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- ItemApplications of quantifier elimination to the proofs of algebraic inequalities in engineering problems related to Todinov's method for risk reduction
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςAn interesting and efficient method based on algebraic inequalities for the reduction of risk and uncertainty as well as for the generation of new knowledge and the optimization of systems and processes has been recently proposed by Todinov. This method requires the proof of the related inequality and several classical approaches were successfully used by Todinov for this task. Here the use of the well-known method of quantifier elimination for the same task, that is for the proof of algebraic inequalities related to Todinov's method, as an additional approach appropriate for use with respect to some algebraic inequalities is proposed and actually used in some problems already proposed and successfully solved by Todinov on the basis of his own method of algebraic inequalities. The present approach is applied to some algebraic inequalities of engineering interest already proved by Todinov, more explicitly (i) to the inequality for the equivalent resistances of two resistors in an electrical circuit, (ii) to the inequality for the equivalent spring constants of two, three or four elastic springs, (iii) to inequalities concerning the supply of high-reliability components, (iv) to inequalities concerning ranking systems, (v) to the construction of a system with superior reliability and (vi) to the accumulated strain energy in bars under tension or in cantilevers under bending. Naturally, because of the well-known doubly-exponential computational complexity of quantifier elimination the present approach is applicable only when the related universally quantified formula contains a small total number of variables (free and quantified). Yet, the method of quantifier elimination can also be combined with the method of proof by induction and this additional possibility is also illustrated in two inequalities related to Todinov's method for their proof in the general case.
- ItemProblems under uncertainty : quantifier elimination to universally–existentially (AE) quantified formulae related to two or more horizons of uncertainty
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςProblems under uncertainty appear frequently in practical applications. Ben-Haim's IGDT (info-gap decision theory) constitutes a very efficient method for the study of such problems. The three components (or elements) of Ben-Haim's IGDT are (i) the system model, (ii) the info-gap uncertainty model and (iii) the performance requirement(s). Appropriate (mainly positivity) assumptions can also be made. Here we use the IGDT only partially by restricting our attention to its first component, the system model, and to its second component, the info-gap uncertainty model, but paying no attention to the performance requirement(s) also very important in the IGDT. Here an emphasis is put on the use of (mixed) universally–existentially (AE) quantified formulae assuring the validity of the system model (under the assumptions made) for all values of the universally quantified uncertain variable(s) and for at least one value (or a set of values) of the existentially quantified uncertain variable(s) of course provided that these quantified variables satisfy the adopted info-gap uncertainty model here the popular fractional-error model. On the other hand, here we also assume that each uncertain variable (either universally or existentially quantified) has its own uncertainty parameter (or horizon of uncertainty). Next, by using the method of quantifier elimination in its powerful implementation in the computer algebra system Mathematica we transform the quantified formula to an equivalent QFF (quantifier-free formula) free from the quantifiers and the quantified variables, but, evidently, including the horizons of uncertainty. Two simple applications concerning (i) a product/quotient and (ii) the buckling load of a fixed–free column illustrate the present approach with the derivation of the related QFFs, some of which can also be verified manually.
- ItemQuantifier elimination and quantifier-free formulae for universally–existentially (AE) quantified formulae in Ben-Haim's info-gap model of uncertainty
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe method of quantifier elimination with implementations in some computer algebra systems already proved useful for the computation of both the robustness and the opportuneness (or opportunity) functions in Ben-Haim's info-gap (or information-gap) model of uncertainty. As is well known, this model constitutes an interesting and practical tool in decision theory. Moreover, quantifier elimination concerning the robustness/opportuneness functions can be performed to the related universally/existentially quantified formulae. Here we proceed to the consideration of the additional mixed (AE) case, where both the universal and the existential quantifiers are present in the quantified formula related to Ben-Haim's info-gap model of uncertainty. In this mixed (AE) case, evidently now with more than one uncertain variable, the universal quantifier concerns one (or more than one) uncertain variable and similar is the case with the existential quantifier. After performing quantifier elimination to this quantified formula (here by using the computer algebra system Mathematica), we derive the related QFF (quantifier-free formula) that concerns the horizon of uncertainty. The case of more than one horizon of uncertainty can also be similarly studied. In this way, an expression for the horizon of uncertainty in a logical form with the appropriate inequalities is derived. From this form it is observed that additional immunity functions (beyond the classical robustness and opportuneness functions) appear in the mixed universal–existential (AE) case. The present approach is applied to four uncertainty problems which are based on info-gap models and concern (i) the area of a rectangle, (ii) the buckling load of a fixed–free column, (iii) the volume of a rectangular cuboid and (iv) the reactions at the ends of a fixed beam loaded by a concentrated load.
- ItemRobust reliability under uncertainty conditions by using modified info-gap models with two to four horizons of uncertainty and quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςQuantifier 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).
