Giordano et al., 2005; Teissandier et al., 1999); to calculate the intersection of domains for parallel kinematic chain; and to verify the inclusion of a domain inside an other one. The methods based on this mathematical representation of tolerances are very efficient for tolerance analysis.
However, these two approaches do not take into account the quantifier notion. This notion translates the concept that a functional requirement must be respected in at least one acceptable configuration of gaps (existential quantifier “there exists”), or that a functional requirement must be respected in all acceptable configurations of gaps (universal quantifier “for all”) (Dantan et al., 2003a, 2005). A configuration is a particular relative position of parts in an assembly featuring gaps without interference between parts. The current methods also do not take into account the gaps in a mechanism that have an effect on the performance of the mechanism. In presence of variations, each feature of all the individual components of a given assembly will exhibit departure from its ideal dimension. These departures from nominal will uniquely affect each assembly and each configuration resulting into a unique set of deviations and gaps arising from these variations. The evaluation of these deviations and gaps is necessary to establish the conformance of the assembly in terms of the cumulative effect of these variations on the performance of the product and its conformity to the functional conditions. For this purpose, it is necessary to quantify the deviations and gaps in a given assembly components. The quantifier notion impacts the result of the tolerance analysis (Dantan et al., 2003a, 2005). Therefore, we propose a mathematical formulation of tolerance analysis which simulates the in- fluences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion.

Expression of product design in terms of constraint satisfaction

As discussed earlier, design is a decision making problem, in which the designer needs to take the decisions to move from one design step to another throughout the product design life cycle (Mistree et al., 1990). These decisions are taken while considering different conditions specified by the client requirements. The client requirements are translated in the design process in terms of functional requirements. These functional requirements are the first level constraints that dictate the key requirements and product performance expectations. As the design process proceeds, these requirements are translated in a quantifiable form in terms of the key parameters and related design constraints. Key parameters are the main qualitative and quantitative parameters that have a profound effect on the product performance. These may be involved in univariate or multivariate relations that dictate the fundamental properties related to product physics and performance. The design constraints are seldom simple and explicit and are often complex, consisting of implicit multiple simultaneous requirements. The resulting set of parameters and related design constraints forms a mathematical model of the product design. All engineering design processes must, at some level, pass through this stage. The main objective of the designer, thus, is to focus on finding the feasible solution belonging to the design spaces for the key parameters such that the imposed constraints are satisfied. Using the above approach, the product design process can be transformed into a process of constraint identification and satisfaction.
Tsang (1993) defines a constraints satisfaction problem as: “A CSP is a prob- lem composed of a finite set of variables, each of which is associated with a finite domain, and a set of constraints that restricts the values the variables can simul- taneously take. The task is to assign a value to each variable satisfying all the constraints.”

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Main Theoretical Basis in Terms of Logic

Hurley (2008) defines Logic as the science of evaluating the arguments. Logic has been used since the ancient times in Greek philosophy and mathematics. Numerous works of Plato include the use of logic to establish the syntactic relation between principles and their verity or falsity. The first known systematic study in Logic inference was carried out by Aristotle and assembled by his students in 322 B.C. based on Aristotle’s Syllogisms known more popularly as inference rules. Since then Logic has developed over the period of time to encompass the fields of Philosophy, Mathematics, Psychology, Cognitive behavior, and recently in the last century towards applied the fields of Artificial intelligence, Computer science and Engineering.
In engineering, popular application of Logic remains in the fields of electronics engineering as well as in mechanical and industrial engineering, where under the sub domain of industrial and manufacturing engineering, operations research and planning, logic is implemented in applied form in the shape of different programming and algorithmic techniques from its sub domain, artificial intelligence.
Most of the work done in the domain of Mechanical and industrial engineering is of applied nature and no formal logical syntactic expression is used to describe or formalize an engineering problem. Instead, the Algorithmic and artificial intelligence routines are used directly in most of the cases to resolve a problem  according to a given set of constraints, putting the practice near the domain of applied constraint satisfaction or constraint programming with the help of search routines.

Table of contents :

