Sparsity optimized high order finite element functions for H(curl) on tetrahedra

AbstractH(curl) conforming finite element discretizations are a powerful tool for the numerical solution of the system of Maxwellʼs equations in electrodynamics. In this paper we construct a basis for conforming high-order finite element discretizations of the function space H(curl) in 3 dimensions. We introduce a set of hierarchic basis functions on tetrahedra with the property that both the L2-inner product and the H(curl)-inner product are sparse with respect to the polynomial degree. The construction relies on a tensor-product based structure with properly weighted Jacobi polynomials as well as an explicit splitting of the basis functions into gradient and non-gradient functions. The basis functions yield a sparse system matrix with O(1) nonzero entries per row.The proof of the sparsity result on general tetrahedra defined in terms of their barycentric coordinates is carried out by an algorithm that we implemented in Mathematica. A rewriting procedure is used to explicitly evaluate the inner products. The precomputed matrix entries in this general form for the cell-based basis functions are available online

Casus

Ecvet Şeci'nin Saadet'te tefrika edilen Casus adlı romanıTefrikanın devamına rastlanmamış, tefrika yarım kalmıştır

Resolving the sign conflict problem for hp–hexahedral Nédélec elements with application to eddy current problems

The eddy current approximation of Maxwell’s equations is relevant for Magnetic Induction Tomography (MIT), which is a practical system for the detection of conducting inclusions from measurements of mutual inductance with both industrial and clinical applications. An MIT system produces a conductivity image from the measured fields by solving an inverse problem computationally. This is typically an iterative process, which requires the forward solution of a Maxwell’s equations for the electromagnetic fields in and around conducting bodies at each iteration. As the (conductivity) images are typically described by voxels, a hexahedral finite element grid is preferable for the forward solver. Low order Nédélec (edge element) discretisations are generally applied, but these require dense meshes to ensure that skin effects are properly captured. On the other hand, hp–Nédélec finite elements can ensure the skin effects in conducting components are accurately captured, without the need for dense meshes and, therefore, offer possible advantages for MIT. Unfortunately, the hierarchic nature of hp–Nédélec basis functions introduces edge and face parameterisations leading to sign conflict issues when enforcing tangential continuity between elements. This work describes a procedure for addressing this issue on general conforming hexahedral meshes and an implementation of a hierarchic hp–Nédélec finite element basis within the deal.II finite element library. The resulting software is used to simulate Maxwell forward problems, including those set on multiply connected domains, to demonstrate its potential as an MIT forward solver

Технология и техника сооружения поисково-оценочных скважин на участке "Водораздельный" Алгаинско-Березовского золоторудного проявления (Кузбасс)

Целью работы является геологическое изучение объекта, разработка технологии и техники сооружения скважины. Объектом исследования являются поисковые скважины на Алгаинско-Березовском золоторудном проявлении. Специальная часть заключается в анализе критериев оптимизации параметров бурения.The purpose of the work is geological study of the facility, development of technology and technology of the well construction. The object of the study are exploratory wells at the Algaino-Berezovsky gold ore occurrence. A special part is to analyze the criteria for optimizing drilling parameters

Accelerating magnetic induction tomography‐based imaging through heterogeneous parallel computing

Magnetic Induction Tomography (MIT) is a non‐invasive imaging technique, which has applications in both industrial and clinical settings. In essence, it is capable of reconstructing the electromagnetic parameters of an object from measurements made on its surface. With the exploitation of parallelism, it is possible to achieve high quality inexpensive MIT images for biomedical applications on clinically relevant time scales. In this paper we investigate the performance of different parallel implementations of the forward eddy current problem, which is the main computational component of the inverse problem through which measured voltages are converted into images. We show that a heterogeneous parallel method that exploits multiple CPUs and GPUs can provide a high level of parallel scaling, leading to considerably improved runtimes. We also show how multiple GPUs can be used in conjunction with deal.II, a widely‐used open source finite element library

High Order Nédélec Elements with local complete sequence properties

The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order H(curl) finite elements. We discuss a systematic strategy for the realization of arbitrary order hierarchic H(curl)conforming finite elements for triangular and tetrahedral element geometries. The shape functions are classified as lowestorder Nédélec, higher-order edge-based, face-based (only in 3D) and element-based ones. Our new shape functions provide not only the global complete sequence property, but also local complete sequence properties for each edge-, face-, element-block. This local property allows an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second advantage of this construction is that simple block-diagonal preconditioning gets efficient. Our high order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic boundary value problem, the gradient basis functions can be skipped, which reduces the problem size, and improves the condition number. We successfully apply the new high order elements for a 3D magnetostatic boundary value problem, and a Maxwell eigenvalue problem showing severe edge and corner singularities

High Order Finite Element Methods for . . .

This thesis deals with the higher-order Finite Element Method (FEM) for computational electromagnetics. The hp-version of FEM combines local mesh refinement (h) and local increase of the polynomial order of the approximation space (p). A key tool in the design and the analysis of numerical methods for electromagnetic problems is the de Rham Complex relating the function spaces H 1 (Ω), H(curl,Ω), H(div,Ω), and L2(Ω) and their natural differential operators. For instance, the range of the gradient operator on H 1 (Ω) is spanned by the space of irrotional vector fields in H(curl), and the range of the curl-operator on H(curl,Ω) is spanned by the solenoidal vector fields in H(div,Ω). The main contribution of this work is a general, unified construction principle for H(curl)and H(div)-conforming finite elements of variable and arbitrary order for various element topologies suitable for unstructured hybrid meshes. The key point is to respect the de Rham Complex already in the construction of the finite element basis functions and not, as usual, only for the definition of the local FE-space. A short outline of the construction is as follows. The gradient fields of higher-order H 1-conforming shape functions are H(curl)-conformin