Second-order Time Derivatives Research Articles

Numerical simulation of wave propagation by using a hybrid method with an arbitrary order accuracy in both spatial and temporal approximations

This paper introduces an innovative numerical methodology designed to achieve high precision solution of acoustic wave propagation problem in isotropic material. During the temporal discretization process, the Krylov deferred correction (KDC) technique is employed, wherein a new variable is introduced to handle the second-order time derivative in the governing equations. An improved strategy is adopted for precisely implementing boundary conditions. Following this, the arbitrary-order generalized finite difference method (GFDM) is employed to simulate the transformed boundary value problem at each time step, enabling our framework to select the Taylor series expansion order arbitrarily. Ultimately, a hybrid numerical approach for wave problems is developed to achieve arbitrary-order accuracy in both spatial and temporal approximations. The developed scheme undergoes validation through four different numerical experiments in two- or three-dimension, wherein the numerical solutions obtained are compared with either analytical solutions or results from COMSOL software.

Variance sum rule: proofs and solvable models

We derive, in more general conditions, a recently introduced variance sum rule (VSR) (Di Terlizzi et al 2024 Science 383 971) involving variances of displacement and force impulse for overdamped Langevin systems in a nonequilibrium steady state (NESS). This formula allows visualising the effect of nonequilibrium as a deviation of the sum of variances from normal diffusion 2Dt, with D the diffusion constant and t the time. From the VSR, we also derive formulas for the entropy production rate σ that, differently from previous results, involve second-order time derivatives of position correlation functions. This novel feature gives a criterion for discriminating strong nonequilibrium regimes without measuring forces. We then apply and discuss our results to three analytically solved models: a stochastic switching trap, a Brownian vortex, and a Brownian gyrator. Finally, we compare the advantages and limitations of known and novel formulas for σ in an overdamped NESS.

Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems

An essential tool in data-driven modeling of dynamical systems from frequency response measurements is the barycentric form of the underlying rational transfer function. In this work, we propose structured barycentric forms for modeling dynamical systems with second-order time derivatives using their frequency domain input-output data. By imposing a set of interpolation conditions, the systems’ transfer functions are rewritten in different barycentric forms using different parametrizations. Loewner-like algorithms are developed for the explicit computation of second-order systems from data based on the developed barycentric forms. Numerical experiments show the performance of these new structured data-driven modeling methods compared to other interpolation-based data-driven modeling techniques from the literature.

Theory of moment propagation for quantum dynamics in single-particle description.

We present a novel theoretical formulation for performing quantum dynamics in terms of moments within the single-particle description. By expressing the quantum dynamics in terms of increasing orders of moments, instead of single-particle wave functions as generally done in time-dependent density functional theory, we describe an approach for reducing the high computational cost of simulating the quantum dynamics. The equation of motion is given for the moments by deriving analytical expressions for the first-order and second-order time derivatives of the moments, and a numerical scheme is developed for performing quantum dynamics by expanding the moments in the Taylor series as done in classical molecular dynamics simulations. We propose a few numerical approaches using this theoretical formalism on a simple one-dimensional model system, for which an analytically exact solution can be derived. The application of the approaches to an anharmonic system is also discussed to illustrate their generality. We also discuss the use of an artificial neural network model to circumvent the numerical evaluation of the second-order time derivatives of the moments, as analogously done in the context of classical molecular dynamics simulations.

Observer-Based Order-Reduction Speed Control for Converter-Fed BLDC Motors With Current-Loop Adaptation

This research considers the speed control problem for brushless DC motors (BLDCMs) fed by a buck converter subject to the BLDCM parameter and load uncertainty. This study makes three important contributions that distinguish it from previous results: (a) the plant model-free observer estimates the first-and second-order time derivatives for BLDCM speed by incorporating the disturbance-observing mechanism as the subsystem, (b) the proposed control law leads to the first-order closed-loop transfer function from the converter current reference to the DCM speed using observer-based active damping injection, and (c) the analytic variable feedback gain implements converter current-loop adaptation based on the size of the current error. Experimental analysis demonstrates the actual advantages of the proposed technique using a 420 W BLDCM fed by a DC/DC buck converter.

Hermite Finite Element Method for Variable Coefficient Damping Beam Vibration Problem

Beam is the most common component in mechanical equipment and construction. The heterogeneous beam and the variable cross-section beam both have good structural performance. In this study, a Hermite finite element method is proposed for variable coefficient damping beam vibration. The first-order time derivative is approximated by the Richardson format, the second-order time derivative is approximated by the central difference method, and the fully discrete scheme is obtained. The variable coefficients are processed, the error estimate is analyzed in detail. Finally, we verify the validity of the scheme by Matlab and observe influence of damping coefficient variation on beam vibration.

