Garry Rodrigue

Improved vector FEM solutions of Maxwell's equations using grid pre-conditioning

The Time Domain Vector Finite Element Method is a promising new approach for solving Maxwells equations on unstructured triangular grids. This method is sensitive to the quality, or condition, of the grid. In this study grid pre-conditioning techniques, such as edge swapping, Laplacian smoothing, and energy minimization, are shown to improve the accuracy of the solution and also reduce the overall computational effort.

Study of optically induced effects due to bending and twisting using the vector finite-element method

We study the effects of macroscopic bends and twists in an optical waveguide and how they influence the transmission capabilities of a waveguide. These mechanical stresses and strains distort the optical indicatrix of the medium, producing optical anistropy. The spatially varying refractive indices are incorporated into the full-wave Maxwells equations. The governing equations are discretized by using a vector finite-element method cast in a high-order finite element approximation. This approach allows us to study the complexities of the mechanical deformation within a framework of a high-order formulation that can, in turn, reduce the computational requirement without degrading its performance. The optical activities generated, total energy produced, and power loss due to the mechanical stresses and strains are reported and discussed.

A generalized front marching algorithm for the solution of the eikonal equation

A new front marching algorithm for solving the eikonal equation is presented. An important property of the algorithm is that it can be used on nodes that are located on highly distorted grids or on nodes that are randomly located. When the nodes are located on an orthogonal grid, the method is first-order accurate and is shown to be a generalization of the front marching algorithm in (Proc. Natl. Acad. Sci. 93 (4) (1996) 1591). The accuracy of the method is also shown to be dependent on the principle curvature of the wave front solution. Numerical results on a variety of node configurations as well as on shadow, nonconvex and nondifferentiable solutions are presented.

An efficient vector finite element method for nonlinear electromagnetic modeling

We have developed a mixed Vector Finite Element Method (VFEM) for Maxwells equations with a nonlinear polarization term. The method allows for discretization of complicated geometries with arbitrary order representations of the B and E fields. In this paper we will describe the method and a series of optimizations that significantly reduce the computational cost. Additionally, a series of test simulations will be presented to validate the method. Finally, a nonlinear waveguide mode mixing example is presented and discussed.

Domain Decomposition Models for Parallel Monte Carlo Transport

We present a strategy for parallelizing computations that use the transport method. It combines spatial domain decomposition with domain replication to realize the scaling benefits of replication while allowing for problems whose computational mesh will not fit in a single processors memory. The mesh is decomposed into a small number of spatial domains—typically fewer domains than there are processors—and heuristics are used to estimate the computational effort required to generate the solution in each subdomain using Monte Carlo. That work estimate determines the number of times a subdomain is replicated relative to the others. Timing of runs for two problems show that the new method scales better than traditional domain decomposition.

Large eddy simulation and ALE mesh motion in Rayleigh-Taylor instability simulation ✩

Abstract Many large eddy simulation (LES) techniques have been developed for stationary computational meshes. This study applies a single equation LES to Arbitrary Lagrangian–Eulerian (ALE) simulations of Rayleigh–Taylor instability and investigates its effects. Behavior of LES is similar for Eulerian and ALE simulations for the test problem studied. However, the motion of the mesh can be tied to the subgrid scale model in the form of a relaxation weight based on subgrid scale energy. This increases mesh resolution in areas of high subgrid scale energy.

A domain decomposition method for solving a two-dimensional viscous Burgers' equation

Abstract We present an efficient finite difference numerical solution for a convection equation that includes diffusion terms with small coefficients. The equation is first advanced on a coarse mesh. Regions of significant diffusion activity are identified using a threshold criterion. The coarse mesh is dynamically decomposed and the mesh is refined in these regions. The decomposition is heterogeneous in the sense that the problem formulations as well as the solution methods may vary from mesh to mesh. In particular, a mixed Euler-Lagrange method is developed that explicitly advances the coarse mesh relative to an Eulerian reference frame, and implicitly advances the refined meshes relative to moving Lagrangian reference frames. On the refined meshes the method is second-order accurate in space. Asynchronous Schwarz iterations are used between overlapping refined meshes, when needed, to communicate the data dependence of the implicit refined solution. The computation is distributed across the four processors of the shared memory CRAY-2.

A parallel DSDADI method for solution of the steady state diffusion equation

A parallel diagonally scaled dynamic alternating-direction-implicit (DSDADI) method is shown to be an effective algorithm for solving the 2D and 3D steady-state diffusion equation on large uniform Cartesian grids. Empirical evidence from the parallel solution of large gridsize problems suggests that the computational work done by DSDADI to converge over an Nd grid with continuous diffusivity is of lower order than O(Nd+α) for any fixed α > 0. This is in contrast to the method of diagonally scaled conjugate gradients (DSCG), for which the computational work necessary for convergence is O(Nd+1). Furthermore, the combination of diagonal scaling, spatial domain decomposition (SDD), and distributed tridiagonal system solution gives the DSDADI algorithm reasonable scalability on distributed-memory multiprocessors such as the CRAY T3D. Finally, an approximate parallel tridiagonal system solver with diminished interprocessor communication exhibits additional utility for DSDADI.

Convergence rate estimates for iterative solutions of the biharmonic equation

Abstract A technique is developed whereby one can obtain asymptotic estimates of eigenvalues of first-order iteration matrices. The technique is applied to iteration matrices arising from the numerical solution of the 1- and 2-dimensional biharmonic equation. The eigenvalue estimates are computationally verified.

Solving the Eikonal equation on an adaptive mesh

A fast marching method (FMM) using the line-of-site calculation to solve the eikonal equation is applied to an adaptive mesh. The criteria for refinement are the curvature of the propagating front. It is shown empirically that for cases involving an initial front initiated from a single point in an open three dimensional domain and constant front propagating speed that the FMM with adaptive mesh refinement (AMR) uses roughly an order of magnitude less CPU time and an order of magnitude less CPU memory than the non-AMR FMM to attain a similar level of accuracy. It is also shown that the AMR-FMM refines when the curvature is caused by boundary irregularities and also non-constant front propagating speed.