Complex-time-step Newmark methods with controllable numerical dissipation

In this paper, unconditionally stable higher-order accurate time-step integration algorithms with controllable numerical dissipation are presented. The algorithms are based on the Newmark method with complex time steps. The ultimate spectral radius (μ), the sub-step locations (β_j) and the weighting factors (α_j) are the algorithmic parameters. For an algorithm that is (2n−1)th order accurate, the sub-step locations which may be complex, are shown to be the roots of an nth degree polynomial. The polynomial is given explicitly in terms of n and μ. The weighting factors are then obtained by solving a system of n simultaneous equations. It is further shown that the order of accuracy is increased by one for the non-dissipative algorithms with μ=1. The stability properties of the present algorithms are studied. It is shown that if the ultimate spectral radius is set between −1 and 1, the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity. The algorithms are therefore unconditionally C-stable. When the ultimate spectral radius is set to 0 or 1, the algorithms are found to be equivalent to the first sub-diagonal and diagonal Padé approximations, respectively. The present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers. The accuracy of the excitation responses is found to be enhanced by the present complex-time-step procedure. To maintain high-order accuracy, the excitation may need some modifications. © 1998 John Wiley & Sons, Ltd.     

UNCONDITIONALLY STABLE HIGHER-ORDER ACCURATE HERMITIAN TIME FINITE ELEMENTS

In this paper, single step time finite elements using the cubic Hermitian shape functions to interpolate the solution over a time interval are considered. The second-order differential equations are manipulated directly. Both the effects of modal damping and external excitation are considered. The accuracy of the solutions at the end of the time interval and the interpolated solutions within the time interval is investigated. The weighted residual approach is adopted to derive the time-integration algorithms. Instead of specifying the weighting functions, the weighting parameters are used to control the characteristics of the time finite elements. The weighting parameters are chosen to eliminate the higher-order truncation error terms or to enforce the asymptotic annihilation condition. A one-parameter family of third-order accurate asymptotically annihilating algorithms and another one-parameter family of fourth-order accurate non-dissipative algorithms are presented. The ranges of the weighting parameters for unconditionally stable algorithms are given. It is found that one of the members in each family corresponds to the Padé approximants of the exponential function in solving the first-order differential equations. Some of the existing unconditionally stable higher-order accurate algorithms are re-derived by the present unified approach.     

Command filter-based adaptive neural network controller design for uncertain nonsmooth nonlinear systems with output constraint

This paper investigates the command filter-based adaptive neural network tracking control problem for uncertain nonsmooth nonlinear systems. First, an integral barrier Lyapunov function is introduced to deal with the symmetric output constraint and make the output comply with prescribed restrictions. Second, by the Filippov's differential inclusion theory and approximation theorem, the considered nonsmooth nonlinear system is converted to an equivalent smooth nonlinear system. Third, the Levant's differentiator is used to deal with the “explosion of complexity” problem. An error compensation mechanism is established to attenuate the effect of the filtering error on control performance. Then, an adaptive neural network controller is set up by resorting to the backstepping technique. It is strictly mathematically proved that the tracking error can converge to an arbitrarily small neighborhood of the origin and all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. Finally, a numerical example and an application example of the robotic manipulator system are provided to demonstrate the availability of the proposed control strategy.     

Transient natural convective heat transfer and fluid flow in an undulated cavity: Effects of localized heat sources

The purpose of this research work is to investigate two-dimensional transient natural convective heat transfer and fluid flows in an undulated cavity by placing solid objects with isolated heated surfaces on the bottom wall. We discretize the coupled nonlinear transport equations using a higher-order compact finite-difference scheme. First, we test our scheme using existing experimental and numerical data. Then, we analyze the transient and steady-state natural convective flow phenomena for distributed heat sources on corrugations on the lower wall for a range of the Rayleigh number $\left(Ra=10^3-10^6\right)$ and Prandtl number $\left(Pr=0.71\right)$. These simulated outcomes are presented in the form of central-line velocity $\left(u,v\right)$, local $(N{u}_h^{*},N{u}_v^{*})$ and averaged $(\overline{N{u}_h^{*}},\overline{N{u}_v^{*}})$ Nusselt numbers, streamlines $\left(\psi \right)$, dispersion of isotherms $\left(T\right)$, and so forth. It is found that the transient fluid flow behavior is more magnificent than the steady-state solutions and shows the dominant behavior of the prominent primary cells over secondary cells, where it influences the heat transfer rates inside the entire enclosure. In steady states, at high Rayleigh numbers; convection dominates, formation of thermal boundary layers, compression of isotherms, and stratification of isotherms are significantly observed. Our results show many interesting flow phenomena that have not been analyzed previously.     

Magnetic Second-Order Topological Insulator: An Experimentally Feasible 2D CrSiTe3

2D second-order topological insulators (SOTIs) have sparked significant interest, but currently, the proposed realistic 2D materials for SOTIs are limited to nonmagnetic systems. In this study, for the first time, a single layer of chalcogenide CrSiTe₃—an experimentally realized transition metal trichalcogenide is proposed with a layer structure—as a 2D ferromagnetic (FM) SOTI. Based on first-principles calculations, this study confirms that the CrSiTe₃ monolayer exhibits a nontrivial gapped bulk state in the spin-up channel and a trivial gapped bulk state in the spin-down channel. Based on the higher-order bulk–boundary correspondence, it demonstrates that the CrSiTe₃ monolayer exhibits topologically protected corner states with a quantized fractional charge ($\frac{e}{3}$) in the spin-up channel. Notably, unlike previous nonmagnetic examples, the topological corner states of the CrSiTe₃ monolayer are spin-polarized and pinned at the corners of the sample in real space. Furthermore, the CrSiTe₃ monolayer retains SOTI features when the spin–orbit coupling (SOC) is considered, as evidenced by the corner charge and corner states distribution. Finally, by applying biaxial strain and hole doping, this study transforms the magnetic insulating bulk states into spin-gapless semiconducting and half-metallic bulk states, respectively. Importantly, the topological corner states persist in the spin-up channel under these conditions.