In this paper, a 3-D hybrid Maxwell's equations finite-difference time-domain (ME-FDTD)/wave equation based finite element time-domain (WE-FETD) method is proposed. This method retains the ...non-conformal mesh and the implicit-explicit time integration scheme. The WE-FETD region is based on the wave equation rather than the Maxwell's curl equations. This method needs to store all the electric fields in the entire region and only the magnetic fields on the interface, which can prominently reduce degrees of freedom (DoFs) and save calculation time. The ME-FDTD region follows the Yee's scheme. A Maxwell's equations spectral element time-domain (ME-SETD) region and a virtual region are used to combine the ME-FDTD and WE-FETD regions. Consequently, a WE-FETD/ME-SETD/Virtual/ME-FDTD framework is formed. Hybrid Newmark-beta (NB) and Crank-Nicolson (CN) time stepping are employed for implicit WE-FETD and ME-SETD regions. The leapfrog (LF) time integration is used for the explicit Virtual and FDTD regions. At the interface, it employs upwind flux in the discontinuous Galerkin (DG) method to couple neighboring regions. Numerical examples are included to demonstrate the accuracy of the proposed method. Several cases exhibit the improved efficiency compared with the hybrid FDTD/FETD method only based on the Maxwell's equations and the pure FETD method.
In this paper, finite-time synchronization and H∞ synchronization of coupled complex-valued memristive neural networks (CCVMNNs) with or without parameter uncertainty are analyzed. First, a ...finite-time synchronization (FTS) condition is presented for CCVMNNs by means of deploying Lyapunov stability theory and developing suitable controllers. Then, we utilize the similar method to derive a criterion of robust finite-time synchronization (RFTS) for the proposed CCVMNNs with uncertain parameter. Furthermore, we establish some criteria for the sake of ensuring that the considered network can reach finite-time H∞ synchronization and robust finite-time H∞ synchronization. At last, two numerical examples with simulations demonstrate the validity of the acquired results.
Summary
Finite elements of class 𝒞1 are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment ...of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). We here explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh‐Clough‐Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two‐dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).
Finite elements of class C1 are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). Here we explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bezier polynomials on a structured mesh of curved quadrangular elements, which allow to remain in the physical space. Reduced cubic Hsieh‐Clough‐Tocher finite elements are then adopted on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. The Poisson problem on a two‐dimensional bounded regular domain is considered as a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio). Numerical results on the accuracy of the local interpolation and of the coupled problem are provided together with an analysis of the algorithm convergence. This analysis is a first step to the adoption of the presented methodology in the industrial code such as JOREK.
A new family of hybrid/mixed finite elements optimized for numerical stability is introduced. It comprises a linear hexahedral and quadratic hexahedral and tetrahedral elements. The element ...formulation is derived from a consistent linearization of a well-known three-field functional and related to Simo–Taylor–Pister (STP) elements. For the quadratic hexahedral and tetrahedral elements we derive (static reduced) discontinuous hybrid elements, as well as continuous mixed finite elements with additional primary unknowns for the hydrostatic pressure and the dilation, whereas the linear hexahedral element is of the discontinuous type. The elements can readily be used in combination with any isotropic, invariant-based hyperelastic material model and can be considered as being locking-free. In a representative numerical benchmark test the elements numerical stability is assessed and compared to STP-elements and the family of discontinuous hybrid elements implemented in the commercial finite element code Abaqus/Standard. The new elements show a significant advantage concerning the numerical robustness.
Extensions of finite-difference time-domain (FDTD) and finite-element time-domain (FETD) algorithms are reviewed for solving transient Maxwell equations in complex media. Also provided are a few ...representative examples to illustrate the modeling capabilities of FDTD and FETD for complex media. The term complex media refers here to media with dispersive, (bi)anisotropic, inhomogeneous, and/or nonlinear properties present in the constitutive tensors.
We propose a finite-difference time-domain (FDTD) scheme based on generalized sheet transition conditions (GSTCs) for the simulation of polychromatic, nonlinear, and space-time varying metasurfaces. ...This scheme consists in placing the metasurface at virtual nodal plane introduced between the regular nodes of the staggered Yee grid and inserting fields determined by GSTCs in this plane in the standard FDTD algorithm. The resulting update equations are an elegant generalization of the standard FDTD equations. Indeed, in the limiting case of a null surface susceptibility (χ surf = 0), they reduce to the latter, while in the next limiting case of a time-invariant metasurface χ surf ≠ χ surf (t), they split in two terms, one corresponding to the standard equations for a one-cell (Δx) thick slab with diluted volume susceptibility (χ = χ surf /(2Δx)), and the other one reducing that slab to a quasi-zero-thickness mesh-less sheet. The proposed scheme is fully numerical and very easy to implement. Although it is explicitly derived for a monoisotropic metasurface, it may be straightforwardly extended to the bianisotropic case. Except for some particular cases, it is not applicable to dispersive metasurfaces, for which an efficient auxiliary different equation extension of the scheme is currently being developed by the authors. The scheme is validated and illustrated by five representative examples.
This article proposes a fault-tolerant finite-time controller for attitude tracking control of rigid spacecraft using intermediate quaternion in the presence of external disturbances, uncertain ...inertia parameter, and actuator faults. First, a novel nonsingular fast terminal sliding mode control law is derived using intermediate quaternion, free of singularity, ambiguity, and unwinding phenomenon. Second, a proposed controller is developed by combining the continuous nonsingular fast terminal sliding mode method with the finite-time disturbance observer. The key feature of the proposed control strategy is that it globally stabilizes the system in finite time, even in the presence of actuator faults, inertia uncertainty, and external disturbances. Simulation results are presented under actuator constraint to show the desirable properties and the superiority of the proposed controller compared to an asymptotic controller and nonsingular fast terminal sliding mode controller.
In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. ...We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in 25,45, could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.
•A new class of high order finite difference and finite volume WENO schemes are constructed.•These schemes are based on the multi-resolution idea, and a series of unequal-sized hierarchical central spatial stencils.•These schemes can use arbitrary positive linear weights, and are easy to implement for one and multi-dimensions.•These schemes have a gradual degrading of accuracy near discontinuities.
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