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  • ItemOpen Access
    Computation of approximate intervals for real roots of transcendental equations with one interval parameter by using the method of quantifier elimination
    (2024-04-08) Ioakimidis, Nikolaos; Ιωακειμίδης Νικόλαος
    The method of quantifier elimination over the reals is an interesting approach in computer algebra with many practical applications. The methods of cylindrical algebraic decomposition (CAD) devised by Collins in 1973 and of virtual term substitution (VTS) devised by Weispfenning in 1988 constitute two efficient algorithms repeatedly and successfully used for quantifier elimination over the reals. Beyond the classical Hong's QEPCAD package in the SACLIB library efficient implementations of quantifier elimination over the reals are available in the computer algebra systems Reduce, Maple and Mathematica. This method was already used for the computation of sharp enclosures of the two real roots of the classical parametric quadratic equation with only one interval coefficient. Here we apply the same method, quantifier elimination (by using Mathematica for our computations), to the computation of approximate intervals for real roots of elementary transcendental equations with only one uncertain, interval parameter. The computed intervals are approximate simply because polynomial approximations (such as Taylor–Maclaurin, Taylor and minimax approximations) are used for the transcendental functions that appear in the transcendental equations, which are transformed to approximate polynomial equations. Nevertheless, very satisfactory numerical results with accuracies of 15 up to 50 significant digits for the computed uncertainty intervals of the sought roots can easily be derived. Four applications are studied in detail by using this methodology including the problem of buckling of a simple frame in the theory of elastic stability.
  • ItemOpen Access
    On the convergence of the direct quadrature method for Cauchy type singular integral equations of the first kind
    (1983-02-10) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The convergence of the direct quadrature method (based on the Gauss–Chebyshev quadrature rule) for Cauchy type singular integral equations of the first kind is demonstrated under appropriate conditions. The rate of convergence is also established.
  • ItemOpen Access
    Application of the method of quantifier elimination to Ben-Haim's info-gap decision theory (IGDT) under the presence of both horizon-of-uncertainty-related and ordinary interval uncertain variables
    (2023-06-13) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Problems under uncertainty conditions appear frequently in practice. The use of classical interval analysis constitutes an interesting tool for the study of such problems with the uncertain variables assumed to be interval variables. Ben-Haim's info-gap (or information-gap) decision theory (IGDT) constitutes an interesting method for the study of such problems. The determination of the maximum value of the uncertainty parameter (or horizon of uncertainty) for the uncertain variables so that the performance requirement(s) is (are) satisfied is of primary importance in the IGDT. On the other hand, the method of quantifier elimination in computer algebra permits the transformation of quantified formulae to equivalent formulae, but free from the quantified variables. Here the method of quantifier elimination is applied to the mixed case with two or three uncertain variables where one (or two) of these variables is (are) ordinary interval variable(s) whereas the remaining uncertain variable(s) satisfies (satisfy) the popular fractional-error model of uncertainty in the IGDT. Therefore, here the horizon of uncertainty concerns only the latter variable(s). The present method is illustrated in the following five simple applications: (i) the problem of the area of a rectangle, (ii) the problem of the volume of a rectangular cuboid, (iii) the problem of the buckling load of a fixed–free column, (iv) the problem of the equivalent spring constants of two elastic springs connected in series and in parallel and (v) the similar problem for the resistances of three resistors.
