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  • ItemOpen Access
    A note on the closed-form determination of zeros and poles of generalized analytic functions
    (Elsevier Science Publishing Co., Inc., 1989-12) Ioakimidis, Nikolaos
    The generalized method of Burniston and Siewert for the derivation of closed-form formulae for the zeros (and/or poles) of analytic functions inside a closed contour in the complex plane is further extended to the case of generalized analytic functions with real and imaginary parts satisfying homogeneous generalized Cauchy–Riemann equations. Two special cases and one generalization of this approach are also considered in brief.
  • ItemOpen Access
    The crack tip elastic stress field using computer algebra software
    (Pergamon Press, 1991) Ioakimidis, Nikolaos
    In various cases, we are interested in the equations for the elastic stress field components near a crack tip expressed in series forms. Frequently, the dominant terms in these series are not sufficient and additional terms are taken into account. Here we use the computer algebra system DERIVE in order to derive these series symbolically. (Additional popular computer algebra systems can also be successfully used.) The case of a finite straight crack under normal loading at infinity was selected for the illustration of the approach. The obtained symbolic results for the Westergaard complex potential, its derivative and the stress components are displayed in detail. The present results clearly show that the automatic symbolic series expansion of the components of the elastic field of interest near a crack tip is just a routine process if computer algebra software facilities are used.
  • ItemOpen Access
    Orders of singularity at wedge apices: the computer algebra approach
    (Pergamon Press, 1991) Ioakimidis, Nikolaos
    We illustrate the possibilities of computer algebra systems in performing the necessary tedious and, manually, error-prone symbolic computations during the determination of the orders of singularity at wedge apices in plane elasticity. The cases of the first, second and mixed fundamental problems for a simple wedge as well as the case of the first fundamental problem for a double wedge were used as examples of the approach (together with two special cases in crack problems). The rather elementary (yet powerful) computer algebra system DERIVE was used in the present formal computations.
  • ItemOpen Access
    Application of computer algebra to the iterative solution of singular integral equations
    (North-Holland (an imprint of Elsevier Science Publishers), 1992-01) Ioakimidis, Nikolaos
    The use of computer algebra systems in the approximate iterative solution of Cauchy-type singular integral equations (SIEs) appearing in crack problems of plane elasticity and in many additional branches of applied mechanics and engineering is proposed. The resulting method combining numerical and symbolic computations is called a SAN (semi-analytical/numerical) method. The computer algebra system DERIVE is used in the present computations, where the problem of a periodic array of collinear cracks has been selected for the illustration of the present approach. Extensive SAN results are displayed and their convergence is shown as well as their agreement (for the stress intensity factor) with the already available theoretical formula in this concrete application. Generalizations of the present SAN approach are completely possible.
  • ItemOpen Access
    Application of DERIVE to conformal mapping techniques in plane elasticity problems
    (Pergamon Press, 1991) Ioakimidis, Nikolaos
    Computer algebra systems have been already applied to several applied mechanics and plane elasticity problems. In this paper, we show the possibility of using DERIVE, a simple, but also powerful and modern computer algebra system, for the determination of the conformal mapping function in plane elasticity problems associated with holes (including triangles and squares). We display the obtained results in sufficient detail for the equilateral triangle and the square holes and we make several comments concerning the efficient use of DERIVE in the present application, where computer algebra systems seem not having been used up to now.
  • ItemOpen Access
    Application of complex path-independent integrals to problems of bending of thin elastic plates
    (Springer-Verlag, 1992-08) Ioakimidis, Nikolaos
    Complex path-independent integrals have been already widely applied to problems of plane and antiplane elasticity for the determination of a variety of quantities of interest including stress intensity factors, loading intensities and the positions of geometrical characteristic lengths of singularities in the elastic field (like cracks, holes and inclusions). In this paper, we show that the same results also apply to the case of problems of thin isotropic elastic plates under bending, where the complex-variable formulation is also valid. We make reference to the experimental methods which are appropriate for these integrals in an engineering environment and, finally, we apply this approach to the location of a circular hole in the problem of bending of a thin plate. Numerical results are also presented.