Space-Time Mixed System Formulation of Phase-Field Fracture Optimal Control Problems

In this work, space-time formulations and Galerkin discretizations for phase-field fracture optimal control problems are considered. The fracture irreversibility constraint is formulated on the time-continuous level and is regularized by means of penalization. The optimization scheme is formulated in terms of the reduced approach and then solved with a Newton method. To this end, the state, adjoint, tangent, and adjoint Hessian equations are derived. The key focus is on the design of appropriate function spaces and the rigorous justification of all Fréchet derivatives that require fourth-order regularizations. Therein, a second-order time derivative on the phase-field variable appears, which is reformulated as a mixed first-order-in-time system. These derivations are carefully established for all four equations. Finally, the corresponding time-stepping schemes are derived by employing a dG(r) discretization in time.

An improved theory of thermoelasticity for ultrafast heating of materials using short and ultrashort laser pulses

This paper introduces an improved thermoelastic theory designed specifically to address the challenges posed by fast and ultrafast heating of materials. The proposed theory incorporates a time delay in the temperature gradient and introduces the rate of temperature gradient as an additional term in the heat equation. This term accounts for the phase lag and the precedence of the temperature gradient and heat flux vector. By leveraging the sign of the temperature gradient rate, the theory automatically determines the precedence of either the temperature gradient or the heat flux vector. To verify the effectiveness of the proposed theory, experiments reported in the literature were simulated using several theories, namely the classical, Lord-Schulman (LS), Green-Lindsay (GL), dual-phase-lag (DPL), and the presented theory. The simulations considered the effects of relaxation times and pulse duration on the results. The findings reveal that for pulse durations shorter than 50 ps, the second-order time derivative of the temperature gradient becomes a crucial term in the energy equation for accurately predicting temperature profiles. Additionally, as the pulse duration decreases, the sensitivity to relaxation times increases. Regarding the surface temperature corresponding to the material ablation threshold, it is observed that for an 80 fs laser pulse, the ablation temperature falls between the boiling and critical temperatures, significantly higher than the boiling temperature associated with the ablation by a 270 ps pulse duration. Lastly, the simulation results exhibit a high level of agreement with the experimental data, particularly in terms of displacements, further validating the accuracy of the presented simulations.

A Recursive Lie-Group Formulation for the Second-Order Time Derivatives of the Inverse Dynamics of Parallel Kinematic Manipulators

Series elastic actuators (SEA) were introduced for serial robotic arms. Their model-based trajectory tracking control requires the second time derivatives of the inverse dynamics solution, for which algorithms were proposed. Trajectory control of parallel kinematics manipulators (PKM) equipped with SEAs has not yet been pursued. Key element for this is the computationally efficient evaluation of the second time derivative of the inverse dynamics solution. This has not been presented in the literature, and is addressed in the present paper for the first time. The special topology of PKM is exploited reusing the recursive algorithms for evaluating the inverse dynamics of serial robots. A Lie group formulation is used and all relations are derived within this framework. Numerical results are presented for a 6- DOF Gough-Stewart platform (as part of an exoskeleton), and for a planar PKM when a flatness-based control scheme is applied.

Massive Quantum Vortices in Superfluids

We consider the dynamical properties of quantum vortices with filled massive cores, hence the term “massive vortices”. While the motion of massless vortices is described by first-order motion equations, the inclusion of core mass introduces a second-order time derivative in the motion equations and thus doubles the number of independent dynamical variables needed to describe the vortex. The simplest possible system where this physics is present, i.e. a single massive vortex in a circular domain, is thoroughly discussed. We point out that a massive vortex can exhibit various dynamical regimes, as opposed to its massless counterpart, which can only precess at a constant rate. The predictions of our analytical model are validated by means of numerical simulations of coupled Gross-Pitaevskii equations, which indeed display the signature of the core inertial mass. Eventually, we discuss a nice formal analogy between the motion of massive vortices and that of massive charges in two-dimensional domains pierced by magnetic fields.

Limits of solutions to the semilinear plate equation with small parameter

We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.

A numerical approach for 2D time-fractional diffusion damped wave model

<abstract><p>In this article, we introduce an approximation of the rotated five-point difference Crank-Nicolson R(FPCN) approach for treating the second-order two-dimensional (2D) time-fractional diffusion-wave equation (TFDWE) with damping, which is constructed from two separate sets of equations, namely transverse electric and transverse magnetic phases. Such a category of equations can be achieved by altering second-order time derivative in the ordinary diffusion damped wave model by fractional Caputo derivative of order $ \alpha $ while $ 1 < \alpha < 2 $. The suggested methodology is developed from the standard five-points difference Crank-Nicolson S(FPCN) scheme by rotating clockwise $ 45^{o} $ with respect to the standard knots. Numerical analysis is presented to demonstrate the applicability and feasibility of the R(FPCN) formulation over the S(FPCN) technique. The stability and convergence of the presented methodology are also performed.</p></abstract>

Second-order flows for computing the ground states of rotating Bose-Einstein condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modeled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. In comparison with the gradient flow type approaches: When explicit temporal discretization strategies are adopted, the proposed methods allow for larger stable time step sizes; While for semi-implicit discretization, using the same step size, a much smaller number of iterations are needed for the proposed methods to reach the stopping criterion, and every time step encounters almost the same computational complexity. Rich and detailed numerical examples are documented for verification and comparison.

Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation

Hyperbolic reaction–diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction–diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction–diffusion systems which are distinct from the well studied Turing case.

Sound and soliton wave propagation in homogeneous and heterogeneous mediums with the new two-derivative implicit–explicit Runge–Kutta–Nyström method

This paper derives a new family of implicit–explicit time-marching methods for PDEs with the second-order derivative in time. The present implicit method is based on the two-derivative Runge–Kutta–Nyström methods, which use a third-order time derivative of the solution. Although the current approach is implicit, it does not need to invert the coefficient matrix of the discretized system of equations. The stability properties are assessed using Fourier analysis for the model test problems by considering space–time discretizations together. The present methods are validated by comparing to some of the most widely used time-marching methods available in the literature. In addition, to assess the robustness and efficiency of the present methods, we have also performed numerical simulations of acoustic wave propagation in two- and three-layered heterogeneous media and sine-Gordon solitons for damped and undamped cases. Computed results match very well with the exact and numerical solutions noted in the literature.

The BLUES function method for second-order partial differential equations: Application to a nonlinear telegrapher equation

An analytic iteration sequence based on the extension of the BLUES (Beyond Linear Use of Equation Superposition) function method to partial differential equations (PDEs) with second-order time derivatives is studied. The original formulation of the BLUES method is modified by introducing a matrix formalism that takes into account the initial conditions for higher-order time derivatives. The initial conditions of both the solution and its derivatives now play the role of a source vector. The method is tested on a nonlinear telegrapher equation, which can be reduced to a nonlinear wave equation by a suitable choice of parameters. In addition, a comparison is made with three other methods: the Adomian decomposition method, the variational iteration method (with Green function) and the homotopy perturbation method. The matrix BLUES function method is shown to be a worthwhile alternative for the other methods.

New Formulations on Kinetic Energy and Acceleration Energies in Applied Mechanics of Systems

Multibody mechanical systems (i.e., serial, and parallel robots) have a wide range of applications in the industrial field. In technological processes, these systems perform mechanical movements, in which the active forces have a certain time variation law and, hence, induce higher-order accelerations in the mechanical system, which become central functions in acceleration energies. The advanced dynamics study of multibody systems, often characterized by symmetry, is conducted by applying the differential and variational principles. Lagrange–Euler equations and their time derivatives are commonly used. Here, the central function is the kinetic energy and its higher-order time derivatives. Additionally, the generalization of Gibbs–Appell equations, where the central function is represented by the first and higher-order acceleration energy, can be applied. This paper aims to establish a relation between the kinetic energy and acceleration energy for different material systems. This purpose is achieved by applying the absolute second-order time derivative on the expressions of kinetic energy, corresponding to different material systems. Following this differential calculation and by applying some constraints, the relationship between kinetic energy and acceleration energy is obtained. For validating the relation between kinetic energy and acceleration energy of the first, second and third order, an application is presented.

Second-Order Flows for Computing the Ground States of Rotating Bose-Einstein Condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modelled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. This is judged by the fact that larger size of time steps can be applied in explicit methods, and much smaller number of time steps are needed in both explicit and semi-implicit discretizations, while every time step encounters almost the same computational complexity in comparison with the respective gradient flow type approaches. Rich and detailed numerical examples are documented for verification and comparison.

Ostrogradsky–Hamilton approach to geodetic brane gravity

In this paper, we develop the Ostrogradsky–Hamilton formalism for geodetic brane gravity, described by the Regge–Teitelboim geometric model in higher codimension. We treat this gravity theory as a second-order derivative theory, based on the extrinsic geometric structure of the model. As opposed to previous treatments of geodetic brane gravity, our Lagrangian is linearly dependent on second-order time derivatives of the field variables, the embedding functions. The difference resides in a boundary term in the action, usually discarded. Certainly, this suggests applying an appropriate Ostrogradsky–Hamiltonian approach to this type of theories. The price to pay for this choice is the appearance of second-class constraints. We determine the full set of phase space constraints, as well as the gauge transformations they generate in the reduced phase space. Additionally, we compute the algebra of constraints and explain its physical content. In the same spirit, we deduce the counting of the physical degrees of freedom. We comment briefly on the naive formal canonical quantization emerging from our development.

Bi-Variationality, Symmetries and Approximate Solutions

By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator S to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples.