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http://hdl.handle.net/10889/1353
2021-11-29T06:02:35ZRobust reliability under uncertainty conditions by using modified info-gap models with two to four horizons of uncertainty and quantifier elimination
http://hdl.handle.net/10889/15176
Title: Robust reliability under uncertainty conditions by using modified info-gap models with two to four horizons of uncertainty and quantifier elimination
Authors: Ioakimidis, Nikolaos
Abstract: Quantifier elimination for real variables constitutes an interesting computational tool with efficient implementations in some popular computer algebra systems and many applications in several disciplines. On the other hand, many practical problems concern situations under uncertainty, where uncertainty intervals and, more generally, reliability regions of uncertain quantities have to be computed. Here the interest is in the popular Ben-Haim's IGDT (info-gap or information-gap decision theory) for problems under severe uncertainty based on info-gap models, where quantifier elimination already proved to constitute a possible tool for the computation of the related reliability regions and robustness functions. Here Ben-Haim's IGDT is considered again, but now in a modified form, where more than one horizon of uncertainty is present (here two, three or four). More explicitly, here each uncertain quantity is assumed to have its own horizon of uncertainty contrary to the usual case in the IGDT, where only one horizon of uncertainty is present in the related info-gap model. Six applications are presented showing the usefulness of the present computational approach. These applications (mainly based on fractional-error info-gap models) concern (i) a linear system, (ii) a sum, (iii) the area of a rectangle, (iv) the volume of a rectangular cuboid, (v) the buckling load of a fixed-free column and (vi) the von Mises yield criterion in two-dimensional elasticity. Beyond the uncertain quantities (here two, three or four) one, two or three parameters may also be present and appear in the derived QFFs (quantifier-free formulae). Of course, it is noted that quantifier elimination generally has a doubly-exponential computational complexity and this restricts its applicability to problems with a small total number of variables (quantified and free).Application of quantifier elimination to robust reliability under severe uncertainty conditions by using the info-gap decision theory (IGDT)
http://hdl.handle.net/10889/14899
Title: Application of quantifier elimination to robust reliability under severe uncertainty conditions by using the info-gap decision theory (IGDT)
Authors: Ioakimidis, Nikolaos
Abstract: Ben-Haim's info-gap (or information-gap) decision theory (IGDT) constitutes a very interesting and popular method for the study of problems in engineering and in many other scientific disciplines under severe uncertainty conditions. On the other hand, quantifier elimination constitutes an equally interesting approach implemented in some computer algebra systems and aiming at the transformation of quantified formulae (i.e. formulae including the universal and/or the existential quantifiers) to logically equivalent formulae but free from these quantifiers and the related quantified variables. Here we apply the method of quantifier elimination (by using its implementation in Mathematica) to the info-gap decision theory and we compute the related reliability regions and, next, the related robustness functions. The computation of the opportuneness (or opportunity) functions is also considered in brief. More explicitly, the four problems studied here concern: (i) the Hertzian contact of two isotropic elastic spheres, (ii) a spring with a linear stiffness but also with an uncertain cubic non-linearity in its stiffness, (iii) the robust reliability of a project with uncertain activity (task) durations and (iv) a gap-closing electrostatic actuator. In all these problems here under uncertainty conditions, the present results are seen to be in complete agreement with the results already derived for the same problems by Ben-Haim and his collaborators (who used appropriate more elementary methods) with respect to the robustness and/or opportuneness functions, but here the reliability regions are also directly computed. Moreover, the present approach permits the study of some difficult parametric cases (e.g. in the problem of the gap-closing electrostatic actuator with a non-linearity in its stiffness), where the help of a computer algebra system seems to be necessary.Uncertainty intervals/regions for the stress intensity factors at crack tips under uncertain loading by using the ellipsoidal model and numerical integration
http://hdl.handle.net/10889/14848
Title: Uncertainty intervals/regions for the stress intensity factors at crack tips under uncertain loading by using the ellipsoidal model and numerical integration
Authors: Ioakimidis, Nikolaos
Abstract: Quantifier elimination constitutes an interesting computational approach in computer algebra already successfully applied to several disciplines. Here we apply this approach to crack problems in fracture mechanics with respect to the two stress intensity factors at the crack tips, but under uncertainty conditions as far as the loading of the crack(s) is concerned. At first, a single straight crack loaded by two uncertain concentrated normal loads satisfying an ellipsoidal inequality constraint is studied. Next, the more interesting case of an uncertain distributed normal load on the crack(s) is also considered in the problems of (i) a single straight crack, (ii) a periodic array of collinear cracks and (iii) a periodic array of parallel cracks. In these problems, the inequality constraint satisfied by the loading is assumed to have a quadratic (`energy'-type) integral form. Beyond quantifier elimination the computational approach consists in using either (i) the closed-form formulae for the stress intensity factors (for a single crack) or (ii) the method of Cauchy-type singular integral equations and, next, the quadrature method for their numerical solution, more explicitly, the Lobattoâ€“Chebyshev method (for all three aforementioned crack problems). Moreover, for the integral inequality constraint the Gaussâ€“Chebyshev quadrature rule is used. By performing quantifier elimination to the relevant existentially quantified formulae and computing the related QFFs (quantifier-free formulae), we were able to derive both (i) the uncertainty intervals (or uncertainty ranges) for the stress intensity factors and (ii) the related uncertainty regions. These results show the uncertainty propagation from the loading of the crack(s) to the resulting stress intensity factors.Application of the method of quantifier elimination to the determination of intervals when the uncertain parameters satisfy an ellipsoidal inequality constraint
http://hdl.handle.net/10889/14403
Title: Application of the method of quantifier elimination to the determination of intervals when the uncertain parameters satisfy an ellipsoidal inequality constraint
Authors: Ioakimidis, Nikolaos
Abstract: Quite frequently, problems that appear in applied mechanics should be solved under uncertainty conditions. Among the related non-probabilistic methods that based on interval analysis constitutes a very popular model. Here we consider another popular model: that based on an ellipsoidal inequality constraint among the uncertain parameters. This is the so-called ellipsoidal convex model. Generalized ellipsoidal convex models are also frequently adopted. Here the aim is to use the interesting computational method of quantifier elimination for the solution of such an uncertainty problem generally for the determination of the intervals of the responses of the system under consideration of course under the restriction that the total number of variables and the degrees of the polynomials involved are small. The present approach is applied to the problems of (i) a three-parametric cubic equation with respect to its real root, (ii) a two-storey shear frame building with non-linear stiffness, (iii) a three-member truss (with the adoption of several uncertainty models), (iv) a simple structural mechanics problem with symbolic intervals, (v) the correlation propagation in a system involving three uncertain parameters and (vi) a problem with a complicated uncertainty region for the uncertain parameters. The alternative, but essentially not so different, approach based on minimization and maximization is also considered in brief. The present results show us that the method of quantifier elimination can be successfully applied to simple systems with uncertain parameters satisfying an inequality constraint (such as an ellipsoidal constraint) and provide us the exact intervals of the responses of the system or even the exact regions showing their correlations.