Analysis on Boundary Conditions for Elastic Structures

With the rapid development of modern industry, elastic materials and structures have been widely used in all walks of life, such as construction, machinery and so on. The traditional elastic mechanics method generally adopts the semi inverse method. However, this method ignores the local deformation caused by boundary constraints and can only find part of the solution of the problem. In this paper, the basic equations of symplectic system are established, and the solutions of various boundary condition problems are given.

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

In this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

Multi-symplectic magnetohydrodynamics

A multi-symplectic formulation of ideal magnetohydrodynamics (MHD) is developed based on the Clebsch variable variational principle in which the Lagrangian consists of the kinetic minus the potential energy of the MHD fluid modified by constraints using Lagrange multipliers that ensure mass conservation, entropy advection with the flow, the Lin constraint, and Faraday's equation (i.e. the magnetic flux is Lie dragged with the flow). The analysis is also carried out using the magnetic vector potentialÃwhere α=Ã⋅dxis Lie dragged with the flow, andB=∇×Ã. The multi-symplectic conservation laws give rise to the Eulerian momentum and energy conservation laws. The symplecticity or structural conservation laws for the multi-symplectic system corresponds to the conservation of phase space. It corresponds to taking derivatives of the momentum and energy conservation laws and combining them to producen(n−1)/2 extra conservation laws, wherenis the number of independent variables. Noether's theorem for the multi-symplectic MHD system is derived, including the case of non-Cartesian space coordinates, where the metric plays a role in the equations.

SYMPLECTIC METHOD FOR DYNAMIC BUCKLING OF CYLINDRICAL SHELLS UNDER COMBINED LOADINGS

A symplectic system is developed for dynamic buckling of cylindrical shells subjected to the combined action of axial impact load, torsion and pressure. By introducing the dual variables, higher-order stability governing equations are transformed into the lower-order Hamiltonian canonical equations. Critical loads and buckling modes are converted to solving for the symplectic eigenvalues and eigensolutions, respectively. Analytical solutions are presented under various combinations of the in-plane and transverse boundary conditions. The results indicated that in-plane boundary conditions have a significant influence on this problem, especially for the simply supported shells. For the shell with a free impact end, buckling loads should become much lower than others. And the corresponding buckling modes appear as a "bell" shape at the free end. In addition, it is much easier to lose stability for the external pressurized shell. The effect of the shell thickness on buckling results is also discussed in detail.

An Eigenexpansion Method in 2D Viscoelastic Materials

The two-dimensional viscoelastic solid is considered in symplectic system. The general solutions of the governing equations include zero eigensolutions and non-zero eigensolutions. Zero eigensolutions can describe all the Saint-Venant problems, and non-zero ones are local effect solutions. Via this analytical approach, the final solution of the problem can be expressed by the linear combination of the general eigensoutions. In this paper, the local effects are described by employing this analytical approach.

The Symplectic System for the Analytical Solutions of Skew Plate

The Hamiltonian Systems is applied for the bending of skew plate.In contrast to the traditional technique using only one kind of variables, the symplectic dual variables include displacement components as well as stress components. The analytic solutions are obtained under simple support which shows the effectiveness of the symplectic dual and indicates that this method is good in speed of convergence and reliability.