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A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment for the singular logarithmic terms and the surface diffusion term, and an explicit update for the expansive concave term. The convective term for the phase variable, as well as the coupled term in the Stokes equation, is approximated in a semi-implicit manner. In the spatial discretization, the marker and cell difference method is applied, which evaluates the velocity components, the pressure and the phase variable at different cell locations. Such an approach ensures the divergence-free feature of the discrete velocity, and this property plays an important role in the analysis. The positivity-preserving property and the unique solvability of the proposed numerical scheme are theoretically justified, utilizing the singular nature of the logarithmic term as the phase variable approaches the singular limit values. An unconditional energy stability analysis is standard, as an outcome of the convex-concave decomposition technique. A convergence analysis with accompanying error estimate is provided for the proposed numerical scheme. In particular, a higher order consistency analysis, accomplished by supplementary functions, is performed to ensure the separation properties of numerical solution. In turn, using the approach of rough and refined error estimates, we are able to derive an optimal rate convergence. To conclude, several numerical experiments are presented to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

A second‐order accurate, linear numerical method is analyzed for the Landau–Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non‐convexity constraint of unit length of the magnetization. The numerical method is based on the second‐order backward differentiation formula in time, combined with an implicit treatment for the linear diffusion term from the harmonic mapping part and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point‐wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1)$$ {\ell}^{\infty}\left(0,T;{\ell}^2\right)\cap {\ell}^2\left(0,T;{H}_h^1\right) $$ norm, under suitable regularity assumptions and reasonable ratio between the time step size and the spatial mesh size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3. [ABSTRACT FROM AUTHOR]

Highlights • We propose a new ASGD method, Acc-Prox-SVRG-BB, which uses the BB method to automatically compute step size for Acc-Prox-SVRG. We prove the convergence of Acc-Prox-SVRG-BB and show that its complexity achieves the same level as the best known stochastic gradient methods. • To reduce the difficulty of choosing the parameters in Acc-Prox-SVRG, we propose a new method, FISTA-Prox-SVRG, which incorporates Beck and Teboulle's APG (a.k.a FISTA) and Prox-SVRG in a mini-batch setting. • To further validate the effectiveness of the BB method in the ASGD methods, we combine the BB method with FISTA-Prox-SVRG. Numerical experiments show the efficacy of our proposed methods. Abstract Accelerated stochastic gradient descent (ASGD) methods, which incorporate accelerated proximal gradient (APG) and stochastic gradient (SG), have received considerable attention recently for solving regularized risk minimization problems in signal/image processing, statistics and machine learning. However, there has been a paucity of practical guidance proposed for resolving one of the major issues in ASGD: how to choose an appropriate step size. To solve this problem, we propose to use the Barzilai-Borwein (BB) method to automatically compute step size for the accelerated mini-batch Prox-SVRG (Acc-Prox-SVRG) method (the state of the art ASGD method), thereby obtaining a new accelerated method: Acc-Prox-SVRG-BB. We prove the convergence of Acc-Prox-SVRG-BB and show that its complexity is comparable with the best known stochastic gradient methods. In addition, we incorporate Beck and Teboulle's APG (FISTA) and Prox-SVRG in a mini-batch setting and obtain another new accelerated gradient descent method, FISTA-Prox-SVRG, which requires the selection of fewer unknown parameters than those required in Acc-Prox-SVRG. Finally, we introduce the BB method into FISTA-Prox-SVRG to further show the efficacy of the BB method. Numerical results demonstrate the advantage of our algorithms. [ABSTRACT FROM AUTHOR]

Highlights • A bore-center annular shaped charge was proposed and validated to improve the penetration diameter and depth of the annular shaped charge on hard targets. • An X-ray experiment was conducted to validate the reliability of numerical results. • The influence of liner configurations on radial and axial velocity of bore-center annular shaped charges was investigated. • The penetrating performance of the different bore-center annular shaped charges were evaluated in terms of size of penetration hole diameter and depth. Abstract The classical shaped charges are well known for their capability to yield a small ratio of the hole diameter to charge diameter in metallic targets. However, in specific situations, for instance the follow-through charge of tandem warheads and the fast establishment of a rapid rescue channel, the ratio of the hole diameter to charge diameter of 1:1 is demanded that cannot be achieved by normal shaped charges. To improve the penetration diameter and depth of the annular shaped charge on hard targets, in this work, a bore-center annular shaped charge (BCASC) is proposed and validated. An X-ray experiment is conducted to validate the reliability of numerical results. The maximal deviation of the head and tail diameters, length, axial and radial velocity of the BCASC projectiles at different times, between the simulation and X-ray experimental results, is only 4.9%. The numerical simulation results of penetrating into steel targets are also compared with the terminal ballistic performance including the diameter and depth of the penetration boreholes. Furthermore, based on the validated numerical models and the corresponding parameters, a series of numerical simulations are performed to investigate the influence of liner configurations (the radial distance (f) of circle centers, the thickness (b) of upper wall, the diameter (d) of center bore and the location of maximal wall thickness (θ)) on radial and axial velocity of BCASCs. Then, a series of BCASC penetration tests on steel targets are conducted. By assessing the penetration hole diameter and depth, the influences the key parameters on the target damage are numerically and experimentally discussed. The results indicate that, compared with other liner parameters (f, b, d), the location where the maximal wall thickness (θ) is placed has the most significant effect on both the radial and axial velocity of the projectile. As the value of θ increases, the radial velocity of the projectile head decreases, while the axial velocity of the projectile head increases. BCASCs with different parameter values can all penetrate into the steel targets and form annular bullet hole at a certain penetration depth. Within our research scope, the θ values imposes more influence on the penetration hole diameter and the depth. As the value of θ increases, a penetration hole diameter decreases, while penetration depth increases gradually. At θ = 30°, the minimal distance between the outer wall of the penetration access is 0.92 D (D indicates the charge diameter), and the mean value of the penetration access depth is 0.56 D. [ABSTRACT FROM AUTHOR]

