Variational Method Research Articles

Discrete Schrödinger–Newton equations with power nonlinearity

We study Schrödinger–Newton equations with general power nonlinearity on the lattice Z d . After the proof of well-posedness of the Cauchy problem, our interest is focused on the existence of stable standing (stationary) solutions. Employing variational methods we prove the existence of a ground state solution solving the associated stationary system. Orbital stability as well as exponential localization of the ground state is verified. Finally, we discuss the existence of excitation thresholds for ground state solutions in dependence of the conserved mass ( l 2 -norm).

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

Assessing the Accuracy of Quantum Dynamics Performed in the Time-Dependent Basis Representation.

A full quantum-mechanical (QM) description of large amplitude nuclear motion, associated with chemical reactions or isomerization of high-dimensional molecular systems, is inherently challenging due to the exponential scaling of the QM complexity with system size. To ameliorate the scaling bottleneck in studies of realistic systems, typically modeled in the configuration space, the nuclear wave functions are represented in terms of time-dependent basis functions. Such bases are expected to give an accurate description with a modest number of basis functions employed, by adapting them to the wave function solving the time-dependent Schrödinger equation. It is not, however, straightforward to estimate the accuracy of the resulting solution: in QM the energy conservation, a convenient such measure for a classical trajectory evolving in a time-independent potential, is not a sufficient criterion of the dynamics' accuracy. In this work, we argue that the expectation value of the Hamiltonian's "variance", quantifying the basis completeness, is a suitable practical measure of the quantum dynamics' accuracy. Illustrations are given for several chemistry-relevant test systems, modeled employing time-independent as well as time-dependent bases, including the coupled and variational coherent states methods and the quantum-trajectory guided adaptable Gaussians (QTAG) as the latter basis type. A novel semilocal definition of the QTAG basis time-evolution for placing the basis functions "in the right place at the right time" is also presented.

Variational analysis of sensory feedback mechanisms in powerstroke-recovery systems.

Although the raison d'etre of the brain is the survival of the body, there are relatively few theoretical studies of closed-loop rhythmic motor control systems. In this paper we provide a unified framework, based on variational analysis, for investigating the dual goals of performance and robustness in powerstroke-recovery systems. To demonstrate our variational method, we augment two previously published closed-loop motor control models by equipping each model with a performance measure based on the rate of progress of the system relative to a spatially extended external substrate-such as a long strip of seaweed for a feeding task, or progress relative to the ground for a locomotor task. The sensitivity measure quantifies the ability of the system to maintain performance in response to external perturbations, such as an applied load. Motivated by a search for optimal design principles for feedback control achieving the complementary requirements of efficiency and robustness, we discuss the performance-sensitivity patterns of the systems featuring different sensory feedback architectures. In a paradigmatic half-center oscillator-motor system, we observe that the excitation-inhibition property of feedback mechanisms determines the sensitivity pattern while the activation-inactivation property determines the performance pattern. Moreover, we show that the nonlinearity of the sigmoid activation of feedback signals allows the existence of optimal combinations of performance and sensitivity. In a detailed hindlimb locomotor system, we find that a force-dependent feedback can simultaneously optimize both performance and robustness, while length-dependent feedback variations result in significant performance-versus-sensitivity tradeoffs. Thus, this work provides an analytical framework for studying feedback control of oscillations in nonlinear dynamical systems, leading to several insights that have the potential to inform the design of control or rehabilitation systems.

Single-Layer Digitized-Counterdiabatic Quantum Optimization for p-spin Models

Abstract Quantum computing holds the potential for quantum advantage in optimization problems, which requires advances in quantum algorithms and hardware specifications. Adiabatic quantum optimization is conceptually a valid solution that suffers from limited hardware coherence times. In this sense, counterdiabatic quantum protocols provide a shortcut to this process, steering the system along its ground state with a fast-changing Hamiltonian. In this work, we take full advantage of a digitized- counterdiabatic quantum optimization (DCQO) algorithm to find an optimal solution for the p-spin model up to 4-local interactions. We choose a suitable scheduling function and initial Hamiltonian such that a single-layer quantum circuit suffices to produce a good ground-state overlap. By further optimizing parameters using variational methods, we solve with unit accuracy 2-spin, 3-spin, and 4-spin problems for 100%, 93%, and 83% of instances, respectively. As a particular case of the latter, we also solve factorization problems involving 5, 9, and 12 qubits. Due to the low computational overhead, our compact approach may become a valuable tool towards quantum advantage in the NISQ era.

