Dielectric boundary condition for the embedded curved boundary (ECB) method

摘要

A new version of ECB has been completed that allows nonuniform grid spacing and a new dieledric boundary condition. ECB was developed to retain the simplicity and speed of an orthogonal mesh while capturing much of the fidelity of adaptive, unstructured finite element meshes. Codes based on orthogonal meshes are easy to work with and lead to well-posed elliptic and parabolic problems that are comparatively easy to solve. Generally, othogonal mesh representations lead to banded matrices while unstructured representations lead to more complicated sparse matrices. Recent advances in adapting banded linear systems to massively parallel computers reinforce our opinion that iterative field solutions utilizing banded matrix methods will continue to be competitive. Unfortunately, the underlying ''stair-step'' boundary representation in simple orthogonal mesh (and recent Adaptive Mesh Refinement) applications is inadequate. With ECB, the curved boundary is represented by piece-wise-linear representations of curved internal boundaries embedded into the orthogonal mesh. ECB structures are constructed out of the superposition of analytically prescribed building blocks. A link-list AIRFOIL has also been constructed. In the ECB scheme, we first find each intercept of the structure boundary with an I or J grid line is assigned an index K. We store the actual z,y value at the intercept, and the slope of the boundary at that intercept, in arrays whose index K is associated with the corresponding mesh point just inside the structure. In 2-D, a point just outside a structure may have up to 4 intercepts associated with it. We now construct PieceWise Linear Segments (PWLS)-from the slope and intercept- that will serve as the computational boundary-a boundary constructed as a superposition of the analytic elements given as input. The intersection of these PWLSs with other nearby PWLSs define the end- points. We insist that there be an endpoint within each cell for each sequence of PWLSs coming through that cell. We give directionality to a PWLS by the convention that looking from point 1 to point 2 will place the interior of the structure on the left. This scheme allows the generalization to those boundary structures that do not actually have any mesh points inside a structure. Since these structures are defined by their analytically defined bounding walls, points can always be found that are ''inside'' a given wall segment-even though it may, in fact, be outside the opposite bounding segment. Such subgrid geometry is useful, for example, in a gun configuration with a thin electrode, perhaps not parallel to any axes of the orthogonal mesh, in which the electrode simply contributes to an external focusing field.