Expansion of periodic boundary condition for 3-D FEM analysis using edge elements

The application of periodic boundary conditions in the analysis of three-dimensional magnetic fields by finite element methods (FEMs) leads to a substantial reduction of computation labor and storage. The expansion of the condition for magnetostatic curl-curl formulation with the magnetic vector potential employing edge tetrahedral elements is discussed. Differences between the definitions of the condition for nodal and edge elements are examined. The vectorial nature of edge elements is emphasized and associated difficulties in the formulation and application of the condition are carefully analyzed and overcome. Details for computer implementation are given and a simple test problem to verify the validity of the software is proposed. Advantages gained when the condition is used for TEAM Workshop problem 13 as an example are shown </p

Analysis and simulations of multifrequency induction hardening

We study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwell's equations coupled with an internal energy balance and an ODE for the volume fraction of {\sl austenite}, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable a-priori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system

3D Modeling of the Magnetization of Superconducting Rectangular-Based Bulks and Tape Stacks

In recent years, numerical models have become popular and powerful tools to investigate the electromagnetic behavior of superconductors. One domain where this advances are most necessary is the 3D modeling of the electromagnetic behavior of superconductors. For this purpose, a benchmark problem consisting of superconducting cube subjected to an AC magnetic field perpendicular to one of its faces has been recently defined and successfully solved. In this work, a situation more relevant for applications is investigated: a superconducting parallelepiped bulk with the magnetic field parallel to two of its faces and making an angle with the other one without and with a further constraint on the possible directions of the current. The latter constraint can be used to model the magnetization of a stack of high-temperature superconductor tapes, which are electrically insulated in one direction. For the present study three different numerical approaches are used: the Minimum Electro-Magnetic Entropy Production (MEMEP) method, the $H$-formulation of Maxwell's equations and the Volume Integral Method (VIM) for 3D eddy currents computation. The results in terms of current density profiles and energy dissipation are compared, and the differences in the two situations of unconstrained and constrained current flow are pointed out. In addition, various technical issues related to the 3D modeling of superconductors are discussed and information about the computational effort required by each model is provided. This works constitutes a concrete result of the collaborative effort taking place within the HTS numerical modeling community and will hopefully serve as a stepping stone for future joint investigations

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

A FIC-based stabilized finite element formulation for turbulent flows

We present a new stabilized finite element (FEM) formulation for incompressible flows based on the Finite Increment Calculus (FIC) framework. In comparison to existing FIC approaches for fluids, this formulation involves a new term in the momentum equation, which introduces non-isotropic dissipation in the direction of velocity gradients. We also follow a new approach to the derivation of the stabilized mass equation, inspired by recent developments for quasi-incompressible flows. The presented FIC-FEM formulation is used to simulate turbulent flows, using the dissipation introduced by the method to account for turbulent dissipation in the style of implicit large eddy simulation.Peer ReviewedPostprint (author's final draft