Zichen Deng

Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

Nonlinear wave equations, such as Burgers equation and compound KdV-Burgers equation, are a class of partial differential equations (PDEs) with dissipation in Hamiltonian space, the numerical method of which plays an important role in complex fluid analysis. Based on the multi-symplectic idea, a new theoretical framework named generalized multi-symplectic integrator for a class of nonlinear wave PDEs with small damping is proposed in this paper. The generalized multi-symplectic formulation is introduced, and a twelve-point generalized multi-symplectic scheme, which satisfies two discrete modified conservation laws approximately as well as the local momentum conservation law accurately, is constructed to solve the first-order PDEs that derived from the compound KdV-Burgers equation. To test the generalized multi-symplectic scheme, several numerical experiments on the travelling front solution are carried out, the results of which imply that the generalized multi-symplectic scheme can simulate the travelling front solution accurately and satisfy the modified conservation laws well when step sizes and the damping parameter satisfy the inequality (41). It is more remarkable that the scheme (36) can be used to capture the shock wave structure of the compound KdV-Burgers equation within one data point, which can further illustrate the good structure-preserving property of the generalized multi-symplectic scheme (36). From the results of this paper, we can conclude that, similar to the multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property. For the existing of the excellent numerical properties, the generalized multi-symplectic method can be used to exposit some specific phenomena in the complex fluid.

Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory

Variational principles for the buckling and vibration of multi-walled carbon nanotubes (MWCNTs) are established with the aid of the semi-inverse method. They are used to derive the natural and geometric boundary conditions coupled by small scale parameters. Hamilton’s principle and Rayleigh’s quotient for the buckling and vibration of the MWCNTs are given. The Rayleigh-Ritz method is used to study the buckling and vibration of the single-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotubes (DWCNTs) with three typical boundary conditions. The numerical results reveal that the small scale parameter, aspect ratio, and boundary conditions have a profound effect on the buckling and vibration of the SWCNTs and DWCNTs.

Band gap analysis of star-shaped honeycombs with varied Poisson’s ratio

In this paper, star-shaped honeycombs are analyzed in terms of their equivalent mechanical behaviors and band gap properties. Firstly, by applying Castiglianos second theorem, the effective Youngs modulus and Poissons ratio are derived by an analytical method used in structural mechanics. On the basis of Blochs theorem, the dispersion characteristics are then analyzed by the dynamic matrix in conjunction with the Wittrick–Williams (W–W) algorithm. It should be noted that the presented method can form a more simple stiffness and mass matrices of the proposed structures, compared with the traditional finite element (FE) method. Thereafter, the effects of the geometrical parameters on the effective constants and band gaps are investigated and discussed. Numerical results demonstrate that the negative Poissons ratio provides an enhanced effective Youngs modulus of the considered honeycombs. Furthermore, the band gap exists in a much lower frequency region with an unchanged summing band gap width when the Poissons ratio is in negative values. In general, the work can serve as a guide for the optimal design of cellular structures.

AN IMPLICIT DIFFERENCE SCHEME FOCUSING ON THE LOCAL CONSERVATION PROPERTIES FOR BURGERS EQUATION

Focusing on the local conservation properties, a nine-point implicit difference scheme derived from the multi-symplectic idea, which is named as a generalized multi-symplectic integrator, is presented for the Burgers equation firstly. And then, the associated proofs needed on the local conservation properties of the scheme are given in detail. Finally, to illustrate the high accuracy, the good local conservation properties as well as the excellent long-time numerical behavior of the scheme, the numerical experiments on the single-front solution of the Burgers equation by the implicit scheme are reported.

Closed solutions for the electromechanical bending and vibration of thick piezoelectric nanobeams with surface effects

In this paper, a more accurate model is established to study the influences of surface effects (SEs) including the surface elasticity, residual surface stress and surface piezoelectricity, on the electromechanical bending and vibration of the piezoelectric nanobeam (PNB) in the presence of shear deformation and rotary inertia. Analytical solutions are obtained for the electromechanical bending deflection, resonant frequency and mode shape of the PNB for three typical boundary conditions, demonstrating the significance of incorporation of shear deformation, rotary inertia and the surface parameters at different aspect ratios. The analytical solutions are found to be in good agreement with both molecular dynamics results and experimental data. The numerical results reveal that the surface elasticity plays a less significant role on the electromechanical bending and vibration than that of surface piezoelectricity and residual surface stress, and can be neglected for PNB with small aspect ratio and stiffer boundary conditions. In addition, the shear deformation and rotary inertia are found to play a larger impact effect than SEs for a stubby beam at higher vibration mode (i.e., aspect ratio less than 14 for doubly clamped beam). Moreover, the SEs are found to be increasing notably as aspect ratio increases. The continuum model established in this study will be useful for characterizing the mechanical properties of size-dependent piezoelectric structures and the design, calibration and application of PNB-based devices.

