Solution of initial–boundary-value problems of electroelasticity revisited

The Hamilton–Ostrogradsky principle is used to substantiate the statement of initial–boundary-value problems of electroelasticity. The free vibrations of a piezoceramic layer are used as an example to illustrate some features of solving nonstationary problems of electroelasticity Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 62–69, December 2008.     

Further improvement on fundamental solutions of plane problems for orthotropic materials

On the basis of the ecisting fundamental solutions of displacements, further improvement is made, and then the general fundamental solutions of both plane elastic and plane plastic problems for orthotropic materials are obtained. Two parameters based on material constants α₁ = α₁ are used to derive the relevant expressions in a real variable form. Additionally an analytical method of solving the singular integral for the internal stresses is introduced, and the corresponding results are given. If α₁ = α₁ = 1, all the expressions obtained for orthotropy can be reduced to the corresponding ones for isotropy. Because all these expressions and results can be directly used for both isotropic problems and orthotropic problems, it is convenient to use them in engineering with the boundary element method (BEM).     

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     

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     

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.     

Optimal design of a torsional shaft system using Pontryagin’s Maximum Principle

Optimal problems are investigated in the present work in order to control the natural frequencies of a torsional shaft system including the total weight constraint and effects of tuned mass dampers. Maier objective functional is used. Pontryagin’s Maximum Principle is employed to derive necessary optimality conditions of the optimal problems. Numerical simulations are performed to study effects of tuned mass dampers on controlling natural frequencies as well as minimizing the system’s weight. Advantages of the proposed method are also discussed.     

A Hybrid Nonlinear Programming Method for Design Optimization*

ABSTRACT

Solutions to engineering design problems formulated as nonlinear programming (NLP) problems usually require the use of more than one optimization technique. Moreover, the interaction between the user (analysis/synthesis) program and the NLP system can lead to interface, scaling, or convergence problems. An NLP solution system is presented that seeks to solve these problems by providing a programming system to ease the user-system interface. A simple set of rules is used to select an optimization technique or to switch from one technique to another in an attempt to detect, diagnose, and solve some potential problems. Numerical examples involving finite element based optimal design of space trusses and rotor bearing systems are used to illustrate the applicability of the proposed methodology.     

Incremental analysis of the nonlinear behavior of thin shells

The paper proposes a method of incremental loading for solving nonlinear problems of shell theory. A feature of this method is that the same algorithm is used before buckling, at the limit point, and after buckling. The method is based on the assumption that all the unknowns, including the load parameter, are on an equal footing. It is shown that the method can be used to solve algebraic equations with Cramer's rule involved to avoid numerical instability in the neighborhood of the limit point. Test problems confirm the validity of the approach Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 85–93, September 2008.     

Effective properties of a nonuniform beam under torsion

Effective characteristics are considered in the pure torsion problem for a nonuniform beam. The Saint-Venant semi-inverse method is used. A torsion stress function is introduced; this function can be found by solving a cross-sectional boundary value problem for a partial differential equation with variable coefficients. Two special boundary value problems are formulated for such an equation; after solving these problems, some effective characteristics are calculated in the case of torsion. It is shown that these effective characteristics satisfy the conditions of symmetry and positive definiteness. The case of an infinite in-plane layer of nonuniform thickness is discussed.     

Vibrations of Reinforced Cylindrical Shells with Initial Deflections under Nonstationary Loads

The paper addresses dynamic problems for discretely reinforced shells with initial deflections. Timoshenko theory is used. A numerical method of solving such problems is developed and theoretically justified. Numerical results for a specific problem are presented__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 60–68, January 2005.     

Run-up and spin-up in a viscoelastic fluid. IV

Three problems are discussed — run-up when the fluid is contained between infinite parallel plates and longitudinal run-up and spin-up when it is contained in an infinitely long circular cylinder. The procedure adopted for solving these problems differs from that employed in Part I of this series, where these three problems were previously discussed, and yields results for the velocity fields in quite different forms. It is similar to that used in Part III in the context of the problem of run-up between parallel plates.     

