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- ItemOpen AccessA closed-form formula for the critical buckling load of a bar with one end fixed and the other pinned
Γενικό Τμήμα (Τεχνικές Αναφορές)Anastasselou, Eleni; Ioakimidis, Nikolaos; Αναστασέλου, Ελένη; Ιωακειμίδης, ΝικόλαοςThe classical problem of elastic buckling of a bar with one end fixed and the other pinned is reconsidered and a closed-form formula for the critical buckling load is derived. This is achieved through the closed-form solution (in terms of two regular integrals) of the transcendental equation tan u = u, to which this problem is reduced. The method of solution of this equation is too simple and based on a generalized form of the Cauchy theorem in complex analysis; yet the sought root of this equation does not contain complex quantities. Finally, numerical results verifying the validity of the derived formula are presented.
- ItemOpen AccessA hybrid method for the solution of problems of bending of thin plates
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςAn extremely elementary hybrid method for the solution of the problem of thin plates (either finite or infinite with holes) under the action of bending moments and normal forces along the boundary of the plate is proposed. This method is based on the optical method of studying the deformed shape of the boundary of the plate (method of reflected light or `pseudocaustics') and, subsequently, on the application of elementary analytical techniques from the theory of complex variables. All quantities of interest (deflections, displacements, bending and torsional moments and shear stresses) are determined completely in the whole plate by the present hybrid method.
- ItemOpen AccessA modification of the generalized airfoil equation and the corresponding numerical methods
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe two-dimensional problem of steady, inviscid, irrotational, subsonic flow around a straight or curvilinear thin airfoil or an array of such airfoils inside a wind tunnel is generally reduced to a one-dimensional Cauchy type real or complex singular integral equation called generalized airfoil equation. Here a new form of this equation is suggested (with no change in the unknown function) with index equal to 1 instead of 0. The new equation is supplemented by an integral condition assuring the uniqueness of its solution. This modification of the generalized airfoil equation permits the application of the theoretical results and the algorithms for the numerical solution of Cauchy type singular integral equations with index equal to 1 (mainly appearing in crack problems in the theory of plane elasticity) to the generalized airfoil equation and it establishes the relationship between crack and airfoil problems. Moreover, it permits the utilization of the classical Chebyshev polynomials instead of the airfoil polynomials. Three applications are also made and numerical results are presented.
- ItemOpen AccessA new approach to the construction of some path-independent integrals about crack tips
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςA new method, based on the complex potentials of Kolosov–Muskhelishvili and Cauchy's theorem in complex analysis, is applied to the establishment of path-independence of some integrals along a curve surrounding the tip of a crack in plane isotropic or anisotropic elasticity. The cases considered are (i) of loaded straight cracks in plane isotropic elasticity, (ii) of unloaded cracks having the shape of a circular arc (circular-arc-shaped cracks) in plane isotropic elasticity and (iii) of unloaded straight cracks in plane anisotropic elasticity. Further generalizations of the proposed method can be easily made.
- ItemOpen AccessA new approach to the derivation of exact integral formulae for zeros of analytic functions
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςA new method for the reduction of the problem of locating the zeros of an analytic function inside a simple closed contour to that of locating the zeros of a polynomial is proposed. The new method (exactly like the presently used classical relevant method) permits in this way the derivation of exact integral formulae for these zeros if they are no more than four. The present approach is based on the solution of a simple homogeneous Riemann–Hilbert boundary value problem. An application to a classical problem in physics concerning neutron moderation is also made and numerical results obtained by using the trapezoidal quadrature rule are presented.
- ItemOpen AccessA new method for the computation of the zeros of analytic functions
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Anastasselou, Eleni; Ιωακειμίδης, Νικόλαος; Αναστασέλου, ΕλένηA new method for the computation of the zeros of analytic functions (or the poles of meromorphic functions) inside or outside a closed contour C in the complex plane is proposed. This method is based on the Cauchy integral formula (in generalized forms) and leads to closed-form formulae for the zeros (or the poles) if they are no more than four. In general, for m zeros (or poles) these can be evaluated as the zeros of a polynomial of degree m. In all cases, complex contour integrals have to be evaluated numerically by using appropriate numerical integration rules. Several practical algorithms for the implementation of the method are proposed and the method of Abd-Elall, Delves and Reid is rederived by two different approaches as one of these algorithms. A numerical application to a transcendental equation appearing in the theory of neutron moderation is also made and numerical results of high accuracy are easily obtained.
- ItemOpen AccessA new quadrature method for locating the zeros of analytic functions with applications to engineering problems
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςA new method for the computation of real or complex zeros of analytic functions and/or poles of meromorphic functions outside a fundamental interval [a,b] of the real axis is proposed. This method is based on appropriately taking into account the error terms in the Gauss– and Lobatto–Chebyshev quadrature rules for ordinary integrals and it leads to a very simple non-iterative algorithm for the computation of these zeros. The results obtained by this algorithm with very few functional evaluations are of a very good accuracy and they can further be improved, if required, by local methods, which are generally inappropriate for the original localization of the zeros. The proposed method was tested in two engineering problems: a problem of neutron moderation in nuclear reactors and a problem of determining the critical buckling load of an elastic frame. The corresponding transcendental equations were solved by this method and numerical results for their zeros are presented. In all equations solved, numerical values for their zeros accurate to at least five significant digits were obtained by the present method with no more than thirteen functional evaluations.
