Verification of high-order mixed FEM solution of transient Magnetic diffusion problems

We develop and present high order mixed finite element discretizations of the time dependent electromagnetic diffusion equations for solving eddy current problems on 3D unstructured grids. The discretizations are based on high order H(grad), H(curl) and H(div) conforming finite element spaces combined with an implicit and unconditionally stable generalized Crank-Nicholson time differencing method. We develop three separate electromagnetic diffusion formulations, namely the E (electric field), H (magnetic field) and the A-{phi} (potential) formulations. For each formulation, we also provide a consistent procedure for computing the secondary variables F (current flux density) and B (magnetic flux density), as these fields are required for the computation of electromagnetic force and heating terms. We verify the error convergence properties of each formulation via a series of numerical experiments on canonical problems with known analytic solutions. The key result is that the different formulations are equally accurate, even for the secondary variables J and B, and hence the choice of which formulation to use depends mostly upon relevance of the Natural and Essential boundary conditions to the problem of interest. In addition, we highlight issues with numerical verification of finite element methods which can lead to false conclusions on the accuracy of the methods

GMOs: Non-Health Issues

The controversy over genetically modified [GM] organisms is often framed in terms of possible hazards for human health. Articles in a previous volume of this *Encyclopedia* give a general overview of GM crops [@Mulvaney2014] and specifically examine human health [@Nordgard2014] and labeling [@Bruton2014] issues surrounding GM organisms. This article explores several other aspects of the controversy: environmental concerns, political and legal disputes, and the aim of "feeding the world" and promoting food security. Rather than discussing abstract, hypothetical GM organisms, this article explores the consequences of the GM organisms that have actually been deployed in the particular contexts that they have been deployed, on the belief that there is little point in discussing GM organisms in an idealized or context-independent way

Coupling Magnetic Fields and ALE Hydrodynamics for 3D Simulations of MFCG's

We review the development of a full 3D multiphysics code for the simulation of explosively driven Magnetic Flux Compression Generators (MFCG) and related pulse power devices. In a typical MFCG the device is seeded with an initial electric current and the device is then detonated. The detonation compresses the magnetic field and amplifies the current. This is a multiphysics problem in that detonation kinetics, electromagnetic diffusion and induction, material deformation, and thermal effects are all important. This is a tightly coupled problem in that the different physical quantities have comparable spatial and temporal variation, and hence should be solved simultaneously on the same computational mesh

Vector Finite Element Modeling of the Full-Wave Maxwell Equations to Evaluate Power Loss in Bent Optical Fibers

We measure the loss of power incurred by the bending of a single mode step-indexed optical fiber using vector finite element modeling of the full-wave Maxwell equations in the optical regime. We demonstrate fewer grid elements can be used to model light transmission in longer fiber lengths by using high-order basis functions in conjunction with a high order energy conserving time integration method. The power in the core is measured at several points to determine the percentage loss. We also demonstrate the effect of bending on the light polarization

Development and Application of Compatible Discretizations of Maxwell's Equations

We present the development and application of compatible finite element discretizations of electromagnetics problems derived from the time dependent, full wave Maxwell equations. We review the H(curl)-conforming finite element method, using the concepts and notations of differential forms as a theoretical framework. We chose this approach because it can handle complex geometries, it is free of spurious modes, it is numerically stable without the need for filtering or artificial diffusion, it correctly models the discontinuity of fields across material boundaries, and it can be very high order. Higher-order H(curl) and H(div) conforming basis functions are not unique and we have designed an extensible C++ framework that supports a variety of specific instantiations of these such as standard interpolatory bases, spectral bases, hierarchical bases, and semi-orthogonal bases. Virtually any electromagnetics problem that can be cast in the language of differential forms can be solved using our framework. For time dependent problems a method-of-lines scheme is used where the Galerkin method reduces the PDE to a semi-discrete system of ODE's, which are then integrated in time using finite difference methods. For time integration of wave equations we employ the unconditionally stable implicit Newmark-Beta method, as well as the high order energy conserving explicit Maxwell Symplectic method; for diffusion equations, we employ a generalized Crank-Nicholson method. We conclude with computational examples from resonant cavity problems, time-dependent wave propagation problems, and transient eddy current problems, all obtained using the authors massively parallel computational electromagnetics code EMSolve