  • ItemOpen Access
    An application of Ben-Haim's info-gap decision theory (IGDT) to Todinov's method of algebraic inequalities by employing the method of quantifier elimination
    (Κανένας, 2022-10-10) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Problems under uncertainty conditions can be studied by using the very interesting and popular Ben-Haim's info-gap (or information-gap) decision theory (IGDT). On the other hand, recently, Todinov proposed an 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. One of the main problems where Todinov applied his new method is the problem concerning the equivalent resistances of n resistors in an electrical circuit connected both in series and in parallel. Here we consider the same problem, but now with the related algebraic inequality used as the performance requirement in Ben-Haim's IGDT. The methodology used here is based on the computational method of quantifier elimination. This method constitutes a very interesting approach for the transformation of quantified formulae to logically equivalent formulae, but now free from the quantifiers and the quantified variables. The same method is implemented in some computer algebra systems including Mathematica, which is used here. The problems studied here and related to the equivalent resistances of two or three resistors concern (i) two resistors with one horizon of uncertainty including the cases of parametric nominal value(s) of one resistance or both resistances here by using a fractional-error uncertainty model in Ben-Haim's IGDT, (ii) two resistors again, but with two horizons of uncertainty, (iii) three resistors with one horizon of uncertainty and (iv) two resistors again, but with the use of an ellipsoidal uncertainty model. The use of negated existentially quantified formulae instead of universally quantified formulae is also studied.
  • ItemOpen Access
    Applications of quantifier elimination to the proofs of algebraic inequalities in engineering problems related to Todinov's method for risk reduction
    (Κανένας) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    An 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.
  • ItemOpen Access
    Problems under uncertainty : quantifier elimination to universally–existentially (AE) quantified formulae related to two or more horizons of uncertainty
    (Κανένας) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Problems 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.
  • ItemOpen Access
    Quantifier elimination and quantifier-free formulae for universally–existentially (AE) quantified formulae in Ben-Haim's info-gap model of uncertainty
    (Κανένας) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The 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.
  • ItemOpen Access
    Robust reliability under uncertainty conditions by using modified info-gap models with two to four horizons of uncertainty and quantifier elimination
    (Κανένας) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    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).
  • ItemOpen Access
    Application of quantifier elimination to robust reliability under severe uncertainty conditions by using the info-gap decision theory (IGDT)
    (Αυτο-έκδοση) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    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.
  • ItemOpen Access
    Uncertainty intervals/regions for the stress intensity factors at crack tips under uncertain loading by using the ellipsoidal model and numerical integration
    (Αυτο-έκδοση) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    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 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.
  • ItemOpen Access
    Application of the method of quantifier elimination to the determination of intervals when the uncertain parameters satisfy an ellipsoidal inequality constraint
    (Κανένας) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    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.
  • ItemOpen Access
    Generalized interval-based polynomial approximations to functions in applied mechanics by using the method of quantifier elimination
    (Αυτο-έκδοση) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The 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.
  • ItemOpen Access
    Quantifier-elimination-based interval computations in beam problems studied by using the approximate methods of finite differences and of finite elements
    (Αυτο-έκδοση) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The 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.
  • ItemOpen Access
    Determination of intervals in systems of parametric interval linear equilibrium equations in applied mechanics with the method of quantifier elimination
    Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The 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.
  • ItemOpen Access
    Intervals for the resultants of interval forces with existentially and/or universally quantified formulae with the help of the method of quantifier elimination
    Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The problem of the computation of the interval of the resultant of collinear uncertain forces represented by intervals without overestimation has been recently studied in two papers (i) by Elishakoff, Gabriele and Wang (2016) and (ii) by Popova (2017). In the first paper, a modification of classical interval arithmetic is proposed whereas the methodology proposed in the second paper is based on the algebraic extension of classical interval arithmetic. Here the general case of the computation of the interval of this resultant is studied in detail on the basis of the use of quantified formulae including the existential and/or the universal quantifiers with respect to the interval forces. Many quantified formulae are possible in a resultant problem and the method of quantifier elimination in its implementation in the computer algebra system Mathematica is used for the derivation of the related quantifier-free formulae. After the illustration of the present approach in the elementary subtraction problem, which is well known for the overestimation phenomenon, the same approach is illustrated in problems (originally studied in the above papers) concerning the resultants of two, three and four collinear forces with different directions as well as in the problem of three collinear forces acting on a box. Symbolic intervals with parameters one or two of the forces are also computed. The case of the resultant of many collinear interval forces is also successfully studied. The conclusion drawn is that several overestimation-free, exact intervals can be computed for the resultant of interval forces (frequently including a degenerate interval: sharp resultant) and the derived interval (if it exists) strongly depends on the quantifiers used for the interval forces.