  • ItemOpen Access
    The energy method in problems of buckling of bars with quantifier elimination
    (Elsevier Science and The Institution of Structural Engineers, 2018-02) Ioakimidis, Nikolaos
    The classical energy method for the approximate determination of critical buckling loads of bars is revisited. This method is based on the stability condition of the bar and on the appropriate selection of an approximation to the deflection of the bar. Moreover, it is frequently related to the Rayleigh quotient or to the Timoshenko quotient for the determination of the critical buckling load. Here we will use again the energy method for the determination of critical buckling loads of bars but now on the basis of a new computational approach. This new approach consists in using the modern computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica instead of partial differentiations when we use the stability condition of the bar or essentially equivalently when we minimize the Rayleigh quotient or the Timoshenko quotient. This approach, which avoids partial differentiations, is also more rigorous than the classical approach based on partial derivatives because it does not require the use of the conditions for a minimum based on second partial derivatives, which are generally ignored in practice. Moreover, it is very simple to use inside the powerful computational environment offered by Mathematica. The present approach is illustrated in several buckling problems of bars including parametric buckling problems. Buckling problems of bars with two internal unilateral constraints, where the classical energy method is difficult to apply, are also studied. Even in this rather difficult application the critical buckling load is directly determined with a sufficient accuracy.
  • ItemOpen Access
    Supplementing the numerical solution of singular/hypersingular integral equations/inequalities with parametric inequality constraints with applications to crack problems
    (Institute of Fundamental Technological Research of the Polish Academy of Sciences, 2017) Ioakimidis, Nikolaos
    Singular and hypersingular integral equations appear frequently in engineering problems. The approximate solution of these equations by using various numerical methods is well known. Here we consider the case where these equations are supplemented by inequality constraints mainly parametric inequality constraints, but also the case of singular/hypersingular integral inequalities. The approach used here is simply to employ the computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica and derive the related set of necessary and sufficient conditions for the validity of the singular/hypersingular integral equation/inequality together with the related inequality constraints. The present approach is applied to singular integral equations/inequalities in the problem of periodic arrays of straight cracks under loading- and fracture-related inequality constraints by using the Lobatto–Chebyshev method. It is also applied to the hypersingular integral equation/inequality of the problem of a single straight crack under a parametric loading by using the collocation and Galerkin methods and parametric inequality constraints.
  • ItemOpen Access
    Application of quantifier elimination to mixed-mode fracture criteria in crack problems
    (Springer-Verlag Germany, 2017-10) Ioakimidis, Nikolaos
    Several criteria for mixed-mode fracture in crack problems are based on the maximum of a quantity quite frequently related to stress components. This quantity should not reach a critical value. Computationally, this approach requires the use of the first and the second derivatives of the above quantity although frequently the use of the second derivative is omitted because of the necessary complicated computations. Therefore, mathematically, the determination of the maximum of the quantity of interest is not assured when the classical approach is used without the second derivative. Here a completely different and more rigorous approach is proposed. The present approach is based on symbolic computations and makes use of modern quantifier elimination algorithms implemented in the computer algebra system Mathematica. The maximum tangential stress criterion, the generalized maximum tangential stress criterion (with a T-stress term), the T-criterion and the modified maximum energy release rate criterion are used for the illustration of the present new approach in the mode I/II case. Beyond the conditions of fracture initiation, the determination of the fracture angle is also studied. The mode I/III case is also considered in brief. The present approach completely avoids differentiations, similarly the necessity of a distinction between maxima and minima, always leads to a global (absolute) and not to a local (relative) maximum and frequently to closed-form formulae and automatically makes a distinction of cases in the final formula whenever this is necessary. Moreover, its use is easy and direct and the maximum of the quantity of interest is always assured.