As a typical latent factor model, Matrix Factorization (MF) has demonstrated its great effectiveness in recommender systems. Users and items are represented in a shared low-dimensional space so that the user preference can be modeled by linearly combining the item factor vector $V$ using the user-specific coefficients $U$ . From a generative model perspective, $U$ and $V$ are drawn from two independent Gaussian distributions, which is not so faithful to the reality. Items are produced to maximally meet users’ requirements, which makes $U$ and $V$ strongly correlated. Meanwhile, the linear combination between $U$ and $V$ forces a bijection (one-to-one mapping), which thereby neglects the mutual correlation between the latent factors. In this paper, we address the upper drawbacks, and propose a new model, named Correlated Matrix Factorization (CMF). Technically, we apply Canonical Correlation Analysis (CCA) to map $U$ and $V$ into a new semantic space. Besides achieving the optimal fitting on the rating matrix, one component in each vector ($U$ or $V$ ) is also tightly correlated with every single component in the other. We derive efficient inference and learning algorithms based on variational EM methods. The effectiveness of our proposed model is comprehensively verified on four public datasets. Experimental results show that our approach achieves competitive performance on both prediction accuracy and efficiency compared with the current state of the art. [ABSTRACT FROM AUTHOR]

In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an ℓ ∞ ( 0 , T ; H h 2 ) stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented. [ABSTRACT FROM AUTHOR]

Document network is a kind of intriguing dataset which can provide both topical (textual content) and topological (relational link) information. A key point in modeling such datasets is to discover proper denominators beneath the text and link. Most previous work introduces the assumption that documents closely linked with each other share common latent topics. However, the heterophily (i.e., tendency to link to different others) of nodes is neglected, which is pervasive in social networks. In this paper, we simultaneously incorporate community detection and topic modeling in a unified framework, and appeal to Canonical Correlation Analysis (CCA) to capture the latent semantic correlations between the two heterogeneous factors, community and topic. Despite of the homophily (i.e., tendency to link to similar others) or heterophily, CCA can properly capture the inherent correlations which fit the dataset itself without any prior hypothesis. We also impose auxiliary word embeddings to improve the quality of topics. The effectiveness of our proposed model is comprehensively verified on three different types of datasets which are hyperlinked networks of web pages, social networks of friends, and coauthor networks of publications. Experimental results show that our approach achieves significant improvements compared with the current state of the art. [ABSTRACT FROM AUTHOR]

We present an error analysis for a fully discrete finite difference scheme for the three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn–Hilliard system with a Flory–Huggins–deGennes free energy potential. The numerical scheme was recently proposed, and the positivity-preserving property and unconditional energy stability were theoretically established. In this paper, we rigorously prove first order convergence in time and second order convergence in space for the numerical scheme, in the L Δ t ∞ (0 , T ; H h − 1) ∩ L Δ t 2 (0 , T ; H h 1) norm. Many highly non-standard estimates have to be involved, due to the nonlinear and singular nature of the surface diffusion coefficients. The combination of (i) a higher order asymptotic expansion of the numerical solution (up to second order temporal accuracy); (ii) a rough error estimate (to establish the L Δ t ∞ bound for the phase variables); (iii) and a refined error estimate have to be carried out to conclude such a convergence result. To our knowledge, it will be the first work to provide an optimal rate convergence estimate for a ternary phase field system with singular energy coefficients. [ABSTRACT FROM AUTHOR]

