Convergence analysis on Browder-Tikhonov regularization for second-order evolution hemivariational inequality

This paper studies the Browder-Tikhonov regularization of a second-order evolution hemivariational inequality （SOEHVI） with non-coercive operators. With duality mapping, the regularized formulations and a derived first-order evolution hemivariational inequality （FOEHVI） for the problem considered are presented. By applying the Browder-Tikhonov regularization method to the derived FOEHVI, a sequence of regularized solutions to the regularized SOEHVI is constructed, and the strong convergence of the whole sequence of regularized solutions to a solution to the problem is proved.

SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.

Adiabatic Shear Localization for Steels Based on Johnson-Cook Model and Second-and Fourth-Order Gradient Plasticity Models

To consider the effects of the interactions and interplay among microstructures, gradient-dependent models of second- and fourth-order are included in the widely used phenomenological Johnson-Cook model where the effects of strain-hardening, strain rate sensitivity, and thermal-softening are successfully described. The various parameters for 1006 steel, 4340 steel and S-7 tool steel are assigned. The distributions and evolutions of the local plastic shear strain and deformation in adiabatic shear band （ASB） are predicted. The calculated results of the second- and fourth- order gradient plasticity models are compared. S-7 tool steel possesses the steepest profile of local plastic shear strain in ASB, whereas 1006 steel has the least profile. The peak local plastic shear strain in ASB for S-7 tool steel is slightly higher than that for 4340 steel and is higher than that for 1006 steel. The extent of the nonlinear distribution of the local plastic shear deformation in ASB is more apparent for the S-7 tool steel, whereas it is the least apparent for 1006 steel. In fourth-order gradient plasticity model, the profile of the local plastic shear strain in the middle of ASB has a pronounced plateau whose width decreases with increasing average plastic shear strain, leading to a shrink of the portion of linear distribution of the profile of the local plastic shear deformation. When compared with the sec- ond-order gradient plasticity model, the fourth-order gradient plasticity model shows a lower peak local plastic shear strain in ASB and a higher magnitude of plastic shear deformation at the top or base of ASB, which is due to wider ASB. The present numerical results of the second- and fourth-order gradient plasticity models are consistent with the previous numerical and experimental results at least qualitatively.

Distributed Second-Order Continuous-Time Optimization via Adaptive Algorithm with Nonuniform Gradient Gains

This paper addresses the distributed adaptive optimization problem over second-order multi-agent networks(MANs)with nonuniform gradient gains.A general convex function consisting of a sum of local differentiable convex functions is chosen as the team objective function.First,based on the local information of each agent’s neighborhood,a novel distributed adaptive optimization algorithm with nonuniform gradient gains is designed,where these gains only have relations with agents’own states.And then,the original closed-loop system is changed into an equivalent one by taking a coordination transformation.Moreover,it is proved that the states including positions and velocities of all agents are bounded by constructing a Lyapunov function provided that the initial values are given.By the theory of Lyapunov stability,it is shown that all agents can finally reach an agreement and their position states converge to the optimal solution of the team objective function asymptotically.Finally,the effectiveness of the obtained theoretical results is demonstrated by several simulation examples.

Second-order magnetic field gradient-induced strong coupling between nitrogen-vacancy centers and a mechanical oscillator

We consider a cantilever mechanical oscillator(MO) made of diamond. A nitrogen-vacancy(NV) center lies at the end of the cantilever. Two magnetic tips near the NV center induce a strong second-order magnetic field gradient. Under coherent driving of the MO, we find that the coupling between the MO and the NV center is greatly enhanced. We studied how to generate entanglement between the MO and the NV center and realize quantum state transfer between them. We also propose a scheme to generate two-mode squeezing between different MO modes by coupling them to the same NV center. The decoherence and dissipation effects for both the MO and the NV center are numerically calculated using the present parameter values of the experimental configuration. We have achieved high fidelity for entanglement generation, quantum state transfer, and large twomode squeezing.

Hilbert-Schmidt-Hankel norm model reduction for matrix second-order linear systems

This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original systems. The expressions of the error function and its gradient are derived. Two numerical examples are given to illustrate the presented model reduction technique.

Micro-Scanning Error Correction Technique for an Optical Micro-Scanning Thermal Microscope Imaging System

An error correction technique for the micro-scanning instrument of the optical micro-scanning thermal microscope imaging system is proposed. The technique is based on micro-scanning technology combined with the proposed second-order oversampling reconstruction algorithm and local gradient image reconstruction algorithm. In this paper, we describe the local gradient image reconstruction model, the error correction technique, down-sampling model and the error correction principle. In this paper, we use a Lena original image and four low-resolution images obtained from the standard half-pixel displacement to simulate and verify the effectiveness of the proposed technique. In order to verify the effectiveness of the proposed technique, two groups of low-resolution thermal microscope images are collected by the actual thermal microscope imaging system for experimental study. Simulations and experiments show that the proposed technique can reduce the optical micro-scanning errors, improve the imaging effect of the system and improve the system's spatial resolution. It can be applied to other electro-optical imaging systems to improve their resolution.

Continuous-time Distributed Heavy-ball Algorithm for Distributed Convex Optimization over Undirected and Directed Graphs

This paper proposes second-order distributed algorithms over multi-agent networks to solve the convex optimization problem by utilizing the gradient tracking strategy, with convergence acceleration being achieved. Both the undirected and unbalanced directed graphs are considered, extending existing algorithms that primarily focus on undirected or balanced directed graphs. Our algorithms also have the advantage of abandoning the diminishing step-size strategy so that slow convergence can be avoided. Furthermore, the exact convergence to the optimal solution can be realized even under the constant step size adopted in this paper. Finally, two numerical examples are presented to show the convergence performance of our algorithms.

Work conjugate pair of stress and strain in molecular dynamics

Certain stress and strain form a thermodynamic conjugate pair such that their strain energy equals to a scalar-valued potential energy.Different atomistic stresses and strains are analytically derived based on the work conjugate relation.It is numerically verified with both two-body and three-body potentials that the atomistic Kirchhoff stress,first-order Piola–Kirchhoff stress and second-order Piola–Kirchhoff stress are conjugates to atomistic logarithmic strain,deformation gradient and Lagrangian strain,respectively.Virial stress at 0 K based on original volume is the special form of atomistic Kirchhoff stress for pair potential.It is numerically verified that Hencky strain is not conjugate to any stress.