Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

Anderson Acceleration of Gradient Methods with Energy for Optimization Problems

Anderson acceleration(AA)is an extrapolation technique designed to speed up fixed-point iterations.For optimization problems,we propose a novel algorithm by combining the AA with the energy adaptive gradient method(AEGD)[arXiv:2010.05109].The feasibility of our algorithm is ensured in light of the convergence theory for AEGD,though it is not a fixed-point iteration.We provide rigorous convergence rates of AA for gradient descent(GD)by an acceleration factor of the gain at each implementation of AA-GD.Our experimental results show that the proposed AA-AEGD algorithm requires little tuning of hyperparameters and exhibits superior fast convergence.

Improvement of the Energy Stability of the Single Ion Microbeam

Energy instability strongly affects the state and the beam size of the single ion microbeam. A facility based on the Generating Voltmeter was developed to improve the energy stability of the CAS-LIBB （Chinese Academy of Sciences, key laboratory of ion beam bioengineering） single ion microbeam. This paper presents the analysis of the energy＇ instability of the single ion microbeam. A simplified theoretical model is set up to calculate the relationship between the energy instability and the beam spot size. By using this technique, the energy instability is adjusted to about 1%. Stable run-time is over 6 hours. The radius of the single ion beam is reduced by 10% compared to the previous olin.

Stability Analysis of the Viscous Polytropic Dark Energy Model in Einstein Cosmology

The viscous polytropic gas model as one model of dark energy is hot-spot and keystone to the modern cosmology. We study the evolution of the viscous polytropic dark energy model interacting with the dark matter in the Einstein cosmology. Setting the autonomous dynamical system for the interacting viscous polytropic dark energy with dark matter and using the phase space analysis method to investigate the dynamical evolution and its critical stability, we find that the viscosity property of the dark energy creates a benefit for the stable critical dynamical evolution of the interaction model between dark matter and dark energy in the flat Friedmann-Robertson-Walker universe and the viscosity of dark energy will soften the coincidence problem just like the interacting dark energy model.

A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS

This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations.In this method,the system is decoupled and linearized to avoid solving the non-linear equation at each step.The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability,and this is confirmed by the numerical result,and also shows that the numerical algorithm is stable.

Hybrid Seagull and Whale Optimization Algorithm-Based Dynamic Clustering Protocol for Improving Network Longevity in Wireless Sensor Networks

Energy efficiency is the prime concern in Wireless Sensor Networks(WSNs) as maximized energy consumption without essentially limits the energy stability and network lifetime. Clustering is the significant approach essential for minimizing unnecessary transmission energy consumption with sustained network lifetime. This clustering process is identified as the Non-deterministic Polynomial(NP)-hard optimization problems which has the maximized probability of being solved through metaheuristic algorithms.This adoption of hybrid metaheuristic algorithm concentrates on the identification of the optimal or nearoptimal solutions which aids in better energy stability during Cluster Head(CH) selection. In this paper,Hybrid Seagull and Whale Optimization Algorithmbased Dynamic Clustering Protocol(HSWOA-DCP)is proposed with the exploitation benefits of WOA and exploration merits of SEOA to optimal CH selection for maintaining energy stability with prolonged network lifetime. This HSWOA-DCP adopted the modified version of SEagull Optimization Algorithm(SEOA) to handle the problem of premature convergence and computational accuracy which is maximally possible during CH selection. The inclusion of SEOA into WOA improved the global searching capability during the selection of CH and prevents worst fitness nodes from being selected as CH, since the spiral attacking behavior of SEOA is similar to the bubble-net characteristics of WOA. This CH selection integrates the spiral attacking principles of SEOA and contraction surrounding mechanism of WOA for improving computation accuracy to prevent frequent election process. It also included the strategy of levy flight strategy into SEOA for potentially avoiding premature convergence to attain better trade-off between the rate of exploration and exploitation in a more effective manner. The simulation results of the proposed HSWOADCP confirmed better network survivability rate, network residual energy and network overall throughput on par with the competitive CH selection schemes under different number of data transmission rounds.The statistical analysis of the proposed HSWOA-DCP scheme also confirmed its energy stability with respect to ANOVA test.