A second-order accurate (in time) and linear numerical scheme is proposed and analyzed for the nonlocal Cahn–Hilliard equation. The backward differentiation formula is used as the temporal discretization, while an explicit extrapolation is applied to the nonlinear term and the concave expansive term. In addition, an |$O (\varDelta {t}^2)$| artificial regularization term, in the form of |$A \varDelta _N (\phi ^{n+1} - 2 \phi ^n + \phi ^{n-1})$|⁠ , is added for the sake of numerical stability. The resulting constant-coefficient linear scheme brings great numerical convenience; however, its theoretical analysis turns out to be very challenging, due to the lack of higher-order diffusion in the nonlocal model. In fact, a rough energy stability analysis can be derived, where an assumption on the |$\ell ^\infty $| bound of the numerical solution is required. To recover such an |$\ell ^\infty $| bound, an optimal rate convergence analysis has to be conducted, which combines a high-order consistency analysis for the numerical system and the stability estimate for the error function. We adopt a novel test function for the error equation, so that a higher-order temporal truncation error is derived to match the accuracy for discretizing the temporal derivative. Under the view that the numerical solution is actually a small perturbation of the exact solution, a uniform |$\ell ^\infty $| bound of the numerical solution can be obtained, by resorting to the error estimate under a moderate constraint of the time step size. Therefore, the result of the energy stability is restated with a new assumption on the stabilization parameter |$A$|⁠. Some numerical experiments are carried out to display the behavior of the proposed second-order scheme, including the convergence tests and long-time coarsening dynamics. [ABSTRACT FROM AUTHOR]

In this article, we develop a combined finite element-weighted upwind finite volume method for convection-dominated diffusion problems in two dimensions, which discretizes the diffusion term with the standard finite element scheme, and the convection and source terms with the weighted upwind finite volume scheme. The developed method leads to a totally new scheme for convection-dominated problems, which overcomes numerical oscillation, avoids numerical dispersion, and has high-order accuracy. Stability analyses of the scheme are given for the problems with constant coefficients. Numerical experiments are presented to illustrate the stability and optimal convergence of our proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 799-818, 2016 [ABSTRACT FROM AUTHOR]

In this paper, by adopting a fifth-order weighted essentially non-oscillatory (WENO) scheme with a third-order TVD Runge-Kutta time stepping method for two-dimensional reactive Euler equations, a parallel code is developed, and reignition behavior after a self-sustaining detonation from the tube into free space filled with H2/O2 mixtures is investigated. The numerical results show that the initial pressure has a great influence on the detonation cellular width, and that as the initial pressure increases, the cellular width gradually decreases and the cellular shape changes from irregular structure to regular structure, demonstrating the detonation instability to stability transition. When the initial pressure is larger than 1.2 atm, the detonation wave expands over the edge of the splitter plate, reignition can come into being because enough transverse waves collide with each other at the leading edge of the expanding front. When the initial pressure is 1.2 atm, hot spots appear on the front, and ignite the combustible gas near the hot spots after detonation diffraction. When the initial pressure is 1.0 atm, reignition fails. These findings hint that a critical initial pressure exists between 1.0-1.2 atm for direct reignition after detonation diffraction. [ABSTRACT FROM AUTHOR]

It is known that real‐time power quality monitoring is usually rather difficult to achieve due to the requirement of expensive equipments and necessary of more signals to be captured. To solve this problem, this study combines fractional order Sprott chaos synchronisation system with extension theory to detect power signal disturbance including voltage swell, sag, interruption, and harmonics for real‐time power quality monitoring. The use of fractional order chaotic systems can significantly improve the situations of misjudgement due to the dynamic error is too large by using general integer order chaotic systems. Otherwise, the three‐dimensional (3D) Sprott error dynamics can be reduced to 2D error system. It will become easier and cheaper to implement this scheme in the portable device. The results of numerical simulation show that the detection accuracy rate is 100% and it is better than the methods in previous studies. It should be able to get a very high diagnostic accuracy if this method can be applied to the actual power system. [ABSTRACT FROM AUTHOR]

In this paper, we study the residual-based a posteriori error estimates of h p -discontinuous Galerkin finite element methods for strongly nonlinear elliptic boundary value problems. Computable upper and lower bounds on the error are derived in a natural mesh-dependent energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. Numerical experiments are also provided to illustrate the performance of the proposed estimators. [ABSTRACT FROM AUTHOR]

Abstract: A numerical investigation is performed into the nonlinear dynamic behavior of a clamped–clamped micro-beam actuated by a combined DC/AC voltage and subject to a squeeze-film damping effect. An analytical model based on a nonlinear deflection equation and a linearized Reynolds equation is proposed to describe the deflection of the micro-beam under the effects of the electrostatic actuating force. The deflection of the micro-beam is investigated under various actuating conditions by solving the analytical model using a hybrid numerical scheme comprising the differential transformation method and the finite difference approximation method. It is shown that the numerical results for the dynamic pull-in voltage of the clamped–clamped micro-beam deviate by no more than 2.04% from those presented in the literature based on the conventional finite difference scheme. The effects of the AC voltage amplitude, excitation frequency, residual stress, and ambient pressure on the center-point displacement of the micro-beam are systematically explored. Moreover, the actuation conditions which ensure the stability of the micro-beam are identified by means of phase portraits. Overall, the results presented in this study confirm that the hybrid numerical method provides an accurate means of analyzing the complex nonlinear behavior of common electrostatically-actuated microstructures. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a three-dimensional parallel solver using the fifth order high-resolution weighted essentially non-oscillatory (WENO) finite difference scheme to perform extensive simulation for three-dimensional gaseous detonations. A careful study is conducted for the propagation modes of three-dimensional gaseous detonation wave-front structures in a long square duct. The numerical results indicate that, the instability of detonation, overdrive factor, transverse dimension and initial perturbation are the main factors that influence the appearance of spinning detonation. For unstable detonation with an overdrive factor f =1.0, with the duct width decreasing, the transverse modes tend to be the lower ones. When the duct width is sufficiently small, detonation propagation is coupled to the lowest transverse mode in order to sustain itself. There is a maximum duct width for the appearance of spinning detonation. When the duct width is smaller than this value, the unstable detonation propagates in a spinning mode. When the duct width is larger than this value, spinning detonation disappears. For highly unstable detonation with an overdrive factor f =1.2, the formation of the spinning detonation does not depend on the initial perturbation and it only depends on the high instability of detonation and channel dimension in the transverse directions. Both initial rectangular modes and initial diagonal modes can eventually evolve into spinning modes. For mildly unstable detonation with an overdrive factor f =1.6, it is easier for the initial diagonal mode to trigger the spinning mode than for the initial rectangular mode. The formation of spinning detonation thus not only depends on the high instability of detonation and channel dimension in the transverse directions, but also depends on the initial perturbation. For the stable detonation with an overdrive factor f =1.1, when the initial perturbation is in the transverse cosine mode, the front of the stable detonation has a rectangular structure and slapping waves appear on the walls. When the initial perturbation is in the symmetrical mode along the diagonals of the detonation front, the diagonal detonation is formed and the slapping waves on the walls disappear. [Copyright &y& Elsevier]