Variational method for solving problems of the plasticity of compressible media

Variational methods used in the theory of plastic flow are formulated on the assumption of the incompressibility of the deformable medium. In solving problems of the mechanics of soils and friable media and technological problems of the plastic shaping of uncompacted materials it is very important to take account of irreversible volumetric change. Extremum and variational theorems are proved in [1, 2] for rigid-plastic and viscoplastic expanding bodies. A variational equation equivalent to a complete system of differential equations is derived for a compressible plastic body.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 153–155, September–October, 1977.     

Design Sensitivity Analysis in Structural Mechanics. I. Static Response Variations

ABSTRACT

The dependence of the solution of boundary-value problems of structural mechanics on design variables that specify material properties and distribution is characterized. Prototype problems treated include beams, plates, and plane elastic solids. Symmetry and positive definiteness properties of the elliptic differential operators that govern system response are used to show that their inverses, hence the displacement fields, are Frechet differentiable with respect to design variables. Formulas for the derivatives are given and used to obtain computable formulas for design sensitivity coefficients (first variation) of integrals that arise in optimal design formulations. The results establish an extension of the concept of “well-posed” problems of structural mechanics to include continuity (in fact, differentiability” of static structural response with respect to distributed design variables and design parameters     

A conservative and shape-preserving semi-Lagrangian method for the solution of the shallow water equations

Semi-Lagrangian methods are now perhaps the most widely researched algorithms in connection with atmospheric flow simulation codes. In order to investigate their applicability to hydraulic problems, cubic Hermite polynomials are used as the interpolant technique. The main advantage of such an approach is the use of information from only two points. The derivatives are calculated and limited so as to produce a shape-preserving solution. The lack of conservation of semi-Lagrangian methods, however, is widely regarded as a serious disadvantage for hydraulic studies, where non-linear problems in which shocks may develop are often encountered. In this work we describe how to make the scheme conservative using an FCT approach. The method proposed does not guarantee an unconditional shock-capturing ability but is able to correctly reproduce the discontinuous flows common in open channel simulation without any shock-fitting algorithm. It is a cheap way to improve existing 1D semi-Lagrangian codes and allows stable calculations beyond the usual CFL limits. A basic semi-Lagrangian method is presented that provides excellent results for a linear problem: the new techniques allow us to tackle non-linear cases without unduly degrading the accuracy for the simpler problems. Two one-dimensional hydraulic problems are used as test cases, water hammer and dam break. In the latter case, because of the non-linearity, special care is needed with the low-order solution and we show the advantages of using Leveque's large-time step version of Roe's scheme for this purpose.     

Asymptotic analysis of perforated plates and membranes. Part 2: Static and dynamic problems for large holes

Static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi-scale asymptotic homogenization method. The problems of bending and vibration of perforated plates are considered. Using the asymptotic homogenization method the original boundary-value problems are reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. In the present paper the perforated plates with large holes are considered, and the singular perturbation method is used to solve the pertinent unit cell problems. Matching of limiting solutions for small and large holes using the two-point Padé approximants is also accomplished, and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained.     

On the use of characteristic‐based split meshfree method for solving flow problems

This study presents characteristic‐based split (CBS) algorithm in the meshfree context. This algorithm is the extension of general CBS method which was initially introduced in finite element framework. In this work, the general equations of flow have been represented in the meshfree context. A new finite element and MFree code is developed for solving flow problems. This computational code is capable of solving both time‐dependent and steady‐state flow problems. Numerical simulation of some known benchmark flow problems has been studied. Computational results of MFree method have been compared to those of finite element method. The results obtained have been verified by known numerical, analytical and experimental data in the literature. A number of shape functions are used for field variable interpolation. The performance of each interpolation method is discussed. It is concluded that the MFree method is more accurate than FEM if the same numbers of nodes are used for each solver. Meshfree CBS algorithm is completely stable even at high Reynolds numbers. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of the angular superposition method to the contact problem on the compression of an elastic cylinder

