Variational Principle Research Articles

Determination of Chip Compression Ratio for the Orthogonal Cutting Process

The chip compression ratio is the most important characteristic of various machining processes with chip generation. This characteristic enables the determination of kinetic and other energy loads on the tool and the machined material. This provides an overall evaluation of the machining process and the possibility of its subsequent optimization. This paper presents the results of determining this cutting characteristic by experimental method, analytical calculation, and numerical modeling. For the analytical calculation of the chip compression ratio, an analytical cutting model developed based on the variational principle of the minimum potential energy was used. A finite element model of orthogonal cutting was used for the numerical simulation of the above process characteristic. Experimentally, the chip compression ratio was determined by the ratio of the chip thickness to the cutting depth (undeformed cutting thickness). The chip thickness was determined by direct measurement using chip slices obtained during the cutting process. The Johnson–Cook constitutive equation was used as the machined material model and the Coulomb model was used as the friction model. The generalized parameters’ determination of the constitutive equation was performed through a DOE (Design of Experiment) sensitivity analysis. The variation range of these parameters was chosen based on the analysis of the effect of individual parameters of the constitutive equation on the chip compression ratio value. The largest deviations between the experimental and analytically calculated values of the chip compression ratio did not exceed 21%. At the same time, the largest deviations of simulated values of the indicated cutting characteristic and its experimental values did not exceed 20%. When comparing the experimental values of the chip compression ratio with the corresponding calculated and simulated values, the deviations were within 22%.

Multiconfigurational Surface Hopping: a Time-Dependent Variational Approach with Momentum-Jump Trajectories.

The Ehrenfest mean field dynamics and trajectory surface hopping have been widely used in nonadiabatic dynamics simulations. Based on the time-dependent variational principle (TDVP), the multiconfigurational Ehrenfest (MCE) method has also been developed and can be regarded as a multiconfigurational extension of the traditional Ehrenfest dynamics. However, it is not straightforward to apply the TDVP to surface hopping trajectories because there exists momentum jump during surface hops. To solve this problem, we here propose a multiconfigurational surface hopping (MCSH) method, where continuous momenta are obtained by linear interpolation and the interpolated trajectories are used to construct the basis functions for TDVP in a postprocessing manner. As demonstrated in a series of representative spin-boson models, MCSH achieves high accuracy with only several hundred trajectory bases and can uniformly improve the performance of surface hopping. In principle, MCSH can be combined with all kinds of mixed quantum-classical trajectories and thus has the potential to properly describe general nonadiabatic dynamics.

Vibration-attitude integrated control of large membrane diffraction space telescope

The development of telescopes with large apertures is driven by the increasing demand for higher imaging resolution and coverage. Large membrane diffraction space telescopes are particularly attractive due to their lightweight nature, ease of folding and unfolding, and high-precision imaging capabilities. However, their substantial size and flexibility make them susceptible to long-duration, low-frequency vibrations during attitude maneuvers, which degrade attitude and imaging accuracy and risk instrument damage. Consequently, an urgent need exists for a high-precision integrated control scheme. In this paper, the integrated control of vibration and attitude in a large space telescope is studied using cable actuators and control moment gyroscopes (CMGs). Since cables can only withstand limited tension, the unilateral and constrained characteristics of the control input are taken into account during the control design process. Firstly, a rigid-flexible coupling dynamic model of the space telescope is established using the velocity variation principle with hybrid coordinates. Additionally, a two-body astrodynamic model is developed to assess the impact of gravity gradient torque. Next, combining the computational torque method and H∞ control theory, an integrated control scheme is proposed, which achieves vibration and attitude control during maneuvers. Then, this paper optimizes the cable actuator positions via the discrete particle swarm optimization (PSO) algorithm and plans various attitude maneuver paths. Finally, numerical simulations are adopted to demonstrate the effectiveness of the control scheme. The results show that this integrated control scheme swiftly suppresses structural vibrations during attitude maneuvers, enabling high-precision attitude control of the space telescope.

