Higher Order Vector Finite Elements Research Articles

HPC Geophysical Electromagnetics: A Synthetic VTI Model with Complex Bathymetry

We introduce a new synthetic marine model for 3D controlled-source electromagnetic method (CSEM) surveys. The proposed model includes relevant features for the electromagnetic geophysical community such as large conductivity contrast with vertical transverse isotropy and a complex bathymetry profile. In this paper, we present the experimental setup and several 3D CSEM simulations in the presence of a resistivity unit denoting a hydrocarbon reservoir. We employ a parallel and high-order vector finite element routine to perform the CSEM simulations. By using tailored meshes, several scenarios are simulated to assess the influence of the reservoir unit presence on the electromagnetic responses. Our numerical assessment confirms that resistivity unit strongly influences the amplitude and phase of the electromagnetic measurements. We investigate the code performance for the solution of fundamental frequencies on high-performance computing architectures. Here, excellent performance ratios are obtained. Our benchmark model and its modeling results are developed under an open-source scheme that promotes easy access to data and reproducible solutions.

Land CSEM Simulations and Experimental Test Using Metallic Casing in a Geothermal Exploration Context: Vallès Basin (NE Spain) Case Study

Controlled-source electromagnetic (CSEM) measurements are complementary data for magnetotelluric (MT) characterization although its methodology on land is not sufficiently developed and tested as in marine environments. Acquiring expertise in CSEM is crucial for surveys in places where MT cannot be performed due to high levels of cultural noise. To acquire that expertise, we perform CSEM experiments in the Vallès fault [Northeast (NE), Spain], where MT results have been satisfactory and allow us to verify the CSEM results. The Vallès basin is relevant for potential heat generation because of the presence of several geothermal anomalies and its nearby location in urban areas. In this article, we present the experimental setup for that region, a 2-D joint MT+CSEM inverse model, several 3-D CSEM simulations in the presence of metallic casing, and its comparison with real data measurements. We employ a parallel and high-order vector finite element algorithm to discretize the governing equations. By using an adapted meshing strategy, different scenarios are simulated to study the influence of the source position/direction and the conductivity model in a metallic casing presence. An excellent agreement between the simulated data and analytical/real field data demonstrates the feasibility of study metallic structures in realistic configurations. Our numerical results confirm that metallic casing strongly influences electromagnetic (EM) responses, making surface measurements more sensitive to resistivity variations near the metallic structure. It could be beneficial getting higher signal-to-noise ratios and sensitivity to deep targets. However, such a casing effect depends on the input model (e.g., conductivity contrasts, frequency, and geometry).

Full Wave Analysis of RF Signal Attenuation in a Lossy Rough Surface Cave Using a High Order Time Domain Vector Finite Element Method

We present a computational study of signal propagation and attenuation of a 200 MHz planar loop antenna in a cave environment. The cave is modeled as a cylindrical guiding structure with a lossy wall. The wall is nominally circular with a random roughness. To simulate a broad frequency band, the full wave Maxwell equations are solved directly in the time domain via a high order vector finite element discretization using the massively parallel CEM code EMSolve. The numerical technique is first verified against theoretical results for a planar loop antenna in a smooth lossy cave. The simulation is then performed for a series of random rough surface meshes in order to generate statistical data for the propagation and attenuation properties of the antenna in a cave environment. Results for the mean and variance of the power spectral density of the electric field are presented and discussed.

Modelling of Industrial Conveyorized Applicators Using Higher Order Vector Finite Elements

This paper presents a finite element time domain method for the solution of Maxwell’s equations in microwave heating applicators using first and second order vector finite elements. Results are compared with experimental data and it has been shown that second order vector finite elements have many advantages over first order elements. Capitalising on the high accuracy and low computational cost attainable by higher order elements, an industrial conveyor belt system is numerically analyzed.

A mixed high-order vector finite element method for waveguides: convergence and spurious mode studies

In this article we construct and analyse the performance of various combinations of high order edge elements for the transverse field component with high order node elements for the longitudinal field component. The overall efficiency and mode resolution improve dramatically with high order elements. For example, quadratic/cubic elements provide accuracy of 10−4 in eigenvalue 1000 times faster and resolve the discontinuities in the field very accurately. In the mixed FEM for waveguides we observed that edge elements may produce spurious modes. When the order of the edge elements is equal or larger than the order of the node elements, spurious and degenerate modes are present. Any other combination of the orders did not produce spurious modes. We study the spurious modes from the divergence point of view. We also study the convergence properties of high order elements and compare the results with the theory. Copyright © 2005 John Wiley & Sons, Ltd.

