A variational formulation for plate buckling problems by the hybrid finite element method

A functional is derived for development of stress hybrid finite elements for plate buckling problems. The equilibrium equations inside the element are identically satisfied in terms of Southwell stress functionsand the transverse displacement. Along the boundary of the element further displacement and normal slope functions are employed. These functions are so chosen as to satisfy the interelement compatibility requirements when the elements are connected. The boundary and internal displacements are selected entirely independently and comments are made on the choice of interpolation functions for the internal displacement.The stationarity of the functional is shown to lead to satisfaction of the equilibrium conditions along interelement boundaries, and the compatibility conditions inside the elements. The paper includes the details of a simple rectangular element and the results of a number of plate buckling problems analysed by the developed element.     

Simulation of two-dimensional elastodynamic problems using a new adaptive physics-based method

In this paper, an adaptive physics-based method is developed for solving wave motion problems in two dimensions (i.e., lamb waves). The solution of the problem has two main parts. In the first part, after discretization of the domain, a physics-based method is developed considering the conservation of mass and the balance of momentum. In the second part, adaptive points are determined using wavelet theory. This part is well done using Deslauries–Dubuc (D–D) like wavelets. After solving the problem in the first step, the domain of the problem is discretized by the same cells attending loading and characteristics of the structure. After the first trial solution, the D–D interpolation shows the lack and redundancy of points in the domain. These points will be added or eliminated for the next solution. This process may be repeated for achieving adaptive mesh for each step. Finally, the results of the proposed method are compared with the results available in the literature. This comparison shows excellent agreement between the obtained results and those already reported.     

Numerical modeling of functionally graded materials using a variational formulation

An approach of numerical modeling of heterogeneous, functionally graded materials, by using the finite element method, is proposed. The variational formulation is derived from the generic case so that the implementation of material coefficients, which are functions in space, is realized without any further assumptions. An analytical solution for a simple case is presented and used for validation of the numerical model.     

Dynamics of an elastic half-space with a reinforced cylindrical cavity under moving loads

Partial separation of variables and reexpansion of cylindrical and plane waves are used to find the solution describing the uniform motion of a load along a thin circular cylindrical shell in an elastic half-space with the free surface parallel to the axis of the shell. This is a model problem for studying the dynamics of tunnels and shallow-buried pipelines under transport loads. Dispersion curves for the cases of sliding and tight contact between the shell and the half-space are plotted and analyzed. The effect of the shell parameters on the stress–strain state of the half-space is examined     

Fundamental solutions in antiplane elastodynamic problem for anisotropic medium under moving oscillating source

In this paper we study the antiplane problem of concentrated point force moving with constant velocity and oscillating with constant frequency in unbounded homogeneous anisotropic elastic medium.The explicit representation of the elastodynamic Green's function is obtained by using Fourier integral transform techniques for all rates of source motion as a sum of the integrals over the finite interval. The dynamic and quasistatic components of the Green's function are extracted. The stationary phase method is applied to derive an asymptotic approximation at the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too.It is shown that the motion brings some differences in the far field properties, such as, for example, fast and slow waves appearance under superseismic motion and modification of the wave propagation zones and their numbers.The case of isotropic medium is considered separately. For isotropic material all main formulae are obtained in explicit forms.     

Dynamic response of beams under moving loads with finite deformation

Nonlinear Dynamics - A novel material parameter-dependent model is proposed in this work to investigate the nonlinear vibration of a beam under a moving load within a finite deformation framework....     

Numerical solution of plane and axially symmetric jet flow problems based on the variational inequality formulation

The numerical analysis of plane and axially symmetric jet flows of an incompressible inviscid fluid is treated. A new formulation of the variational inequality type is developed from the variational principle associated with jet problems. A successive approximation method is formulated by the combined use of variational inequality and the finite element method. Numerical examples based on the iterative method are presented. The results obtained agree well with those by other methods.     

Application of a generalized variational Hamilton-Ostrogradskii-Reissner principle to the formulation of linearized problems of the mechanics of spatial composite bodies

Kazan' Aviation Institute. Translated from Prikladnaya Mekhanika, Vol. 25, No. 9, pp. 33–39, September, 1989.     

On a formulation of meteor physics problems

The general equation describing the motion of a meteoroid in the Earth’s atmosphere is considered. It is shown that, in this case, the momentum variation of the system consisting of the meteoroid and its separating mass is due to the velocity variation of the meteoroid. The projection of Newton’s equation onto the tangent to the meteoroid trajectory is analyzed. It is also shown that, because of large geocentric velocities (between 11 and 72 km/s), the dominant force is the drag force proportional to the square of the incident air flow velocity. Some numerical results obtained for the Bene>sov bolide registered by the European Fireball Network (May 7, 1991) are given as an illustrative example.     

Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

We propose, in this paper, a unified method of generating a regularized integral equation in the double layer potential approach for 3D anisotropic elastodynamics. Our regularization preserves the causality in the time-domain. The method is based on a special decomposition of the hypersingular kernel which appears in the integral representation of the stress tensor.     

On the matched-asymptotic solution of elastodynamic problems: Loading over a semi-infinite rectangular boundary

The paper discusses the solution to an elastodynamic problem of an infinite medium excited by loads acting over a semi-infinite rectangular boundary whose height 2a is assumed small compared to the wavelengths of interest. The solution is developed using the method of matched-asymptotics in conjunction with an application of the Wiener-Hopf technique for the outer problems and the Muskhelishvili-Kolosov complex functions for the inner problems. The solutions are matched mutually and the resulting stress behaviour, as far as it is described by the inner limit to the outer solution, is also derived.     

Investigation of solution of Navier–Stokes equations using a variational formulation

A variational formulation for the solution of two dimensional, incompressible viscous flows has been developed by one of the authors.¹ The main objective of the present paper is to demonstrate the applicability of this approach for the solution of practical problems and in particular to investigate the introduction of boundary conditions to the Navier-Stokes equations through a variational formulation. The application of boundary conditions for typical internal and external flow problems is presented. Sample cases include flow around a cylinder and flow through a stepped channel. Quadrilateral, bilinear isoparametric elements are utilized in the formulation. A single-step, implicit, and fully coupled numerical integration scheme based on the variational principle is employed. Presented results include sample cases with different Reynolds numbers for laminar and turbulent flows. Turbulence is modelled using a simple mixing length model. Numerical results show good agreement with existing solutions.