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We investigate a spectral Galerkin method for the two-dimensional fractional diffusion-reaction equations on a disk. We first prove regularity estimates of solutions in the weighted Sobolev space. Then we obtain optimal convergence orders of a spectral Galerkin method for the fractional diffusion-reaction equations in the L 2 L^2 and energy norm. We present numerical results to verify the theoretical analysis.

In this work, we revisit the spectral Petrov-Galerkin method for fractional elliptic equations with the general fractional operators. To prove the optimal convergence of the method, we first present the ultra-weak formulation and establish its well-posedness. Then, based on such a novel formulation, we are able to prove the discrete counterpart and obtain the optimal convergence of the spectral method in the weighted L 2 -norm. For simple and easy implementation of the method, we also describe the fast solver with linear storage and quasilinear complexity. To support our theory, we carry out the numerical experiments and provide several numerical results to show the accuracy and efficiency of our method.

Nickel based single crystal alloys have excellent properties such as heat resistance, corrosion resistance and creep resistance, which are widely used in aerospace and other national defense fields. Severe work hardening occurs in the process of cutting nickel based single crystal alloy. How to improve the machining quality and grasp the cutting deformation mechanism has become the research focus. In this paper, the effect of work hardening on the surface of workpiece during the atomic and close-to-atomic scale (ACS) cutting process is studied. The model of S i 3 N 4 ceramic tool cutting the nickel based single crystal alloy was established, and the ACS cutting process was simulated by the molecular dynamics method. The existing strain rate conversion model was modified to make it suitable for the process of ACS cutting into nano compression with the same strain rate. The results show that the dislocation density of Ni-based single crystal alloy workpiece changes greatly with the change of cutting distance. According to the change of microstructure in the workpiece, a new staged work hardening mechanism is proposed. The development of work hardening in the cutting process is divided into three stages, and the transition node of each hardening stage is defined. An important sign of the transformation from the first stage to the second stage of work hardening is the occurrence of a large number of dislocation pile-up group, dislocation tangles and the appearance of non-basal slip lines. The distinctive feature of the transformation from the second stage to the third stage of work hardening is that a large number of screw dislocations are cross-slip and the dislocation pile-up group is destroyed. At the same time, the different hardening mechanisms in each stage and the reasons for the change of work hardening mechanism in different stages are summarized. The research content is believed to be helpful to understand the mechanism of significant work hardening effect in nickel-based alloys.

Nickel-based single crystal alloys are widely used in aerospace and other important fields of national defense due to their excellent properties. Phase transformation occurs during high-speed cutting of nickel-based single crystal alloy, which seriously affects the surface quality. It is of great significance to carry out theoretical research on phase transformation for improving the machining quality of nickel-based alloy. In this paper, molecular dynamics method is used to study the nano-cutting of single crystal nickel-based alloy with silicon nitride ceramic tool. The mechanism of phase transformation and the effect of cutting speed on phase transformation in workpieces are studied in detail. The nano-cutting model is established. Morse potential functions for molecular dynamics simulation are calculated, and EAM and Tersoff potential functions are selected. The effect of cutting speed on phase transformation was studied by using radial distribution function, coordination number analysis, common neighbor analysis, and the deep reasons for the sharp change of lattice structure were analyzed from many aspects. Finally, in order to verify the universality of the research results and explore the new properties of compression, nano compression (the same strain rate as the nano cutting process) was simulated. The results show that the increase of cutting speed leads to the increase of hydrostatic stress, the increase of energy in crystal and the rise of cutting temperature. As a result, the change of lattice structure becomes more and more intense, and the conversion rate of different crystal structures increases greatly.

In this paper, an efficient algorithm is presented by adopting the extrapolation technique to improve the accuracy of finite difference schemes for two-dimensional space-fractional diffusion equations with non-smooth solution. The popular fractional centered difference scheme is revisited and the stability and error estimation of numerical solution are given in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, the extrapolation technique and numerical correction method are exploited to enhance the accuracy and convergence rate of the computation. Two numerical examples are provided to validate the theoretical prediction and efficiency of the algorithm. It is shown that, by using the proposed algorithm, both accuracy and convergence rate of numerical solutions can be significantly improved and the second-order accuracy can even be recovered for the equations with large fractional orders.

Digital image correlation (DIC) is a typical non-contact full-field deformation parameters measurement technique based on image processing technology and numerical computation methods. To obtain the displacements of each point of interrogation in DIC, subsets surrounding the point must be chosen in the reference image and deformed image before correlating. In the existing DIC techniques, the size of subset is always pre-defined by users manually according to their experiences. However, the subset size has proven to be a critical parameter for the accuracy of computed displacements. In the present paper, a self-adaptive selection of subset size method based on Shannon entropy is proposed to overcome the deficiency of existing DIC methods. To verify the effectiveness and accuracy of the proposed algorithm, a numerical translated test is performed on four actual speckle patterns with different entropies, and then another test is performed on four computer-generated speckle patterns with non-uniform displacement field. All the results successfully demonstrate that the proposed algorithm can significantly improve displacement measurement accuracy without reducing too much computational efficiency. Finally, a practical application of the proposed algorithm to micro-tensile of Q235 steel is conducted.