We investigate the evolution of cylindrical cellular detonation with different instabilities. The numerical results show that with decreasing initial temperature, detonation becomes more unstable and the cells of the cylindrical detonation tend to be irregular. For stable detonation, a divergence of cylindrical detonation cells is formed eventually due to detonation instability resulting from a curved detonation front. For mildly unstable detonation, local overdriven detonation occurs. The detonation cell diverges and its size decreases. For highly unstable detonation, locally driven detonation is more obvious and the front is highly wrinkled. As a result, the diverging cylindrical detonation cell becomes highly irregular. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the bifurcation and nonlinear behaviors of a united gas-lubricated bearing (UGB) system by a hybrid numerical method combining the differential transformation method and the finite difference method. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, quasi-periodic and chaotic responses of the rotor center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those of other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of UGB systems. [Copyright &y& Elsevier]

Abstract: In , two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax–Wendroff (ILW) procedure for inflow boundary conditions and high order extrapolation for outflow boundary conditions. However, the algebra of the ILW procedure is quite heavy for two dimensional (2D) hyperbolic systems, which makes it difficult to implement the procedure for order of accuracy higher than three. In this paper, we first discuss a simplified and improved implementation for this procedure, which uses the relatively complicated ILW procedure only for the evaluation of the first order normal derivatives. Fifth order WENO type extrapolation is used for all other derivatives, regardless of the direction of the local characteristics and the smoothness of the solution. This makes the implementation of a fifth order boundary treatment practical for 2D systems with source terms. For no-penetration boundary condition of compressible inviscid flows, a further simplification is discussed, in which the evaluation of the tangential derivatives involved in the ILW procedure is avoided. We test our simplified and improved boundary treatment for Euler equations with or without source terms representing chemical reactions in detonations. The results demonstrate the designed fifth order accuracy, stability, and good performance for problems involving complicated interactions between detonation/shock waves and solid boundaries. [Copyright &y& Elsevier]

Abstract: One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied, e.g., . For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter in can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently in . In this paper, we first discuss an extension of the technique in to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one in . Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter. [Copyright &y& Elsevier]

Abstract: This paper deals with the robust stability analysis for uncertain systems with time-varying delay. New delay-dependent robust stability criteria of uncertain time-delay systems are proposed by exploiting appropriate Lyapunov functional candidate. These developed results have advantages over some previous ones in that they have fewer matrix variables yet less conservatism, due to the introduction of a method to estimate the upper bound of the derivative of Lyapunov functional candidate without ignoring the additional useful terms. Numerical examples are given to demonstrate the effectiveness and the advantage of the proposed method. [Copyright &y& Elsevier]

In this paper, we consider a kernel-based online learning algorithm for regression when the sampling process is unbounded. Under a moment hypothesis on the sampling outputs, we provide a confidence-based bound for the error in the corresponding reproducing kernel Hilbert space. [ABSTRACT FROM AUTHOR]

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the bifurcation behavior of a high speed spindle air bearing (HSSAB) system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, and quasi-periodic responses of the rotor center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those of other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of HSSAB systems. [ABSTRACT FROM PUBLISHER]

Abstract: To study the behavior of the high speed spindle air bearing (HSSAB) system, we conduct the research by means of a hybrid numerical method which combines the differential transformation method and the finite difference method in this paper. According to the results of the research, the flexible rotor center is found to include a complex dynamic behavior that comprises periodic, sub-harmonic and quasi-periodic responses. In addition, as the rotor mass and the bearing number are increased, there will be some changes taking place in the dynamic behavior of the bearing system. The results are proven to have no conflict with those of the other numerical methods, which enables an effective means in gaining insights into the nonlinear dynamics of HSSAB systems. [ABSTRACT FROM AUTHOR]

In order to obtain the flow field characteristics and the influence of boundary layer, numerical simulations and wind tunnel tests are conducted for two streamline traced Jaws inlets at Mach number 7. The inlets are designed based on a flow field with 8-7 planar shock wave (the ramp in pitch plane is inclined at 8° to the free stream and in yaw plane is inclined at 7° to the free stream, yielding planar shocks). In the study, the static pressure distributions were measured and analyzed along the plane-symmetric centerline of the inlet with and without the boundary layer correction, respectively. Results show that boundary layer correction can obviously weaken the viscous influence to the inlet, increasing the mass flow coefficient and improving total pressure recovery. [ABSTRACT FROM AUTHOR]

Abstract: We consider a quantum optics model where the cavity interacts with two-coupled atoms. The atom–atom entanglement, atoms–cavity entanglement and the mixture for the two atoms are investigated, and discuss the effects of the initial conditions, atom–atom coupling and the mean number of photons on the entanglements and mixture. We find that atom–atom coupling plays an important role in the entanglement and mixture. Numerical results show that under some conditions the phenomena of “entanglement sudden death” and “entanglement collapse and revival” emerge. [Copyright &y& Elsevier]

In recent years, image coding based on wavelet transform has made rapid progress. In this paper, quincunx lifting scheme in wavelet transform is introduced and all phase interpolation filter banks which can be used in the lifting scheme for prediction and update are designed. Based on the basic idea of set partitioning in hierarchical trees (SPIHT) algorithm, the binary tree image coding algorithm is proposed. Just like SPIHT, the encoding algorithms can be stopped at any compressed file size or let run until the compressed file is a representation of a nearly lossless image. The experimental results on test images show that compared with SPIHT algorithm, the PSNRs of the proposed algorithm are superior by about 0.5 dB at the same bit rates and the subjective quality of reconstructed images is also better. [ABSTRACT FROM AUTHOR]

Abstract: This paper analyses the large deflections of an orthotropic rectangular clamped and simply supported thin plate. A hybrid method which combines the finite difference method and the differential transformation method is employed to reduce the partial differential equations describing the large deflections of the orthotropic plate to a set of algebraic equations. The simulation results indicate that significant errors are present in the numerical results obtained for the deflections of the orthotropic plate in the transient state when a step force is applied. The magnitude of the numerical error is found to reduce, and the deflection of the orthotropic plate to converge, as the number of sub-domains considered in the solution procedure increases. The deflection of the simply supported orthotropic plate is great than the clamped orthotropic plate. The current modeling results confirm the applicability of the proposed hybrid method to the solution of the large deflections of a rectangular orthotropic plate. [Copyright &y& Elsevier]

Abstract: In this paper, we present and analyze two-grid P 1 nonconforming finite element method for second order elliptic problems. The two-grid method involves solving one small origin problem on coarse grid with grid size H, and some correction problems on local fined grids with grid size h. If h =O(H 2) is chosen, we show that the convergence rate of this two-grid method is optimal in broken H 1-norm. Numerical results conforming the theory are also provided. [Copyright &y& Elsevier]

Abstract: A numerical analysis of a rigid rotor supported by relative short aerodynamic journal bearings is presented for nonlinear dynamic behaviors and bifurcation. The compressible Reynolds'' equation is solved by the finite differences method and the successive over relation method and a time-dependent mathematical model for aerodynamic journal bearings is studied. A comparison of the results for the system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams is made and dynamic behavior of the rotor center in the horizontal and vertical directions under different operating conditions is analyzed. The analysis shows how the existence of a complex dynamic behavior comprises periodic and subharmonic response of the rotor center. This paper shows how the dynamic behavior of this type of system varies with changes in bearing number and squeeze number. [Copyright &y& Elsevier]

Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to develop a novel analytical finite element for the fictitious crack model in fracture mechanics of concrete. The new analytical element can be implemented into FEM program systems to solve fictitious crack propagation problems for concrete cracked plates with arbitrary shapes and loads. Numerical results indicate that the method is more efficient and accurate than ordinary finite element method. [ABSTRACT FROM AUTHOR]

In this paper we present an error analysis for a second order accurate numerical scheme for the 2-D and 3-D Cahn–Hilliard (CH) equation, with an improved convergence constant. The unique solvability, unconditional energy stability, and a uniform-in-time H 2 stability of this numerical scheme have already been established. However, a standard error estimate gives a convergence constant in an order of exp (C T ε − m 0 ) , with m 0 a positive integer and the interface width parameter ε being small, which comes from the application of discrete Gronwall inequality. To overcome this well-known difficulty, we apply a spectrum estimate for the linearized Cahn–Hilliard operator (Alikakos and Fusco, 1993; Chen, 1994; Feng and Prohl, 2004), perform a detailed numerical analysis, and get an improved estimate, in which the convergence constant depends on 1 ε only in a polynomial order, instead of the exponential order. [ABSTRACT FROM AUTHOR]

