Application of the liquid bridges theory to find the melt menisci shape when growing cylindrical crystals

A vertical liquid bridge of small volume between a conical shaper and a convex crystallization front was investigated. Two variants of the front crystallization shape selection are considered: a conical and a spherical fronts. A variational statement of the original problem is given. As the boundary conditions we used the condition of engagement at the edge of the shaper and a given growth angle at the crystallization front. The Bond number was assumed to be small, and to find a solution of the problem the asymptotic approach was applied. The calculations are carried out for small diameter cylindrical sapphire crystals, grown from the melt by the Stepanov method. The results of the menisci shapes calculations are presented. The comparison of the results of calculations for conical and spherical crystallization fronts is carried out.

Geometrically exact, intrinsic mixed variational formulation for smart, slender multilink composite structures

Flexible links are often part of massive aerospace structures like helicopter or wind turbine blades, satellite bae, airplane wings, and space stations. In the present work, a mixed variational statement based on intrinsic variables is derived for multilinked smart slender structures. Equations involved in the derivation do not involve approximations of kinematical variables to describe the deformation of the reference line or the rotation of the deformed cross-section of the slender links resulting in a geometrically exact formulation. Finite element equations are derived from weak formulation, which can analyze large geometrically non-linear problems. The weakest possible variational statement provides greater flexibility in the choice of shape functions, therefore reducing the associated numerical complexities. The present work focuses on developing a single integrated computational platform which can study multibody, multilink, lightweight composite, structural system built with both embedded actuations, sensing, as well as passive links. Validation of static mechanical and electrical outputs from 3D FE simulation and literature proves the efficacy of the computational platform. Dynamic results will be communicated in future correspondence. The computational platform developed here can be applied for monitoring and active control applications of flexible smart multilink structures like swept wings, multi-bae space structures, and helicopter blades.

Numerical Computation of Circumferential Waves in Cylindrical Curved Waveguides

We present a complex field general variational statement to study the waves propagating in the circumferential directions of cylindrical curved waveguides. A semi-analytical technique that has been applied on straight waveguides in the literature is reformulated to adapt to the circumferential directions and used for constructing the trial wavefunction in complex field. The method requires the waveguide to be analyzed to be geometrically and physically uniform along its circumferential axis; however, its circumferential cross-section can be arbitrarily complex. The formulation is verified using various examples, which were examined previously by other numerical or analytical solutions. Different cases were studied and comparisons with those published are also performed show the utility and advantages of present method.

Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff–Love plates with a crack

A non-linear model describing the equilibrium of a cracked plate with a volume rigid inclusion is studied. We consider a variational statement for the Kirchhoff–Love plate satisfying the Signorini-type non-penetration condition on the crack faces. For a family of problems, we study the dependence of their solutions on the location of the inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on a suitable Sobolev space. For this problem, the location parameter of the inclusion serves as a control parameter. We prove continuous dependence of the solutions with respect to the location parameter and the existence of a solution of the optimal control problem.

Variational Statement and Domain Decomposition Algorithms for Bitsadze-Samarskii Nonlocal Boundary Value Problem for Poisson’s Two-Dimensional Equation

The Bitsadze-Samarskii nonlocal boundary value problem is considered. Variational formulation is done. The domain decomposition and Schwarz-type iterative methods are used. The parallel algorithm as well as sequential ones is investigated.

Mixed variational potentials and inherent symmetries of the Cahn–Hilliard theory of diffusive phase separation

This work shows that the Cahn–Hilliard theory of diffusive phase separation is related to an intrinsic mixed variational principle that determines the rate of concentration and the chemical potential. The principle characterizes a canonically compact model structure, where the two balances involved for the species content and microforce appear as the Euler equations of a variational statement. The existence of the variational principle underlines an inherent symmetry in the two-field representation of the Cahn–Hilliard theory. This can be exploited in the numerical implementation by the construction of time- and space-discrete incremental potentials , which fully determine the update problems of typical time-stepping procedures. The mixed variational principles provide the most fundamental approach to the finite-element solution of the Cahn–Hilliard equation based on low-order basis functions, leading to monolithic symmetric algebraic systems of iterative update procedures based on a linearization of the nonlinear problem. They induce in a natural format the choice of symmetric solvers for Newton-type iterative updates, providing a speed-up and reduction of data storage when compared with non-symmetric implementations. In this sense, the potentials developed are believed to be fundamental ingredients to a deeper understanding of the Cahn–Hilliard theory.

Size Effect on Buckling Behaviors of Gradient Elastic Slender Columns

For the buckling problems of slender columns, size effects on buckling behaviors have been studied based on the modified couple stress theory. The governing equations are obtained by using variational statement and the buckling loads of slender columns are assessed. The results show that the buckling loads predicted by the new model are size-dependent.