《KeepingTimewithMaxwellx27sEquations(HFSS时域求解器HFSS-TD)》由会员分享，可在线阅读，更多相关《KeepingTimewithMaxwellx27sEquations(HFSS时域求解器HFSS-TD)（8页珍藏版）》请在金锄头文库上搜索。
1、Harald Songoro,Martin Vogel, and Zoltan CendesW *7Keeping Time withEquationsEYEWIREarald Sonoro (HS), Martin Vogel, and Zoltan Cendesarc with Ansoft LLC, Pittsburgh, Pcnnsyknia, U.S.A.Digital Object Idnilificr 10.1109/MMM.2010.935779# ieee mcrov/ave magazine1527-3342/10/$26.002010 IEEEApril 2010Duri
2、ng the 1980s, advances in meshing technology together with the finite element method (FEM) made it possible to solve Maxwells equations in complex geometries by using an unstructured mesh based on a tetrahedral tessellation. In such a mesh, rather than forming a regular grid, the elements have a ran
3、ge of sizes and orientations that conform to the geometry of the mod el and respect all geometrical details. This novel approach, initially pioneered in two-dimension (2-D) in 1J and three-dimension (3-D) in 2, contrasted with cube-based space partitioning in its ability to fa让hfully represent compl
4、ex geometries The use of unstructured meshes combined with automatic mesh adaptation has been instrumental in delivering accurate, dependable, and consis tent results for real-world geometries.Historically, the prominent industrial need in microwave engineering has been to calculateS-parameters and
5、radiation patterns in the frequencydomain. However, in certain applications, it is helpful to visualize transient field data, for instance to distinguish locally incident fields from scattered fields. This capability is provided more directly and efficiently by using time-domain methods So far, comm
6、ercially available solvers based on time-domain methods have relied on the finite difference time-domain (FDTD) method 3 or on the closely related finite integration technique (FIT) 4|. The use of unstructured meshes for both schemes presents well-known technical d让ficul- ties that have yet to be ov
7、ercome and thus both FDTD and FIT continue to rely on brick-based space partitioning Time-domain methods developed for unstructured meshes such as the finite volume timedomain (FVTD) 5z 6 or the finite element time-domain (FETD) 7 methods do not offer anmemory usage However, recent advances infinite
8、-element techniques have led to a new approach based on the discontinuous Galerkin 8family of numerical methods. This approach retains the flexibil让y, accuracy, and reliabil让y ofunstructuredmesh finite-element methods but, as is the case with FIT and FDTD, discontinuous Galerkin timedomnin (DGTD) me
9、thods avoid the solution of a large linear system at each time step. Instead, in the DGTD method, each mesh element advances in time using its own time step in a synchronous manner. This provides a significant speed-up on unstructured meshes since the largest mesh elements typically take time steps
10、that are two to five orders of magnitude larger than those of the smallest mesh elementsThis article presents the application of the DGTD method to Maxwells equations as well as a description of the first commercial solver high-frequency structure simulator-time domain (HFSS-TD) to use the FEM with
11、unstructured meshes in the time domain.Discontinuous Galerkin Finite Element Method+/=a/r/(H) =-curl(E);div(D) = p div(B) = 0.The mathematical framework of discontinuous Galerkin methods for solving Maxwells equations has been presented by Jan Hesthaven and Tim Warburton 9, 10J. In a similar way to
12、the FE4, we choose local basis and test functions in each mesh element /April 2010ieee microwave magazine #=1TTcundB畔0=卜曲(g)0atFigure 1. The time steps in discontinuous Galcrkin time domain are local. Lare elements are colored yellow and use larger time-steps than the smaller elements colored reen o
13、r the even smaller elements in blue and dark blue.PIFA AntennaMetal Housi ngMonitorKeyboardFigure 2. Laptop computer model.Differential Lines Between the Monitor, Traces, and PCB1(f) H- dig x fl) -/ ?curl(p) E + Jx E).TAs is the case w让h the frequency-domain fin让e- element solver, the DGTD method is
14、 free of spurious solutions 11 and supports higher-order basis functions. In the time domain, the use of high orders is crucial to minimize dispersion errors occurring during the simulation of electrically large problems. The DGTD solver not only employs high-order elements but allows mixed element
15、orders in the same meshz known as hp-convergence, to optimize accu racy and solution times.Following the approach used in the finite volume method, the divergence term containing volume integrals is transformed into surface integralsThe tangential fields n X E and n X H are then evaluated as upwind fluxes on the surface of 1 by using a Riemann solver 12. Note that with discontinuous ele merits, the approximation inside the element is treated separately from its surface and provides a robust and flexible