A nonlinear rheological model of backfill material for retaining roadways and the analysis of its stability

The long-term stability of backfill material is the key to retaining roadways successfully. In order to study the rheological deformation of backfill material and its long-term stability, given the visco-elastoplastic properties of this material, we introduced a softening and a hardening function for a new nonlinear the- ological model with time-varying parameters. Based on this, we presented the instability condition of this model by using the principle of minimum potential energy. Combined with engineering practice, we cal- culated the urlstable time period of backfill material. The results show that the time of instability of the backfill material relate to the initial parameters of the material, ＂the coefficients decided by temperature and the ratio of the plastic zone of the backfill material. Based on the results of our analysis from the point of view of energy, we can quickly obtain the time of instability of this model from our graphical analysis. The time of instability of the backfill material obtained from our investigation coincides with an actual project.

Noise-induced bistable switching dynamics through a potential energy landscape

Interlinked positive feedback loops,an important building block of biochemical systems,can induce bistable switching,leading to long-lasting state changes by brief stimuli.In this work,prevalent mutual activation between two species as another positive feedback is added to a generic interlinked positive-feedback-loop model originating from many realistic biological circuits.A stochastic fluctuation of the positive feedback strength is introduced in a bistable interval of the feedback strength,and bistability appears for the moderate feedback strength at a certain noise level.Stability analysis based on the potential energy landscape is further utilized to explore the noise-induced switching behavior of two stable steady states.

Stability and reinforcement analysis of rock slope based on elasto-plastic finite element method

The rigid body limit equilibrium method(RBLEM) and finite element method(FEM) are two widely used approaches for rock slope's stability analysis currently. RBLEM introduced plethoric assumptions; while traditional FEM relied on artificial factors when determining factor of safety(FOS) and sliding surfaces. Based on the definition of structure instability that an elasto-plastic structure is not stable if it is unable to satisfy simultaneously equilibrium condition, kinematical admissibility and constitutive equations under given external loads, deformation reinforcement theory(DRT) is developed. With this theory, plastic complementary energy(PCE) can be used to evaluate the overall stability of rock slope, and the unbalanced force beyond the yield surface could be the identification of local failure. Compared with traditional slope stability analysis approaches, the PCE norm curve to strength reduced factor is introduced and the unbalanced force is applied to the determination of key sliding surfaces and required reinforcement. Typical and important issues in rock slope stability are tested in TFINE(a three-dimensional nonlinear finite element program), which is further applied to several representatives of high rock slope's stability evaluation and reinforcement engineering practice in southwest of China.

Relation between stabilization energy, crystal field coefficient and the magnetic exchange interaction for Tb^(3+) ion

Based on a single ion model, Hamiltonian of the simplest form about magnetocrystalline anisotropy for Tb3+ ion was solved by using the numerical method. The relation between the stabilization energy, crystal field coefficient B20 and the magnetic exchange interaction was studied as temperature approaches to 0 K. The results show that the stabilization energy contributed by Tb3+ is linear with crystal field coefficient B20 approximately, but it is insensitive to the change of magnetic exchange interaction for the strong magnetic substances such as TbCo5, Tb2Co17 and Tb2Fe14B compounds.

Grid tie inverter energy stabilizing in smart distribution grid with energy storage

With the growing deployment of smart distribution grid,it has become urgent to investigate the smart distribution grid behavior during transient faults and improve the system stability.The feasibility of segmenting large power grids and multiple smart distribution grids interconnections using energy storage technology for improving the system dynamic stability was studied.The segmentation validity of the large power grids and smart distribution grid inverter output interconnections power system using energy storage technology was proved in terms of theoretical analysis.Then,the influences of the energy storage device location and capacity on the proposed method were discussed in detail.The conclusion is obtained that the ESD optimal locations are allocated at the tie line terminal buses in the interconnected grid,respectively.The effectiveness of the proposed method was verified by simulations in an actual power system.

Theoretical Study on the Molecular Orbital and Stabilization Energy of Substituted Lithium Carbene Cations

The stabilization energies of substituted lithium carbene cations were calculated by using ab initio molecular orbital theory, and the relationship between the stabilization energies and molecular orbitals was discussed. The substituents with pi donor engender strong stabilization to CH2Li+. The calculations show the Y-C* bond lengths of cations become shorter and H-Y bond lengths longer than those of corresponding neutral molecules.

Relations Between Stabilities and Structures of Closo Borane Dianions

An effective method to investigate the stabilities of a series of new closo-BnHn2- （n = 12, 14, 16, 18, 20, 22, 24, 30） was put forward with the aid of G96PW91/SHC calculations. Stabilities are related to the relative stabilized energies （RSE） and the 2e3c bound geometries of closo-BnHn2-. The structures in which a boron atom connects to four atoms up to seven are stable and appear in many borides because of the lower relative stabilized energy. In geometries, both triangular and quadrangular faces are in favor of forming the structures of closo-BnHn2-. The energies of optimized geometries support the existence of these new compounds. By employing both RSE and AE per boron atom in cage, the stabilities were studied to predict the probabilities of unknown clusters in existence. The electron-deficient clusters can be understood that the positive holes should be disperse to every triangular face and lead to share the holes, wherever there are not enough electrons to occupy them. The negative charges which anions carry distribute to 2e3c bonds to increase the stabilities.