In this work, we consider numerical methods for the Poisson–Nernst–Planck–Cahn–Hilliard (PNPCH) equations with steric interactions, which correspond to a non-constant mobility H − 1 gradient flow of a free-energy functional that consists of electrostatic free energies, steric interaction energies of short range, entropic contribution of ions, and concentration gradient energies. We propose a novel energy stable numerical scheme that respects mass conservation and positivity at the discrete level. Existence and uniqueness of the solution to the proposed nonlinear scheme are established by showing that the solution is a unique minimizer of a convex functional over a closed, convex domain. The positivity of numerical solutions is further theoretically justified by the singularity of the entropy terms, which prevents the minimizer from approaching zero concentrations. A further numerical analysis proves discrete free-energy dissipation. Extensive numerical tests are performed to validate that the numerical scheme is first-order accurate in time and second-order accurate in space, and is capable of preserving the desired properties, such as mass conservation, positivity, and free energy dissipation, at the discrete level. Moreover, the PNPCH equations and the proposed scheme are applied to study charge dynamics and self-assembled nanopatterns in highly concentrated electrolytes that are widely used in electrochemical energy devices. Numerical results demonstrate that the PNPCH equations and our numerical scheme are able to capture nanostructures, such as lamellar patterns and labyrinthine patterns in electric double layers and the bulk, and multiple time relaxation with multiple time scales. The multiple time relaxation dynamics with metastability take long time to reach an equilibrium, highlighting the need for robust, energy stable numerical schemes that allow large time stepping. In addition, we numerically characterize the interplay between cross steric interactions of short range and the concentration gradient regularization, and their impact on the development of nanostructures in the equilibrium state. • A positivity-preserving energy stable numerical scheme is proposed for a PNPCH model. • Existence and uniqueness of numerical solutions are established. • Positivity is theoretically justified using singularity of entropy terms. • Nanostructures and multiple time relaxation are captured. [ABSTRACT FROM AUTHOR]

• A nonlinear numerical analysis for sandwich panels subjected to blast load. • Sandwich panels consisted of tubular cores. • The effects of tube spacing/number on energy absorption capacity was investigated. • The analysis presented is useful for the design of the blast mitigator. This article presents the results of a numerical study on the response of cladding sandwich panels with tubular cores that can be used as blast mitigators. The core of the cladding sandwich panels consisted of empty thin-walled aluminum circular tubes (38 mm in diameter) riveted laterally between the skin plates in different configurations (varying number of tubes and spacing between the tube). The skin of cladding sandwich panels consisted of a "rigid" DOMEX700 steel front plate which was exposed to the blast load and a back plate made of mild steel. The uniform blast load was achieved by detonating varying charge masses of explosive (ranging from 6 g to 50 g) at a stand-off distance of 200 mm down a square mild steel tube. The numerical model, developed in ANSYS AutoDYN 15.0, was validated for average deflection of the front plate and crush mode of the tubular cores using the experimental results that were presented in a previous paper [1]. The results of the simulations provided insights into the transient response of the cladding panels. The effects of the spacing between the tubular cores which also influenced the interaction between the tubular cores were also investigated. The number of plastic hinges formed was increased whilst the rotation angle of plastic hinges decreased with increasing interaction between the tubular cores. [ABSTRACT FROM AUTHOR]

In this paper an analysis of the nonlinear dynamic behavior and control of an atomic force microscope system is described. Phase plane trajectories, spectrum analysis, bifurcation analysis, the Poincare cross-section, the maximal Lyapunov exponent, and other numerical analysis methods were used to observe and verify the dynamic characteristics and the differential equations for the system. The results showed that at specific excitation frequencies, the system will exhibit nonlinear behavior that may be cyclic, multi-cyclic or acyclic. A Psim circuit simulation using the same parameters showed the same nonlinear behavior, as did laboratory circuit implementations. The results also showed than an understanding and control of the dynamic characteristics would not be an easy task. However, they could be used as a basis for the suppression and control of nonlinear behavior and vibration in atomic force microscope systems. A proportional-derivative system was also used, with particle swarm optimization, to find the control parameters Kp and KD and a fuzzy controller was used to compare the results. The controller simulation and hardware implementation both effectively inhibited the nonlinear behavior and were most helpful for the control and enhancement of measurement accuracy. [ABSTRACT FROM AUTHOR]

Highlights • The development of a second-order scheme that describes the annealing of Metal-Intermetallic Laminate composites. • Numerical analysis on the convergence order is rigorously established, confirming expected discretization accuracy. • The formation of Al-rich and Ni-rich layers and diffusion paths in a ternary phase diagram show nice agreements with experimental data. • The numerical simulation results show nice agreements with experimental data. Abstract Metal–Intermetallic Laminate (MIL) composites are laminate structures that are fabricated by optimizing unique benefits of constituent components to have attractive physical and mechanical features, such as high strength, stiffness, and toughness. The synthesis of MIL composites involves annealing reaction of multiple metallic elements at high temperatures. In this work, we propose, analyze, and implement a second-order semi-implicit scheme for the annealing of a ternary metallic reaction system. The robustness of such a numerical scheme is demonstrated by various numerical experiments, as well as the expected accuracy tests. At the theoretical level, we provide a detailed convergence analysis, which confirms the second-order accuracy in both the temporal and spatial discretization. Moreover, this numerical scheme is used to study the annealing process of the Al–Fe–Ni ternary system. The computation reproduces the formation of the Al-rich and Ni-rich layers in the annealing process. We present the computational results as diffusion paths in a ternary phase diagram, with a nice agreement with that of experimental data. We also study the growth kinetics of the two layers by calculating the rate constant and kinetic exponent of the reaction. The morphology of interfaces between different layers is thoroughly investigated as well. The computational results indicate that the numerical scheme is an effective, useful tool for predicting the microstructure evolution in the annealing process of a ternary reaction system. [ABSTRACT FROM AUTHOR]

