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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
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.
2 Thales Airborne Systems
"... Abstract. The modal intervals theory deals with quantified propositions in AEform, i.e. universal quantifiers precede existential ones, where variables are quantified over continuous domains and with equality constraints. It allows to manipulate such quantified propositions computing only with bo ..."
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Abstract. The modal intervals theory deals with quantified propositions in AEform, i.e. universal quantifiers precede existential ones, where variables are quantified over continuous domains and with equality constraints. It allows to manipulate such quantified propositions computing only with bounds of intervals. A simpler formulation of this theory is presented. Thanks to this new framework, a meanvalue extension to generalized intervals (intervals whose bounds are not constrained to be ordered) is defined. Its application to the validation of quantified propositions is illustrated. 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.
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"... design under fuzzy poleplacement specifications: an interval arithmetic approach ..."
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design under fuzzy poleplacement specifications: an interval arithmetic approach
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.