Anamaria Gomide

Interpolating Wavelets and Adaptive Finite Difference Schemes for Solving Maxwell's Equations: The Effects of Gridding

In this paper, we discuss the use of the sparse point representation (SPR) methodology for adaptive finite-difference simulations in computational electromagnetics. The principle of the SPR method is to represent the solution only through those point values indicated by the significant wavelet coefficients, which are used as local regularity indicators. Recently, two kinds of SPR schemes have been considered for solving Maxwells equations: 1) staggered grids in the time-space domain are used for the discretization of the magnetic and electrical fields, as in the finite-differences time-domain (FDTD) scheme and 2) nonstaggered grids are used in combination with Runge-Kutta ODE solvers. In both cases, 1-D simulations of the SPR method leads to sparse grids that adapt in space to the local smoothness of the fields, and, at the same time, track the evolution of the fields over time with substantial gain in memory and computational speed. However, in the latter case, we found spurious oscillations in the simulations. Therefore, before extending the implementation of the SPR method to higher dimensions, we wanted to evaluate which of these two SPR strategies is more convenient. After a careful theoretical analysis of stability and numerical dispersion comparing the schemes in staggered and nonstaggered grids, we conclude that schemes for staggered grids seem to be preferable from the dispersion viewpoint, especially for low-order schemes and coarse grids. However, by adapting the grid density and increasing the order, SPR schemes for nonstaggered grids also show good performance. In our experiments, no spurious oscillations were detected. We observed that, for a given accuracy, the adaptive scheme on a nonstaggered grid requires less computational effort. Since the use of nonstaggered grids increases the stability range and facilitates the implementation of adaptive strategies, we believe that the SPR method in nonstaggered grids has a very good potential for computational electromagnetics

Finite elements on dyadic grids with applications

A dyadic grid is a d-dimensional hierarchical mesh where a cell at level k is partitioned into two equal children at level k + 1 by a hyperplane perpendicular to coordinate axis (k mod d). We consider here the finite element approach on adaptive grids, static and dynamic, for various functional approximation problems.We review here the theory of adaptive dyadic grids and splines defined on them. Specifically, we consider the space Pcd[G] of all functions that, within any leaf cell of an arbitrary finite dyadic grid G, coincide with a multivariate polynomial of maximum degree d in each coordinate, and are continuous to order c.We describe algorithms to construct a finite-element basis for such spaces. We illustrate the use of such basis for interpolation, least-squares approximation, and the Galerkin-style integration of partial differential equations, such as the heat diffusion equation and two-phase (oil/water) flow in porous media.Compared to tetrahedral meshes, the simple topology of dyadic grids is expected to compensate for their limitations, especially in problems with moving fronts.

Grid structure impact in sparse point representation of derivatives

In the Sparse Point Representation (SPR) method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors by means of an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Illustrating results are presented for 2D Maxwells equation numerical solutions.

Bases for Non-homogeneous Polynomial Ck Splines on the Sphere

We investigate the use of non-homogeneous spherical polynomials for the approximation of functions defined on the sphere S2. A spherical polynomial is the restriction to S2 of a polynomial in the three coordinates x,y,z of ℝ3. Let P d be the space of spherical polynomials with degree ≤ d. We show that P d is the direct sum of P d and H d−1, where H d denotes the space of homogeneous degree-d polynomials in x,y,z.

A User-editable C1-Continuous 2.5D Space Deformation Method For 3D Models

Shape deformation methods are important in such fields as geometric modeling and computer animation. In biology, modeling of shape, growth, movement and pathologies of living microscopic organisms or cells require smooth deformations, which are essentially 2D with little change in depth. In this paper, we present a 2.5D space deformation method. The 3D model is modified by deforming an enclosing control grid of prisms. Spline interpolation is used to satisfy the smoothness requirement. We implemented this method in an editor which makes it possible to define and modify the deformation with the mouse in a user-friendly way. The experimental results show that the method is simple and effective.

Comparison of Finite Element Bases for Global Illumination in Image Synthesis

Finite element bases defined by sampling points were used by J. Lehtinen in 2008 for the efficient computation of global illumination in virtual scenes. The bases provide smooth approximations for the radiosity and spontaneous emission functions, leading to a discrete version of Kajiyas rendering equation. Unlike methods that are based on surface subdivision, Lehtinens method can cope with arbitrarily complex geometries. In this paper we present an experimental validation of Lehtinens meshless method by comparing its results with an independent numerical solution of the rendering equation on a simple three-dimensional scene. We also compare Lehtinens special finite-element basis with two other similar bases that are often used for meshless data interpolation, namely a radial basis with a Gaussian mother function, and Shepards inverse-square-distance weighted interpolation. The results confirm the superiority of Lehtinens basis and clarify why the other two bases provide inferior-looking results.

Adaptive finite difference schemes based on interpolating wavelets for solving 2D Maxwell’s equations

This paper describes a 2D application of interpolating wavelets and recursive interpolation schemes with thresholding, aiming the representation of the electric and magnetic fields in nonuniform, adaptive grids. Applied to Maxwells equations, the method leads to sparse grids that adapt in space to the local smoothness of the fields, and at the same time track the evolution of the fields over time. In general, the number of points in the grid, Ns, is below the maximum number of points, N. It is possible to control Ns, by trading off representation accuracy and data compression, and therefore speed. A numerical example is presented showing the propagation of a Gaussian pulse within a 2D horn.