Hierarchical functions for multiscale problems

The authors' teaching experiences with numerical techniques in computational electromagnetics are described. Course descriptions and objectives are discussed, as well as their relationship to engineering practice.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper presents the author's teaching experiences on Numerical Techniques in Computational Electromagnetics for years, which includes course descriptions and objectives as well as the relation with engineering practice, etc.

The application of p- or hp-refinement techniques on an adaptive finite element procedure requires the use of hierarchical basis functions in order to facilitate the tangential continuity of the solution on the interface between two elements with different order. Interpolatory and hierarchical are nowadays well-established families of basis functions. Hierarchical basis functions have been used already in antenna analysis. In this paper, this kind of basis functions is employed in the full-wave analysis of closed waveguiding structures, which requires scalar/vector interpolation and is compared with interpolatory basis functions, mainly in terms of their convergence.

Spatial Sampling and Interpolation Techniques in Computational Electromagnetics and Beyond

In this paper, the time domain integral equation is solved by marching-on-in-degree method with near-orthogonal higher order hierarchical Legendre basis as spatial basis functions and causal weighted Laguerre polynomials as temporal basis functions. In the traditional marching-on-in-degree solver of time domain integral equation, RWG basis functions are used as spatial basis functions. The memory requirement and time consuming is very large, which becomes a bottleneck of marching-on-in-degree method. In order to solve this problem, the object is meshed with second-order nine-node curved quadrilateral elements and near-orthogonal higher order hierarchical Legendre basis functions are adopted as spatial basis functions. Numerical results show that this method can greatly reduce the unknowns of the problem, by which it can save memory and CPU time.

This paper identifies several important aspects of current Feature Selective Validation (FSV) methodologies that are embodied in IEEE Standard 1597.1 for the Validation of CEM Computer Modeling and Simulations. The FSV method facilitates comparisons of sets of electromagnetic (EM) observable data for a given problem to determine “levels of agreement” across amplitude and feature variables. Areas of future revision to this standard are presented that will further enhance the standard's utility for performing Computational Electromagnetic (CEM) technique validation for a wide range of problems. In particular, we consider the utility of the N-dimensional FSV and revisit applications of the Amplitude Difference Measure (ADM), Feature Difference Measure (FDM) and the Global Difference Measure (GDM). This is discussed within the context of large complex system problems that present interesting challenges to the FSV method due to the potentially wide dynamic range of the data. Certain use cases for scattering cross section, system-level coupling, and large system-level EMC problems require a somewhat modified approach in computing the GDM based on how the FDM and ADM are weighted. For the current 1-D FSV, unweighted or incorrectly weighted amplitude and feature measures can potentially lead to inconclusive or even misleading results. These issues are addressed and future revisions to the IEEE Standard 1597.1 are highlighted.

V-cycle and non-standard W-cycle versions of a multiplicative Schwarz preconditioner based on hierarchical basis functions are presented. It is used together with a Krylov subspace method to efficiently solve the linear system that results from higher-order finite element (FE) discretisations of time-harmonic Maxwell equations. A recently developed hierarchical basis for H(curl)-conforming (vector-valued and tangentially continuous) FE spaces on tetrahedral meshes is also briefly presented. On this basis, a certain amount of orthogonality between basis functions of different orders is obtained through the requirement that the Nédélec interpolation of higher-order basis functions vanishes in lower-order FE spaces. Numerical experiments are used to show the good performance of the presented schemes. In these experiments, the performance obtained with the presented basis is compared with the performance obtained by several other hierarchical bases found in the literature. For third-order elements, it is observed that most bases give very similar performance and that the V-cycle preconditioner typically requires about 30% more computing time than the W-cycle one. For fourth-order elements, the new basis combined with the non-standard W-cycle preconditioner leads to the best performance. The computing times for the other combinations are about 40% longer, at best.

Inhomogeneous waveguiding structures containing corners, even dielectric corners, cannot be modeled efficiently with finite-element procedures based on conventional polynomial basis functions. Recently proposed singular vector basis functions for triangular cells are used in a formulation that combines transverse and longitudinal components for dielectric waveguide analysis. The additive hierarchical basis functions combine singular bases with a full set of existing hierarchical polynomial basis functions to form the representation. The improvement in the satisfaction of boundary conditions resulting from the use of the bases is illustrated.