  • ItemOpen Access
    Interval computations in various direct and inverse applied mechanics problems related to quantifiers by using the method of quantifier elimination
    Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Quantifier elimination offers an interesting computational tool in many research areas including applied mechanics long ago. For example, quantifier elimination was recently applied to the computation of ranges of functions in problems of applied mechanics. Here we modify this approach by using the existential quantifier instead of the universal quantifier in the quantified formulae. This approach permits the reduction (by two) of the number of free variables. Yet, what is more important is that here we also extend this method based on quantifier elimination from the purely existential case to the mixed universal–existential case. The latter case is related to the classical interval tolerance and control problems so popular in interval analysis. Among the few implementations of quantifier elimination (in classical real analysis) in computer algebra systems again we selected the computer algebra system Mathematica for use in the present computations because it seems to offer the most efficient and user-friendly related implementation. Three applied mechanics problems are studied in detail: (i) a classical beam problem (beam fixed–simply-supported at its ends) under a uniform loading, (ii) a problem of a beam on a Winkler elastic foundation and (iii) the problem of free vibrations of the classical damped harmonic oscillator under critical damping. In these three problems, several quantified formulae were considered (of course, under appropriate assumptions) and the related QFFs (quantifier-free formulae) were easily derived. Moreover, the cases of (i) three interval variables and no parameter in the QFF, (ii) two interval variables and one parameter in the QFF and (iii) one interval variable and two parameters in the QFF were studied.
  • ItemOpen Access
    Symbolic intervals for the unknown quantities in simple applied mechanics problems with the computational method of quantifier elimination
    Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Quantifier elimination constitutes an interesting computational method in computer algebra for real polynomial and rational functions permitting the elimination of the universal quantifier "for all'' and/or the existential quantifier "exists'' in quantified formulae. This method was successfully applied to several applied mechanics problems during the last twenty-five years. On the other hand, interval analysis was also successfully employed in a very large number of applied mechanics problems under uncertainty conditions mainly during the same period permitting the determination of intervals for quantities of interest. Recently, the efficient implementation of quantifier elimination in the computer algebra system Mathematica was used in some applied mechanics problems for the computation of such intervals mainly for forces and for displacements in simple truss problems. On the other hand, because computer algebra systems are mainly used for symbolic computations, it seems natural to extend the application of quantifier elimination to the determination of intervals for quantities of mechanical interest from numerical intervals to symbolic intervals. This seems to be a simple task and, hence, it can be successfully applied to simple applied mechanics problems. Here a classical problem of a six-member truss is used for the illustration of this approach and sharp, exact symbolic intervals including one or two parameters of the problem (one or two cross-sectional parameters and/or a loading parameter) are derived of course based on numerical intervals again for one or two interval parameters. Next, the case of symbolic intervals for one or two interval parameters is also considered. In both cases, the derivation of symbolic intervals significantly extends the derivation of numerical intervals as far as the generality of the computed intervals is concerned.
  • ItemOpen Access
    Sharp bounds based on quantifier elimination in truss and other applied mechanics problems with uncertain, interval forces/loads and other parameters
    Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The computational method of quantifier elimination in computer algebra for real numbers related to the elimination of the universal quantifier "for all" and/or the existential quantifier "exists" in quantified formulae has already been efficiently implemented in few computer algebra systems and also applied to several interesting problems of mathematics, physics and engineering. Here this method is applied to some simple problems of applied mechanics including truss problems in structural mechanics in the case of uncertain, interval parameters related to the applied loads and/or to the parameters of the structure. Therefore, the present results are related to classical interval analysis and they permit the determination of sharp bounds for the mechanical quantities of interest such as resultants of forces, reactions and displacements in truss problems. The implementation of quantifier elimination in the powerful and user-friendly computer algebra system Mathematica has been selected as the most efficient and appropriate tool for the present computational tasks and it offers one more computational possibility in simple applied mechanics problems under uncertainty that is described by interval parameters. The five applied mechanics problems studied in detail here concern (i) the resultant of three forces acting on a box, (ii) the resultant of four forces acting on a particle, (iii) a block resting and sliding on a horizontal plane, (iv) a three-member truss and, finally, (v) a six-member truss. All these problems were already proposed and solved with interval parameters by other researchers. The present results are in agreement with the already available related results and, moreover, they always provide sharp bounds for the interval quantities of interest.
