Hierarchical Vector Bases for Pyramid Cells

This presentation summarizes a very simple and straightforward new procedure to build hierarchical vector bases for the pyramid that conform to those used on adjacent differently shaped cells (tetrahedra, hexahedron and triangular prisms). Our new curl- and divergence-conforming bases, together with the corresponding curls and divergences, have simple and easy to implement mathematical expressions. Results confirming faster convergence and avoidance of spurious modes/solutions will be reported at the conference

Finite element differential forms on curvilinear cubic meshes and their approximation properties

We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence.Comment: 17 pages, 1 figure; v2: changes in response to referee reports; v3: minor additional changes, this version accepted for Numerische Mathematik; v3: very minor updates, this version corresponds to the final published versio

Adaptive Semi-Structured Mesh Refinement Techniques for the Finite Element Method

The adaptive mesh techniques applied to the Finite Element Method have continuously been an active research line. However, these techniques are usually applied to tetrahedra. Here, we use the triangular prismatic element as the discretization shape for a Finite Element Method code with adaptivity. The adaptive process consists of three steps: error estimation, marking, and refinement. We adapt techniques already applied for other shapes to the triangular prisms, showing the differences here in detail. We use five different marking strategies, comparing the results obtained with different parameters. We adapt these strategies to a conformation process necessary to avoid hanging nodes in the resulting mesh. We have also applied two special rules to ensure the quality of the refined mesh. We show the effect of these rules with the Method of Manufactured Solutions and numerical results to validate the implementation introduced.This work has been financially supported by TEC2016-80386-

On a general implementation of hh- and pp-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., $p$-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., $h$-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation

High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems

Multi-dimensional modeling and simulation of semiconductor nanophotonic devices

Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonic devices is an important tool in the development of future integrated light sources and quantum devices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically explore their optimization potential. The efficient implementation of multi-dimensional numerical simulations for computer-aided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in device-scale modeling of quantum dot based single-photon sources and laser diodes by self-consistently coupling the optical Maxwell equations with semiclassical carrier transport models using semi-classical and fully quantum mechanical descriptions of the optically active region, respectively. For the simulation of realistic devices with complex, multi-dimensional geometries, we have developed a novel hp-adaptive finite element approach for the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties of the photonic structures. For electrically driven devices, we introduced novel discretization and parameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on various applications, including vertical-cavity surface-emitting lasers, grating couplers and single-photon sources

3 Dimensional Electromagnetic Analysis of an Axial Active Magnetic Bearing

In the rotating electrical machines, active magnetic bearing are basically performing the same role like mechanical bearings to support rotor. The function is based on the principle of magnetic levitation. The idea behind this involves creation of a magnetic field by supplying controlled currents in the bearing coil through amplifiers and complex power electronics. The accurate design of a magnetic bearing system incorporates many parameters before its implementation. The current work of the thesis encircles only the three dimensional (3D) modeling of axial active magnetic bearing (AMB). The static and dynamic models are analyzed for the bearing with a consideration of nonlinear material. In the study, the major emphasis is on the magnetic field, eddy current behavior and exerted magnetic forces in the magnetic bearing. The required input parameters for simulation are considered from the available two dimensional (2D) analysis for the same axial actuator. Elmer open source finite element tool is used in the entire work for making 3D simulations. Finally, the computed results are compared with the 2D case. As a part of the thesis work, a modified geometry is simulated to analyze eddy currents. The hypothesis in later task is the reduction of eddy current losses by providing a radial cut in the bearing ferromagnetic path. The radial cut brings asymmetry in the bearing and the three dimensional analysis provides the possibility to analyze the complete model. The results obtained in the above work provide a good understanding of 3D fields in axial AMB and the computed magnetic forces are in good agreement with the 2D results