Determining the effect of longitudinal stiffeners on the deformation of multilayer ellipsoidal shells under non-stationary loads

The object of this study is the deformation of multilayer ellipsoidal shells under non-stationary distributed load. It is proposed to reinforce the shell with longitudinal stiffeners to enhance the strength of it. The application of Timoshenko's theory of shells and cores enabled the investigation of the influence of longitudinal stiffeners on the deformation of multilayer ellipsoidal shells under non-stationary loads with the discrete positioning of the stiffeners. A mathematical model of shell oscillations under various types of short-term non-stationary loads was based on the Hamilton-Ostrogradsky variational principle. The numerical algorithm based on the application of the integral-interpolation approach to the construction of finite difference schemes in spatial coordinates, combined with an explicit finite-difference scheme for the time coordinate, was used to solve the task set. It has been determined that reinforcing stiffeners significantly affect the deformed state of the multilayer ellipsoidal shell. The "shell-stiffeners" relation was taken into account as a base for research. The maximum deformation ε11 of the smooth three-layer ellipsoidal shell was on average 1.4 times greater than the deformation ε11 of the reinforced three-layer ellipsoidal shell throughout the entire studied time interval. It was determined that over time the presence of reinforcing stiffeners has a more significant impact on the deformed state of the reinforced ellipsoidal shell. A distinctive feature of this research is the consideration of the discrete placement of reinforcing stiffeners which made it possible to investigate the impact of longitudinal stiffeners on the deformation of multilayer ellipsoidal shells under non-stationary loads. The results could be used for the investigation of applied problems in research organizations and design bureaus when designing shell structures

Analysis and optimal control of a hierarchical age-structured population model in a polluted environment

For individual biological populations, examining the impacts of competition within the species and the presence of toxic substances in the environment can effectively facilitate control measures. However research in this area remains limited. This article introduces a toxicant-population model incorporating a hierarchical age-structured and examines the minimum deviation problem within this model. The aim is to explore effective control strategies that strike a balance between improving the environment and population benefits. Through a novel approach, we reformulate the problem as an abstract Cauchy problem on a non-dense domain and then apply integrated semigroup theory to establish the existence, uniqueness, and positivity of solutions to the model. The existence of optimal control is derived using Ekeland's variational principle in functional analysis. The expression for optimal control is given by the Pontryagin's minimum principle. Lastly, we conducted numerical experiments on the model, applied optimality conditions to delineate optimal control, and illustrated the comparison between crucial population parameters under optimal control and those without any control.

MPSDynamics.jl: Tensor network simulations for finite-temperature (non-Markovian) open quantum system dynamics.

The MPSDynamics.jl package provides an easy-to-use interface for performing open quantum systems simulations at zero and finite temperatures. The package has been developed with the aim of studying non-Markovian open system dynamics using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm based on environment chain mapping. The simulations rely on a tensor network representation of the quantum states as matrix product states (MPS) and tree tensor network states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle method for time evolution (including novel bond-adaptive one-site algorithms). The package also provides strong support for the measurement of single and multi-site observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics. It currently handles long-range interactions, time-dependent Hamiltonians, multiple environments, bosonic and fermionic environments, and joint system-environment observables.

Pareto optimality in cooperative differential game of nonlinear mean-field backward stochastic system

This paper is concerned with a necessary condition and a sufficient condition for the existence of Pareto optimal strategy in a cooperative game driven by a nonlinear mean-field backward stochastic system. First, the game problem is equivalently converted into a set of constrained single objective optimal control problems. Next, we utilise Ekeland's variational principle to dispose constraints, and propose a necessary condition in the form of stochastic maximum principle. Meanwhile, we point out that under certain convex assumptions, the necessary condition we proposed is also sufficient. Finally, we give an example to explain the above theoretical results.