INVESTIGATION OF FRACTIONAL-ORDERED TUMOR-IMMUNE INTERACTION MODEL VIA FRACTIONAL-ORDER DERIVATIVE

Dynamical analysis of system of ordinary differential equations (ODEs) is an interesting topic among researchers and several tools have been discovered since so far to deal with its several other features. There are many built-in functions designed in MATLAB that researchers can use as a tool for the system of ODEs with integer order but in the similar systems have fractional-order derivative is a difficult task. Therefore, the purpose of this work is to use the Caputo fractional-order derivative to qualitatively analyze and then numerically simulate the tumor-immune system interaction model. Moreover, the Schauder fixed point theorem is used to establish the condition for the existence of at least one solution, while the Banach fixed point theorem is utilized to guarantee the existence of a unique solution. Moreover, the stability in the trajectories of considered system achieved with the aid of Ulam–Hyers (UH) stability. In addition, the numerical computations are performed using Variational Iteration Method (VIM) and the visualized results are shown using MATLAB.

Identification of Single-Blade Angle Variation in Axial Flow Pumps Based on the Variational Mode Decomposition Method

Pressure pulsations are crucial data within the flow field of a pump, and the characteristics of these pulsations can reveal changes in the internal flow. Based on model experiments, this paper obtained pressure pulsation data under two blade conditions and compared direct time-domain observations, peak-to-peak value changes, and the VMD decomposition method. The results show that even when it is known that one blade condition has changed, it is not possible to determine this through direct observation of pressure pulsation changes. The peak-to-peak value changes indicate that under special flow conditions, they are easily affected by different operating conditions, which can interfere with the results. In contrast, the VMD method, which decomposes the signal into low-frequency components, can better display anomalies within the pressure pulsation cycle and is less susceptible to the interference of flow conditions, offering some reference significance for diagnosing the blade operating conditions of the main pump.

HIGH-ORDER ZIG-ZAG THEORIES TO STATIC LAMINATED COMPOSITE BEAM ANALYSIS BY THE RAYLEIGH-RITZ METHOD

Technological advancements across various engineering fields have grown demand for highly efficient materials for specific applications. Laminated composite materials have emerged as a promising solution, offering a combination of favorable mechanical properties by layering different materials. Theories, such as zig-zag theories, have been developed to predict the mechanical behavior of these materials under external stresses. This study investigates the outcomes provided by the Rayleigh-Ritz variational method, employing different approximation functions, particularly focusing on unified higher-order zig-zag kinematics. All proposed shape functions agree to the reference values, but the model with polynomial shape function shows more applications possibilities. Keywords: composite materials, composite laminated beams, zig-zag theory, Rayleigh-Ritz method.

ISSUES OF PREVENTING GROUND WATER POLLUTION BY OIL AND HYDROCARBON WASTES USING PERMEABLE REACTIVE BARRIERS

The issues of using permeable reactive barriers to prevent ground water pollution by oil and hydrocarbon wastes are analyzed. A model of adsorption treatment of groundwater from pollutants such as petroleum hydrocarbons, chlorocarbons and heavy metals is proposed.The formulated problem of synthesizing the optimal model was solved by applying the variational optimization method. Solving the problem taking into account the corresponding limiting condition made it possible to determine the condition for the adsorbent maximum consumption, under which the adsorbent maximum amount will be required per unit cross-sectional area of the adsorption shaft.

Extended B‐spline‐based implicit material point method for saturated porous media

AbstractThe large deformation and fluidization process of a solid–fluid mixture includes significant changes to the temporal scale of the phenomena and the shape and properties of the mixed material. This paper presents an extended B‐spline (EBS)‐based implicit material point method (EBS‐MPM) for the coupled hydromechanical analysis of saturated porous media to enhance the overall versatility of MPM in addressing such diverse phenomena. The proposed method accurately represents phenomena such as high‐speed motion in both the quasi‐static and dynamic states by employing a full formulation of coupled hydromechanical modeling. The weak imposition of boundary conditions based on Nitsche's method allows representing the boundary conditions independent of the relative position of the particles and computational grid. In addition, it enables dynamic changes in the boundary domain based on the deformation. The robustness of this boundary representation is reinforced using EBS basis functions, which prevent the degradation of the condition number of the system matrices regardless of the position of the boundary domain with respect to the computational grid. Furthermore, a stabilization method based on a variational multiscale method (VMS) approach is employed to provide the flexibility in choosing arbitrary basis functions for spatial discretization, facilitating the effective construction of EBS. Numerical examples including comparisons between a full formulation and a simplified formulation are presented to demonstrate the performance of the developed method under various boundary conditions and loading states across different time scales.

