Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms

A variant of the refined geometric nonlinear theory is suggested for nonshallow shells with a transversely soft core of medium thickness with regard to modifications of metric characteristics across the core thickness. The kinematic relations for the core are derived by sequential integration of the initial three-dimensional equations of elasticity theory along the transverse coordinate. The equations are preliminarily simplified by the assumption that the tangential stress components are equal to zero. With the example of sandwich plates, it is shown that these equations allow us to investigate synphasic, antiphasic, mixed flexural, and mixed flexural-shear buckling forms of load-bearing layers and the core depending on the precritical stress-strain state. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 95–108, January–February, 2000.     

Buckling forms of homogeneous and sandwich plates in pure shear in tangential directions

The mathematical problem of the plane shear buckling form (BF) of sandwich plates and plates homogeneous across the thickness in pure shear is considered. The solution to this problem is compared with the solution to the problem of a flexural BF which is realized in these plates with the formation of oblique waves. It is established that, in the case of plates homogeneous across the thickness, the critical loads corresponding to the plane shear BF are maximum for isotropic ones. In real one-layer structural elements manufactured both of isotropic homogeneous and orthotropic composite materials, these critical loads cannot be reached since they exceed considerably the critical loads for the flexural BFs with oblique waves. The critical loads corresponding to the two BFs are comparable only for relatively thick plates. However, the plane shear BF can be realized in sandwich plates earlier than the flexural one even if the plates are thin. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 215–228, 2000.     

A stability theory of sandwich structural elements 1. Analysis of the current state and a refined classification of buckling forms

The main stages of development of the stability theory of sandwich structural elements are considered. The mechanism of their stability loss is revealed using the experimental data and theoretical solutions obtained on the basis of refined statements of problems. A classification of all possible forms of stability loss is given within the limits of continuum representation of load-bearing layers and the core of these structures.Center for Study of Dynamics and Stability, Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 707–716, November–December, 1999.     

Mixed finite element analysis of a thermally nonlinear coupled problem

In Loula and Zhou [Comput Appl Math 20 (2001), 321–339], a thermally coupled nonlinear elliptic system modeling a large class of engineering problems was considered, and some mathematical and numerical analyses (C⁰ Lagrangian finite elements combined with a fixed point algorithm) were given. To continue our work, we propose in this article a mixed method for the potential equation and present the corresponding analyses and numerical implementations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Shear buckling modes of cylindrical sandwich shells under external pressure, tension-compression of bearing layers with unequal forces, and inhomogeneous across-the-thickness temperature

Equations are set up for describing, in a correct statement and with an accuracy sufficient in actual practice, the shear buckling modes (BMs) of cylindrical sandwich shells with a transversely soft core of arbitrary thickness. Based on them, solutions are obtained to a number of problems on the buckling instability according to shear modes under some force and thermal loadings. It is found that the BMs occur in the shell along the circumferential and axial directions if, in the precritical state, a normal compressive stress arises in the transverse direction. It is shown that this condition is fulfilled in the following cases: in axial tension of the shell with unequal forces applied to the end faces of bearing layers (the parameter of critical load is maximum if the tensile forces are equal); under external (internal) pressure; on cooling the outer and heating the inner layers. The results obtained are presented in the form of simple analytical formulas for determining the corresponding critical parameters of the force and thermal actions.Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 37–48, January–February, 2005.     

Optimal error estimates of mixed finite element methods for a fourth-order nonlinear elliptic problem

A mixed finite element method is developed for a nonlinear fourth-order elliptic problem. Optimal L² error estimates are proved by using a special interpolation operator on the standard tensor-product finite elements of order k?1. Then two iterative schemes are presented and proved to keep the same optimal error estimates. Three numerical examples are provided to support the theoretical analysis.     

Research in the flexural buckling modes of a cylindrical sandwich shell under the action of a temperature field inhomogeneous across its thickness

A solution to the problem on the stability according to the flexural buckling mode is given for a cylindrical sandwich shell with a transversely soft core of arbitrary thickness. The shell is under the action of a temperature field inhomogeneous across the thickness, and its end faces are fastened in such a way (in the axial direction, the face sections of the external layer are fixed, but of the internal one are free) that an inhomogeneous subcritical stress-strain state arises in the shell across the thickness of its layers. It is shown that, under such conditions, the buckling mode of the shell is mixed flexural. To reveal and investigate this mode, equations of subcritical equilibrium and stability of a corresponding degree of accuracy are needed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 715–730, November–December, 2004.     

Stability problem of a circular sandwich ring under uniform external pressure

The solution of the stability problem of a circular sandwich ring under uniform external pressure is given in a refined statement. The need to determinate the precritical stresses in load-bearing layers in the refined statement with regard to the transverse compression of the core is established, which is the basis for the detection of the mixed flexural buckling forms (BFs) with more than two half-waves along the circumferential coordinate (n>2). It is found that sandwich structures with a determining parameter of transverse compression corresponding to the limit of transition from the mixed BFs to synphasic ones are the most efficient from the weight viewpoint. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 317–328, May–June, 2000.     

Finite element analysis of parabolic integro-differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.     

