Structure-preserving Method Research Articles

Structure-preserving methods for Marcus stochastic Hamiltonian systems with additive Lévy noise

A general structure-preserving method is proposed for a class of Marcus stochastic Hamiltonian systems driven by additive Lévy noise. The convergence of the symplectic Euler scheme for this systems is investigated by Generalized Milstein Theorem. Realizable numerical implementation of this scheme is also provided in details. Numerical experiments are presented to illustrate the effectiveness and superiority of the proposed scheme. Applications of the method to solve two mathematical physical problems are provided.

Structure-preserving analysis on chaotic characteristics of transverse vibration for embedded double-walled carbon nanotube

Abstract Analysing on the ultra-high frequency vibrational characteristics of carbon nanotubes, especially on the chaotic characteristics, is a key scientific problem in the dynamic design of the carbon nanotube devices. Considering the van der Waals force between the inner layer and the outer layer of the embedded double-walled carbon nanotube, and the effects of the elastic medium as well as the effects of the simple harmonic external excitation, the coupling dybamic model describing the transverse vibration of the embedded double-walled carbon nanotube is presented. The generalized multi-symplectic formulations with an explicit multi-symplectic structure residual are deduced by introducing the dual momenta. The Preissmann approach, which has been proved to be a structure-preserving method that can be used to reproduce the chaotic characteristics of carbon nanotubes, is employed to discrete the generalized multi-symplectic formulations. The numerical results imply that, the transverse vibration of the embedded double-walled carbon nanotube subjected to the external excitation larger than the critical external excitation will enter the chaotic state through a period-doubling bifurcation path. In addition, the critical external excitation for the chaos of the inner layer carbon nanotube’s transverse vibration is larger than that of the outer layer carbon nanotube’s transverse vibration. The above findings reported in this paper provide some guidance for the dynamic design of the carbon nanotube devices directly.

A structure preserving numerical method for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

Effective numerical simulations of synchronous generator system

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods.

A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics

Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Coupling dynamic behaviors of axially moving cracked cantilevered beam subjected to transverse harmonic load

Considering the existence of the opening crack, the axially motion as well as the transverse harmonic excitation, the dynamic response of the beam is studied by a structure-preserving method in this paper. With the equivalent bending stiffness introduced to describe the effect of the crack on the flexibility of the beam, the vibration model of the axially moving cracked cantilevered beam subjected to a transverse harmonic load at its free end is presented. A structure-preserving method in the generalized multi-symplectic framework is developed subsequently. Finally, the effects of the crack’s parameters, the axially moving speed and the circular frequency of the transverse harmonic load on the steady vibration amplitude of the cracked beam are investigated by the structure-preserving method in detail. From numerical results, it can be found that when the beam moves with the high speed, the appearance of the crack on the beam will intensify the vibration of the beam sharply. In addition, the vibration amplitude of the cracked beam is suddenly larger in five narrow circular frequency bands. Comparing with the results of the wavelet spectral finite element method, the center circular frequencies of the narrow five circular frequency bands obtained by the generalized multi-symplectic method are highly agreed with the first five natural frequencies of the cracked beam, which illustrates the precision of the numerical results reported in this paper directly.

Meshless structure-preserving GRBF collocation methods for stochastic Maxwell equations with multiplicative noise

In this paper, motivated by the principle that numerical methods should preserve the intrinsic properties of the original system as much as possible, we propose two novel classes of structure-preserving methods for the stochastic Maxwell equations with multiplicative noise. More precisely, due to the advantages of high-order accuracy and the simplicity to deal with high-dimensional problems, the meshless global radial basis function (GRBF) collocation method is firstly utilized to discretize the stochastic Maxwell equations in space, the resulting semi-discretization preserves energy and symplectic structure. Then we apply Padé approximation, the splitting technique and the Runge–Kutta method to propose two kinds of efficient fully-discrete methods that are proved to be symplectic, multi-symplectic and energy-preserving simultaneously. Numerical experiments are given to indicate the validity of the proposed methods.

