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Modal Intervals Revisited Part 1: A Generalized Interval Natural Extension
 Reliable Computing
"... Modal interval theory is an extension of classical interval theory which provides richer interpretations (including in particular inner and outer approximations of the ranges of real functions). In spite of its promising potential, modal interval theory is not widely used today because of its origin ..."
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Modal interval theory is an extension of classical interval theory which provides richer interpretations (including in particular inner and outer approximations of the ranges of real functions). In spite of its promising potential, modal interval theory is not widely used today because of its original and complicated construction. The present paper proposes a new formulation of modal interval theory. New extensions of continuous real functions to generalized intervals (intervals whose bounds are not constrained to be ordered) are defined. They are called AEextensions. These AEextensions provide the same interpretations as the ones provided by modal interval theory, thus enhancing the interpretation of the classical interval extensions. The construction of AEextensions strictly follows the model of classical interval theory: starting from a generalization of the definition of the extensions to classical intervals, the minimal AEextensions of the elementary operations
On the Axiomatization of Interval Arithmetic
"... We investigate some abstract algebraic properties of the system of intervals with respect to the arithmetic operations and the relation inclusion and derive certain practical consequences from these properties. In particular, we discuss the use of improper intervals (in addition to proper ones) and ..."
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We investigate some abstract algebraic properties of the system of intervals with respect to the arithmetic operations and the relation inclusion and derive certain practical consequences from these properties. In particular, we discuss the use of improper intervals (in addition to proper ones) and of midpointradius presentation of intervals. This work is a theoretical introduction to interval arithmetic involving improper intervals. We especially stress on the existence of special “quasi”multiplications in interval arithmetic and their role in relevant symbolic computations. 1.
Use of Modal Interval Analysis in Early Engineering Design 1 Part I: Modal Interval Analysis
"... In order to describe possible applications of modal interval analysis in early engineering design, let us start by briefly explaining what is interval computation and what is modal interval analysis. Direct and indirect measurements: uncertainty is ubiquitous. In practice, how do we obtain the numer ..."
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In order to describe possible applications of modal interval analysis in early engineering design, let us start by briefly explaining what is interval computation and what is modal interval analysis. Direct and indirect measurements: uncertainty is ubiquitous. In practice, how do we obtain the numerical values of different physical quantities? For some quantities, we can obtain these values directly, either by performing a measurement or by eliciting the value from an expert. Measurements are never absolutely accurate; as a result, the result ˜x of the measurement is somewhat different from the actual (unknown) value x of the desired physical quantity: ˜x ̸ = x. In other words, from measurements, we can only determine the value x with uncertainty: the approximation error ∆x def = ˜x − x is, in general, different from 0. Expert estimates are usually even less accurate than measurements, so the values ˜x obtained from the experts also always contain uncertainty.
Modal Intervals Revisited Part 2: A Generalized Interval MeanValue Extension ∗
"... In Modal Intervals Revisited Part 1, new extensions to generalized intervals (intervals whose bounds are not constrained to be ordered), called AEextensions, have been defined. They provide the same interpretations as modal intervals and therefore enhance the interpretations of classical interval e ..."
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In Modal Intervals Revisited Part 1, new extensions to generalized intervals (intervals whose bounds are not constrained to be ordered), called AEextensions, have been defined. They provide the same interpretations as modal intervals and therefore enhance the interpretations of classical interval extensions (for example, both inner and outer approximations of function ranges are in the scope of AEextensions). The construction of AEextensions is similar to the cnstruction of classical interval extensions. In particular, a natural AEextension has been defined from Kaucher arithmetic which simplified some central results of modal interval theory. Starting from this framework, the meanvalue AEextension is now defined. It represents a new way to linearize a real function, which is compatible with both inner and outer approximations of its range. With a quadratic order of convergence for realvalued functions, it allows one to overcome some difficulties which were encountered using a preconditioning process together with the natural AEextensions. Some application examples are finally presented, displaying the application potential of the meanvalue AEextension.
ClosedLoop Analysis in Semantic Tolerance Modeling
"... A semantic tolerance modeling scheme based on generalized intervals was recently proposed to allow for embedding more tolerancing intents in specifications with a combination of numerical intervals and logical quantifiers. By differentiating a priori and a posteriori tolerances, the logic relationsh ..."
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A semantic tolerance modeling scheme based on generalized intervals was recently proposed to allow for embedding more tolerancing intents in specifications with a combination of numerical intervals and logical quantifiers. By differentiating a priori and a posteriori tolerances, the logic relationships among variables can be interpreted, which is useful to verify completeness and soundness of numerical estimations in tolerance analysis. In this paper, we present a semantic tolerance analysis approach to estimate size and geometric tolerance stackups based on closed loops of interval vectors. An interpretable linear system solver is constructed to ensure interpretability of numerical results. A direct linearization method for nonlinear systems is also developed. This new approach enhances traditional numerical analysis methods by preserving logical information during computation such that more semantics can be derived from variation estimations. 1.
DETC2006/DAC99609 SEMANTIC TOLERANCE MODELING
"... A significant amount of research efforts has been given to explore the mathematical basis for 3D dimensional and geometric tolerance representation, analysis, and synthesis. However, engineering semantics is not maintained in these mathematic models. It is hard to interpret calculated numerical resu ..."
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A significant amount of research efforts has been given to explore the mathematical basis for 3D dimensional and geometric tolerance representation, analysis, and synthesis. However, engineering semantics is not maintained in these mathematic models. It is hard to interpret calculated numerical results in a meaningful way. In this paper, a new semantic tolerance modeling scheme based on modal interval is introduced to improve interpretability of tolerance modeling. With logical quantifiers, semantic relations between tolerance specifications and implications of tolerance stacking are embedded in the mathematic model. The model captures the semantics of physical property difference between rigid and flexible materials as well as tolerancing intents such as sequence of specification, measurement, and assembly. Compared to traditional methods, the semantic tolerancing allows us to estimate true variation ranges such that feasible and complete solutions can be obtained. 1.
Interpretable Interval Constraint Solvers in Semantic Tolerance Analysis
"... A semantic tolerance modeling scheme based on generalized intervals was recently proposed to allow for embedding more tolerancing intents in specifications with a combination of numerical intervals and logical quantifiers. By differentiating a priori and a posteriori tolerances, the logic relationsh ..."
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A semantic tolerance modeling scheme based on generalized intervals was recently proposed to allow for embedding more tolerancing intents in specifications with a combination of numerical intervals and logical quantifiers. By differentiating a priori and a posteriori tolerances, the logic relationships among variables can be interpreted, which is useful to verify completeness and soundness of numerical estimations in tolerance analysis. In this paper, we present a semantic tolerance analysis approach to estimate tolerance stackups. New interpretable linear and nonlinear constraint solvers are developed to ensure interpretability of variation estimations. This new approach enhances traditional numerical analysis methods by preserving logical information during computation such that more semantics can be derived from numerical results.
Generalized Interval Projection: A New Technique for Consistent Domain Extension
, 2010
"... This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of ..."
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This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of a single punctual solution is of particular interest, because it gives greater freedom for choosing a solution. Our approach is able to build an inner box for the problem starting with a single point solution, by consistently extending the domain of every variable. The key point is a new method called generalized projection. The requirements are that each parameter must occur only once in the system, variable domains must be bounded, and each variable must occur only once in each constraint. Our extension is based on an extended algebraic structure of intervals called generalized intervals, where improper intervals are allowed (e.g. [1,0]). 1