Complete system of two-dimensional invariants for formulation of triangular finite element of composite shells

The paper describes a system of invariants of symmetric two-dimensional tensors defined on a plane or a surface. The system comprises the well-known first and second invariants and a new quantity called the combined invariant of two tensors. The focus is on the expression for the invariants in terms of normal components of the tensors determined in three different directions on the surface. The system of invariants is used to construct a triangular finite element for geometrically nonlinear analysis of shear deformable anisotropic shells subject to the Reissner–Mindlin assumptions. The relations obtained allow one to readily determine the strain energy of the element for the normal components of the stress and strain tensors in the direction of the element edges. Numerical examples are given to demonstrate some nonlinear capabilities of the element.     

Convergence analysis of a finite volume method via a new nonconforming finite element method

The article is devoted to the study of convergence properties of a Finite Volume Method (FVM) using Voronoi boxes for discretization. The approach is based on the construction of a new nonconforming Finite Element Method (FEM), such that the system of linear equations coincides completely with that for the FVM. Thus, by proving convergence properties of the FEM, we obtain similar ones of the FVM. In this article, the investigations are restricted to the Poisson equation. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:213–231, 1998     

Global superconvergence of finite element methods for parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we study finite element methods and present an immediate analysis for global superconvergence for these problems, on basis of which we obtain a posteriori error estimators. This research was supported in part by the Shahid Beheshti University, the National Basic Research Program of China (2007CB814906), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300).     

Transient response of sandwich viscoelastic beams, plates, and shells under impulse loading

The transient response of sandwich beams, plates, and shells with viscoelastic layers under impulse loading is studied using the finite element method. The viscoelastic material behavior is represented by a complex modulus model. An efficient method using the fast Fourier transform is proposed. This method is based on the trigonometric representation of the input signals and the matrix of the transfer functions. The present approach makes it possible to preserve exactly the frequency dependence of the storage and loss moduli of viscoelastic materials. The logarithmic decrements are determined using the steady state vibrations of sandwich structures to characterize their damping properties. Test problems and numerical examples are given to demonstrate the validity and application of the approach suggested in this paper. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 367–378, March–April, 2000.     

Stabilized nodal‐based finite element methods for the magnetic induction problem

We consider the time‐dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal‐based finite element approximations of the induction model with inf‐sup stable finite elements for the magnetic field and the magnetic pseudo‐pressure are investigated. Based on a residual‐based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal‐based finite element method for the numerical solution. Error estimates are given for the semi‐discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.     

On the problem of superconvergence of finite element method algorithms

The coincidence of an approximate solution to the boundary value problem for an ordinary differential equation with the exact solution at mesh nodes is proved for a certain class of the generalized finite element methods.     

Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems

Some least-squares mixed finite element methods for convection-diffusion problems, steady or nonstationary, are formulated, and convergence of these schemes is analyzed. The main results are that a new optimal a priori error estimate of a least-squares mixed finite element method for a steady convection-diffusion problem is developed and that four fully-discrete least-squares mixed finite element schemes for an initial-boundary value problem of a nonlinear nonstationary convection-diffusion equation are formulated. Also, some systematic theories on convergence of these schemes are established.

     

Exact analytic solutions to the problem on plane buckling modes of rectangular orthotropic plates with free edges under biaxial loading

A two-dimensional linearized problem on plane buckling modes (BMs) of a rectangular plate with free edges, made of an elastic orthotropic material, underbiaxial tension-compression is considered. With the use of double trigonometric basis functions, displacement functions exactly satisfying all static boundary condition on plate edges are constructed. It is shown that the exact analytic solutions found describe only the pure shear BMs, and if the normal stress in one direction is assumed equal to zero, an analog of the solution given by the kinematic Timoshenko model can be obtained. Upon performing the limit passage to the zero harmonic in the displacement functions of one of the directions, the solution to the problem of biaxial compression can be obtained by equating the Poisson ratio to zero; in the case of uniaxial compression, this solution exactly agrees with that following from the classical Bernoulli-Euler model. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 149–170, March–April, 2007.     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.     

A WKB analysis of the buckling condition for a cylindrical shell of arbitrary thickness subjected to an external pressure

We consider the plane-strain buckling of a cylindrical shellof arbitrary thickness which is made of a Varga material andis subjected to an external hydrostatic pressure on its outersurface. The WKB method is used to solve the eigenvalue problemthat results from the linear bifurcation analysis. We show thatthe circular cross-section buckles into a non-circular shapeat a value of µ₁ which depends on A₁/A₂ and a mode number,where A₁ and A₂ are the undeformed inner and outer radii, andµ₁ is the ratio of the deformed inner radius to A₁. Inthe large mode number limit, we find that the dependence ofµ₁ on A₁/A₂ has a boundary layer structure: it is constantover almost the entire region of 0 < A₁/A₂ < 1 and decreasessharply from this constant value to unity as A₁/A₂ tends tounity. Our asymptotic results for A₁ – 1 = O(1) and A₁– 1 = O(1/n) are shown to agree with the numerical resultsobtained by using the compound matrix method.     

Superconvergence of finite volume element method for a nonlinear elliptic problem

In this article, we consider the finite volume element method for the second‐order nonlinear elliptic problem and obtain the H¹ and W^1,∞ superconvergence estimates between the solution of the finite volume element method and that of the finite element method, which reveal that the finite volume element method is in close relationship with the finite element method. With these superconvergence estimates, we establish the L^p and W^1,p (2 < p ≤ ∞) error estimates for the finite volume element method for the second‐order nonlinear elliptic problem. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007