  • ItemOpen Access
    Sharp enclosures of the real roots of the classical parametric quadratic equation with one interval coefficient by the method of quantifier elimination
    Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The method of quantifier elimination constitutes an interesting rather recent computational method in computer algebra implemented in few computer algebra systems. Here we apply this method to the determination of sharp enclosures of the two real roots (when there exist such roots) of the classical parametric quadratic equation (in its complete form with three parameters) with one interval coefficient, which is here an interval parameter, whereas the remaining two coefficients are crisp (deterministic) parameters. The powerful computer algebra system Mathematica is used in all the present computations. The classical closed-form formulae for the above two roots are not required in the present quantifier-elimination-based approach: only the original quadratic equation is employed during quantifier elimination. All three cases of parametric coefficients in the quadratic equation are studied in detail and sharp enclosures of its roots are derived in parametric forms. The present results are also verified by using minimization and maximization commands directly on the closed-form formulae for these two roots. Several numerical applications now with numerical (instead of parametric) intervals for the interval coefficient are also made. The present results are seen to be in complete agreement with previous related original results by Elishakoff and Daphnis, who appropriately used classical interval analysis and based their results on the classical closed-form formulae for these roots. Finally, the enclosures of the roots derived by the present approach are always sharp without any possibility of overestimation contrary to what happens in the classical interval-analysis-based approach, where overestimation may be present in some cases.
  • ItemOpen Access
    Interval computations in the formulae for the stress intensity factors at crack tips using the method of quantifier elimination
    Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The concept of the stress intensity factor at a crack tip is extremely well known and it plays a very important role in fracture mechanics. On the other hand, uncertainty is often present in engineering problems mainly because of measurement errors and it is frequently represented with the help of interval variables. Here we consider the case of formulae for the computation of stress intensity factors at crack tips with one or more than one variable in such a formula being an interval variable. In this case, we compute the related intervals for the stress intensity factors, which, naturally, are also interval variables. This computation is based on the related existentially quantified formulae and it is made with the help of the interesting computational method of quantifier elimination as this method is efficiently implemented in the computer algebra system Mathematica. More explicitly, here the following four classical crack problems are studied: (i) the problem of a straight crack in an infinite plane isotropic elastic medium under a tensile loading at infinity normal to the crack, (ii) the related problem of a slant straight crack with respect to the loading at infinity, (iii) the problem of a crack in a similar medium now under an exponential normal loading on its edges and (iv) the problem of a periodic array of collinear straight cracks again in an infinite plane isotropic elastic medium under a tensile loading at infinity normal to the cracks. In the third and the fourth problems, approximate formulae for the stress intensity factors are used. The present results permit the efficient evaluation of the intervals (the ranges) for the stress intensity factors at crack tips when interval variables instead of crisp (deterministic) variables are present in the related formulae without any overestimation of the intervals for the stress intensity factors. Naturally, the present method is also applicable to more difficult crack problems provided, of course, that the total number of variables in the existentially quantified formulae used for quantifier elimination is small (generally up to five or six variables); otherwise, quantifier elimination may fail to yield a QFF (quantifier-free formula) at least in a reasonable time interval. The present results constitute one more application of quantifier elimination and interval analysis to applied mechanics, here to fracture mechanics.