In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in Hu et al. (2009), with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation. [ABSTRACT FROM AUTHOR]

Purpose This study aims to investigate the dynamic motion of an ultrasonic machining system comprising two Duffing oscillators, each with a single degree of freedom. After derivation of the differential equations of the system using the Lagrange equations and dimensionless time, numerical analysis was used to observe changes in the system caused by differences in excitation frequency.Design/methodology/approach To suppress this effect and improve performance, proportional differential (PD) control was used. The integral absolute error was used as the fitness function, and particle swarm optimization was used to find the best value for the gain constant of the PD controller.Findings The results showed that with specific changes of excitation frequency, the dynamic motion of the system became nonlinear and chaotic behavior resulted. This made the system unstable and affected performance.Originality/value A range of methods, including fuzzy control, was used to analyze the results, and exhaustive laboratory work was carried out. Means of control were found that were effective in suppressing the chaotic behavior, and differences in response to control were investigated and verified. The findings of this study can be used as a basis for system parameter settings or control circuit design. [ABSTRACT FROM AUTHOR]

It is known that spectral methods offer exponential convergence for infinitely smooth solutions. However, they are not applicable for problems presenting singularities or thin layers, especially true for the ones with the location of singularity unknown. An adaptive domain decomposition method (DDM) integrated with Chebyshev tau method based on the highest derivative (CTMHD) is introduced to solve singular perturbed boundary value problems (SPBVPs). The proposed adaptive algorithm uses the refinement indicators based on Chebyshev coefficients to determine which subintervals need to be refined. Numerical experiments have been conducted to demonstrate the superior performance of the method for SPBVPs with a number of singularities including boundary layers, interior layers and dense oscillations. A fourth order nonlinear SPBVP is also concerned. The numerical results illustrate the efficiency and applicability of our adaptive algorithm to capture the locations of singularities, and the higher accuracy in comparison with some existing numerical methods in the literature. [ABSTRACT FROM AUTHOR]

A class of nonlinear Fokker–Planck equations with nonlocal interactions may include many important cases, such as porous medium equations with external potentials and aggregation–diffusion models. The trajectory equation of the Fokker–Plank equation can be derived based on an energetic variational approach. A structure‐preserving numerical scheme that is mass conservative, energy stable, uniquely solvable, and positivity preserving at a theoretical level has also been designed in the previous work. Moreover, the numerical scheme is shown to satisfy the discrete energetic dissipation law and preserve steady states and has been observed to be second order accurate in space and first‐order accurate time in various numerical experiments. In this work, we give the rigorous convergence analysis for the highly nonlinear numerical scheme. A careful higher order asymptotic expansion is needed to handle the highly nonlinear nature of the trajectory equation. In addition, two step error estimates (a rough estimate and a refined estimate) are necessary in the convergence proof. Different from a standard error estimate, the rough estimate is performed to control the nonlinear term. A few numerical results are also presented to verify the optimal convergence order and the preservation of equilibria. [ABSTRACT FROM AUTHOR]

In this paper, an implicit fully discrete direct discontinuous Galerkin (DDG) finite element method is considered for solving the time fractional diffusion equation. The scheme is based on the Gorenflo–Mainardi–Moretti–Paradisi (GMMP) scheme in time and direct discontinuous Galerkin method in space. Unlike the traditional local discontinuous Galerkin method, the DDG method is based on the direct weak formulation for solutions of parabolic equations in each computational cell, letting cells communicate via the numerical flux u x ^ only. We prove that our scheme is stable and the energy norm error estimate is convergent with O ( ( Δ x ) k + Δ t α + 1 + Δ t α 2 ( Δ x ) k ) by choosing admissible numerical flux. The DDG method has the advantage of easier formulation and implementation as well as the high order accuracy. Finally numerical experiments are presented to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we present interference models for cognitive radio (CR) networks employing various interference management mechanisms including power control, contention control or hybrid power/contention control schemes. For the first case, a power control scheme is proposed to govern the transmission power of a CR node. For the second one, a contention control scheme at the media access control (MAC) layer, based on carrier sense multiple access with collision avoidance (CSMA/CA), is proposed to coordinate the operation of CR nodes with transmission requests. The probability density functions (PDFs) of the interference received at a primary receiver from a CR network are first derived numerically for these two cases. For the hybrid case, where power and contention controls are jointly adopted by a CR node to govern its transmission, the interference is analyzed and compared with that of the first two schemes by simulations. Then, the interference PDFs under the first two control schemes are fitted by log-normal PDFs to reduce computation complexity. Moreover, the effect of a hidden primary receiver on the interference experienced at the receiver is investigated. It is demonstrated that both power and contention controls are effective approaches to alleviate the interference caused by CR networks. Some in-depth analysis of the impact of key parameters on the interference of CR networks is given as well. [ABSTRACT FROM AUTHOR]