  • ItemOpen Access
    Application of quantifier elimination to inverse buckling problems
    (Springer-Verlag Austria, 2017-10) Ioakimidis, Nikolaos
    The inverse buckling problem for a column is the problem where both the loading and the buckling mode are defined in advance (the latter generally in a polynomial form) and the flexural rigidity of the column is sought in a similar form with the help of the related ordinary differential equation. This problem was proposed and studied in many buckling problems by Elishakoff and his collaborators. A serious difficulty in its solution is that the resulting flexural rigidity should be positive along the column. Here in order to check this positivity the modern computational method of quantifier elimination is proposed and used inside the computational environment offered by the computer algebra system Mathematica and mainly based on the Collins cylindrical algebraic decomposition algorithm. At first, the simple inverse buckling problem of an inhomogeneous column under a concentrated load is studied with respect to the aforementioned positivity requirement. Next, the much more difficult problem concerning a variable distributed loading is also studied both in the case of one parameter and in the case of two parameters in this loading. Parametric rational and trigonometric forms of the flexural rigidity are also studied. Naturally, the resulting conditions for the positivity of the flexural rigidity are rather simple for one loading parameter, but they may become sufficiently complicated for two loading parameters. The present computational approach constitutes a simple, efficient and mathematically rigorous way for the derivation of positivity conditions for the flexural rigidity of a column in a variety of inverse buckling problems.
  • ItemOpen Access
    Caustics, pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination
    (Elsevier, 2017-01) Ioakimidis, Nikolaos
    The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future.
  • ItemOpen Access
    Derivation of conditions of complete contact for a beam on a tensionless Winkler elastic foundation with Mathematica
    (Elsevier Science, 2016-03) Ioakimidis, Nikolaos
    The classical problem of a beam on a tensionless Winkler elastic foundation is reconsidered for the derivation of the conditions of complete contact between the beam and the foundation. This is achieved through the application of modern quantifier elimination software included in the computer algebra system Mathematica together with Taylor–Maclaurin series approximations to the deflection of the beam. Four particular beam problems have been considered in detail and the related QFFs (quantifier-free formulae) have been obtained for several values of the order in the series approximations. Additional approximation possibilities have also been investigated with an emphasis put on the use of the Galerkin method based on weighted residuals. The present results seem to constitute one more interesting application of modern quantifier elimination algorithms and the related software (here in Mathematica) to applied and engineering mechanics.
  • ItemOpen Access
    Finite differences/elements in classical beam problems: derivation of feasibility conditions under parametric inequality constraints with the help of Reduce and REDLOG
    (Springer-Verlag, 2001-02) Ioakimidis, Nikolaos
    The solution of the classical fourth-order ordinary differential equation for static beam problems by using the finite difference method is reconsidered, but this time for the derivation of feasibility conditions in cases of validity of parametric linear inequality constraints with respect to the loading/geometry of the beam. To this end, the computer algebra system Reduce has been used, but supplemented by its recent REDLOG (REDuce LOGic) package incorporating the efficient Weispfenning computational quantifier elimination algorithms. A particular problem for a finite beam loaded by a triangular loading has been employed as the vehicle for the illustration of the present approach and the derived feasibility conditions are displayed. The finite element method has also been used (instead of the finite difference method) in the same problem. The present results can also be generalized to problems of beams on an elastic foundation, to two-dimensional problems, to optimization problems, etc.
  • ItemOpen Access
    Derivation of feasibility conditions in engineering problems under parametric inequality constraints with classical Fourier elimination
    (John Wiley & Sons, 2000-08) Ioakimidis, Nikolaos
    Fourier (or Motzkin or even Fourier–Motzkin) elimination is the classical and equally old analogue of Gaussian elimination for the solution of linear equations to the case of linear inequalities. Here this approach (and two of its standard improvements) is applied to two engineering problems (involving numerical integration in fracture mechanics as well as finite differences in heat transfer in the case of existence of parameters) with inequality constraints. The results (solvent systems of inequalities including only the related parameters) concern the feasibility conditions (existential quantifier-free formulae) so that the satisfaction of the original system of linear inequality constraints can be possible (for appropriate values of the variables in it). Further applications, e.g. to Cauchy-type singular integral equations and to the boundary and finite element techniques in computational mechanics and engineering, are also possible. The powerful computer algebra system Maple V has been used and a related elementary procedure for Fourier elimination was prepared and is displayed. The competitive Weispfenning elimination approach is also reported in brief and several comments aiming at the understanding and further applicability of the present results in engineering are made. The present results constitute an extension of the already available applications of computer algebra software to the classical approximate–numerical methods traditionally employed in engineering and are also related to computational quantifier elimination techniques in computer algebra and applied logic.