The angular superposition method is used to construct an approximate solution of the contact problem on the compression of an elastic cylinder by two rigid plates. The solution thus obtained has a closed-form analytic expression and can be used in the entire domain of the cylinder cross-section. We analyze the absolute error, which takes the largest value near the points of contact between the plates and the cylinder, where the boundary conditions are discontinuous. According to the von Mises criterion, when moving into the depth of the cylinder from the contact site along the symmetry axis, the second invariant J ₂ of the stress deviator tensor first decreases and then, after attaining a minimum, increases and attains the largest value at a small depth, which agrees with Johnson’s photoelastic experiments and Dinnik’s computations. We present the graphs of the displacement and normal stress distributions over the contact site, the dependence of the compressing force on the displacements of rigid plates, and the dependence of the invariant J ₂ on the coordinate along the symmetry axis. If 640 computation points are chosen on the cylinder boundary and the Hertz law for the normal pressure on the contact site is used, then the error in the approximate solution near the endpoint of the contact site is approximately 55%, and if the proposed two-parameter normal law is used, then the error is of the order of 4%. On the free lateral surface of the cylinder boundary, we find the critical pointM*, which separates the cylinder contraction and extension parts.The contact problems are the most difficult problems, and their solution is complicated by the discontinuous boundary conditions [1–5]. In [6], the contact problem is solved by the Fourier method, which can be used only for bodies of classical shapes. In such cases, the problem can be reduced to solving coupled integral equations [7]. The interaction between the bandage and a cylindrical body is considered in [2, 6, 7]. In [8], the possibility of using the finite element method is investigated in the case of contact problems for a differential wheel with roughness of the contacting surfaces taken into account. In [9, 10], the method of homogeneous solutions is used to consider contact problems for a finite-dimensional elastic cylinder loaded on its end surfaces. Note that only error estimates are given in the literature cited above; the absolute error over the entire domain of the elastic body is not studied, although this is one of the important characteristics of the obtained approximate solution. A sufficiently complete survey of the literature in the field of contact interactions of elastic bodies is given in [3–5].In what follows, we propose to solve contact problems by the angular superposition method [11]. This method can be used for bodies of nonclassical shapes, which can be multiply connected, and the friction on the contact site can be taken into account. In the present paper, as a first example of applied character, we show how this method can be used in the simplest case. The multiple connectedness and the curvilinearity of the shape of the body, as well as taking into account the friction on the boundary, do not create new essential difficulties in this method.     

Numerical simulation of solidification processes in enclosures

The present work deals with the development and application of numerical models for the simulation of solidification problems liquid/solid taking diffusion and convection into account. For the calculation of the thermal coupled flow process the finite element method is applied. In order to improve the numerical stability of the free convection problems, the streamline-upwind/Petrov–Galerkin method is used. Solidification processes are moving boundary problems. Three different models are set up which consider latent heat at the solidification front respectively in the mixed zone during the phase transition. Moreover, numerical methods are investigated in order to describe the behaviour of the flow at the boundary of the moving phase. Three examples serve illustrations; the technical example – casting of a transport and storage container – was provided by the company Siempelkamp Gießerei GmbH.     

Antiplane strain in a nonlinearly elastic incompressible body

The stress-strain state of an incompressible cylindrical elastic body with antiplane strain under the action of potential forces and surface loading constant along the body is considered in a nonlinear formulation in actual variables. The stresses are expressed via the pressure and independent strains, the pressure is expressed via the force and elastic potentials, and nonlinear boundary-value problems are posed for strains (and displacements). Various methods for solving these problems are developed. For the nonlinear equations obtained, some analytical solutions containing free parameters are given, which can be used as a basis for solving particular problems. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 93–101, November–December, 2006.