High-order Runge–Kutta structure-preserving methods for the coupled nonlinear Schrödinger–KdV equations

A novel class of high-order Runge–Kutta structure-preserving methods for the coupled nonlinear Schrödinger–KdV equations is proposed and analyzed. With the aid of the quadratic auxiliary variable, an equivalent system is obtained from the original problem. The Fourier pseudo-spectral method is employed in spatial discretization and the symplectic Runge–Kutta method is utilized for the resulting semi-discrete system to arrive at a high-order fully discrete scheme. Simultaneously, the conservation of the original multiple invariants for the schemes are rigorously proven. Numerical experiments are performed to verify the theoretical analysis.

$$\mathcal {L_C}$$ structure-preserving method based on semi-tensor product of matrices for the QR decomposition in quaternionic quantum theory

$$\mathcal {L_C}$$ structure-preserving method based on semi-tensor product of matrices for the QR decomposition in quaternionic quantum theory

Dynamical Symmetry Breaking of Infinite-Dimensional Stochastic System

The mapping relationship between the symmetry and the conserved quantity inspired researchers to seek the conserved quantity from the viewpoint of the symmetry for the dynamic systems. However, the symmetry breaking in the dynamic systems is more common than the symmetry in the engineering. Thus, as the supplement of our previous work on the symmetry breaking of infinite-dimensional deterministic dynamic systems, the dynamical symmetry breaking of infinite-dimensional stochastic systems is discussed in this paper. Following a brief review of the stochastic (multi-)symplectic for the dynamic system excited by stochastic white noise, two types of stochastic symmetry breaking factors, including the general stochastic excitation and the general stochastic parameters of the infinite-dimensional dynamic systems, are investigated in detail. We find that both the general stochastic excitation and the general stochastic parameters will not break the local multi-symplectic structure of the dynamic systems. However, the local energy conservation law will be broken by the general stochastic excitation, as well as by the stochastic parameters, which are given by the local energy dissipation in this paper. To illustrate the validity of the analytical results, the stochastic vibration of a clamped single-walled carbon nanotube is investigated and the critical condition of the appearance of chaos is obtained. The theoretical results obtained can be used to guide us to construct the structure-preserving method for the stochastic dynamic systems.

Dynamic analysis on flexible hub-beam with step-variable cross-section

Decreasing the vibration amplitude of the manipulator in the Space Manipulator Systems (SMS) with a given mass limitation is an effective way to reduce the control costs for the SMS. In this paper, the dynamic response of the SMS modeled by a flexible hub-beam with a step-variable cross-section excited by an external torque is investigated employing a new complex structure-preserving method. The dynamic model of the flexible hub-beam with a step-variable cross-section is deduced firstly. Then, a new structure-preserving method connecting the symplectic precise integration method and the generalized multi-symplectic method is proposed, in which, the time-consuming iteration is avoided and the simulation time is saved. In the numerical simulations, several simple cases for the step-variable cross-section are considered and the stable vibration amplitudes of the flexible beam at the free end are reported. The main contribution of this work is proposing a structure-preserving method for the solving the coupling dynamic problems of the hub-beam system to provide some suggestions on the structure design for the SMS directly.

変分的積分法を用いたベイジアンニューラルネットワークの応用

In numerical analysis, the conservation of structure, i.e., whether the essential properties of physical phenomena can be reproduced numerically, is of great importance. On the other hand, neural networks that can learn conservation laws has recently received great attention. These models take important structure of physical problems into consideration, which are often ignored in traditional models that directly learn using data, consequently exhibiting long-term accurate prediction and interpretability. In the current study, we construct neural network models based on variational integrator, a typical kind of structure-preserving method. In addition, we introduce Bayesian framework to measure uncertainty of the model and evaluate the models quantitively. Illustrative examples are studied.

Coupling Dynamic Behavior of Space Flexible Hollow Beam

The coupling between the plane motion and transverse vibration is a challenge for the dynamic analysis of space structures. The coupling dynamic behavior of the on-orbit space flexible hollow beam is studied in this paper. As a common configuration of spacecraft and their components, the space structure with a large slenderness ratio and hollow structure is considered. With weak damping and variable structural parameters, the dynamic model is derived for the space flexible hollow beam by the Hamilton least-action principle. Then, a complex structure-preserving method that has been proved to be feasible for the infinite-dimensional dynamic system is constructed. Subsequently, effects of the hollow structure, weak damping and variable structural parameters on the orbit-attitude-vibration coupling dynamics are discussed comprehensively. Meanwhile, the evolutions of energy of the coupling dynamic system with/without damping are reproduced. The results show that the effects of the hollow structure, damping and variable structural parameters are all notable for the dynamic behavior of the space coupling dynamic systems. The constructed complex structure-preserving method can effectively characterize the coupling dynamic behavior and energy evolution of the space system. The numerical examples may serve as practical references for the design of space flexible hollow structures.

