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

A parallel computer implementation of fast low-rank QR approximation of the Biot-Savart law

In this paper we present a low-rank QR method for evaluating the discrete Biot-Savart law on parallel computers. It is assumed that the known current density and the unknown magnetic field are both expressed in a finite element expansion, and we wish to compute the degrees-of-freedom (DOF) in the basis function expansion of the magnetic field. The matrix that maps the current DOF to the field DOF is full, but if the spatial domain is properly partitioned the matrix can be written as a block matrix, with blocks representing distant interactions being low rank and having a compressed QR representation. The matrix partitioning is determined by the number of processors, the rank of each block (i.e. the compression) is determined by the specific geometry and is computed dynamically. In this paper we provide the algorithmic details and present computational results for large-scale computations

Investigation of Radar Propagation in Buildings: A 10 Billion Element Cartesian-Mesh FETD Simulation

In this paper large scale full-wave simulations are performed to investigate radar wave propagation inside buildings. In principle, a radar system combined with sophisticated numerical methods for inverse problems can be used to determine the internal structure of a building. The composition of the walls (cinder block, re-bar) may effect the propagation of the radar waves in a complicated manner. In order to provide a benchmark solution of radar propagation in buildings, including the effects of typical cinder block and re-bar, we performed large scale full wave simulations using a Finite Element Time Domain (FETD) method. This particular FETD implementation is tuned for the special case of an orthogonal Cartesian mesh and hence resembles FDTD in accuracy and efficiency. The method was implemented on a general-purpose massively parallel computer. In this paper we briefly describe the radar propagation problem, the FETD implementation, and we present results of simulations that used over 10 billion elements

An Explicit Time-Domain Hybrid Formulation Based on the Unified Boundary Condition

An approach to stabilize the two-surface, time domain FEM/BI hybrid by means of a unified boundary condition is presented. The first-order symplectic finite element formulation [1] is used along with a version of the unified boundary condition of Jin [2] reformulated for Maxwell's first-order equations in time to provide both stability and accuracy over the first-order ABC. Several results are presented to validate the numerical solutions. In particular the dipole in a free-space box is analyzed and compared to the Dirchlet boundary condition of Ziolkowski and Madsen [3] and to a Neuman boundary condition approach

Electrostatic Breakdown Analysis using EMsolve and BEMSTER

Computer simulations modeling electrostatic behavior were used to simulate dielectric breakdown problems. These simulations modeled composite dielectric and conducting structures to see how much voltage difference or charge accumulation could occur before dielectric breakdown occurred in an air region. Two different computer codes were used for the analysis; EMSolve and BEMSTER. EMSolve, an existing LLNL internal finite element code, requires that a complete volume mesh of the problem be constructed. BEMSTER, a boundary-element code, was developed from an extension of the FEMSTER libraries which power EMSolve. The boundary-integral code offers the advantages of solving for accumulated charge and maximum electric field directly, and of only requiring a surface mesh. However, because it does not automatically solve for the voltage and electric field everywhere in space, post-processing and visualization are slightly more difficult than with EMSolve. Both codes were compared to several analytical solutions, and then applied to the structures of interest. Both codes showed good agreement with the analytic solution and with each other

Performance of low-rank QR approximation of the finite element Biot-Savart law

We are concerned with the computation of magnetic fields from known electric currents in the finite element setting. In finite element eddy current simulations it is necessary to prescribe the magnetic field (or potential, depending upon the formulation) on the conductor boundary. In situations where the magnetic field is due to a distributed current density, the Biot-Savart law can be used, eliminating the need to mesh the nonconducting regions. Computation of the Biot-Savart law can be significantly accelerated using a low-rank QR approximation. We review the low-rank QR method and report performance on selected problems

Fiscal Year 2006

This is the final report for LDRD 01-ERD-005. The Principle Investigator was Robert Sharpe. Collaborators included Niel Madsen, Benjamin Fasenfest, John D. Rockway, of the Defense Sciences Engineering Division (DSED), Vikram Jandhyala and James Pingenot from the University of Washington, and Mark Stowell of the Center for Applications Development and Software Engineering (CADSE). It should be noted that Benjamin Fasenfest and Mark Stowell were partially supported under other funding. The purpose of this LDRD effort was to enhance LLNL's computational electromagnetics capability in the area of broadband radiation and scattering. For radiation and scattering problems our transient EM codes are limited by the approximate Radiation Boundary Conditions (RBC's) used to model the radiation into an infinite space. Improved RBC's were researched, developed, and incorporated into the existing EMSolve finite-element code to provide a 10-100x improvement in the accuracy of the boundary conditions. Section I provides an introduction to the project and the project goals. Section II provides a summary of the project's research and accomplishments as presented in the attached papers