Exact solution of the problem of plane bending of a layered ring

An exact solution is given for the problem of a closed cylindrical transversely isotropic ring subjected to bending in its own plane under an arbitrary external load; it is assumed that a fiber straight before deformation and orthogonal to the neutral surface remains straight and undeformed but may no longer be orthogonal to the neutral surface. A variant of the theory developed by Korolev is employed.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 1, pp. 168–171, January–February, 1973.     

A method for the approximate solution of quasi-static problems for hardening bodies

A method for the approximate solution of quasi-static problems for hardening elastoplastic bodies is proposed. The constitutive relation of the model is taken in the form of a variational inequality. An approximate solution of the initial problem is constructed in time steps and, by means of the finite element method, is reduced to the solution of a system of two variational inequalities in corresponding finite-dimensional space. It is shown that the solution of this system is equivalent to finding the saddle point of the corresponding quadratic functional. To find the saddle point, Udzawa's algorithm is used, by means of which the process of finding the velocity vector and stress tensor reduces to the successive calculation of these quantities: the velocity vector is determined from the variational inequality corresponding to the equilibrium equation, and the stress tensor is determined from the variational inequality corresponding to the constitutive relation. The latter inequality is reduced to a certain non-linear equation containing the operation of projection onto a closed convex set corresponding to the elastic strains of the medium. In turn, the solution of the non-linear equation is constructed using the method of successive approximations. To illustrate the use of the proposed method, the one-dimensional problem of the quasi-static deformation of a cylindrical tube under a load applied to its internal surface is considered.     

An exact finite deformation elasto-plastic solution for the outside-in free eversion problem of a tube of elastic linear-hardening material

An exact solution is obtained in this paper for the elasto-plasticoutside-in free eversion problem of a tube of elastic linear-hardeningmaterial using a tensorial formulation. The solution is basedon a finite-strain version of Hencky's deformation theory, thevon Mises yield criterion, and the assumptions of volume incompressibilityand axial length constancy. All expressions for the stress,strain distributions and the eversion load are derived in anexplicit form. In addition, with both the linear-elastic andstrain-hardening-plastic responses of the material being includedand with the thickness effect of the tube being incorporated,this solution provides a rigorous and complete theoretical analysisof the elasto-plastic eversion problem, unlike existing solutions.Two specific solutions are also presented as limiting casesof the solution. Also provided are some numerical results andthe related observations to show quantitatively applicationsof the solution.     

A homotopy analysis solution to large deformation of beams under static arbitrary distributed load

In this article, homotopy analysis method (HAM) is employed to investigate non-linear large deformation of Euler–Bernoulli beams subjected to an arbitrary distributed load. Constitutive equations of the problem are obtained. It is assumed that the length of the beam remains constant after applying external loads. Different auxiliary parameters and functions of the HAM and the extra auxiliary parameter, which is applied to initial guess of the solution, are employed to procure better convergence rate of the solution. The results of the solution are obtained for two different examples including constant cross sectional beam subjected to constant distributed load and periodic distributed load. Special base functions, orthogonal polynomials e.g. Chebyshev expansion, are employed as a tool to improve the convergence of the solution. The general solution, presented in this paper, can be used to attain the solution of the beam under arbitrary distributed load and flexural stiffness. Ultimately, it is shown that small deformation theory overestimates different quantities such as bending moment, shear force, etc. for large deflection of the beams in comparison with large deformation theory. Finally, it is concluded that solution of small deformation theory is far from reality for large deflection of straight Euler–Bernoulli beams.     

On parallel solution of linear elasticity problems. Part III: higher order finite elements

This is the third part of a trilogy on parallel solution of the linear elasticity problem. We consider the separate displacement ordering for a plain isotropic problem with full Dirichlet boundary conditions. The parallel solution methods presented in the first two parts of the trilogy are here generalised to higher order by using hierarchical finite elements. We discuss node numberings on regular grids for high degree of parallelism and even processor load as well as the problem of stability of the modified incomplete Cholesky factorisations used. Several preconditioning techniques for the conjugate gradient method are studied and compared for quadratic finite elements. Bounds for the condition numbers of the corresponding preconditioning methods are derived, and computer experiments are performed in order to confirm the theory and give recommendations on the choice of method. The parallel implementation is performed by message passing interface. Copyright © 2012 John Wiley & Sons, Ltd.     

