Some problems in boundary element programming of axisymmetric elasticity

Axisymmetric problems in elasticity can be reduced to two dimensional ones, but they are a little more complicated than plane problems. Therefore, some special problems will be encountered in the boundary element programming of axisymmetric elasticity. In this paper, the methods to treat these problems and some remarks are given according to our experience in programming. Numerical examples are presented for the checking of these treatments.     

Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems

Abstract

This article is concerned with penalty methods for solving optimal Dirichlet control problems governed by the steady-state and time-dependent Navier-Stokes equations. We present, in two different versions, the penalized methods for solving the steady-slate Dirichlet control problems. These approaches are implemented and compared numerically. We also generalize the penalty methods to the time-dependent case. Scmidiscrete and fully discrete approximations of time-dependent Dirichlet control problems are discussed and implemented. Numerical results for solving both the steady-state and the time dependent Dirichlet control problems are reported.     

Parametrization of sets of stabilizing controllers in mechanical systems

Procedures to parametrize a set of stabilizing controllers are reviewed. These procedures are the key ones in the frequency-domain synthesis of the optimal (minimum H ₂-and H _∞-norms) controller or filter for a linear stationary system. A relationship between the parametrization procedures proposed by different authors is shown. Examples of parametrization procedures in synthesis problems (delay problems, multichannel filtering problems, etc.) are given __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 3–27, June 2008.     

Analytical solutions of dynamic mode I crack under the conditions of displacement boundary

By the approaches of the theory of complex variable functions, the problems of dynamic mode I crack under the condition of displacement boundary are investigated. For this kind of dynamic crack extension problems with arbitrary index of self-similarity, the universal representations of analytical solutions are facilely deduced by the methods of self-similar functions. Analytical solutions of the stresses, displacements and stress intensity factors are readily acquired using the methods of self-similar functions. The problems studied can be very easily translated into Riemann–Hilbert problems and their closed solutions are gained rather straightforward in terms of this technique. According to corresponding material properties, the mutative rule of stress intensity factor was illustrated very well. Using those solutions and superposition theorem, the solutions of arbitrarily complex problems can be attained.     

Numerical Analysis of the Branched Forms of Bending for a Rod

Nonlinear boundary–value problems of plane bending of elastic arches subjected to uniformly distributed loading are solved numerically by the shooting method. The problems are formulated for a system of sixth–order ordinary differential equations that are more general than the Euler equations. Four variants of rod loading by transverse and longitudinal forces are considered. Branching of the solutions of boundary–value problems and the existence of intersected and isolated branches are shown. In the case of a translational longitudinal force, the classical Euler elasticas are obtained. The existence of a unique (rectilinear) form of equilibrium upon compression of a rod by a following longitudinal force is shown.     

Various methods of reconstruction of a cavity in an orthotropic layer

Direct and inverse problems of oscillations of an anisotropic layer with a cylindrical cavity of an arbitrary cross-sectional shape under the action of a load applied to the layer surface are considered. An asymptotic approach to solving these problems with cavities of small relative sizes is proposed. Numerical results of solving direct and inverse problems are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 181–189, May–June, 2009.     

On solution of dynamic problems of elastic-stress concentration near defects on cylindrical surfaces

The results obtained in [11] are extended to the case of unsteady dynamic loads and the integral equations to which these problems are reduced are solved by methods that are illustrated by the example of the corresponding static problems. Without fundamental complications, they can be applied to the dynamic problems examined here. Institute of Mathematics, Economics, and Mechancis, Odessa State University, Ukraine. Translated from Priknadnaya Mekhanika, Vol. 35, No. 1, pp. 28–36, January, 1999.     

Spatial Thermoviscoplastic Problems

Methods and results of studies of the three-dimensional viscoplastic stress–strain state of engineering structures under thermomechanical loading are presented. The following classes of thermoviscoplastic problems are considered: axisymmetric problems, nonaxisymmetric problems for bodies of revolution, three-dimensional problems for bodies of arbitrary shapes, and three-dimensional problems for anisotropic bodies of revolution     

Solution of Some Structural Analysis Problems by Orthogonal Collocation

Abstract

Application of the method of orthogonal collocation to boundary value problems in structural and applied mechanics is investigated. Typical boundary value problems, such as the torsion of rectangular bars and the bending of plates, are employed as illustrative examples. Simplicity in application and good accuracy of orthogonal collocation are demonstrated by the solution of such complex problems as the large deflection analysis of rectangular isotropic, orthotropic, and sandwich plates. Results are compared wherever possible with existing solutions based on much more laborious and lengthy methods of computation. Excellent agreements are obtained.     

