We study the extensions of the Bogner-Fox-Schmit element to the whole family of Q_k continuously differentiable finite elements on rectangular grids, for all k≥3, in 2D and 3D. We show that the newly defined C₁ spaces are maximal in the sense that they contain all C₁-Q_k functions of piecewise polynomials. We give examples of other extensions of C₁-Q_k elements. The result is consistent with the Strang's conjecture (restricted to the quadrilateral grids in 2D and 3D). Some numerical results are provided on the family of C₁ elements solving the biharmonic equation.

A class of steady-state metal-forming problems, with rigid-plastic, incompressible, strain-rate dependent material model and nonlocal Coulomb's friction, is considered. Primal, mixed and penalty variational formulations, containing variational inequalities with nonlinear and nondifferentiable terms, are derived and studied. Existence, uniqueness and convergence results are obtained and shortly presented. A priori finite element error estimates are derived and an algorithm, combining the finite element and secant-modulus methods, is utilized to solve an illustrative extrusion problem.

The paper introduces the gas-kinetic scheme for three-dimensional (3D) flow simulation. First, under a unified coordinate transformation, the 3D gas-kinetic BGK equation is transformed into a computational space with arbitrary mesh moving velocity. Second, based on the Chapman-Enskog expansion of the kinetic equation, a local solution of gas distribution function is constructed and used in a finite volume scheme. As a result, a Navier-Stokes flow solver is developed for the low speed flow computation with dynamical mesh movement. Several test cases are used to validate the 3D gas-kinetic method. The first example is a 3D cavity flow with up-moving boundary at Reynolds number $3200$, where the periodic solutions are compared with the experimental measurements. Then, the flow evolution inside a rotating 3D cavity is simulated with the moving mesh method, where the solution differences between 2D and 3D simulation are explicitly presented. Finally, the scheme is applied to the falling plate study, where the unsteady plate tumbling motion inside water tank has been studied and compared with the experimental measurements.

A numerical investigation of laminar natural double diffusive convection in an open ended vertical cylindrical annulus with unheated entry and unheated exit is performed. Both boundary conditions of uniform wall temperature/uniform wall concentration (UWT/UWC) and uniform heat flux/uniform mass flux (UHF/UMF) are considered. Results of dimensionless induced volume rate $Q$, average Nusselt number $\overline{Nu}$ and Sherwood number $\overline{Sh}$ are obtained for air flow under various buoyancy ratio $N$, Grashof numbers due to heat and mass transfer $Gr_T$ and $Gr_M$, Schmidt number $Sc$ and combinations of unheated entry, heated section and unheated exit length. Since the flow under consideration is a boundary layer type, the governing partial differential equations was discretized to a linear system of equations by the use of an implicit finite difference method. The nonlinear convective terms are approximated by second upwind difference method for the numerical stability. The numerical results reveal that the presence of unheated entry and unheated exit severely affects the heat and mass transfer rates. The numerical solutions are found to approach asymptotically the closed form solutions for fully developed flow. Further, the present numerical results are validated with the existing solutions for pure thermal convection and are found to be in good agreement.

This paper analyzes a nonconforming 5-node quadrilateral transition finite element for Poisson equation. This element was originally proposed by Choi and Park [Computers and Structures, 32 (1989), pp. 295–304 and Thin-Walled Structures, 28 (1997), pp. 1–20] for the analysis of Mindlin plates. We show the consistency error of this element is only $\mathcal{O}(h^{1/2})$ over the transition edges of the quadrilateral subdivision. By modifying the shape functions with respect to mid-nodes, we get an improved version of the element for which the consistency error is $\mathcal{O}(h)$. Numerical examples are provided to verify the theoretical results.

This paper is concerned with a stabilized approach to low-order mixed finite element methods for the Stokes equations. We will provide a posteriori error analysis for the method. We present two a posteriori error indicators which will be demonstrated to be globally upper and locally lower bounds for the error of the finite element discretization. Finally two numerical experiments will be carried out to show the efficiency on constructing adaptive meshes.

The present problem is concerned with the study of deformation of a rotating generalized thermoelastic medium under the influence of gravity. The components of displacement, force stress and temperature distribution are obtained in Laplace and Fourier domain by applying integral transforms. These components are then obtained in the physical domain by applying a numerical inversion method. Some particular cases are also discussed in context of the problem. The results are also presented graphically to show the effect of rotation and gravity in the medium.