- ItemApplication of quantifier elimination to robust reliability under severe uncertainty conditions by using the info-gap decision theory (IGDT)
Γενικό Τμήμα (Τεχνικές Αναφορές)(Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςBen-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.
- ItemUncertainty intervals/regions for the stress intensity factors at crack tips under uncertain loading by using the ellipsoidal model and numerical integration
Γενικό Τμήμα (Τεχνικές Αναφορές)(Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςQuantifier 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 Lobatto–Chebyshev method (for all three aforementioned crack problems). Moreover, for the integral inequality constraint the Gauss–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.
- ItemApplication of the method of quantifier elimination to the determination of intervals when the uncertain parameters satisfy an ellipsoidal inequality constraint
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςQuite 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.
- ItemGeneralized interval-based polynomial approximations to functions in applied mechanics by using the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)(Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe method of quantifier elimination constitutes an interesting computational approach in computer algebra already implemented in few computer algebra systems. In applied mechanics, this method was already used for the determination of ranges of functions. Here the application of the same method, quantifier elimination, is generalized to the determination of generalized interval-based polynomial approximations to functions again in applied mechanics. The main idea behind the present application is the use of linear interval enclosures for the approximation to functions and, more generally, the use of parameterized solutions to parametric interval systems of linear algebraic equations. This idea is mainly due to Lubomir V. Kolev. Here the present method is at first applied to two simple examples concerning (i) a rational function and (ii) the exponential function with their variables lying in intervals. Next, the same method is also applied to functions in applied-mechanics problems with variables also lying in intervals: (i) the problem of a beam on a Winkler elastic foundation with related function the dimensionless deflection of the beam, (ii) the problem of free vibrations of an oscillator with critical damping with related function the dimensionless displacement of the oscillator and (iii) the problem of a seven-member truss with related functions the nodal displacements. In this application, the stiffness of a bar is an uncertain, interval variable and, moreover, the classical perturbation method is also used. From the present results it is concluded that the method of quantifier elimination constitutes a useful tool for the derivation of simple parameterized interval-based polynomial approximations to functions in applied mechanics.
- ItemQuantifier-elimination-based interval computations in beam problems studied by using the approximate methods of finite differences and of finite elements
Γενικό Τμήμα (Τεχνικές Αναφορές)(Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe rather recent interesting computational method of quantifier elimination already implemented in four computer algebra systems has been already used in many problems of engineering interest including several problems of applied and computational mechanics. Among the previous applications of interest here is mainly the problem of a beam with parametric inequality constraints and under the presence of a loading parameter. This problem was solved by the popular methods (i) of finite differences and (ii) of finite elements in combination with the method of quantifier elimination. Here the same approach is generalized to the case where the loading parameter belongs to an interval. The methods (i) of finite differences and (ii) of finite elements are used again (leading to parametric systems of linear equations) with the computation of the approximate intervals concerning (i) the dimensionless deflection, (ii) the rotation and (iii) the dimensionless bending moment on the whole beam computed on the basis of their values at the nodes used on the beam. In the application of the finite difference method both (i) the purely existential case and (ii) a mixed universal–existential case are considered evidently with respect to the interval loading parameter. The REDLOG computer logic package of the REDUCE computer algebra system has been used again in the present interval computations and the excellent convergence of the obtained approximate intervals computed with the finite difference method is observed. In the purely existential case, up to 3072 intervals on the beam have been successfully used and this is an extremely satisfactory situation in quantifier elimination because it concerns a total number of 3076 variables.
- ItemDetermination of intervals in systems of parametric interval linear equilibrium equations in applied mechanics with the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe method of quantifier elimination is an interesting computational tool in computer algebra with many practical applications including problems of applied mechanics. Recently, this method was used in applied mechanics problems with uncertain parameters varying in known intervals (interval parameters) including systems of parametric interval linear equilibrium equations, direct and inverse problems and the computation of resultants of interval forces. Here the case of systems of parametric interval linear equilibrium equations is further considered by using related quantified formulae including not only the existential quantifier (as is the case with the united solution set of such a system), but also both the universal and the existential quantifiers in the quantified formula (a more general case) with respect to the parameters of the problem including the external loads applied to the mechanical system. Two problems of applied mechanics related to systems of parametric interval linear equilibrium equations are studied in detail: (i) the problem of a simply-supported truss with two external loads recently studied under uncertainty (interval) conditions by E. D. Popova and (ii) the problem of a clamped bar with a gap subjected to a concentrated load recently studied again under uncertainty (interval) conditions by E. D. Popova and I. Elishakoff. Here, in both these problems, by using the method of quantifier elimination both (i) complete solution sets for the unknown quantities (here mainly reactions) and (ii) separate intervals for each unknown quantity are computed on the basis of related quantified formulae. The present results are compared to the results obtained by E. D. Popova and I. Elishakoff on the basis of both the classical interval model and the new algebraic interval model, the latter recently proposed by E. D. Popova.