We investigate a spectral Galerkin method for the fractional advection-diffusion-reaction equations in one dimension. We first prove sharp regularity estimates of solutions in nonweighted and weigh...

The traditional material constitutive model can effectively simulate the mechanical properties during the cutting process. However, the scale characteristics contained in materials are not considered in the traditional cutting model, and the inherent scale effect of materials is also ignored. Therefore, the traditional cutting constitutive model cannot effectively reflect the size effect in the cutting process, and then cannot obtain the accurate stress, strain, and temperature. In this present paper, a material constitutive model which can reflect the scale effect is established based on the strain gradient plasticity theory. Through the established model and secondary development of abaqus, the two-dimensional dynamic finite element simulation model of cutting Inconel 718 is established. By comparing the cutting experiment results with the simulation results, the established simulation model can more accurately reflect the effects of temperature, strain gradient effect, equivalent stress, and its scale effect on cutting deformation during the machining process.

This paper develops a Galerkin approach for two-sided fractional differential equations with variable coefficients. By the product rule, we transform the problem into an equivalent formulation which additionally introduces the fractional low-order term. We prove the existence and uniqueness of the solutions of the Dirichlet problems of the equations with certain diffusion coefficients. We adopt the Galerkin formulation, and prove its error estimates. Finally, several numerical examples are provided to illustrate the fidelity and accuracy of the proposed theoretical results.

In this paper, we propose a three-level linearized finite difference scheme for the high-dimensional nonlinear Ginzburg–Landau equation with fractional Laplacian. The Crank–Nicolson scheme is used for time discretization, and the fractional Laplacian is discretized by the fractional centered difference scheme. The proposed difference scheme (i.e. a linear system) can be solved efficiently by fast Fourier transform (FFT) and complex conjugate gradient method, since the coefficient matrix is a multi-level block Toeplitz matrix. Furthermore, we analyze the unique solvability and boundedness of solution of the difference scheme by the discrete energy method. It is also proved that the difference scheme is unconditionally stable and second-order accurate in time and space with respect to l ∞ -norm. Finally, several numerical examples are provided to validate the theoretical results.

Nickel-based superalloy Alloy 718 is widely used in aerospace and power industry due to its mechanical and physical properties at elevated temperature. In the process of high-speed cutting Alloy 718, plastic behavior of material in the cutting zone is investigated by quick-stop tests and split-Hopkinson pressure bar (SHPB) test at elevated temperatures with strain rates ranging from 5 × 103 s−1 to 10.5 × 103 s−1 and temperatures between 500 and 800 °C. The results show that a significant thermal softening effect occurs in the process of material deformation. Based on the kinematics of the dynamic recrystallization, the combined flow stress constitutive equations of the work hardening and softening are established for describing the plastic behavior of the material in high speed cutting of Alloy 718. Finally, the predicted results of the model quite coincide with the experimental values, showing that the established constitutive model can well reflect the deformation characteristics in high-speed machining of Alloy 718.

In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with nonsmooth solution. Two popular finite difference schemes, the weighted shifted Grunwald difference (WSGD) scheme and the fractional centered difference (FCD) scheme, are revisited and stability of the schemes is shown in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, it is demonstrated that, with the use of the proposed algorithm, the improved WSGD and FCD schemes can achieve higher accuracy than the original ones for nonsmooth solution. To further improve the accuracy for solving problems with small fractional order, an extended algorithm dealing with two-term singularities correction is also developed. Several numerical examples are given to validate our theoretical prediction. It is shown that both accuracy and convergence rate of numerical solutions can be significantly improved by using the proposed algorithms.

In order to investigate the antiferromagnetic coupling of Co and Mn atoms in Cu2−xCoxMn1+yAl1−y, a Heusler alloy, under Mn rich condition, our basic research objects are Cu2MnAl with only one kind of magnetic atoms and Co2MnAl which remain ferromagnetic in Mn rich conditions. After a large number of calculations, we found that because the Co atom is higher than the d-band center of Cu atom, then the anti-bonding state of Mn atom is pushed above Fermi level, thus changing the d-orbital electron arrangement of Mn atom, making the original antiferromagnetic Mn atom become ferromagnetic. Based on this theory, a method of applying stress to the alloy to adjust the lattice size to change the magnetic moment of Mn atom is proposed to achieve the same effect as doping special atoms. In addition, it was also inferred that the ferromagnetism of the parent phase and the ferromagnetism of the martensite phase are closely related to the d-band center caused by the change of lattice size.