Quantum neural network approach to Markovian dissipative dynamics of many-body open quantum systems.

Numerous variational methods have been proposed for solving quantum many-body systems, but they often face exponentially increasing computational complexity as the Hilbert space dimension grows. To address this, we introduce a novel approach using quantum neural networks to simulate the dissipative dynamics of many-body open quantum systems. This method combines neural-network quantum state representation with the time-dependent variational principle, both implemented via quantum algorithms. This results in accurate open quantum dynamics described by the Lindblad quantum master equation, exemplified by the spin-boson and transverse field Ising models. Our approach avoids the computational expense of classical algorithms and demonstrates the potential advantages of quantum computing for many-body simulations. To reduce measurement errors, we introduce a projection reset procedure, which could benefit other quantum simulations. In addition, our approach can be extended to simulate non-Markovian quantum dynamics.

Low depth virtual distillation of quantum circuits by deterministic circuit decomposition

Virtual distillation using measurements of multiple copies of a quantum circuit have recently been proposed as a method of noise mitigation of expectation values. Circuit decompositions known as B gates were found only for 1-local Hamiltonians, however practical problems for chemistry require n-local Hamiltonians, which cannot be corrected with B gates. We discover low depth circuit decompositions for expectation values for n-local Pauli strings by combining multiple projections to recover the correct measurement statistics or expectation values. Our method adds linear entangling gates with the number of qubits, but it requires extra measurements. Furthermore, in applications to find ground states such as the variational quantum eigensolver (VQE) algorithm, the variational principle is required which states that the energy cannot go below the ground-state energy. We find that the variational principle is violated when using B gates and is preserved if we use our low depth decomposition on all expectation values. We perform a demonstration on real devices, and we show that our decomposition can mitigate real experimental noise in VQE for the H2 molecule with a two-qubit tapered mapping, H3 with three qubits, and H2 with four qubits. Our decomposition provides a way to perform duplicate circuit virtual distillation on real devices at significantly lower depth and for arbitrary observables. Published by the American Physical Society 2024

Variational consistent one-point integration with Taylor's expansion-based stabilization in the second-order meshfree Galerkin method for strain gradient elasticity

A generalized variational principle with five independent variables is proposed for strain gradient elasticity, including displacement, strain, strain gradient, stress, and double stress. Based on the principle, a one-point integration scheme is designed for the second order meshfree Galerkin method through nodal smoothed derivatives and their high order derivatives by Taylor's expansion. Since the proposed integration scheme meets the orthogonality conditions, it is variational consistent. The weak form expanded with Taylor's polynomials can be well evaluated by nodal smoothed derivatives and their high order derivatives on one quadrature point. Numerical one- and two-dimensional case studies show that the proposed integration scheme performs better than the standard Gaussian integration method in terms of accuracy, convergence, efficiency, and stability.

Measurement-induced phase transitions by matrix product states scaling

We study the time evolution of long quantum spin chains subjected to continuous monitoring projected on matrix product states (MPS) with fixed bond dimension, by means of the Time-Dependent Variational Principle (TDVP) algorithm. The latter gives an effective unitary evolution which approximates the real quantum evolution up to the projection error. We show that such error displays, at large times, a phase transition in the monitoring strength, which can be well detected by scaling analysis with relatively low values of bond dimensions. Moreover, in the presence of U(1) global spin charge, we show the existence of a charge-sharpening transition well separated from the entanglement transition which we detect by studying the charge fluctuations of a local subpart of the system at large times. Our work shows that quantum monitored dynamics projected on MPS manifolds contains relevant information on measurement-induced phase transitions and provides a new method to identify measured-induced phase transitions in systems of arbitrary dimensions and sizes. Published by the American Physical Society 2024

A novel successive convexification framework with symplectic pseudospectral solutions for nonlinear optimal control problems of constrained pantograph delay systems

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses.