- ItemOpen AccessA new semi-Gaussian quadrature rule for finite-part integrals in crack problems with a second-order singularity
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςFinite-part (hypersingular or Hadamard-type) integrals appear, naturally, in two-dimensional crack and additional problems in three-dimensional elasticity. Their computation along the radial direction leads to a one-dimensional finite-part integral on the interval [0,1] with a second-order singularity at the left end, x=0, of this interval. Kutt's Gauss–Jacobi-equivalent quadrature rule (on [0,1]) is the natural approach to the computation of this finite-part integral, but the appearance of two complex conjugate nodes (and analogous weights) outside the integration interval is a physically serious disadvantage to its use. Although the subtraction of the singularity is completely possible, here a new, semi-Gaussian quadrature rule is suggested for n = 4, 5, . . . , 10 nodes, where no node lies outside the integration interval [0,1] (exactly as in classical Gaussian quadrature rules) and, moreover, no derivative of the integrand appears in the approximation to the integral on [0,1]. The polynomial accuracy of the present quadrature rule for finite-part integrals is very high and equal to 2n-3. The proposed quadrature rule seems to be a reasonable choice (as an alternative both to Kutt's rule and to the subtraction of the singularity approach) during the evaluation of two-dimensional finite-part integrals in crack and related problems. The computation of the nodes and the weights (for n = 4, 5, . . . , 10 nodes) is described in brief and numerical results for both of these quantities are displayed together with experimental numerical results, which illustrate both the accuracy and the rapid convergence of the present quadrature rule.
- ItemOpen AccessA one-dimensional hypersingular integral equation for axisymmetric planar cracks in three-dimensional isotropic elasticity
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe problem of an axisymmetric planar crack (or a system of such cracks) under an arbitrary axisymmetric normal loading distribution inside an infinite three-dimensional isotropic elastic medium is reduced to a one-dimensional hypersingular integral equation (with a second-order, a first-order and a logarithmic singularity in its kernel). This equation seems to be most appropriate for the solution of the aforementioned crack problem. On the other hand, this equation seems to be unique in physical and engineering problems and may draw the interest in its investigation from the mathematical (theoretical and numerical) point of view.
- ItemOpen AccessA quadrature method for the numerical solution of two real nonlinear equations in two unknowns
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςA quadrature method for the numerical solution of a system of two real nonlinear equations in two unknowns in a region of the xy-plane, based on the use of two appropriate numerical integration rules for ordinary integrals in this region, is proposed. The main advantage of the method, beyond its originality and peculiarity, is the fact that no initial approximation to the sought solution is required. Three numerical applications, where the Gauss– and Lobatto–Chebyshev quadrature rules have been used, show the efficiency and the convergence of the proposed method. A generalization to the case of four real nonlinear equations in four unknowns (by using quaternions) is also suggested in brief.
- ItemOpen AccessA real closed-form integral formula for roots of nonlinear equations
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςAn analytical integral formula for a single simple root of a nonlinear equation in a finite interval is proposed. The derivation of this formula is based on Picard's method for the number of roots of nonlinear equations and on Green's theorem. The present formula is particularly useful for the location of roots of nonanalytic functions. A comparison between the present formula and a related classical formula based on the theory of analytic functions is also made.
- ItemOpen AccessAn application of Ben-Haim's info-gap decision theory (IGDT) to Todinov's method of algebraic inequalities by employing the method of quantifier elimination
Γενικό Τμήμα (Τεχνικές Αναφορές)(Κανένας, 2022-10-10) Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςProblems 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 AccessAn 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.
- ItemOpen AccessApplication of complex path-independent integrals to locating circular holes and inclusions in classical plane elasticity
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςWe propose an elementary method, based on complex path-independent integrals and the classical complex potentials of Kolosov–Muskhelishvili, for the location of the position of the centre and the determination of the radius of circular holes and inclusions of a different material (either simply inserted or attached) in an infinite plane isotropic elastic medium. In practice, the method of pseudocaustics can be successfully used as the related experimental method. Generalizations of the present results follow trivially.
- ItemOpen AccessApplication of Mathematica to the Rayleigh–Ritz method for plane elasticity problems
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe classical Rayleigh–Ritz method for plane isotropic elasticity problems governed by the well-known biharmonic equation (satisfied by the Airy stress function) is revisited. The modern and powerful computer algebra system Mathematica was employed for the symbolic/numerical approximate solution of the biharmonic equation. A related simple procedure was prepared and the classical problem of a rectangular elastic region loaded by a parabolic tensile loading was chosen as an example of the application of the approach. The available symbolic/numerical results in the literature and additional more complicated analogous results were directly derived by using the aforementioned procedure. Further related possibilities and generalizations are also discussed in brief.
- ItemOpen AccessApplication 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.
- ItemOpen AccessApplication 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.
- ItemOpen AccessApplication of the method of caustics to the experimental determination of the Poisson ratio
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe classical optical method of caustics is used for the direct experimental determination of the Poisson ratio of an isotropic elastic material. Only one simple tension experiment and the measurement of two diameters of caustics are required. The numerical values of all other constants in the experimental set-up are not required. This method is applicable both for reflected and for transmitted caustics. The case of the experimental determination of the modulus of elasticity (by a generalization of the above method) is also described in brief. Several additional possibilities are also reported.
- ItemOpen AccessApplication 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.
- ItemOpen AccessApplications 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.