I R´esum´e ´Etendu En Fran¸cais xv
1 Introduction 
1.1 Objectifs
1.2 Situation
1.3 Probl´ematique
1.4 Positionnement De Travaux
2 ´Etat de L’Art 
2.1 Processus De La Conception
2.2 Conception Robuste
2.3 Tol´erancement
2.4 Bilan
3 Formalisation pour la maˆıtrise des variations 
3.1 Formalisation th´eorique
3.2 Notions de base en logique
3.3 Approche g´en´erale
3.4 Formalisation de l’approche ensembliste de la conception robuste
3.5 Formalisation de l’analyse des tol´erances
3.6 Synth`ese
4 Application `a l’approche ensembliste de la conception robuste 
4.1 Consid´erations sur l’application de la formalisation de la conception robuste
4.2 Repr´esentation de l’espace de conception
4.3 Expression de la consistance
4.4 Outils pour l’exploration de l’espace de conception
4.5 Exemple d’illustration
4.6 Application `a d’autres exemples
4.6.1 Structure `a deux poutres
4.6.2 Accouplement `a plateaux viss´es
4.6.3 La presse `a six barres
4.7 Conclusion.
5 Application `a l’analyse des tol´erances 
5.1 Consid´erations pour l’application de la formalisation d’analyse des tol´erances
5.2 Repr´esentation des d´efauts g´eom´etriques
5.3 Description du comportement – expression des contraintes par ≪Hulls ≫
5.4 D´eveloppement des m´ethodes d’analyse
5.4.1 L’approche d’analyse des tol´erances par pire de cas
5.4.2 L’analyse statistique des tol´erances bas´ee sur la formalisation
et la simulation Monte Carlo
5.5 L’application
5.6 Conclusion
6 Conclusion et perspectives 
II English Version 
1 Introduction 
1.1 Placement and Importance of Variation Management in Design Process
1.2 Positioning of Work
1.3 Layout of the Thesis
2 State of the Art 
2.1 Product Design Process
2.1.1 Prevailing design models and their stages
2.1.2 Concurrent design process
2.1.3 Decision based design perspective
2.1.3.1 Point based design
2.1.3.2 Set based concurrent engineering
2.2 Robust Design
2.3 Tolerancing
2.3.1 Displacement accumulation
2.3.2 Tolerance accumulation
2.4 Synthesis
3 Formalization For Variation Management 
3.1 Formalization of Theory
3.1.1 Expression of product design in terms of constraint satisfaction
3.2 Main Theoretical Basis in Terms of Logic
3.2.1 Quantifiers
3.2.2 Constraint satisfaction problem (CSP) limitations .
3.2.3 Quantified constraint satisfaction problem (QCSP)(Verger & Bessiere, 2006)
3.3 General Theory
3.4 Formulation of Set Based Robust Design
3.4.1 Variable definition for Robust Design
3.4.2 Domain
3.4.3 Constraints
3.4.4 Quantifier based expression for the robust set based design exploration
3.4.5 Consistency evaluation for solutions
3.4.5.1 Existence of a solution
3.4.5.2 Existence of a robust solution
3.5 Formulation of Tolerance Analysis
3.5.1 Variable definition for Tolerance analysis
3.5.2 Domain
3.5.3 Constraints
3.5.4 Quantifier based expression for the Tolerance Analysis for Mechanical Assemblies
3.5.4.1 Respect of assemblability of the mechanism
3.5.4.2 Respect of functional requirements
3.6 Synthesis
4 Application to Set Based Robust Design 
4.1 Considerations for Application of Robust Design Formalization
4.2 Design Space Representation
4.3 Consistency Evaluation
4.3.1 Transformation
4.3.2 Basic notations and definitions
4.3.2.1 Interval operations
4.3.2.2 Extension of constraints
4.3.2.3 Interval Analysis
4.3.3 Consistency for the existence of a solution
4.3.4 Consistency for the existence of a robust solution
4.4 Space Exploration Tools
4.5 Illustrative Example
4.5.1 Problem Description
4.5.2 Conversion to interval arithmetic for consistency
4.5.3 Results
4.5.3.1 Results verification
4.6 Application to Examples
4.6.1 Embodiment design of a two bar structure
4.6.1.1 Results
4.6.1.2 Results verification
4.6.2 Embodiment design of a rigid flange coupling
4.6.2.1 Problem description
4.6.2.2 Design constraints
4.6.2.3 Flange design
4.6.2.4 Design model
4.6.2.5 Results
4.6.3 Design of a 6 Bar Mechanism
4.6.3.1 Problem Description
4.6.3.2 Design Constraints
4.6.3.3 Design Model
4.6.3.4 Algorithm improvements
4.6.3.5 Results
4.7 Conclusion and Discussion
5 Application to Tolerance Analysis 
5.1 Considerations for the Application of Tolerance Analysis Formalization
5.2 Representation of the Geometric Variation
5.2.1 Explanation of geometrical description
5.2.1.1 Geometric description in 1D
5.2.1.2 Geometric description in 3D
5.3 Constraint Expression Via Hulls
5.4 Development Of Analysis Methods
5.4.1 Approach for the worst case tolerance analysis
5.4.1.1 QCSP
5.4.1.2 Formulation of tolerance analysis for QCSP solver
5.4.2 Statistical tolerance analysis based on formalization and Monte Carlo simulation
5.4.2.1 Monte Carlo simulation
5.4.2.2 Transformation for statistical tolerance analysis .
5.4.2.3 Results
5.5 Application
5.5.1 Geometric description
5.5.2 Geometric behavior and constraint model
5.5.2.1 Compatibility Hull (Hcompatibility)
5.5.2.2 Interface Hull HInterface
5.5.2.3 Functional Hull HFunctional
5.5.3 Optimization
5.5.4 Assembly Example Results
5.5.5 Error Control and Revalidation
5.6 Conclusion
6 Conclusion And Perspectives 
6.1 Conclusion
6.2 Perspectives
A Appendix 1 
A.1 Description of FOL terms and symbols
A.2 Constraints For Flange Coupling Example
A.2.1 Nomenclature
A.2.2 Constraints for flange coupling design
A.2.2.1 Bolt design
A.3 Constraints For Six Bar Mechanism
A.3.1 General Considerations
A.3.2 Assemblability Constraints
A.3.2.1 Fitting and Framing Constraints
A.3.2.2 Path generation constraints
References 

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