Complex Dispersion Analysis of Flexural Wave Propagation in Periodic Track Structure

The periodic characteristics of the track structure have filtering characteristics for the propagation of vibration waves within it, and the dispersion analysis is of great significance to understand the vibration transmission characteristics of the track structure. The accuracy of the traditional virtual spring method depends on the value of spring stiffness, which leads to poor stability. Therefore, this paper adopts a null space method with both stability, accuracy and efficiency, and takes the CRTSIII ballastless track structure as the research object. Based on the null space method and energy functional variational principle, the damping factor and the material frequency variation effect are considered in detail, and the complex frequency dispersion of the CRTSIII ballastless track structure is analyzed. The accuracy of the proposed method is verified by comparing it with the results of the existing literature, and the influence of fastener stiffness frequency variation, fastener damping and concrete support layer damping on the complex dispersion characteristics of vibration waves of track structures is analyzed by using this method. The results show that the stiffness frequency effect, fastener damping and concrete support layer damping have a great influence on the attenuation domain and attenuation rate of vibration, which must be considered in the vibration transmission analysis of track structures.

Semi-analytical framework for modelling of surface heating and its impact on nonlinear stability of nanotube reinforced doubly curved porous fiber composite panels

In elevated temperatures, the stiffness of the panel deteriorates and can lead to loss of stability of structures. Hence, the stability analysis of thin-walled structures is a critical investigation in the thermal environment. Finding such criticality, the present semi-analytical study investigated the nonlinear stability behaviour of porous doubly curved thin-walled panels subject to surface heating like dome heating and localised heating. To improve the stiffness of the panel, the matrix is reinforced with carbon nanotubes (CNTs). Improper mixing of such nanoparticles can lead to agglomeration or bundling effect, which may reduce the structure's stiffness. Still, its investigation in surface heating can be an essential aspect that is found to be untouched by researchers for porous shell panels. The effective material properties of the three-phase composite panel are determined using the Eshelby-Mori-Tanaka (E-M-T) approach and the Chamis homogenisation technique. Using the variational principle, governing equations are derived and further simplified to non-linear algebraic equations using the Galerkin technique. Gibson and Ashby foam model is used to model cell structure. It is observed that a complete agglomerated state results in higher post-buckling strength, a continuous deformation path due to biaxial compression and curved geometry, and porosity of cellular structure distribution and cell walls are the important parameters to study the stability of panel in a thermal environment are the major conclusions drawn from the current study. Current investigation can be an important contribution towards modelling aircraft structure in supersonic airflow and furnace walls of thermal power plants subjected to localised heating.

Physics and geometry from a Lagrangian: Dirac spinors on a generalised frame bundle

We clarify the structure obtained in Hélein and Vey's proposition for a variational principle for the Einstein-Cartan gravitation formulated on a frame bundle, starting from a structure-less differentiable 10-manifold [17]. The obtained structure is locally equivalent to a frame bundle thus we term it “generalised frame bundle”. In the same time, we enrich the model with a Dirac spinor coupled to the Einstein-Cartan spacetime. The obtained variational equations generalise the usual Einstein-Cartan-Dirac field equations in the sense that they are shown to imply the usual field equations when the generalised frame bundle is a standard frame bundle.

Tight Lipschitz hardness for optimizing mean field spin glasses

AbstractWe study the problem of algorithmically optimizing the Hamiltonian of a spherical or Ising mixed ‐spin glass. The maximum asymptotic value of is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing (AMP) algorithms efficiently optimize up to a value given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property (OGP). However, can also occur, and no efficient algorithm producing an objective value exceeding is known. We prove that for mixed even ‐spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than with non‐negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of . It encompasses natural formulations of gradient descent, AMP, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving mentioned above. To prove this result, we substantially generalize the OGP framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions.

