Simultaneous Optimal Uncertainty Apportionment and Robust Design Optimization of Systems Governed by Ordinary Differential Equations

The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness, suboptimal performance, and higher build costs. Treatment of small geometric uncertainty in the context of manufacturing tolerances is a well studied topic. Traditional sequential design methodologies have recently been replaced by concurrent optimal design methodologies where optimal system parameters are simultaneously determined along with optimally allocated tolerances; this allows to reduce manufacturing costs while increasing performance. However, the state of the art approaches remain limited in that they can only treat geometric related uncertainties restricted to be small in magnitude. This work proposes a novel framework to perform robust design optimization concurrently with optimal uncertainty apportionment for dynamical systems governed by ordinary differential equations. The proposed framework considerably expands the capabilities of contemporary methods by enabling the treatment of both geometric and non-geometric uncertainties in a unified manner. Additionally, uncertainties are allowed to be large in magnitude and the governing constitutive relations may be highly nonlinear. In the proposed framework, uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach allows statistical moments of the uncertain system to be explicitly included in the optimization-based design process. The framework formulates design problems as constrained multi-objective optimization problems, thus enabling the characterization of a Pareto optimal trade-off curve that is off-set from the traditional deterministic optimal trade-off curve. The Pareto off-set is shown to be a result of the additional statistical moment information formulated in the objective and constraint relations that account for the system uncertainties. Therefore, the Pareto trade-off curve from the new framework characterizes the entire family of systems within the probability space; consequently, designers are able to produce robust and optimally performing systems at an optimal manufacturing cost. A kinematic tolerance analysis case-study is presented first to illustrate how the proposed methodology can be applied to treat geometric tolerances. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design at an optimal manufacturing cost, accounting for the entire family of systems within the associated probability space. This case-study highlights the general nature of the new framework which is capable of optimally allocating uncertainties of multiple types and with large magnitudes in a single calculation

A statistical tolerance analysis approach for over-constrained mechanism based on optimization and Monte Carlo simulation

Tolerancing decisions can profoundly impact the quality and cost of the mechanism. To evaluate the impact of tolerance on mechanism quality, designers need to simulate the influences of tolerances with respect to the functional requirements. This paper proposes a mathematical formulation of tolerance analysis which integrates the notion of quantifier: ‘‘For all acceptable deviations (deviations which are inside tolerances), there exists a gap configuration such as the assembly requirements and the behavior constraints are verified’’ & ‘‘For all acceptable deviations (deviations which are inside tolerances), and for all admissible gap configurations, the assembly and functional requirements and the behavior constraints are verified’’. The quantifiers provide a univocal expression of the condition corresponding to a geometrical product requirement. This opens a wide area for research in tolerance analysis. To solve the mechanical problem, an approach based on optimization is proposed. Monte Carlo simulation is implemented for the statistical analysis. The proposed approach is tested on an over-constrained mechanism

Manufacturing variation models in multi-station machining systems

In product design and quality improvement fields, the development of reliable 3D machining variation models for multi-station machining processes is a key issue to estimate the resulting geometrical and dimensional quality of manufactured parts, generate robust process plans, eliminate downstream manufacturing problems, and reduce ramp-up times. In the literature, two main 3D machining variation models have been studied: the stream of variation model, oriented to product quality improvement (fault diagnosis, process planning evaluation and selection, etc.), and the model of the manufactured part, oriented to product and manufacturing design activities (manufacturing and product tolerance analysis and synthesis). This paper reviews the fundamentals of each model and describes step by step how to derive them using a simple case study. The paper analyzes both models and compares their main characteristics and applications. A discussion about the drawbacks and limitations of each model and some potential research lines in this field are also presented

Tolerance analysis by static analogy on 2D assemblies with fits and fasteners

In tolerance analysis, the effect of clearance fits is especially difficult to estimate because the mating parts are not necessarily in actual contact and can take an infinite number of relative positions. The treatment of these situations is allowed in most of the available methods, possibly introducing additional elements in the dimension chains with appropriate statistical assumptions. The paper provides a similar extension for the static analogy, a previously proposed method that converts the tolerance analysis problem into an equivalent problem of force analysis. The procedure represents each fit, possibly between patterns of features (e.g., fasteners and holes), with a proper constraint in the equivalent static model. The ability of the constraint to transmit forces and torques is determined according to the types and directions of misalignments allowed by the joint clearance. With simple rules, this avoids complications in the static model, which must include only the constraint between parts rather than the geometric details of the mating features. The extended method, currently limited to 2D dimension chains, is demonstrated on examples involving both dimensional and geometric tolerances. The comparison with existing methods shows the correctness of the proposed procedure. The simplicity of the workflow confirms the possibility, already demonstrated for the static analogy, of avoiding numerical simulations or even the use of computer-based tools