The numerical solution for the one-dimensional complex fractional Ginzburg–Landau equation is considered and a linearized high-order accurate difference scheme is derived. The fractional centered difference formula, combining the compact technique, is applied to discretize fractional Laplacian, while Crank–Nicolson/leap-frog scheme is used to deal with the temporal discretization. A rigorous analysis of the difference scheme is carried out by the discrete energy method. It is proved that the difference scheme is uniquely solvable and unconditionally convergent, in discrete maximum norm, with the convergence order of two in time and four in space, respectively. Numerical simulations are given to show the efficiency and accuracy of the scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 105–124, 2017

This article is devoted to the study of some high-order difference schemes for the distributed-order time-fractional equations in both one and two space dimensions. Based on the composite Simpson formula and Lubich second-order operator, a difference scheme is constructed with convergence in the L1(L∞)-norm for the one-dimensional case, where τ,h and σ are the respective step sizes in time, space and distributed-order. Unconditional stability and convergence are proven. An ADI difference scheme is also derived for the two-dimensional case, and proven to be unconditionally stable and convergent in the L1(L∞)-norm, where h1 and h2 are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.

Based on the previous research and summary, a new half-metallic Heusler alloy FeCrAs have been proposed. The density functional theory of spin polarization is calculated by using the first-principles calculation software CASTEP to determine the lattice constants of FeCrAs compounds in half-Heusler structure. The effect of each element on its half-metallicity and the formation of band gap are calculated. The total magnetic moment is calculated and verified theoretically. The range of lattice distortion which keeps half -metallic properties was studied to ensure its half-metallic properties. In the calculation process, the role of Fe atom in the crystal cell, based on the theory of level splitting in the crystal field of the complexes, has been studied. According to the calculations, the FeCrAs compound of Half-Heusler phase is a new Heusler alloy.

In order to study the micro-forming mechanism of work hardening in cutting Ni–Fe–Cr series of Ni-based alloy using Cubic Boron Nitride (CBN) tool, the cutting model was established by means of molecular dynamics simulation analysis method. The Morse potential function and other potential functions were calculated to characterize the interaction between atoms. Then, the influence of variation of cutting force, the dislocation density, dislocation pile-up, Lomer-Cottrell dislocation, and solute atoms on work hardening are deeply analyzed. The results show that the metal surface with plastic deformation initiates a variety of internal mechanisms to hinder dislocation movement as dislocation density increases, dislocation motion and interaction between dislocations intensifies. The mechanism of work hardening in Ni–Fe–Cr alloy is not only dislocation pile-up that is not easy to slip or cannot slip or dislocation tangle caused by dislocation intersection, but also a lot of Lomer-Cottrell dislocations. The solute elements that reduce stacking fault energy indirectly affect the generation of Lomer-Cottrell dislocations. In addition, solute atoms in nickel-based alloys can pin dislocations, hinder the movement of dislocations, and promote dislocation tangle in the workpiece. Various mechanisms interact and influence each other, which is a complex whole.

Based on the Lubich's high-order operators, a second-order temporal finite-difference method is considered for the fractional sub-diffusion equation. It has been proved that the finite-difference scheme is unconditionally stable and convergent in norm by the energy method in both one-and two-dimensional cases. The rate of convergence is order of two in temporal direction under the initial value satisfying some suitable conditions. Some numerical examples are given to confirm the theoretical results.

In this paper, a compact algorithm for the fourth-order fractional sub-diffusion equations with first Dirichlet boundary conditions, which depict wave propagation in intense laser beams, is investigated. Combining the average operator for the spatial fourth-order derivative, the L1 formula is applied to approximate the temporal Caputo fractional derivative. A novel technique is introduced to deal with the first Dirichlet boundary conditions. Using mathematical induction method, we prove that the presented difference scheme is unconditionally stable and convergent by the energy method. The convergence order is $$O(\tau ^{2-\alpha }+h^4)$$O(?2-?+h4) in $$L_2$$L2-norm. The outline for the two-dimensional problem is also considered. Finally, some numerical examples are provided to confirm the theoretical results.

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Gr\"{u}nwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time. With the use of the fourth-order compact scheme in space, we give a fully discrete Gr\"{u}nwald-Letnikov-formula-based compact difference scheme and prove its stability and convergence by the energy method under smooth assumptions. In addition, the problem with nonsmooth solution is also discussed, and an improved algorithm is proposed to deal with the singularity of the fractional substantial derivative. Numerical examples show the reliability and efficiency of the scheme., Comment: high-order finite difference method, fractional substantial derivative, weighted average operator, stability analysis, nonsmooth solution