1,909 research outputs found
On a general implementation of - and -adaptive curl-conforming finite elements
Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used
by the computational electromagnetics community. However, its implementation,
specially for high order methods, is not trivial, since it involves many
technicalities that are not properly described in the literature. To fill this
gap, we provide a comprehensive description of a general implementation of edge
elements of first kind within the scientific software project FEMPAR. We cover
into detail how to implement arbitrary order (i.e., -adaptive) elements on
hexahedral and tetrahedral meshes. First, we set the three classical
ingredients of the finite element definition by Ciarlet, both in the reference
and the physical space: cell topologies, polynomial spaces and moments. With
these ingredients, shape functions are automatically implemented by defining a
judiciously chosen polynomial pre-basis that spans the local finite element
space combined with a change of basis to automatically obtain a canonical basis
with respect to the moments at hand. Next, we discuss global finite element
spaces putting emphasis on the construction of global shape functions through
oriented meshes, appropriate geometrical mappings, and equivalence classes of
moments, in order to preserve the inter-element continuity of tangential
components of the magnetic field. Finally, we extend the proposed methodology
to generate global curl-conforming spaces on non-conforming hierarchically
refined (i.e., -adaptive) meshes with arbitrary order finite elements.
Numerical results include experimental convergence rates to test the proposed
implementation
The adaptive computation of far-field patterns by a posteriori error estimations of linear functionals
This paper is concerned with the derivation of a priori and a posteriori error bounds for a class of linear functionals arising in electromagnetics which represent the far-field pattern of the scattered electromagnetic field. The a posteriori error bound is implemented into an adaptive finite element algorithm, and a series of numerical experiments is presented
Parallel 3-D marine controlled-source electromagnetic modelling using high-order tetrahedral Nédélec elements
We present a parallel and high-order Nédélec finite element solution for the marine controlled-source electromagnetic (CSEM) forward problem in 3-D media with isotropic conductivity. Our parallel Python code is implemented on unstructured tetrahedral meshes, which support multiple-scale structures and bathymetry for general marine 3-D CSEM modelling applications. Based on a primary/secondary field approach, we solve the diffusive form of Maxwell’s equations in the low-frequency domain. We investigate the accuracy and performance advantages of our new high-order algorithm against a low-order implementation proposed in our previous work. The numerical precision of our high-order method has been successfully verified by comparisons against previously published results that are relevant in terms of scale and geological properties. A convergence study confirms that high-order polynomials offer a better trade-off between accuracy and computation time. However, the optimum choice of the polynomial order depends on both the input model and the required accuracy as revealed by our tests. Also, we extend our adaptive-meshing strategy to high-order tetrahedral elements. Using adapted meshes to both physical parameters and high-order schemes, we are able to achieve a significant reduction in computational cost without sacrificing accuracy in the modelling. Furthermore, we demonstrate the excellent performance and quasi-linear scaling of our implementation in a state-of-the-art high-performance computing architecture.This project has received funding from the European Union's Horizon 2020 programme under the Marie Sklodowska-Curie grant agreement No. 777778. Furthermore, the research leading to these results has received funding from the European Union's Horizon 2020 programme under the ChEESE Project (https://cheese-coe.eu/ ), grant agreement No. 823844. In addition, the authors would also like to thank the support of the Ministerio de Educación y Ciencia (Spain) under Projects TEC2016-80386-P and TIN2016-80957-P.
The authors would like to thank the Editors-in-Chief and to both reviewers, Dr. Martin Cuma and Dr. Raphael Rochlitz, for their valuable comments and suggestions which helped
to improve the quality of the manuscript.
This work benefited from the valuable suggestions, comments, and proofreading of Dr. Otilio Rojas (BSC). Last but not least, Octavio Castillo-Reyes thanks Natalia Gutierrez (BSC) for her support in CSEM modeling with BSIT.Peer ReviewedPostprint (author's final draft
Implicit a posteriori error estimates for the Maxwell equations
An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases. \u
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