Projected exponential Runge–Kutta methods for preserving dissipative properties of perturbed constrained Hamiltonian systems

Preserving conservative and dissipative properties of dynamical systems is desirable in numerical integration. To this end, we develop and implement numerical methods that preserve the exact rate of dissipation in certain qualitative properties of dissipatively perturbed constrained Hamiltonian systems, which are shown to be conformal symplectic. Projection methods based on exponential Runge–Kutta methods are proposed for such systems. These numerical schemes are shown to be constraint preserving, conformal invariants preserving, symmetric, and second-order accurate. It is shown that these structure-preserving methods can be further composed to obtain higher-order structure-preserving methods. Linear stability analysis is used to derive stability properties and conformal preservation of the phase-space area. Numerical experiments, including constrained oscillators and a damped Korteweg–de Vries partial differential equation, demonstrate the advantages of geometric integration and verify theoretical results.

A new structure-preserving quaternion QR decomposition method for color image blind watermarking

Most of the existing color image watermarking schemes are designed to mark each color channel individually, which ignores the correlation of different color channels and the synchronous embedding of watermarks. In this paper, we propose a robust blind watermarking scheme based on quaternion QR decomposition (QQRD) for color image copyright protection, while using algebraic structure-preserving method to release its computational complexity. In watermark embedding process, QQRD is applied to non-overlapping 4×4 image blocks in a whole manner, whereupon three strong correlation coefficient pairs determined by the Normalized Cross-Correlation (NC) method are modified to embed color watermark within factorized matrices. This avoids layered processing that causes abnormal sensitivity to attacks and less correlation that causes excessive modification of coefficient values. Numerical results show that the newly proposed watermarking scheme has strong ability to resist most image processing attacks and is error-free under no attack, brightness adjustment attack and Poisson noise attack. Its superiority over other state-of-the-art methods is also experimentally demonstrated.

A hybrid mimetic spectral element method for three-dimensional linear elasticity problems

We introduce a domain decomposition structure-preserving method based on a hybrid mimetic spectral element method for three-dimensional linear elasticity problems in curvilinear conforming structured meshes. The method is an equilibrium method which satisfies pointwise equilibrium of forces. The domain decomposition is established through hybridization which first allows for an inter-element normal stress discontinuity and then enforces the normal stress continuity using a Lagrange multiplier which turns out to be the displacement in the trace space. Dual basis functions are employed to simplify the discretization and to obtain a higher sparsity. Numerical tests supporting the method are presented.

Structure-Preserving Layout Optimization on Precision Devices in Spacecraft

The stability and the accuracy of the precision devices installed in spacecraft depend on the local vibration characteristics of the spacecraft. In return, the local vibration characteristics of the spacecraft are influenced by the layout of the precision devices, which implies that, a rational layout of the precision devices in the spacecraft is the precondition for the stable and efficient work of the precision devices. The dynamic model for a flexible panel bearing several precision devices was presented and the structure-preserving method was developed to investigate the local vibration characteristics of the panel. In view of the sizes of the precision devices and the heat dissipation clearances between the precision devices, the layout optimization for the precision devices was performed to minimize the weighted values of the out-of-plane vibration accelerations of the precision devices. The optimization results show that, benefiting from the excellent structure-preserving properties of the numerical method employed during the vibration analysis, the weighted values of the out-of-plane vibration accelerations of the precision devices decrease by about 88.05% through the layout optimization, which provides a useful guide for the layout scheme of the precision devices in the spacecraft and improves the stability of the precision devices.

Structure-Preserving Layout Optimization on Precision Devices in Spacecraft

The stability and the accuracy of the precision devices installed in spacecraft depend on the local vibration characteristics of the spacecraft. In return, the local vibration characteristics of the spacecraft are influenced by the layout of the precision devices, which implies that, a rational layout of the precision devices in the spacecraft is the precondition for the stable and efficient work of the precision devices. The dynamic model for a flexible panel bearing several precision devices was presented and the structure-preserving method was developed to investigate the local vibration characteristics of the panel. In view of the sizes of the precision devices and the heat dissipation clearances between the precision devices, the layout optimization for the precision devices was performed to minimize the weighted values of the out-of-plane vibration accelerations of the precision devices. The optimization results show that, benefiting from the excellent structure-preserving properties of the numerical method employed during the vibration analysis, the weighted values of the out-of-plane vibration accelerations of the precision devices decrease by about 88.05% through the layout optimization, which provides a useful guide for the layout scheme of the precision devices in the spacecraft and improves the stability of the precision devices.