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Numerical modelling of shocks in solids with elastic-plastic conditions

Results are presented for a range of one- and two-dimensional shock-wave problems in elastic-plastic and hydrodynamic metals. These problems were solved numerically using the Flux-Corrected Transport (FCT) technique which achieves high resolution without non-physical oscillations, especially at shock fronts, and has not been used before in elastic-plastic solids. The two-dimensional problems were solved using both operator- and non-operator-split techniques to highlight the significant differences between these techniques when solving shock-wave problems in elastic-plastic solids. Comparisons of the elastic-plastic solutions with the hydrodynamic solutions are made and illustrate the importance of including elastic-plastic conditions when modelling the behaviour of solids. Also, the errors in these solutions that are due to the initial conditions are discussed in detail.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Problems of modeling and optimal stabilization of the gas-lift process

The problems of motion of fluids, gases and gas–liquid mixtures in pipes related to gas-lift oil recovery are mathematically formulated as systems of nonlinear hyperbolic partial differential equations. Optimal-control problems are posed based on the proposed models and some real assumptions. These problems can be used to design programmed paths and controls, which underlie the controllers that stabilize the pressure or volume of injected gas. That the mathematical models agree with available field and laboratory data is demonstrated by examples     

Duality among Optimal Design Problems for Torsional Vibration

ABSTRACT

Four types of mass and frequency optimization problems are stated for free torsional vibration of thin-walled cylinders subject to constraints on wall thickness and frequencies of vibration. It is shown, using Pontryagin's method, that the mathematical structure of all four problems is similar and leads to identical classes of optimal thickness distributions. These duality relations are used in an example to construct an optimal frequency solution from the solutions for both maximum and minimum mass problems. General relations among the governing parameters for the four problems are stated. The results of Grinev and Filippov and of Thermann for the abnormal optimization problems are verfied as a specific limiting example of the general results.     

Development of the Boundary-Element Method for Three-Dimensional Problems of Static and Nonstationary Elasticity

The boundary-element method (BEM) applied to three-dimensional problems in the linear theory of elasticity is analyzed. The solutions of test problems for spherical and cubic cavities are used as examples to consider the basic aspects and difficulties of applying the traditional BEM to static and nonstationary three-dimensional problems. It is established that using Chebyshev polynomials in the Gaussian quadrature formula to evaluate the singular segments of surface integrals reduces the computation time by a factor of 2 to 3 without loss of accuracy compared with the traditional Gauss–Legendre formula. BEM-based approaches are proposed to solve three-dimensional problems in the linear theory of elasticity     

Some Non-linear Elastic Solutions for Masonry Solids*

ABSTRACT

This paper deals with equilibrium problems for solids made of elastic materials of bounded tensile strength and for which exact solutions are achieved. A constitutive equation is adopted and its main properties with regard to uniqueness of the solution to boundary problems are also analyzed. Four distinct equilibrium problems are then considered. The first three are characterized by specific symmetry conditions—polar, spherical, and cylindrical, respectively.     

Study of the Deformation of Anisotropic Viscoelastic Bodies

A study is made of methods for solving linear viscoelastic problems on the basis of the Volterra concept — representation of irrational functions of integral operators as operator power series (analogues of Taylor series). It is pointed out that these series converge weakly. The results of development and substantiation of a new mathematical method for solution of the above problems are summarized. It is based on representing irrational functions of integral operators by operator continued fractions, which converge well. Solutions to certain linear viscoelastic problems for anisotropic bodies are given     

Boundary-value problems on the dynamics of the theory of elastic mixtures for a disk

We consider the first and second dynamic boundary-value problems in the theory of elastic mixtures. These problems are reduced to the corresponding problems for systems of equations for pseudooscillation by Laplace transformation relative to time. The solutions are represented in terms of four metaharmonic functions. It is proved that the problem of pseudooscillation has a unique solution. Conditions are given for existence of inverse transformations that provide solutions for the initial problem. Translated from Prikladnaya Mekhanika, Vol. 34, No. 12, pp. 86–92, December, 1998.     

Local stress singularities in mixed axisymmetric problems of the bending of circular cylinders

A method of analyzing the near-edge stress state in mixed problems of the deformation of an isotropic cylindrical body is proposed. The method is based on the expansion of the solution of three-dimensional problems of elasticity into a series of Lurie–Vorovich homogeneous basis functions. An asymptotic analysis is performed to find the principal part of the solution of the infinite systems of linear algebraic systems to which the problems are reduced. The type of the stress singularity at the edge of the cylinder is the same as in the mixed problems for a quarter plane. Kummer’s convergence acceleration method is used. The obtained results are validated by testing the boundary conditions and by comparing with results obtained by other authors     

A Finite Element Analysis of a Viscoplastic Plate Subjected to Rapid Heating

Abstract

This paper models the response of a thin metallic plate that is subjected to a rapid heat input. In order to accurately model plate response, both the dynamic mechanical and transient heat transfer problems must be solved. The solution is complicated by nonlinearities due to radiation boundary conditions and material inelasticity. Furthermore, the viscoplastic constitutive equations that model the mechanical material behavior are numerically stiff. Nonlinear finite element algorithms are developed for both heat transfer and mechanical analyses. The algorithms are both stable and efficient for solving the problems considered herein. Example problems presented in the paper demonstrate the importance of including material nonlinearity in the model