Two-dimensional complete rational analysis of functionally graded beams within symplectic framework

Exact solutions for generally supported functionally graded plane beams are given within the framework of symplectic elasticity. The Young’s modulus is assumed to exponentially vary along the longitudinal direction while the Poisson’s ratio remains constant. The state equation with a shift-Hamiltonian operator matrix has been established in the previous work, which is limited to the Saint-Venant solution. Here, a complete rational analysis of the displacement and stress distributions in the beam is presented by exploring the eigensolutions that are usually covered up by the Saint-Venant principle. These solutions play a significant role in the local behavior of materials that is usually ignored in the conventional elasticity methods but possibly crucial to the material/structure failures. The analysis makes full use of the symplectic orthogonality of the eigensolutions. Two illustrative examples are presented to compare the displacement and stress results with those for homogenous materials, demonstrating the effects of material inhomogeneity.     

General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.     

基于辛理论的载流碳纳米管能带分析

基于连续介质力学理论和辛弹性理论,将载流碳纳米管等效为铁木辛柯梁,采用哈密顿变分原理建立了载流碳纳米管的振动控制方程;引入对偶变量将振动控制方程从拉格朗日体系导入到哈密顿体系下;通过波传播方法分析了载流碳纳米管的能带结构;研究了流体密度、流速对载流碳纳米管能带结构的影响;同时计算了载流碳纳米管的散射矩阵. 研究发现:管内流速以及流体密度对剪切频率和弯曲频率有着非常重要的影响. 研究结果表明:载流碳纳米管的剪切频率和弯曲频率因流体的加入而减小,并随流速及流体密度的增大而减小;通过对数值结果的分析发现:载流碳纳米管由于管内流体、流速以及流体密度的作用,会使得载流碳纳米管变的更“软”. 其中,哈密顿体系下所得出的载流碳纳米管弯曲频率随管内流体密度的增加而变小,有别于在拉格朗日体系下非局部梁理论所得的结论. 同时,数值结果表明散射矩阵是酉矩阵,辛体系下的入射波功率流与反射波功率流相等,即功率流守恒,体现了辛弹性力学理论的优越性.      

A two-step back–forth algorithm for the solution of the unsteady adjoint equations

A new algorithm for the solution of the unsteady adjoint equations is proposed in this article, aiming at overcoming the excessive computational cost and memory requirements of the conventional adjoint approach for the optimisation of unsteady problems in computational mechanics. The total cost is equal to four times the cost of the unsteady state solution, which is twice the cost of the conventional backward-in-time adjoint calculation but the memory requirements are very small, equivalent to those of a steady-state problem, while stability is acceptable. The proposed algorithm is validated in the case of the 1D unsteady Burgers equation with non-smooth source terms.     

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation,applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.     

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases. Based on the linearizing operator and adjoint linearizing operator for the two real systems, adjoint symmetries can be obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair (SA) method. Relationships between the nonlinear self-adjointness method and the SA method are investigated.     

Optimal control problem for cancer invasion parabolic system with nonlinear diffusion

ABSTRACT

In this paper, we study a distributed optimal control problem of a coupled nonlinear system of reaction–diffusion equations. The system consists of three partial differential equations to represent cancer cell density, matrix-degrading enzymes concentration and oxygen concentration, and an ordinary differential equation to describe the extracellular matrix concentration. Our aim is to minimize the growth of cancer cells by controlling the production of matrix-degrading enzymes. First, we prove the existence and uniqueness of solutions of the direct problem. Then, we prove the existence of an optimal control. Finally, we derive the first-order optimality conditions and prove the existence of weak solutions of the adjoint problem.     

局部β-凸空间L~β(μ,X)(0＜β≤1)的共轭锥的次表示定理

对于0β≤1,有限测度空间(Ω,Σ,μ)与Hilbert空间X,本文研究向量值局部β-凸函数空间L~β(μ,X)的共轭锥[L~β(μ,X)]_β~*的表示问题.在赋范锥(X_β~*,‖-‖)对μ满足Randon-Nikodym性质的条件下,证明次表示定理[L~β(μ,X)]_β~*(?)L~∞(μ,X_β~*).     

Analytic solutions of a class of nonlinear partial differential equations

An approach is presented for computing the adjoint operator vector of a class of nonlinear （that is, partial-nonlinear） operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author. A unified theory is then given to solve a class of nonlinear （partial-nonlinear and including all linear） and non-homogeneous differential equations with a mathematical mechanization method. In other words, a transformation is constructed by homogenization and triangulation, which reduces the original system to a simpler diagonal system. Applications are given to solve some elasticity equations.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper,a new analytical method of symplectic system.Hamiltonian system,is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain.In the system,the fundamental problem is reduced to all eigenvalue and eigensolution problem.The solution and boundary conditions call be expanded by eigensolutions using ad.ioint relationships of the symplectic ortho-normalization between the eigensolutions.A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space.The results show that fundamental flows can be described by zero eigenvalue eigensolutions,and local effects by nonzero eigenvalue eigensolutions.Numerical examples give various flows in a rectangular domain and show effectivenees of the method for solving a variety of problems.Meanwhile.the method can be used in solving other problems.