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 straight and lossy random rough wall. 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

A QR accelerated volume-to-surface boundary condition for finite element solution of eddy current problems

We are concerned with the solution of time-dependent electromagnetic eddy current problems using a finite element formulation on three-dimensional unstructured meshes. We allow for multiple conducting regions, and our goal is to develop an efficient computational method that does not require a computational mesh of the air/vacuum regions. This requires a sophisticated global boundary condition specifying the total fields on the conductor boundaries. We propose a Biot-Savart law based volume-to-surface boundary condition to meet this requirement. This Biot-Savart approach is demonstrated to be very accurate. In addition, this approach can be accelerated via a low-rank QR approximation of the discretized Biot-Savart law

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics

We are interested in the computational study of shock hydrodynamics, i.e. problems involving compressible solids, liquids, and gases that undergo large deformation. These problems are dynamic and nonlinear and can exhibit complex instabilities. Due to advances in high performance computing it is possible to parameterize a hydrodynamic problem and perform a computational study yielding $\mathcal{O}\left({\rm TB}\right)$ of simulation state data. We present an interactive machine learning tool that can be used to compress, browse, and interpolate these large simulation datasets. This tool allows computational scientists and researchers to quickly visualize "what-if" situations, perform sensitivity analyses, and optimize complex hydrodynamic experiments

An Arbitrary Lagrangian-Eulerian Discretization of MHD on 3D Unstructured Grids

We present an arbitrary Lagrangian-Eulerian (ALE) discretization of the equations of resistive magnetohydrodynamics (MHD) on unstructured hexahedral grids. The method is formulated using an operator-split approach with three distinct phases: electromagnetic diffusion, Lagrangian motion, and Eulerian advection. The resistive magnetic dynamo equation is discretized using a compatible mixed finite element method with a 2nd order accurate implicit time differencing scheme which preserves the divergence-free nature of the magnetic field. At each discrete time step, electromagnetic force and heat terms are calculated and coupled to the hydrodynamic equations to compute the Lagrangian motion of the conducting materials. By virtue of the compatible discretization method used, the invariants of Lagrangian MHD motion are preserved in a discrete sense. When the Lagrangian motion of the mesh causes significant distortion, that distortion is corrected with a relaxation of the mesh, followed by a 2nd order monotonic remap of the electromagnetic state variables. The remap is equivalent to Eulerian advection of the magnetic flux density with a fictitious mesh relaxation velocity. The magnetic advection is performed using a novel variant of constrained transport (CT) that is valid for unstructured hexahedral grids with arbitrary mesh velocities. The advection method maintains the divergence free nature of the magnetic field and is second order accurate in regions where the solution is sufficiently smooth. For regions in which the magnetic field is discontinuous (e.g. MHD shocks) the method is limited using a novel variant of algebraic flux correction (AFC) which is local extremum diminishing (LED) and divergence preserving. Finally, we verify each stage of the discretization via a set of numerical experiments

Suppression of Richtmyer-Meshkov instability via special pairs of shocks and phase transitions

The classical Richtmyer-Meshkov instability is a hydrodynamic instability characterizing the evolution of an interface following shock loading. In contrast to other hydrodynamic instabilities such as Rayleigh-Taylor, it is known for being unconditionally unstable: regardless of the direction of shock passage, any deviations from a flat interface will be amplified. In this article, we show that for negative Atwood numbers, there exist special sequences of shocks which result in a nearly perfectly suppressed instability growth. We demonstrate this principle computationally and experimentally with stepped fliers and phase transition materials. A fascinating immediate corollary is that in specific instances a phase transitioning material may self-suppress RMI