  • ItemOpen Access
    On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations
    (John Wiley & Sons, 2000-04) Ioakimidis, Nikolaos
    Complex boundary integral equations (Fredholm-type regular or Cauchy-type singular or even Hadamard–Mangler-type hypersingular) have been used for the numerical solution of general plane isotropic elasticity problems. The related Muskhelishvili and, particularly, Lauricella–Sherman equations are famous in the literature, but several more extensions of the Lauricella–Sherman equations have also been proposed. In this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries (for anyone of the above equations: regular or singular or hypersingular, but with a prerequisite their actual numerical solution) through the appropriate use of the even more classical elementary Cauchy theorem in complex analysis. This approach has been already used for the accurate numerical computation of analytic functions and their derivatives by Ioakimidis, Papadakis and Perdios [BIT, 31, 276–285 (1991)], without applications to elasticity problems, but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non-analytic complex density function (such as an edge dislocation/loading distribution density) is numerically available on the boundary. The present results are also directly applicable to inclusion problems, anisotropic elasticity, antiplane elasticity and classical two-dimensional fluid dynamics, but, unfortunately, not to crack problems in fracture mechanics. Brief numerical results (for the complex potentials), showing the dramatic increase of the computational accuracy, are also displayed and few generalizations are proposed.
  • ItemOpen Access
    Automatic derivation of positivity conditions inside boundary elements with the help of the REDLOG computer logic package
    (Elsevier Science, 1999-12) Ioakimidis, Nikolaos
    The very well known shape functions are used in classical boundary element analysis for the construction of the polynomial interpolation function p for the approximation to the unknown field quantity u. In this note, we show the usefulness of the recent REDLOG Reduce computer logic package of Dolzmann and Sturm for the construction of positivity and analogous parametric feasibility conditions, based on the nodal values of the polynomial p, inside the whole boundary element. A simple one-dimensional case, based on the classical quadratic element, and a more difficult two-dimensional case, based on a quadratic triangular element, are used for the illustration of the approach, whereas several related conclusions are also drawn. The present results are applicable to cases where the positivity or the negativity or just the boundedness of the unknown quantity is required (such as the case of the pressure distribution in contact problems and the opening displacement in crack problems) and significantly extend the already available recent related computational quantifier elimination research results.
  • ItemOpen Access
    Fracture initiation at an elastic crack tip: a computational implementation of the T-criterion
    (Kluwer Academic Publishers, 1999-07) Ioakimidis, Nikolaos
    The classical problem of fracture initiation at an elastic crack tip in two-dimensional isotropic static elasticity and under mixed-mode loading conditions is revisited with the aim to provide simultaneously necessary and sufficient logical–algebraic conditions including all of the material and stress-state parameters involved (the mode I and II stress intensity factors, the Poisson ratio of the material and fracture-related material constants) and concerning either the avoidance or, completely equivalently, the initiation of fracture. Among several popular fracture criteria, the very efficient and sufficiently physically justified Theocaris–Andrianopoulos modified T-criterion, based on the maximum value of the dilatational strain energy density TV along an appropriate elastoplastic boundary about the crack tip of variable polar radius rcr = rcr(θ) (θ denoting the polar angle), determined in accordance with the von Mises classical yield criterion and corresponding to a constant value of the distortional strain energy density TD, has been adopted in the present algebraic approach. The computer algebra methods of Lazard, based on classical considerations about the quartic polynomial, and of González-Vega, based on standard Sturm–Habicht sequences, have been employed as the related computational quantifier elimination algebraic tools for the present computations concerning the positivity of an appropriate quartic polynomial. The present results permit the direct decision about fracture avoidance/initiation under concrete experimental (geometry–material–loading) conditions just by using the final quantifier-free formula, which is independent of the polar angle θ about the crack tip, and, therefore, without the need of numerical investigation of the maximum of the dilatational strain energy density TV along the preselected von Mises elastoplastic boundary. The present results seem also generalizable to several algebraically similar additional fracture criteria in plane elasticity under mixed-mode conditions.