On parallel solution of linear elasticity problems: Part I: theory

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.     

Analytical solution for large deflections of a cantilever beam under nonconservative load based on homotopy analysis method

In this article, large deflection and rotation of a nonlinear beam subjected to a coplanar follower static loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). The present solution can be used in wide range of load and length for beams under large deformations. The results obtained from HAM are compared with those results obtained by fourth order Range Kutta method. Finally, the load‐displacement characteristics of a uniform cantilever under a follower force normal to the deformed beam axis are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:541–553, 2011     

The hub location and network design problem with fixed and variable arc costs: formulation and dual-based solution heuristic

Many air, less-than-truck load and intermodal transportation and telecommunication networks incorporate hubs in an effort to reduce total cost. These hubs function as make bulk/break bulk or consolidation/deconsolidation centres. In this paper, a new hub location and network design formulation is presented that considers the fixed costs of establishing the hubs and the arcs in the network, and the variable costs associated with the demands on the arcs. The problem is formulated as a mixed integer programming problem embedding a multi-commodity flow model. The formulation can be transformed into some previously modelled hub network design problems. We develop a dual-based heuristic that exploits the multi-commodity flow problem structure embedded in the formulation. The test results indicate that the heuristic is an effective way to solve this computationally complex problem.     

On the algorithm of the solution of the signorini problem

An algorithm is proposed for solving the Signorini problem /1/ in the formulation of a unilateral variational problem for the boundary functional in the zone of possible contact /2/. The algorithm is based on a dual formulation of Lagrange maximin problems for whose solution a decomposition approach is used in the following sense: a Ritz process in the basis functions that satisfy the linear constraint of the problem, the differential equation in the domain, is used in solving the minimum problem (with fixed Lagrange multipliers); the maximum problem is solved by the method of descent (a generalization of the Frank-Wolf method) under convexity constraints on the Lagrange multipliers. The algorithm constructed can be conisidered as a modification of the well-known algorithm to find the Udzawa-Arrow-Hurwitz saddle points /3, 4/. The convergence of the algorithm is investigated. A numerical analysis of the algorithm is performed in the example of a classical contact problem about the insertion of a stamp in an elastic half-plane under approximation of the contact boundary by isoparametric boundary elements. The comparative efficiency of the algorithm is associated with the reduction in the dimensionality of the boundary value problem being solved and the possibility of utilizing the calculation apparatus of the method of boundary elements to realize the solution.     

Extension of the solution of nonlinear equations in the neighborhood of a bifurcation point

Using the method of continuous extension with respect to a parameter we develop a method of constructing the load trajectory of a structure having both limit points and bifurcation points. The method is applicable for the systems of nonlinear algebraic equations that describe the family of extremals that minimize the value of the total potential strain energy of the structure, and makes it possible to find all the branches of the load trajectory emanating from a bifurcation point and extend the solution along any of them. The method is based on the fact that the eigenvectors of the augmented Jacobian of the system of equations in the extended space of variables that correspond to zero eigenvalues on the main branch of the load trajectory are bifurcation vectors and form the active subspace of solutions of the equations of the extension. Meanwhile the other eigenvectors form the passive subspace that contains the extension vector with respect to the main branch of the load. As a result the entire process of computing the extension vector of the solution at any point of the load trajectory reduces to determining the eigenvectors of the augmented Jacobian of the original system of nonlinear algebraic equations, identifying them according as they belong to the active or passive subspace, and forming the extension vector of the solution using them and analytic relations Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 35–46.     

On the solution of concave knapsack problems

We consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint. We characterize strict local miniimizers of concave minimization problems subject to linear constraints, and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is NP-hard, it is possible to determine a local minimizer of this problem with at most O(n logn) operations and 1+[logn] evaluations of the function. If the function is quadratic this algorithm requires at most O(n logn) operations.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.Work supported in part by a Fannie and John Hertz Foundation graduate fellowship.     