Convergence-optimized, higher order vector finite elements for microwave simulations

We introduce a general class of higher order parameter-dependent Whitney elements, unlike previous approaches that resulted in specific element definitions. All elements of this kind provide the same solution, but their convergence properties may be significantly different. The most essential fact, though, is the introduction of an optimization procedure, which reveals the existence of an optimal, with respect to convergence, element. The produced second order elements are tested in both two-dimensional (2-D) and three-dimensional (3-D) microwave simulations.

An application of higher order vector finite elements to the frequency-domain analysis of singly and doubly periodic geometries containing fine features

It is known that the finite-element method (FEM) may give rise to inaccurate results when lower order elements are applied to analyze problem geometries that contain very fine features such as narrow conducting strips or thin slots. In this letter, higher order vector finite elements are utilized to significantly improve the numerical accuracy in the case of singly and doubly periodic geometries comprising such fine features.

A Generalized Theory of Higher Order Vector Finite Elements and Its Applications in Three-Dimensional Electromagnetic Scattering Problems

ABSTRACT In this paper we present a generalized theory of higher order vector finite elements. This theory aims at generating tctrahedral Whitney elements with tangential continuity, having better orders of approximation compared to the well-known first order edge elements. The proposed methodology deals with some fundamental theoretical issues about Whitney elements, such as the proper definition of degrees of freedom and the correct modeling of the nullspace of the curl operator, corresponding to the class of irrotational fields. The whole set of properties is enforced to the shape functions, in order to produce their explicit expressions. The new second and third order vector finite elements are implemented in a formulation for 3-D electromagnetic scattering. A second order absorbing boundary condition for rectangular boundaries is used to truncate the infinite space. The numerical results show the improved accuracy of the approximation, obtained by the use of higher order Whitney elements, thus rendering them a promising tool for electromagnetic analysis.

A fully explicit Whitney element-time domain scheme with higher order vector finite elements for three-dimensional high frequency problems

In this paper we propose a new explicit Whitney Element-Time-Domain method for the solution of Maxwell's equations in three dimensions. This scheme could be considered a generalization of the well known Finite Difference Time Domain (FDTD) method for unstructured meshes. However, an entirely different methodology is introduced, based on the use of higher order tangential and normal vector finite elements. The exploitation of higher order approximations in an elaborate leapfrog scheme permits the use of the primary mesh only, thus avoiding the implementation of dual mesh. The new method is, finally, applied to a typical high frequency electromagnetic field formulation.

Development and implementation of second and third order vector finite elements in various 3-D electromagnetic field problems

A generalized theory of higher order vector finite elements is presented. The methodology, that is introduced, is used for the construction of new second and third order generalizations of the concept of tetrahedral edge elements. Basic theoretical issues about their construction, such as the proper choice and placement of degrees of freedom and the correct modeling of irrotational fields are discussed. The presented theory and methodology are also extended to the hexahedral element case. Some practical problems and difficulties related to the numerical implementation of higher order vector finite elements are also dealt with. Finally, both second and third order elements are successfully implemented in several electromagnetic field or potential formulations.

Higher-order vector finite elements for tetrahedral cells

Edge-based vector finite elements are widely used for two-dimensional (2-D) and three-dimensional (3-D) electromagnetic modeling. This paper seeks to extend these low-order elements to higher orders to improve the accuracy of numerical solutions. These elements have relaxed normal-component continuity to prohibit spurious modes, and also satisfy Nedelec's constraints to eliminate unnecessary degrees of freedom while remaining entirely local in character. Element matrix derivations are given for the first two vector finite element sets. Also, results of the application of these basis functions to cavity resonators demonstrate the superiority of the higher-order elements.

Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions

A systematic methodology for the construction of three dimensional tetrahedral higher order vector finite elements is presented. Appropriate degrees of freedom describing tangential field components on interfaces and shape functions depending on a set of unknown coefficients are introduced. A decoupling procedure is applied to determine the shape functions. This leads to a whole family of parameter dependent vector finite elements. However, concerning the irrotational fields to be modeled by the finite element, additional relations among the coefficients are introduced, which fully determines them. The whole procedure is described via the introduction of a new second order tetrahedral finite element. Third order tangential vector finite elements are also derived and the method could be easily generalized to higher orders. A numerical implementation of the second order element in a waveguide scattering problem is finally presented.