Explicitly representing the semantics of composite positional tolerance for patterns of holes

Representing the semantics of the interaction of two or more tolerances (i.e. composite tolerance) explicitly to make them computer-understandable is currently a challenging task in computer-aided tolerancing (CAT). We have proposed a description logic (DL) ontology based approach to complete this task recently. In this paper, the representation of the semantics of the composite positional tolerance (CPT) for patterns of holes (POHs) is used as an example to illustrate the proposed approach. This representation mainly includes: representing the structure knowledge of the CPT for POHs in DL terminological axioms; expressing the constraint knowledge with Horn rules; and describing the individual knowledge using DL assertional axioms. By implementing the representation with the web ontology language (OWL) and the semantic web rule language (SWRL), a CPT ontology is developed. This ontology has explicitly computer-understandable semantics due to the logic-based semantics of OWL and SWRL. As is illustrated by an engineering example, such semantics makes it possible to automatically check the consistency, reason out the new knowledge, and implement the semantic interoperability of CPT information. Benefiting from this, the ontology provides a semantic enrichment model for the CPT information extracted from CAD/CAM systems

A dimensioning and tolerancing methodology for concurrent engineering applications I: problem representation

This paper is the first of two which present a methodology for determining the dimensional specifications of all the component parts and sub-assemblies of a product according to their dimensional requirements. To achieve this goal, two major steps are followed, each of which is described in a paper. In the first paper, all relationships necessary for finding the values of dimensions and tolerances are represented in a matrix form, known as a Dimensional Requirements/Dimensions (DR/D) matrix. In the second paper, the values of individual dimensions and tolerances are determined by applying a comprehensive solution strategy to satisfy all the relationships represented in the DR/D matrix. The methodology is interactive and suitable for use in a concurrent engineering (CE) environment. The graphical tool presented in this paper will assist a CE team in visualizing the overall D&T problem and foreseeing the ramifications of decisions regarding the selection of dimensions and tolerances. This will assist the CE team to systematically determine all the controllable variables, such as dimensions, tolerances, and manufacturing processes

A statistical tolerance analysis approach for over-constrained mechanism based on optimization and Monte Carlo simulation

Tolerancing decisions can profoundly impact the quality and cost of the mechanism. To evaluate the impact of tolerance on mechanism quality, designers need to simulate the influences of tolerances with respect to the functional requirements. This paper proposes a mathematical formulation of tolerance analysis which integrates the notion of quantifier: ‘‘For all acceptable deviations (deviations which are inside tolerances), there exists a gap configuration such as the assembly requirements and the behavior constraints are verified’’ & ‘‘For all acceptable deviations (deviations which are inside tolerances), and for all admissible gap configurations, the assembly and functional requirements and the behavior constraints are verified’’. The quantifiers provide a univocal expression of the condition corresponding to a geometrical product requirement. This opens a wide area for research in tolerance analysis. To solve the mechanical problem, an approach based on optimization is proposed. Monte Carlo simulation is implemented for the statistical analysis. The proposed approach is tested on an over-constrained mechanism

A precision analysis method for the kinematic assembly of complex products based on equivalence of deviation source

Purpose: The assembly quality of complex products is pivotal to their lifecycle performance. Assembly precision analysis (APA) is an effective method used to check the feasibility and quality of assembly. However, there is still a need for a systematic approach to be developed for APA of kinematic mechanisms. To achieve more accurate analysis of kinematic assembly, this paper aims to propose a precision analysis method based on equivalence of the deviation source. Design/methodology/approach: A unified deviation vector representation model is adopted by considering dimension deviation, geometric deviation, joint clearance and assembly deformation. Then, vector loops and vector equations are constructed, according to joint type and deviation propagation path. A combined method, using deviation accumulation and sensitivity modeling, is applied to solve the kinematic APA of complex products. Findings: When using the presented method, geometric form deviation, joint clearance and assembly deformation are considered selectively during tolerance modeling. In particular, the proposed virtual link model and its orientation angle are developed to determine joint deviation. Finally, vector loops and vector equations are modeled to express deviation accumulation. Originality/value: The proposed method provides a new means for the APA of complex products, considering joint clearance and assembly deformation while improving the accuracy of APA, as much as possible.Natural Science Basic Research Project of Shaanxi Province, China (Grant Nos. 2019JM-435 and 2019JM-073), the China Postdoctoral Science Foundation (Grant No. 2018M633439) and the Key Laboratory of Road Construction Technology and Equipment (Chang’an University), MOE (Grant No. 300102258506)