Global Solution for a Fourth-Order Nonlinear Elliptic Equation

Objective: In this work, the existence of a global solution for a fourth-order nonlinear elliptic equation with boundary conditions is studied. Method: Variational methods are applied to the proposed model, naturally using the Banach Fixed Point Contraction Theorem. Results and Discussion: The conclusion of the result obtained is that for the nonlinear elliptic equation of the fourth order they have equivalent norms in the conveniently defined spaces. On the other hand, the existence of a solution exists whenever Banach's fixed-point theorem can be applied. Implications of the research: Apply in the first university years the application of global solution for a fourth-order nonlinear elliptic equation in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of global solution for a fourth-order nonlinear elliptic equation.

Innovative Solutions to the Fractional Diffusion Equation Using the Elzaki Transform

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.

Thermal quench modeling of REBCO racetrack coils under either alternating current or short-circuit voltage

Abstract Racetrack coils in REBCO High-Temperature Superconductor (HTS) motors help to increase the power-over-weight ratio by raising the magnetic flux density and output power. Nevertheless, HTS motors face thermal quench due to self-heating effects when subjected to alternating or short-circuit onset DC voltage. Additionally, thermal events have been observed due to screening currents when motors operate at high frequencies. Therefore, for the safe operation of HTS motors, quench research is crucial. To accurately simulate and analyze quench events in different scenarios, it is imperative to employ fast and precise software to numerically model the electrothermal behavior in racetrack coils. Our contribution involves the development of a novel and efficient model implemented in C++ that takes screening currents into account. This model is designed to conduct coupled electromagnetic and electrothermal analyses, utilizing variational methods. Specifically, the model incorporates both Minimum Electro-Magnetic Entropy Production (MEMEP) and the Finite Difference Method (FDM). In this article, we explore the phenomenon of temperature rise within a racetrack coil subjected to either alternating or DC voltages of magnitudes ranging from 0.1 V to 1000 V. The study encompasses conditions of adiabatic operation and heat exchange with liquid nitrogen (LN2). Among other results, we found that in moderate DC voltages like 10 V, non-uniformity in the AC loss causes highly localized quench at the central turns. Then, screening currents play a key role also for DC voltages. The developed model exhibits the potential to comprehensively and swiftly analyze the electromagnetic and electrothermal characteristics of real-world superconducting applications, including high-field rotating machines, such as motors for aviation.

The Approximate Solutions of 2D- Burger’s Equations

This paper investigates the use of the fractional variational iteration method (FVIM) to obtain approximate analytical solutions to two dimensional Burger’s Equations with the Atangana-Baleanu fractional operator (ABFO). This study provides insight on the fractional variational iteration method's accuracy and reliability while approximating fractional differential equation solutions.

Intrinsic squeezing of mechanical motion of a harmonically trapped atom with spin-orbit coupling

In this paper we investigated one-dimensional harmonically trapped atom with Raman-induced spin-orbit coupling by mapping to quantum-like model. Using displacement Fock states in quantum optics and variational methods we found the odd-parity superposition state of left- and right-displaced oscillator state is a good approximate solution to the ground state, which captures the essential physics of the system. The ground state wave function owes nodes which is remarkably different from the conventional no-node theory in quantum mechanics. The results show the atomic motion of center of mass exhibits intrinsic squeezing properties with relation to the Raman and spin-orbit coupling strength characterized by the variance in position and momentum, information entropy, as well as the Wigner function of the ground state. The dynamics of center of mass are also studied which demonstrates intuitive Zitterbewegung oscillations in momentum space and real space. Our results could be used for observing Zitterbewegung oscillations in neutral atomic gases and engineering nonclassic states.

Fracture and size effect in mechanical metamaterials

We resort to variational methods to evaluate the asymptotic behavior of fine metamaterials as a function of cell size. To zeroth order, the metamaterial behaves as a micropolar continuum with both displacement and rotation degrees of freedom, but exhibits linear-elastic fracture mechanics scaling and therefore no size effect. To higher order, the overall energetics of the metastructure can be characterized explicitly in terms of the solution of the zeroth-order continuum problem by the method of Γ-expansion. We present explicit expressions of the second-order correction for octet frames. As an application, we evaluate the compliance of double-cantilever octet specimens to second order and use the result to elucidate the dependence of the apparent toughness of the specimen on cell size. The analysis predicts the discreteness of the metamaterial lattice to effectively shield the crack-tip, a mechanism that we term lattice shielding. The theory specifically predicts anti-shielding, i. e., coarser is weaker, in agreement with recent experimental observations.

