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Generalized interval-based polynomial approximations to functions in applied mechanics by using the method of quantifier elimination

, 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.

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On the boundary integral equation method for the problem of a plane crack inside a three-dimensional elastic medium

, Ioakimidis, Nikolaos, Ιωακειμίδης, Νικόλαος

The problem of a plane crack of arbitrary shape and under an arbitrary normal pressure distribution inside an infinite three-dimensional isotropic elastic medium is reconsidered by the boundary integral equation method. This method is seen to be capable to produce the singular integral equation of this problem with one unknown function, i.e. the displacements of the points of the crack faces (and not the derivatives of this function), and this is achieved in two ways. This is an alternative and probably interesting new method for the derivation of the aforementioned singular integral equation having been previously derived by two other methods. Generalizations of the present results to more complicated problems follow trivially.

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A new method for the computation of the zeros of analytic functions

, Ioakimidis, Nikolaos, Anastasselou, Eleni, Ιωακειμίδης, Νικόλαος, Αναστασέλου, Ελένη

A 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.

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Stress concentration factors: determination of ranges of values by the method of quantifier elimination

, Ioakimidis, Nikolaos, Ιωακειμίδης, Νικόλαος

The 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.

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Detection of loading/geometrical singularities in isotropic elastic media with the use of Gröbner bases

, Ioakimidis, Nikolaos, Ιωακειμίδης, Νικόλαος

The method of Gröbner bases in computer algebra is applied to the detection and location of loading or geometrical singularities inside two- or three-dimensional isotropic elastic media. The approach consists in using available experimental data concerning the stress/strain components away from the singularity in order to decide whether these components verify the relations corresponding to the particular singularity or not. These conditions can frequently be obtained by the method of Gröbner bases and the related Buchberger algorithm. In the case of an affirmative conclusion, we can further proceed to the location of the singularity. Alternatively, the present approach yields compatibility equations for the stress/strain components in elastic media with loading/geometrical singularities. These equations are displayed in detail for some common stress fields corresponding to concentrated forces as well as to a crack tip.

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Computation of ranges of functions in problems of applied mechanics with the computational method of quantifier elimination

, Ioakimidis, Nikolaos, Ιωακειμίδης, Νικόλαος

The 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.

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Application of Mathematica to the Rayleigh–Ritz method for plane elasticity problems

, Ioakimidis, Nikolaos, Ιωακειμίδης, Νικόλαος

The 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.

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A new approach to the derivation of exact integral formulae for zeros of analytic functions

, Ioakimidis, Nikolaos, Ιωακειμίδης, Νικόλαος

A 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.

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Problems under uncertainty : quantifier elimination to universally–existentially (AE) quantified formulae related to two or more horizons of uncertainty

, 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.

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Application of complex path-independent integrals to locating circular holes and inclusions in classical plane elasticity

, Ioakimidis, Nikolaos, Ιωακειμίδης, Νικόλαος

We 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.