Computationally feasible bounds for the free energy of nonequilibrium steady states, applied to simple models of heat conduction

In this paper, we study computationally feasible bounds for relative free energies between two many-particle systems. Specifically, we consider systems out of equilibrium that admit a nonequilibrium steady state that is reached asymptotically in the long-time limit. The bounds that we suggest are based on the well-known Bogoliubov inequality and variants of Gibbs' and Donsker–Varadhan variational principles. As a general paradigm, we consider systems of oscillators coupled to heat baths at different temperatures. For such systems, we define the free energy of the system relative to any given reference system in terms of the Kullback–Leibler divergence between steady states. By employing a two-sided Bogoliubov inequality and a mean-variance approximation of the free energy (or cumulant generating function), we can efficiently estimate the free energy cost needed in passing from the reference system to the system out of equilibrium (characterised by a temperature gradient). A specific test case to validate our bounds are harmonic oscillator chains with ends that are coupled to Langevin thermostats at different temperatures; such a system is simple enough to allow for analytic calculations and general enough to be used as a prototype to estimate, e.g. heat fluxes or interface effects in a larger class of nonequilibrium particle systems.

On Ekeland Variational Principle and Its Applications Through Fuzzy Quasi Metric Spaces with Non-Archimedean t-norm

The aim of this article is to introduce Ekeland variational principle (EVP) and some results in fuzzy quasi metric space (FQMS) under the non-Archimedean \(t\)-norms. In this article the basic topological properties and a partial order relation are defined on FQMS. Utilizing Brézis-Browder principle on a partial order set, we extend the EVP to FQMS also. Moreover, we derive Takahashi’s minimization theorem, which ensures the existence of a solution of an optimal problem without taking the help of compactness and convexity properties on the underlying space. Furthermore, we give an equivalence chain between these two theorems. Finally, two fixed point results namely the Banach fixed point and the Caristi-Kirk fixed point theorems are described extensively.

Optimal control for nonlinear time-fractional Schrödinger equation: an application to quantum optics

In this research article, a fractional optimal control problem (FOCP) is applied to a nonlinear time-fractional Schrödinger equation (NTFSE) incorporating a trapping potential. The NTFSE is an innovative mathematical advancement in the field of quantum optics, bridging fractional calculus with nonlinear quantum mechanics and addressing the intricacies of systems involving memory and nonlinearity. This exploration helps with potential technological advancements in quantum optics and related domains. Examining the FOCP within this system allows one to design quantum optical systems with enhanced performance, improved precision stability, and robustness against disturbances. In this work, the performance index for the problem is constructed, and then it is reformulated using the fractional variational principle and the Lagrange multiplier method. Additionally, the Jacobi collocation numerical method is employed to solve the FOCP and numerical simulations are demonstrated across various parameters which offer valuable insights into the implemented methodology.

Vibration analysis and suppression of axially moving composite laminated plates with modified energy method and piezoelectric shunt damping circuits

This paper presents a modified variational modeling method and develops an electro-mechanical coupling based vibration control method for moving composite plate structures. Unlike the stationary structures, movement will introduce new challenges for both vibration analysis and control of the laminated plate. Based on the modified variational method, electromechanically coupled dynamic models for the moving plate with piezoelectric patches and control circuits are developed. To release limitations to the admissible functions, modified variational principles are derived for the interface and boundary constraints in both directions of the moving plate. On the basis of first-order shear deformation theory (FSDT), the energy functional expressions of the axially moving laminated plates are formulated and the governing equations are obtained. To suppress the vibration, a piezoelectric shunt damping circuit (PSDC) with negative capacitance is proposed and the control circuit parameters are optimized using the differential evolution (DE) algorithm. Through numerous numerical examples, the high accuracy, good efficiency, and flexibility in constructing admissible functions of the proposed modeling method are demonstrated. The developed PSDC possesses much better vibration suppression properties for the moving laminated plates than the series R-L PSDC. Effects of the moving speed and negative capacitance on the optimal circuit parameters are also examined. The appropriate values of the negative capacitance are shown to be near the capacitance of the patches.