The application of multiresolution analysis to Maxwell's equations results in new multiresolution time-domain (MRTD) schemes with unparalleled inherent properties. In particular, the approach allows the development of MRTD schemes which are based on scaling functions only or on a combination of scaling functions and wavelets leading to a variable mesh grading. The dispersion of the MRTD schemes compared to the conventional Yee finite-difference time-domain (FDTD) scheme shows an excellent capability to approximate the exact solution with negligible error for sampling rates approaching the Nyquist limit. Simple microwave structures including dielectric materials are analyzed in order to illustrate the application of the MRTD schemes and to demonstrate the advantages over Yee's FDTD scheme with respect to memory requirements and execution time.

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MRTD: new time-domain schemes based on multiresolution analysis

Various time-domain finite-element methods for the simulation of transient electromagnetic wave phenomena are discussed. Detailed descriptions of test/trial spaces, explicit and implicit formulations, nodal and edge/facet element basis functions are given, along with the numerical stability properties of the different methods. The advantages and disadvantages of mass lumping are examined. Finally, the various formulations are compared on the basis of their numerical dispersion performance.

Time-domain finite-element methods

Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements.

Fast solution methods in electromagnetics

The various types of edge elements and their useful properties are reviewed. Continuity, singularities, and separation are discussed, with particular reference to tetrahedral elements. The use of triangular and quadrilateral edge elements for 2-D problems is briefly considered. >

Edge elements and what they can do for you

https://ntrs.nasa.gov/api/citations/19950021305/downloads/19950021305.pdf

The Origin of Spurious Solutions in Computational Electromagnetics

The vector parasite problem is studied in the context of finite-element solutions of Maxwell's equations for driven boundary-value problems. An expanded weak form which combines the divergence equation with the conventional weak form of the double-curl equation is introduced. This form is related to penalty methods where the penalty or weighting factor varies with the dielectric constant. The resulting algebraic system is identical to the Galerkin-Helmholtz operator on homogeneous subregions. Normal and tangential boundary conditions arise in terms of the divergence and curl of the field on the boundary which can be reexpressed as equivalent charges and currents. Computational results show the occurrence of two distinct types of parasitic modes in driven problems and their elimination with the formulation presented. Practical observations concerning the conditions which provoke spurious modes in these problems are reported. Spurious solutions arise from improper or unphysical boundary conditions, and the importance of careful specification of boundary-value problems is illustrated. Most conceptual difficulties with boundary conditions per se are removed when hybrid methods are used to couple the interior finite-element solution to the exterior problem. which focuses attention on the physics of the source distribution. >

Elimination of vector parasites in finite element Maxwell solutions

Dispersion relations are derived for conventional finite-element (FE) and finite-difference (FD) approximations for four versions of the Maxwell equations in the plane: the double-curl equation; the vector Helmholtz equation; the penalty equation; and the primitive, coupled Maxwell curl equations. Comparison with their analytic counterparts reveals the presence and origin of vector parasites. For the double-curl and penalty methods, the dispersion relations are double-valued, admitting an extra, spurious dispersion surface of real-valued wavenumbers. As a result, low wavenumbers support well-resolved and poorly resolved vector parasites. The Helmholtz schemes have monotonic, single-valued dispersion relations for divergence-free physical modes. Specification of divergence-free boundary conditions is sufficient to guarantee the absence of parasites. The primitive schemes have single-valued but folded (nonmonotonic) dispersion relations, supporting poorly resolved vector parasites at low wavenumbers. Use of a staggered finite-difference grid eliminates these parasites and results in a dispersion relation identical to that for the Helmholtz scheme. In cases where vector parasites arise, the same essential weakness in the discretized form of either the first or cross-derivative is responsible. >

Origin of vector parasites in numerical Maxwell solutions

The finite-element method with conventional scalar bases is coupled with the moment method to handle the three-dimensional scattering and/or absorption from inhomogeneous, arbitrarily shaped objects. The C/sup 0/ finite-element basis enforces continuity of both normal and tangential E at element boundaries within homogeneous regions. At dielectric interfaces, the continuity of normal D and tangential E are enforced in a strong sense. Excellent agreement between the numerical solution and the Mie series is obtained for both internal and scattered fields for homogeneous and layered spheres under plane wave illumination. Compared to an alternative finite-element method using edge elements which lack strong enforcement of normal field continuity, the present method produces higher-order approximations, especially at dielectric interfaces, with no penalties in computational effort. >

D.R. Lynch

Papers

Elimination of vector parasites in finite element Maxwell solutions

Origin of vector parasites in numerical Maxwell solutions

Importance of normal field continuity in inhomogeneous scattering calculations