Multigrid methods for Maxwell\u27s equations

In this work we study finite element methods for two-dimensional Maxwell\u27s equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell\u27s equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments

Effect of magnetostatic field on microstructure of 5005 aluminum alloy sheet by roll-casting

The roll-casting technology of 5000 series aluminum alloy is one of the difficulties at present. Roll-casting technology was employed in this paper and high quality 5005 aluminum alloy sheets were fabricated successfully under different conditions. The solidification of melt in roll-casting process is the rapid directional solidification. The effect of magnetostatic field on microstructure of 5005 aluminum alloy sheet by roll-casting has been investigated and analyzed. The results indicate that imposing magnetostatic field can refine the grains of 5005 aluminum alloy

Robustness of double-layer group-dependent combat network with cascading failure

The networked combat system-of-system (CSOS) is the trend of combat development with the innovation of technology. The achievement of combat effectiveness requires CSOS to have a good ability to deal with external interference. Here we report a modeling method of CSOS from the perspective of complex networks and explore the robustness of the combat network based on this. Firstly, a more realistic double-layer heterogeneous dependent combat network model is established. Then, the conditional group dependency situation is considered to design failure rules for dependent failure, and the coupling relation between the double-layer subnets is analyzed for cascading failure. Based on this, the initial load and capacity of the node are defined, respectively, as well as the load redistribution strategy and the status judgment rules for the cascading failure model. Simulation experiments are carried out by changing the attack modes and different parameters, and the results show that the robustness of the combat network can be effectively improved by improving the tolerance limit of one-way dependency of the functional net, the node capacity of the functional subnet and the tolerance of the overload state. The conclusions of this paper can provide a useful reference for network structure optimization and network security protection in the military field

Anisotropic Quasi-Wilson Element with Conforming Finite Element Approximation for Coupled Continuum Pipe-Flow/Darcy Model in Karst Aquifers

This paper presents a numerical method for solving systems of partial differential equations describing flow in porous media with an embedded and inclined conduit pipe. This work considers a coupled continuum pipe-flow/Darcy model. The numerical schemes presented are based on combinations of the quasi-Wilson element on anisotropic mesh and the conforming finite element on regular mesh. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in both L2 and H1 norms are obtained independent of the regularity condition on the mesh. Numerical examples show the accuracy and efficiency of the proposed scheme

Precessing jet nozzle connecting to a spinning black hole in M87

The nearby radio galaxy M87 offers a unique opportunity to explore the connections between the central supermassive black hole and relativistic jets. Previous studies of the inner region of M87 revealed a wide opening angle for the jet originating near the black hole. The Event Horizon Telescope resolved the central radio source and found an asymmetric ring structure consistent with expectations from General Relativity. With a baseline of 17 years of observations, there was a shift in the jet's transverse position, possibly arising from an eight to ten-year quasi-periodicity. However, the origin of this sideways shift remains unclear. Here we report an analysis of radio observations over 22 years that suggests a period of about 11 years in the position angle variation of the jet. We infer that we are seeing a spinning black hole that induces the Lense-Thirring precession of a misaligned accretion disk. Similar jet precession may commonly occur in other active galactic nuclei but has been challenging to detect owing to the small magnitude and long period of the variation.Comment: 41 pages, 7 figures, 7 table