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A vectorized particle tracer for unstructured grids is described. The basic approach is to use elementary properties of the linear basis functions to search for particles on the grid using the element last occupied as an initial guess. To permit vectorization, a simple binary sort of the particles is performed every timestep such that all particles that have as yet not found their host element remain at the top of the list. In this way, vector-loops can be easily formed. Timings taken from a numerical example indicate that speed-ups of the order of 1:14 can be obtained on vector-machines when using this algorithm.

Traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences. These so-called finite-difference time-domain (FDTD) methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems. Great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid. An unstructured grid allows one to place the grid points anywhere one chooses, so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh. In this paper we present a technique for solving Maxwell's equations on an unstructured grid based on the Taylor-Galerkin finite-element method. We present several numerical examples which reveal the fundamental accuracy and adaptability of the method. Although our examples are in two dimensions, the techniques and results generalize readily to 3D.

There is an increasing demand to use kinetic plasma simulations to model real plasma devices such as e-beam diodes, plasma torches, opening switches, and so on. With this demands comes the realization that standard simulation tools based on rectangular structured meshes are often too inflexible to accommodate real device geometries. To gain geometric flexibility, one typically must abandon structured rectangular meshes. Some have had considerable success by distorting the grid to conform to boundary-fitted curvilinear coordinates. This gives substantial flexibility while retaining a logically rectangular data structure. However, to achieve the greatest flexibility, an unstructured mesh is necessary. For the last several years, we have been engaged in designing electromagnetic particle-in-cell (PIC) simulations that can be performed on unstructured grids of triangles. We feel this is a powerful strategy for matching kinetic plasma simulations to real problem geometries. Unstructured meshes not only accommodate complicated boundary shapes with ease, but also allow extreme local refinement without affecting resolution elsewhere. The basic challenges to overcome in formulating EM-PIC on these new meshes are: (1) grid generation; (2) particle interpolation and tracking, and (3) field solution. In this paper we briefly describe our techniques and give a simple example.

We describe a pair of finite-element codes which use unstructured meshes to solve the time-dependent Maxwell equations, with particular emphasis on their application to electromagnetic scattering problems. A two-step, flux-corrected transport scheme is used to effect the time integration, while the spatial structure of the field is determined by a Galerkin procedure. The basis functions are piecewise-linear on three-noded triangles in two dimensions and four-noded tetrahedra in three. For the periodic scattering problems with which we are presently concerned, adaptive remeshing is a convenient and powerful method for improving the quality of the solutions. Results for the analytically tractable case of scattering by a perfectly conducting circular cylinder are used to illustrate the performance of the codes.

The most widely used computational model of collisionless plasmas is the Lagrangian-Eulerian hybrid technique known as particle-in-cell or PIC. In the electromagnetic version, Maxwell's equations are solved on an Eulerian grid and electromagnetic forces are interpolated form the grid to particle locations. Particles are then moved in Lagrangian fashion while their currents are interpolated back onto the grid to provide sources for the fields on the next cycle. There are many applications where one needs to model plasmas and electromagnetic waves inside regions of complicated shape. Traditional methods for solving Maxwell's equations employ finite differences on regular grids to replace differential operators. These methods are awkward for complicated boundary shapes, often replacing smoothly curved or slanted boundaries with stairsteps. The desire to incorporate realistic boundaries into plasma simulations is motivated by a host of situations in which proper representation of the boundary shape is expected to be critical. Our approach to solving this problem is to design electromagnetic particle codes based on the use of unstructured grids. The arbitrary connectivity of unstructured grids provides the flexibility to place nodes wherever needed to fit the most complex boundary shapes. The most significant problems that must be addressed as a result of this strategy are: grid generation, field solution, and particle tracking. Our solutions to these problems, along with a few preliminary results, are presented in this paper.