Keywords: electromagnetic scattering, arbitrary order edge elements, output error bounds, goal orientated.

This paper will describe an arbitrary order edge element approach for the solution of 2D electromagnetic problems. The chosen application area is the simulation of the scattering of electromagnetic waves. The numerical solution is obtained, in the frequency domain, on hybrid meshes of triangles and quadrilaterals using the Galerkin method [1] and the shape functions defined by Ainsworth and Coyle are employed [2]. At the truncated far field boundary, the outgoing scattered wave condition is applied by the use of a perfectly matched layer [3,1].

An a-posteriori error estimation procedure is developed that enables the bounding of computational outputs of direct engineering interest. This procedure is an extension of a method that has been presented recently for the Helmholtz equation [4]. In the simulation of electromagnetic wave scattering, the prediction of the distribution of the scattering width is of prime importance and it is demonstrated how bounds may be placed on this quantity at selected viewing angles [5].

Reduced-order models for computational simulations may be constructed by employing full finite element solutions for a small set of the problem parameters. These models can then be applied to enable the rapid prediction of computational outputs for different values of the parameters [6,7]. Initially, a reduced-order model will be developed to enable error bounds to be produced for the complete scattering width distribution [8]. Then, using data from a limited number of finite element computations involving selected wave incidence angles, the approach is also applied to enable the efficient prediction of the scattering width distribution for different angles of incidence.

The numerical performance of the procedures that are described will be illustrated by analysing problems involving the scattering of plane electromagnetic waves by cylindrical and aerofoil shaped objects.

1

P. D. Ledger, O. Hassan, K. Morgan and N. P. Weatherill, Arbitrary order edge elements for electromagnetic scattering simulation using hybrid meshes and a PML, accepted for publication in International Journal of Numerical Methods in Engineering, 2002. doi:10.1002/nme.501

2

M. Ainsworth and J. Coyle, Hierarchic hp-edge element families for Maxwell's equations on hybrid quadrilateral/triangular meshes, Computer Methods in Applied Mechanics and Engineering, 190, 6709-6733, 2001. doi:10.1016/S0045-7825(01)00259-6

3

J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 114, 185-200, 1994. doi:10.1006/jcph.1994.1159

4

J. Sarrate, J. Peraire and A. T. Patera, A-posteriori error bounds for nonlinear outputs of the Helmholtz equation, International Journal for Numerical Methods in Fluids, 31, 17-36, 1999. doi:10.1002/(SICI)1097-0363(19990915)31:1<17::AID-FLD953>3.0.CO;2-X

5

P. D. Ledger, K. Morgan, J. Peraire, O. Hassan and N. P. Weatherill, Efficient, highly accurate hp-adaptive finite element computations of the scattering width output of Maxwell's equations, submitted to International Journal for Numerical Methods in Fluids, 2002.

6

K. E. Wilcox, J. D. Paduano and J. Peraire, Low order aerodynamic models for aeroelastic control of turbomachines, AIAA Paper 99-1467, 1999.

7

K. E. Wilcox, J. Peraire and J. White, An Arnoldi approach for generation of reduced order models for turbomachinery, Computers and Fluids, 31, 369-389, 1999. doi:10.1016/S0045-7930(01)00046-9

8

P. D. Ledger, K. Morgan, J. Peraire, O. Hassan and N. P. Weatherill, The development of an hp-adaptive finite element procedure for electromagnetic scattering problems, submitted to Finite Elements in Analysis and Design, 2002. doi:10.1016/S0168-874X(03)00057-X

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