Existence of nontrivial solutions for fractional Klein–Gordon equation coupled with Born–Infeld theory with critical exponents

In this paper, we investigate the existence of solutions for a fractional Klein–Gordon equation coupled with Born–Infeld theory when the nonlinearity exhibits critical growth. By overcoming the lack of compactness, we obtain that there is a radially symmetric solution by using variational methods.

Research on the influence of unsteady cavitation on the dynamics structures of a twisted hydrofoil based on variational mode decomposition method

Cavitation will produce complex dynamic effects on hydrofoil structures. In this paper, the effects of unsteady cavitation conditions on the dynamics of twisted hydrofoil structures are studied by using variational mode decomposition (VMD) and pearson correlation coefficient (PCC), etc. The research results indicate that there is a certain synergy between the flow field pressure pulsation signal and the structural field stress-strain signal in the frequency distribution of the hydrofoil under cavitation conditions. The second-order center frequency of ΔP1 and the first-order center frequency of ΔP7 are close to the second-order center frequency of the stress-strain signal of the hydrofoil (1202.4Hz). Unsteady cavitation can lead to the increase of the high effective force region of the hydrofoil, especially the high effective force region connecting the leading edge and the root, which is the reason for the degradation of the hydrofoil performance. Cavitation increases the natural frequency of the hydrofoil, but has no obvious effect on the natural mode of the hydrofoil. The research results of this study reveal the effect of unsteady cavitation on the dynamics of hydrofoil structure, and provide important theoretical support for the optimization design and performance improvement of hydrofoil structure.

Agent-SwinPyramidNet: an enhanced deep learning model with AMTCF-VMD for anomaly detection in oil and gas pipelines

PurposeThe anomaly detection task for oil and gas pipelines based on acoustic signals faces issues such as background noise coverage, lack of effective features, and small sample sizes, resulting in low fault identification accuracy and slow efficiency. The purpose of this paper is to study an accurate and efficient method of pipeline anomaly detection.Design/methodology/approachFirst, to address the impact of background noise on the accuracy of anomaly signals, the adaptive multi-threshold center frequency variational mode decomposition method(AMTCF-VMD) method is used to eliminate strong noise in pipeline signals. Secondly, to address the strong data dependency and loss of local features in the Swin Transformer network, a Hybrid Pyramid ConvNet network with an Agent Attention mechanism is proposed. This compensates for the limitations of CNN’s receptive field and enhances the Swin Transformer’s global contextual feature representation capabilities. Thirdly, to address the sparsity and imbalance of anomaly samples, the SpecAugment and Scaper methods are integrated to enhance the model’s generalization ability.FindingsIn the pipeline anomaly audio and environmental datasets such as ESC-50, the AMTCF-VMD method shows more significant denoising effects compared to wavelet packet decomposition and EMD methods. Additionally, the model achieved 98.7% accuracy on the preprocessed anomaly audio dataset and 99.0% on the ESC-50 dataset.Originality/valueThis paper innovatively proposes and combines the AMTCF-VMD preprocessing method with the Agent-SwinPyramidNet model, addressing noise interference and low accuracy issues in pipeline anomaly detection, and providing strong support for oil and gas pipeline anomaly recognition tasks in high-noise environments.

Static and Dynamic Characteristics of 3D-Printed Orthogonal Hybrid Honeycomb Panels with Tunable Poisson’s Ratio

By adjusting the two wall angles of the orthogonal hybrid honeycomb (OHH), the tunable Poisson’s ratio change from negative to positive values and the variation in stiffness can be achieved. To effectively analyze its static and dynamic characteristics, a two-dimensional equivalent Kirchhoff–Love model (2D-EKM) is established based on the variational asymptotic method (VAM).This model aids in effectively addressing the complexity arising from anisotropy. The obtained equivalent orthotropic properties are validated through unit-cell uniaxial compression tests and three-point bending experiments on 3D-printed specimens. The numerical simulation results suggest that the VAM-based 2D-EKM can predict the in-plane and out-of-plane static behaviors of OHH panels, with a maximum error below 10%. Particularly in the dynamic analysis of a four-sided fixed OHH panel, the analysis time required by 2D-EKM is only 0.37% of that needed for the 3D FE model. The OHH-ZPR panel exhibits exceptional resistance to deformation, with a maximum deformation under in-plane tension reaching only 27% of that in the OHH-PPR panel. Moreover, each 1% increase in the height–length ratio results in a respective increase of 275.62% and 281.93% in equivalent bending stiffness along both directions. This highlights that enhancing this ratio effectively boosts the fundamental frequency compared to the elastic modulus ratio, effectively prevents low-frequency resonance occurrences, and offers vital insights for the design and optimization of OHH panels.