We use non-maximally entangled states (NMESs) to simulate an entangling unitary operator (EUO) with a certain probability. Given entanglement resources, the probability of the success we achieve is a decreasing function of the parameters of the EUO. Given an EUO, for certain entanglement resources the result is optimal, i.e., the probability obtains a maximal value, and for optimal result higher parameters of the EUO match more amount of entanglement resources. The probability of the success we achieve is higher than the known results under some condition. [ABSTRACT FROM AUTHOR]

We present a second order accurate numerical scheme for a nonlinear hyperbolic equation with an exponential nonlinear term. The solution to such an equation is proven to have a conservative nonlinear energy. Due to the special nature of the nonlinear term, the positivity is proven to be preserved under a periodic boundary condition for the solution. For the numerical scheme, a highly nonlinear fractional term is involved, for the theoretical justification of the energy stability. We propose a linear iteration algorithm to solve this nonlinear numerical scheme. A theoretical analysis shows a contraction mapping property of such a linear iteration under a trivial constraint for the time step. We also provide a detailed convergence analysis for the second order scheme, in the ℓ ∞ ( 0 , T ; ℓ ∞ ) norm. Such an error estimate in the maximum norm can be obtained by performing a higher order consistency analysis using asymptotic expansions for the numerical solution. As a result, instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O ( Δ t 3 + h 4 ) convergence in ℓ ∞ ( 0 , T ; ℓ 2 ) norm, which leads to the necessary ℓ ∞ error estimate using the inverse inequality, under a standard constraint Δ t ≤ C h . A numerical accuracy check is given and some numerical simulation results are also presented. [ABSTRACT FROM AUTHOR]

We devise second-order accurate, unconditionally uniquely solvable and unconditionally energy stable schemes for the nonlocal Cahn–Hilliard (nCH) and nonlocal Allen–Cahn (nAC) equations for a large class of interaction kernels. We present numerical evidence that both schemes are convergent. We solve the nonlinear equations resulting from discretization using an efficient nonlinear multigrid method and demonstrate the performance of our algorithms by simulating nucleation and crystal growth for several different choices of interaction kernels. [ABSTRACT FROM AUTHOR]

Abstract: This study focuses on developing an adaptive finite difference method oriented by the a posteriori error estimator that is originally derived from the finite element analysis, building a novel type of hybrid mesh adaptive technique that has not been reported elsewhere as of today. The time-dependent heat conduction problem is chosen for numerical investigation. Corresponding to this target problem, three major numerical “black boxes” are established, namely, finite difference solver, a posteriori error estimation framework, and re-meshing via simultaneous local mesh refinement and re-coarsening. Selected transient heat conduction problems are solved using the present hybrid finite difference mesh adaptive tools. The accuracy of the obtained adaptive solutions appears fairly good while the computational resources can be greatly saved, clearly demonstrating the usefulness, effectiveness, and robustness of the hybrid mesh adaptive approach developed through this study. [Copyright &y& Elsevier]

The effects of adding ethanol on the macroscopic and microscopic spray characteristics of biodiesel were investigated by experimental and numerical methods. The spray tip penetration (STP) was recorded with a high-speed camera system, while the statistical size distribution of atomized droplets and the Sauter mean diameter (SMD) were obtained using a Malvern laser particle size analyzer. The spray calculation models were established with Fluent software, and the calculation results were validated by the experimental results. Then the numerical simulation of the spray characteristics for diesel, biodiesel, and biodiesel–ethanol fuel blends were conducted. The results showed that the farther the distance is from the nozzle exit, the lower the concentration and the velocity of atomized droplets. The concentration and the velocity of atomized droplets gradually decrease along the radial direction, extending from the center to the periphery. The turbulent kinetic energy (TKE) at the front and rear of the spray is relatively lower, while the TKE is higher in the middle of the spray, and turbulent vortices appear on both sides of the axial spray center. In addition, the spray characteristics of the different fuels revealed that compared to diesel, the STP and SMD of biodiesel increase by 33.59% and 33.08% on average, respectively. The maximum concentration of biodiesel is higher than that of diesel, and the concentration distribution is more concentrated. The droplet velocity and TKE of biodiesel respectively decrease by 7.38% and 14.49% in comparison to diesel. When ethanol was added to biodiesel by a volume percentage of 30%, the STP and SMD of biodiesel–ethanol fuel blends reduce by 22.05% and 20.88%, respectively. The concentration distribution, the velocity distribution, and the TKE distribution of the fuel blends are similar to that of diesel. This indicates that adding 30% ethanol to biodiesel can effectively improve its spray quality. [ABSTRACT FROM AUTHOR]