A Multiresolution Approach to the Electric Field Integral Equation in Antenna Problems

Andriulli, Francesco P.; Tabacco, Anita; Vecchi, G.

doi:10.1137/050634943

Cited by 40 publications

(25 citation statements)

References 27 publications

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“…Figure 2 shows the number of iterations necessary for solving the EFIE linear system as a function of the fast solver approximation error. All the linear system matrices have been preconditioned with the same quasi-Helmholtz decomposition technique (in this case we have used the Hierarchical scheme in [5], but similar results are obtained with other techniques including loop-tree/star and Calderón). It is clear that, by using a standard fast method, the preconditioning becomes ineffective when the fast solver precision decreases (and the compression efficiency increases).…”

Section: Numerical Resultsmentioning

confidence: 79%

A new fast and rapidly converging method for the solution of the electric field integral equation

Andriulli¹,

Vecchi²

2009

2009 International Conference on Electromagnetics in Advanced Applications

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This paper presents a new fast matrixvector multiplication scheme for the solution of the Electric Field Integral Equation.Similarly to other fast methods, our approach reduces the matrixvector multiplication cost from O(N 2 ) to O(N log N ). Differently from other fast solvers, however, the effectiveness of EFIE preconditioning techniques such as quasi-Helmholtz decompositions or Calderón approaches is maintained by our method even for very high matrix compression rates. This is thanks to the fact that, in the scheme we are proposing, the contribution from the scalar potential when applied to or tested with solenoidal functions is always zero independent of the compression error. In addition, the new method will take advantage of the redundancies of the EFIE matrix in the low-frequency/dense discretization regime, and it will further decrease both the memory storage and the multiplication cost with respect to currently available fast solvers. Numerical results will show the effectiveness of our approach and its impact on the solution of realistic problems.

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Section: Numerical Resultsmentioning

confidence: 79%

A new fast and rapidly converging method for the solution of the electric field integral equation

Andriulli¹,

Vecchi²

2009

2009 International Conference on Electromagnetics in Advanced Applications

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“…Table 1 lists the condition numbers without and with multilevel preconditioning for different system sizes. 3 It clearly conveys the efficacy of the preconditioner, though the condition numbers slightly increase with L, an effect, also observed in the case of multilevel preconditioners for discretized elliptic PDEs, see [8,Sect. 5,Rem.…”

Section: Implementation and Numerical Testmentioning

confidence: 96%

“…Hitherto no multilevel stability theory has been developed for surface edge element spaces and only a few ideas in the direction of multilevel preconditioning have been floated [3,4]. It is the goal of this paper to fill the gap and show the uniform stability of multilevel splitting of edge BEM subspaces of H − 1 2 (div Γ , Γ ) on hierarchies of nested triangular surface meshes created by regular refinement.…”

Section: Introductionmentioning

confidence: 99%

Stable multilevel splittings of boundary edge element spaces

Hiptmair

Mao

2012

Bit Numer Math

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We establish the stability of nodal multilevel decompositions of lowestorder conforming boundary element subspaces of the trace space Hon boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by uniform refinement and the stability bounds are uniform in the number of refinement levels.The main tool is the general theory of P. Oswald (Interface preconditioners and multilevel extension operators, in Proc. 11th Intern. Conf. on Domain Decomposition Methods, London, 1998, pp. 96-103) that teaches, when stability of decompositions of boundary element spaces with respect to trace norms can be inferred from corresponding stability results for finite element spaces. H (curl, Ω)-stable discrete extension operators are instrumental in this.Stable multilevel decompositions immediately spawn subspace correction preconditioners whose performance will not degrade on very fine surface meshes. Thus, the results of this article demonstrate how to construct optimal iterative solvers for the linear systems of equations arising from the Galerkin edge element discretization of boundary integral equations for eddy current problems.

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“…The basic idea in building the gRWG functions is to extend to non-simplex cells the generation process for multi-level RWG functions, described in [11,16,17] for the case of a hierarchic mesh (that has only triangular cells).…”

Section: Generalized Rwgmentioning

confidence: 99%

Two-tier non-simplex grid hierarchic basis for general 3D meshes

Vipiana

Andriulli

Vecchi

2009

Waves in Random and Complex Media

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This paper deals with hierarchical bases for the preconditioning of the Electric Field Integral Equation on arbitrary 3D meshes with triangular cells. The basis functions are constructed on subsequent groupings of the initial triangular cells, which give rise to a nested family of meshes with non-simplex cells. The previously published hierarchic set of basis functions acts as a preconditioner for dense meshes, but it may lose effectiveness when the domain of the most extended basis functions is comparable with the wavelength, thus limiting the application range of the method. The present work undertakes to overcome this limitation by introducing two tiers of basis functions, related to the hierarchic cell groupings. By suitably modifying the grouping algorithm a coarser level is obtained, on which non-hierarchic basis functions are defined. Hierarchy in the basis function defined on detail meshes is preserved instead. Numerical results show the effectiveness for structures with electrical sizes beyond the resonance region.

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A Multiresolution Approach to the Electric Field Integral Equation in Antenna Problems

Cited by 40 publications

References 27 publications

A new fast and rapidly converging method for the solution of the electric field integral equation

A new fast and rapidly converging method for the solution of the electric field integral equation

Stable multilevel splittings of boundary edge element spaces

Two-tier non-simplex grid hierarchic basis for general 3D meshes

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