Large-Scale Structures in Acoustics and Electromagnetics Proceedings of a Symposium (1996) / Chapter Skim
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6 ADAPTIVE FINITE ELEMENT METHODS FOR THE HELMHOLTZ EQUATION IN EXTERIOR DOMAINS
6 ADAPTIVE FINITE ELEMENT METHODS FOR THE HELMHOLTZ EQUATION IN EXTERIOR DOMAINS
Pages 122-142
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From page 122... ...
An hpadaptive strategy, in which the element size h and polynomial order p can be refined simultaneously, is also presented. Through twodimensional simulation of nonuniform radiation from a rigid infinite circular cylinder, it is shown (using in-refinement only)
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have compared the costs of boundary element formulations to the costs of finite element formulations using the fillly coupled DtN boundary condition, and have found that finite element methods are competitive. Burnett (1995)
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In doing so, the adaptive strategy attempts to compute a distribution of degrees of freedom such that the error is equidistributed among the elements of the adaptive mesh. Each of the component technologies-mesh generation, finite element methods, a posterior)
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PROBLEM STATEMENT AND ADAPTIVE SOLUTION METHODOLOGIES In this section we present the problem statement (in the first part) and briefly describe the technologies used to perform the adaptive computation mesh generation, finite element methodologies, a posteriors error estimation, and the tap-adaptive strategy (in the second part)
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The DtN boundary condition is given by (6.4) ' where M is the DtN map and ;?
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This allows us to use the same mesh generator for the initial mesh and the adaptive meshes. All meshes in this paper were generated using a 2D advancing front mesh generator written by Jaime Peraire (Peraire et al., 1987)
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Later we compare GES and Galerkin results. A Posteriori Error Estimation The a posteriors error estimator was derived by Stewart and Hughes (1995)
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h (6.21) where the subscript K refers to quantities in the old mesh, egos is a user-defined element error tolerance for the subsequent adaptive mesh solution, and ,B = 2 for 2D and ,B = 2.5 for 3D .
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2. The tap-adaptive strategy allows for either increasing or decreasing h, but p can only be increased (or held constant)
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The method used to refine the mesh dictates what form this distribution must take, and significantly influences the achievability of the requested adaptive mesh. As given in Box 6.1, the adaptive strategy defines {hneW J Pnew } at discrete locations XK, which may be taken as the centroids of elements K in the old mesh.
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From page 132... ...
The values are stored at the nodes of the old mesh, which in turn provides the necessary input for the mesh generator to build the new, adaptive mesh. NONUNIFORM RADIATION FROM A RIGID INFINITE CIRCULAR CYLINDER In this section we compute nonuniform radiation from a rigid infinite circular cylinder for a wave number of ha = 2~, where a is the radius of the cylinder.
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Alongside the full mesh is an enlargement of the region near the upper boundary condition discontinuity. The elements in the uniform mesh are extremely large in this region compared to the adaptive mesh, leading to a relatively poor resolution of the discontinuity.
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The slopes of these lines, indicated in the figure, show that the convergence rate of adaptive refinement is roughly the same as the convergence rate of uniform refinement. It is easily seen that the uniform mesh requires a much larger number of elements to achieve a given error tolerance, compared to the required adaptive mesh size.
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The definition of h becomes confusing, however, in adaptive meshes where h can vary considerably throughout the mesh and approach O at different rates. For this reason it is more convenient to present convergence studies in terms of ne, .
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Contours of Re(¢h) on a coarse uniform mesh are shown in Figure 6.6, comparing Galerkin and GLS formulations.
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/ ~ ~) ,~ I'- ~.~ Figure 6.6 Nonuniform radiation from an infinite circular cylinder, ka = 27~: comparison of Re(o h )
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.1~ .01 i, I Gal uniform - ^Gal adaptive - zGLS/uniform GLS1adaptive 'I 1 `~11 1 100 1000 10000 net Figure 6.8 Nonuniform radiation from an infinite circular cylinder, ka = 2~: (scaled) estimated error vs.
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, it is clear that the estimated GLS error will be larger than the estimated Galerkin error since Ci Cs + CGLS > CiCs The overprediction of the GLS error estimator is actually greater upon noting, as discussed above, that the exact GLS error is lower than the exact Galerkin error. A common measure of the absolute accuracy of elTor estimators is the global effectivity index, 0, given by ||e~t || lle~xactll (6.31)
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Hughes, 1990, "Design and analysis of finite element methods for the Helmholtz equation in exterior domains," Appl.
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Hughes, 1995, "An a posterior) error estimator and tap-adaptive strategy for finite element discretizations of the Helmholtz equation in exterior domains," to appear in Finite Elements Anal.
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