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- 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.
- ItemComputation of ranges of functions in problems of applied mechanics with the computational method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe method of quantifier elimination constitutes an interesting and relatively modern computational tool in computer algebra. An efficient implementation of quantifier elimination is included in the computer algebra system Mathematica since 2003. Here after an introduction to the approach of quantifier elimination in Mathematica for the determination of intervals in some simple problems in arithmetic and ranges of functions in elementary algebra, we continue with the determination of ranges of functions concerning problems of applied mechanics. Four such kinds of problems are studied here: (i) Two classical beam problems, (ii) A problem of a beam on a Winkler elastic foundation, (iii) The problem of buckling of the Euler classical column and (iv) A problem of free vibrations of an oscillator with critical damping. In all cases, the ranges of the functions of interest are determined. Yet, Taylor–Maclaurin or minimax or similar approximations are necessary in problems where transcendental functions are involved. Naturally, the method is applicable to a variety of additional problems of applied mechanics although, unfortunately, its power is limited to problems with few variables (both quantified variables and free variables) and not very high degree(s) in the polynomial(s) involved. Therefore, the efficiency of the method is particularly clear mainly in problems involving only a single polynomial of degree about up to twenty and with only one parameter. The case of an interval with a parameter as one end is also in principle acceptable.
- 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.
- ItemStress concentration factors: determination of ranges of values by the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe interesting and modern method of quantifier elimination in computer algebra was already applied to a very large number of problems including, e.g., ranges of functions and several applied mechanics problems. Here this method is applied to the determination of ranges of stress concentration factors. The classical related handbook is Peterson's Stress Concentration Factors. Here three classical elasticity problems with stress concentration factors included in this handbook are studied in detail with respect to their ranges of values. These problems concern (i) a tension strip with two opposite semi-circular edge notches, (ii) a tension strip with a circular hole and (iii) an infinite medium with a rectangular hole with rounded corners. In these problems, the ranges of the approximate stress concentration factors on physically appropriate intervals for their variables are determined. Beyond the very simple case of stress concentration factors with no parameters (only their variables), the more interesting cases (i) of stress concentration factors with parameters (here dimensionless length parameters) and (ii) of parametric intervals, i.e. intervals with symbols as their endpoints, are also studied. In all cases, the efficient implementation of quantifier elimination in the popular computer algebra system Mathematica was used for the derivation of the ranges of the corresponding stress concentration factors frequently in parametric forms. Naturally, such a range can directly provide the related numerical range for specific value(s) of the parameter(s) involved. Extensions of the approach, e.g. to stress intensity factors in crack problems, are also possible.
- 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.
- ItemSharp enclosures of the real roots of the classical parametric quadratic equation with one interval coefficient by the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe 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.
- 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.
- 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.
- ItemApplication of quantifier elimination to inverse free vibration problems for inhomogeneous beams and bars
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςInverse free vibration problems for inhomogeneous beams under various boundary conditions were extensively studied by Elishakoff and his collaborators during the last two decades. In these problems, the linear mass density of the beam is assumed to have a polynomial form known in advance. Moreover, the mode shape of the beam is also assumed to have a simple polynomial form known in advance and, evidently, satisfying the four boundary conditions at the ends of the beam. Then, on the basis of the related ordinary differential equation, it is possible to determine the unknown flexural rigidity of the beam, which, naturally, should also have a polynomial form. Obviously, the linear mass density is selected to be a continuously positive function, but the same should also happen for the initially unknown flexural rigidity. The latter positivity requirement is the subject of the present results. This positivity is assured by determining the related necessary and sufficient positivity conditions on the whole vibrating beam by using the modern computational method of quantifier elimination, which is mainly based on the Collins cylindrical algebraic decomposition algorithm. Here the implementation of quantifier elimination in the computer algebra system Mathematica is used as the computational tool for the derivation of the present conditions, which leads to the elimination of the universal quantifier in the positivity condition and constitutes the equivalent quantifier-free formula. At first, the simple inverse vibration problem of a clamped beam is studied with respect to the aforementioned positivity requirement. Next, the inverse vibration problem of a beam clamped at one end and simply-supported at the other end is also studied. The resulting positivity conditions for the flexural rigidity of the beam are rather simple only for one or two parameters in the linear mass density of the beam, but they become sufficiently complicated for three parameters. An inverse problem of free axial vibrations of inhomogeneous bars is also studied in brief. The present computational approach constitutes a simple, efficient and mathematically rigorous way for the derivation of positivity conditions in inverse free vibration problems for the flexural/longitudinal rigidities of beams/bars. On the other hand, it constitutes an extension of previous recent quantifier elimination results concerning the related inverse buckling problem, where the same computational approach, that of quantifier elimination, was also successfully used.
- ItemAn inequality constraint for the deflection of an elastic beam under a uniform distributed loading
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe problem of the deflection of a straight isotropic elastic beam under a uniform distributed loading during bending is reconsidered under the inequality constraint that this deflection should not exceed a critical value because of the existence of a rigid obstacle or because of strength or even aesthetic reasons. This problem reduces to the problem of positivity of an appropriate quartic polynomial along the beam, which is a computational quantifier elimination problem and can further be solved by using classical Sturm–Habicht sequences in the theory of polynomials. The final result is a logical combination of algebraic expressions including the parameters of the present beam problem, that is the deflections and the rotations at the beam ends, the constant distributed loading, the critical/maximum permissible deflection as well as the length and the flexural rigidity of the beam. More complicated loading conditions can also be considered by the same approach, which is also applicable to the classical finite element method in beam problems for each particular finite beam element.