  • ItemOpen Access
    Application of computer-generated finite-difference equations to decision and inverse problems in elasticity
    (Pergamon Press (Elsevier Science), 1998-09) Ioakimidis, Nikolaos
    Differential equations constitute a common tool for the solution of classical elasticity problems. Here we consider the possibility of using finite-difference equations in similar problems on the basis of experimental measurements of an elastic quantity (displacement, strain or stress component) at a small number of equidistant points. These equations can be derived by computer algebra techniques (Gröbner bases, characteristic sets and resultants) and they can be used for arriving at “decisions” whether a “singularity” (concentrated force, hole, inclusion, crack, etc.) exists or not and, further, for the determination of characteristic quantities concerning this singularity (such as the position of the centre and the radius of a circular hole or inclusion, the magnitude and the direction of a concentrated force or the intensity of the tensile loading, its direction at infinity, etc.). Elementary problems as well as the problems of a concentrated force, a crack and a circular hole in an infinite plane isotropic elastic medium (the latter under a tensile loading at infinity) illustrate the present approach. Furthermore, the problem of torsion of an elastic bar with an elliptical cross-section is used for the construction of a finite-difference equation in two dimensions. Futher possibilities for this kind of decision and inverse problems in elasticity are also suggested in brief. The Buchberger algorithm for the derivation of Gröbner bases (as this is implemented in the computer algebra system Maple V) has been used here as the tool for the derivation of the present finite-difference equations.
  • ItemOpen Access
    Classical numerical methods in engineering: a note on existential quantifier elimination under parametric inequality constraints
    (John Wiley & Sons, 1998-02) Ioakimidis, Nikolaos
    In this paper, an attempt is made to show the usefulness of computational quantifier elimination (CQE) techniques in computer algebra inside classical numerical methods in engineering for the derivation of feasibility (consistency) conditions in problems with weakly parametric linear inequality constraints (with the parameters appearing only in their right-hand sides). A simple, but non-trivial, straight edge-crack problem in fracture mechanics under linear inequality constraints both on the applied loading along the crack faces and on the value of the stress intensity factor at the crack tip (associated with the Green/weight function method, numerical approximations and classical numerical integration) is used for an elementary illustration of the proposed approach. In this application, the method used tries to imitate the theoretical principle of the linear programming methods. The manually obtained related result is expressed as a disjunction of conjunctions of inequalities (as is frequently the case in similar CQE problems), and concrete numerical results are also displayed. The related influence of various approximations and the application of the trapezoidal quadrature rule are also considered in some detail. Further possibilities could concern the application of the approach to other numerical methods in engineering (such as to the finite and the boundary element methods, to singular and hypersingular integral equation methods, etc.) combined with efficient algorithms for linear inequality constraints such as the old Fourier and the recent Weispfenning elimination methods.
  • ItemOpen Access
    A numerical replacement of computer algebra methods for the derivation of polynomial equations in mechanics
    (Elsevier Science, 1997-11) Ioakimidis, Nikolaos
    The classical computer algebra methods of Gröbner bases and characteristic sets have been already used for the derivation of polynomial formulae, mainly in classical geometry. Here we derive analogous formulae in applied mechanics by using a completely different approach: the numerical determination of the coefficients in the required polynomial formulae without the use of computer algebra techniques. Two compatibility equations for the stress components in plane elasticity and the equation of caustics in fracture mechanics are used as the vehicle for the illustration of the present elementary numerical approach. Beyond the simplicity of the present method, its powerfulness consists mainly in the fact that the well-known disadvantages of computer algebra methods (in the required computer memory and, mainly, computer time) are now avoided. Generalizations of the present approach follow easily (e.g. from polynomial to differential equations).