Dynamic analysis of embedded curved double‐walled carbon nanotubes based on nonlocal Euler‐Bernoulli Beam theory

Purpose – The aim of this paper is to study the dynamic vibrations of embedded double‐walled carbon nanotubes (DWCNTs) subjected to a moving harmonic load with simply supported boundary conditions.Design/methodology/approach – The model of DWCNTs is considered as an Euler‐Bernoulli beam with waviness along the length, which is more accurate than the straight beam in previous works. Based on the nonlocal beam theory, the governing equations of motion are derived by using the Hamiltons principle, and then the separation of variables is carried out by the Galerkin approach, leading to two second‐order ordinary differential equations (ODEs).Findings – The influences of the nonlocal parameter, the amplitude of the waviness, the surrounding elastic medium, the material length scale, load velocity and van der Waals force on the nonlinear vibration of DWCNTs are important.Originality/value – The dynamic responses of DWCNTs are obtained by using the precise integrator method to ordinary differential equations.

Chaotic region of elastically restrained single-walled carbon nanotube

The occurrence of chaos in the transverse oscillation of the carbon nanotube in all of the precise micro-nano mechanical systems has a strong impact on the stability and the precision of the micro-nano systems, the conditions of which are related with the boundary restraints of the carbon nanotube. To generalize some transverse oscillation problems of the carbon nanotube studied in current references, the elastic restraints at both ends of the single-walled carbon nanotube are considered by means of rotational and translational springs to investigate the effects of the boundary restraints on the chaotic properties of the carbon nanotube in this paper. Based on the generalized multi-symplectic theory, both the generalized multi-symplectic formulations for the governing equation describing the transverse oscillation of the single-walled carbon nanotube subjected to the transverse load and the constraint equations resulting from the elastic restraints are presented firstly. Then, the structure-preserving scheme with discrete constraint equations is constructed to simulate the transverse oscillation process of the carbon nanotube. Finally, the chaotic region of the carbon nanotube is captured, and the oscillations of the two extreme cases (including simply supported and cantilever) are investigated in the numerical investigations. From the numerical results, it can be concluded that the relative bending stiffness coefficient and the absolute bending stiffness coefficients at both ends of the carbon nanotube are two important factors that affect the chaotic region of the carbon nanotube, which provides guidance on the design and manufacture of precise micro-nano mechanical systems. In addition, the different routes to the chaos of the carbon nanotube in two extreme cases are revealed.

Dynamic Crushing Strength Analysis of Auxetic Honeycombs

The in-plane dynamic crushing behavior of re-entrant honeycomb is analyzed and compared with the conventional hexagon topology. Detailed deformation modes along two orthogonal directions are examined, where a parametric study of the effect of impact velocity and cell wall aspect ratio is performed. An analytical formula of the dynamic crushing strength is then deduced based on the periodic collapse mechanism of cell structures. Comparisons with the finite element results validate the effectiveness of the proposed analytical method. Numerical results also reveal higher plateau stress of re-entrant honeycomb over conventional hexagon topology, implying better energy absorption properties. The underlying physical understanding of the results is emphasized, where the auxetic effect (negative Poisson’s ratio) induced in the re-entrant topology is believed to be responsible for this superior impact resistance.

Transient thermal response in thick orthotropic hollow cylinders with finite length: High order shell theory

The transient thermal response of a thick orthotropic hollow cylinder with finite length is studied by a high order shell theory. The radial and axial displacements are assumed to have quadratic and cubic variations through the thickness, respectively. It is important that the radial stress is approximated by a cubic expansion satisfying the boundary conditions at the inner and outer surfaces, and the corresponding strain should be least-squares compatible with the strain derived from the strain-displacement relation. The equations of motion are derived from the integration of the equilibrium equations of stresses, which are solved by precise integration method (PIM). Numerical results are obtained, and compared with FE simulations and dynamic thermo-elasticity solutions, which indicates that the high order shell theory is capable of predicting the transient thermal response of an orthotropic (or isotropic) thick hollow cylinder efficiently, and for the detonation tube of actual pulse detonation engines (PDE) heated continuously, the thermal stresses will become too large to be neglected, which are not like those in the one time experiments with very short time.

GENERALIZED MULTI-SYMPLECTIC METHOD FOR DYNAMIC RESPONSES OF CONTINUOUS BEAM UNDER MOVING LOAD

Based on the multi-symplectic idea, a generalized multi-symplectic integrator method is presented to analyze the dynamic response of multi-span continuous beams with small damping coefficient. Focusing on the local conservation properties, the generalized multi-symplectic formulations are introduced and a fifteen-point implicit structure-preserving scheme is constructed to solve the first-order partial differential equations derived from the dynamic equation governing the dynamic behavior of continuous beams under moving load. From the results of the numerical experiments, it can be concluded that, for the cases considered in this paper, the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3751 when the continuous beam is under a constant-speed moving load and the structure-preserving scheme is generalized multi-symplectic if the viscous damping c ≤ 0.3095 when the continuous beam is under a variable-speed moving load with fixed step lengths Δt = 0.05 and Δx = 0.025. Similar to a multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property.