First-order system least squares (FOSLS) is a methodology that offers an alternative to standard methods for solving partial differential equations. This paper studies the first-order system least-squares approach for scalar second-order elliptic boundary value problems with discontinuous coefficients. In a companion paper [M. Berndt, T. A. Manteuffel, S. F. McCormick, and G. Starke, Analysis of first-order system least squares ({FOSLS}) for elliptic problems with discontinuous coefficients: Part I, SIAM J. Numer. Anal., 43 (2005), pp. 386--408], ellipticity of an appropriately scaled least-squares bilinear form is established in a natural norm. For some geometries this ellipticity is independent of the size of the jumps in the coefficients. The occurrence of singularities at interface corners, cross-points, reentrant corners, and irregular boundary points is discussed, and a basis of singular functions with local support around singular points is established. This paper describes a method for including discrete versions of the singular basis functions together with standard finite element spaces in a least-squares format at little additional computational cost. The singular basis functions are constructed to match the jump conditions that arise at interfaces between regions of continuity of the diffusion coefficient. Because these basis functions must be approximated in practice, the resulting discretization is by nature nonconforming. This necessitates the establishment here of a general error estimate for FOSLS L2 minimization problems discretized by nonconforming finite elements. An advantage of the FOSLS formulation is that this estimate does not involve the consistency error term usually present in bounds for other methods. Based on this general estimate, error bounds are derived for the finite element space that includes singular basis functions. Numerical tests are included that confirm these discretization error bounds. Finally, a multilevel method is developed for solving the discrete system that uses singular basis functions on all levels, and its efficiency is demonstrated by the numerical results.

We present a multidisciplinary scientific analysis combining aerodynamics, flight mechanics and electromagnetics aiming at the design of a new marine distress signaling device. We show how computational fluid dynamics (CFD) and computational electromagnetics (CEM) techniques have been used to assist in the design of both the flight characteristics and the radar performance of the device, as well as how its radar signature compares to popular radar reflectors used on yachts and sailboats.

The Directed Energy Directorate of the Air Force Research Laboratory has the Air Force Responsibility for the development of high power microwave (HPM) weapons. HPM devices tend to fall within the category of ultrawideband or narrowband, each having advantages and difficulties in development and application. The Center for Plasma Theory and Computation has developed a suite of scientific software to aid in the development of the components of such devices. A typical narrowband high power microwave device is made up of 3 components: a source of pulsed power which releases stored energy in the form of a fast (~nsec to msec) applied voltage, a beam/cavity interaction region in which the kinetic energy of a beam is transformed to microwave radiation, and an antenna which is used to direct the microwave radiation. These components may be categorized by their density of charged particles; the pulsed power devices often involve high-density plasmas, the beam/cavity sources have a low density of charged particles, and it is desired that the antennas have no charged particles. These regimes of charged particle density are most efficiently simulated with magnetohydrodynamic (MHD), particle-in-cell (PIC), and computational electromagnetic (CEM) techniques, respectively.

: The low-grazing-angle electromagnetic backscattering from small breaking water waves has been analyzed using computational electromagnetics (CEM) techniques. A new approach based on the multilevel fast multipole algorithm was implemented for 3-D surface crest scattering to supplement existing 2-D techniques. Multipath scattering between the wave crest and the front face can be modeled using physical optics on the face. A new method to model the specular reflection from jetting crests, termed extended gemetrical optics (EGO), was developed. The numerical calculations were shown to respond to the same measured surface features as experimental results, although polarization ratio differences demonstrate the importance or multipath scattering.

Singular divergence-conforming bases have been proposed for the solution of integral equations although they have seen only occasional use in practical applications. The existing singular bases are not hierarchical, which prevents their use in adaptive p-refinement applications. In this paper, a new family of singular hierarchical basis functions is proposed for quadrilateral cells. These functions model the singularities associated with current and charge density at edges and are more convenient for modeling such singularities than triangular bases of the same kind. The basis functions are of the additive kind and combine a hierarchical polynomial representation on quadrilaterals with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest. Moreover, the added singular basis functions are computed on the fly. On the basis of various reported numerical results, this paper also illustrates the difficulties, the advantages, the accuracy, and the cost of using such bases in the method of moment solutions of integral equations.

In this paper, phase-extracted (MPE) basis functions with both the traveling wave terms and the standing wave terms have been proposed to analyze the scattering from the electrically large targets. This kind of basis functions, called modified phase-extracted (MPE) basis function, can be defined on the relatively large patches and constructed with the higher order hierarchical vector basis functions. The constructing idea of the MPE basis function is first introduced. Then, their characteristics were compared with the relevant basis functions. It has been demonstrated that the MPE basis function is suitable for the scattering analysis of the perfect electric conductors (PECs) with smooth convex surfaces as well as the concave structures, such as the electrically large cavities. The numerical solutions to the electromagnetic scattering from both the convex targets and the cavity-like targets with strong mutual couplings have been given to show the capability of the MPE basis functions in the description of the complex distribution of the induced current on the PEC surfaces. Some numerical examples have been given to demonstrate the validation of this kind of basis functions.

High order scalar and vector basis functions have been developed that permit accurate modeling of singularities in electromagnetic fields at geometric edges, corners, and tips. At the Conference we summarize what philosophies have been used in their development and the results obtained so far, and will identify future work needed in this area. A bibliography that can help the interested readers to deepen this subject is provided.