Axiomatizations of the Euclidean compromise solution

A multicriteria optimization problem is one of choosing an alternative that optimizes several—possibly conflicting—objective functions simultaneously. The utopia point of a multicriteria optimization problem is the vector that specifies for each objective function the most favorable feasible value. The Euclidean compromise solution in multicriteria optimization is a solution that selects from a feasible set the alternative such that its vector of criteria values has minimal Euclidean distance to the utopia point. This paper provides several axiomatic characterizations of the Euclidean compromise solution that are based on consistency properties.     

Nonclassical pseudospectral method for the solution of brachistochrone problem

In this paper, nonclassical pseudospectral method is proposed for solving the classic brachistochrone problem. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Properties of nonclassical pseudospectral method are presented, these properties are then utilized to reduce the computation of brachistochrone problem to the solution of algebraic equations. Using this method, the solution to the brachistochrone problem is compared with those in the literature.     

Approximation solution of the supply management problem

The problem is considered of optimizing the product delivery from suppliers to consumers. The size of each open supply is bounded both below and above, the size of consumption for each consumer is bounded below, the supply cost functions are linear for nonzero volumes of supply. A fully polynomial time approximation scheme is proposed for this problem in the case of one consumer, and the complexity of the problem is studied in the general case.     

A complete solution of the normal Hankel problem

The normal Hankel problem is the one of characterizing the matrices that are normal and Hankel at the same time. We give a complete solution of this problem.     

Numerical solution of the Falkner-Skan equation based on quasilinearization

We present two iterative methods for solving the Falkner-Skan equation based on the quasilinearization method. We formulate the original problem as a new free boundary value problem. The truncated boundary depending on a small parameter is an unknown free boundary and has to be determined as part of solution. Using a change of variables, the free boundary value problem is transformed to a problem defined on [0, 1]. We apply the quasilinearization method to solve the resulting nonlinear problem. Then we propose two different iterative algorithms by means of a cubic spline solver. Numerical results for various instances are compared with those reported previously in the literature. The comparisons show the accuracy, robustness and efficiency of the presented methodology.     

Approximate method for the numerical solution of singular perturbation problems

We present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is divided into inner and outer region problems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem. In turn, the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm. The proposed method is iterative on the terminal point. Some numerical experiments have been included to demonstrate its applicability.     

Numerical solution of the linear inverse problem for the Euler-Darboux equation

An inverse problem of the reconstruction of the right-hand side of the Euler-Darboux equation is studied. This problem is equivalent to the Volterra integral equation of the third kind with the operator of multiplication by a smooth nonincreasing function. Numerical solution of this problem is constructed using an integral representation of the solution of the inverse problem, the regularization method, and the method of quadratures. The convergence and stability of the numerical method is proved.     

A solution of the Blasius problem

The classical Blasius boundary layer problem in its simplest statement consists in finding an initial value for the function satisfying the Blasius ODE on semi-infinite interval such that a certain condition at infinity be satisfied. Despite an apparent simplicity of the problem and more than a century of effort of numerous scientists, this elusive constant is determined at present numerically and not much better than it was done by Töpfer in 1912. Here we find this (Blasius) constant rigorously in closed form as a convergent series of rational numbers. Asymptotic behaviour, and lower and upper bounds for the partial sums of the series are also given.     

Preconditioning techniques for the iterative solution of scattering problems

We consider a time-harmonic electromagnetic scattering problem for an inhomogeneous medium. Some symmetry hypotheses on the refractive index of the medium and on the electromagnetic fields allow to reduce this problem to a two-dimensional scattering problem. This boundary value problem is defined on an unbounded domain, so its numerical solution cannot be obtained by a straightforward application of usual methods, such as for example finite difference methods, and finite element methods. A possible way to overcome this difficulty is given by an equivalent integral formulation of this problem, where the scattered field can be computed from the solution of a Fredholm integral equation of second kind. The numerical approximation of this problem usually produces large dense linear systems. We consider usual iterative methods for the solution of such linear systems, and we study some preconditioning techniques to improve the efficiency of these methods. We show some numerical results obtained with two well known Krylov subspace methods, i.e., Bi-CGSTAB and GMRES.