Stabilization of coordinated motion for underwater vehicles

This paper presents a coordinating and stabilizing control law for a group of underwater vehicles with unstable dynamics. The coordinating law is derived from a potential that only depends on the relative configuration of the underwater vehicles. Being coordinated,the group behaves like one mechanical system with symmetry,and we focus on stabilizing a family of coordinated motions,called relative equilibria. The stabilizing law is derived using energy shaping to stabilize the relative equilibria which involve each vehicle translating along its longest（unstable） axis without spinning,while maintaining a relative configuration within the group. The proposed control law is physically motivated and avoids the linearization or cancellation of nonlinearities.

Delocalizedπ_(3)^(6) Bond in OX_(2) (X=Halogen) Molecules

OX_(2)(X=halogen)molecules was studied theoretically.Calculation results show that delocalizedπ_(3)^(6) bonds exist in their electronic structures and O atoms adopt the sp^(2) type of hybridization,which violates the prediction of the valence shell electron pair repulsion theory of sp^(3) type.Delocalization stabilization energy is proposed to measure the contribution of delocalizedπ_(3)^(6) bond to energy decrease and proves to bring extra-stability to the molecule.These phenomena can be summarized as a kind of coordinating effect.

Theoretical Study of Hydrated Cd^(2+) Interactions with Guanine

Theoretical study was performed to investigate how the hydration of cadmium ca-tion influences the structure and properties of guanine.The aqueous environment was simulated by both explicit solvent（1-5 water molecules） model and implicit solvent model.For complexes in which Cd2＋ attached to the N（7） and O（6） sites of guanine,energy analysis together with the Natural Bonding Orbital（NBO） analysis were performed to elucidate the bonding characteristics in detail.The most stable structures are penta-coordinate complexes without aqua ligand located at the guanine site.Higher number of water ligands corresponds to higher stabilization energies.Average bonding energies of G-Cd increase with the number of water molecules.Bonding energies of water ligands depend on its position in the complexes.The charge distribution of guanine changed with increasing the number of water ligands,which may also influence the base-pairing pattern of guanine.There is positive charge transfer from guanine to aqua ligand as the number of the hydration waters increases.IEFPCM optimization has results comparable to the [CdG（H2O）5]2＋ structure 5a.

EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS

In this paper,we study linearly first and second order in time,uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel"scalar auxiliary variable"(SAV)approach,a new developed efficient and accurate method for a large class of gradient flows.The schemes are based on the first order Euler method and the second order backward differential formulas(BDF2)for time discretization,and finite element methods for space discretization.The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps.Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.

Convergence Analysis of the Fully Decoupled Linear Scheme for Magnetohydrodynamic Equations

In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.

IMPROVED COVARIANCE DRIVEN BLIND SUBSPACE IDENTIFICATION METHOD

An improved covariance driven subspace identification method is presented to identify the weakly excited modes. In this method, the traditional Hankel matrix is replaced by a reformed one to enhance the identifiability of weak characteristics. The robustness of eigenparameter estimation to noise contamination is reinforced by the improved Hankel matrix, in combination with component energy index （CEI） which indicates the vibration intensity of signal components, an alternative stabilization diagram is adopted to effectively separate spurious and physical modes. Simulation of a vibration system of multiple-degree-of-freedom and experiment of a frame structure subject to wind excitation are presented to demonstrate the improvement of the proposed blind method. The performance of this blind method is assessed in terms of its capability in extracting the weak modes as well as the accuracy of estimated parameters. The results have shown that the proposed blind method gives a better estimation of the weak modes from response signals of small signal to noise ratio （SNR）and gives a reliable separation of spurious and physical estimates.

Analysis of a Linearized Energy Stable Numerical Scheme for a Modified Incompressible Cahn-Hilliard-Navier-Stokes System

In this paper,a linearized energy stable numerical scheme is used to solve the modified Cahn-Hilliard-Navier-Stokes model,which is a phase-field model for two-phase incompressible flows.The time discretization is based on the convex splitting of the energy functional,which leads to a linearized system.In order to maintain the energy stability,the definition domain of energy function is extended to infinity.The stability of the scheme is proved and the error estimate is given.Numerical experiments are done to demonstrate the effectiveness for the proposed scheme.