Adi Ditkowski

Staircase-free finite-difference time-domain formulation for general materials in complex geometries

A stable Cartesian grid staircase-free finite-difference time-domain formulation for arbitrary material distributions in general geometries is introduced. It is shown that the method exhibits higher accuracy than the classical Yee (1966) scheme for complex geometries since the computational representation of physical structures is not of a staircased nature. Furthermore, electromagnetic boundary conditions are correctly enforced. The method significantly reduces simulation times as fewer points per wavelength are needed to accurately resolve the wave and the geometry. Both perfect electric conductors and dielectric structures have been investigated. Numerical results are presented and discussed.

On Error Bounds of Finite Difference Approximations to Partial Differential Equations—Temporal Behavior and Rate of Convergence

This paper considers a family of spatially semi-discrete approximations, including boundary treatments, to hyperbolic and parabolic equations. We derive the dependence of the error-bounds on time as well as on mesh size.

Bounded Error Schemes for the Wave Equation on Complex Domains

This paper considers the application of the method of boundary penalty terms (SAT) to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell’s equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g., the staggered Yee scheme)—we achieve a decrease of two orders of magnitude in the level of the L2-error.

Intrinsic compressive stress in polycrystalline films with negligible grain boundary diffusion

The model developed here describes compressive stress evolution during the growth of continuous, polycrystalline films (i.e., beyond the point where individual islands have coalesced into a continuous film). These stresses are attributed to the insertion of excess adatoms at grain boundaries. Steady state occurs when the strain energy at the top of the film is balanced by the local excess chemical potential of surface adatmos. Strain gradients associated with this compressive stress mechanism depend on the kinetics of the process. In the absence of grain boundary diffusion, these strain profiles are determined by the ratio of the atom insertion and growth rates. The steady-state strain and the strain evolution kinetics also depend on the two key length scales, the grain size, and the film thickness. The ratio of these two lengths (i.e., the grain aspect ratio) can also have a significant influence on the thermodynamic driving force for strain evolution if the grain sizes are sufficiently small. The model ...

MULTI-DIMENSIONAL ASYMPTOTICALLY STABLE FINITE DIFFERENCE SCHEMES FOR THE ADVECTION-DIFFUSION EQUATION

An algorithm is presented which solves the multi-dimensional advection-diffusion equation on complex shapes to

A grid redistribution method for singular problems

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Computing Derivatives of Noisy Signals Using Orthogonal Functions Expansions

-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by tradition definitions, fail. It gives accurate, non oscillatory results even when boundary layers are not resolved.

Optimization of chemical vapor infiltration with simultaneous powder formation

Many physical phenomena develop singular, or nearly singular behavior in localized regions, e.g. boundary layers or blowup solutions. Using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity. Ren and Wang developed a semi-static adaptive grid method [W. Ren, X.P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions, J. Comput. Phys. 159 (2000) 246-273] for the solution of these problems, known as the iterative grid redistribution (IGR) method. In this study we develop a theoretical basis for semi-static adaptive grid method for singular problems. Based on this theory, we obtain the key result of this study - a methodology for designing robust weight functionals which ensures grid resolution in the singular region, as well as control of the maximal grid spacing in the outer region. Using this methodology, we introduce a semi-static adaptive grid method, which does not involve an iterative procedure for grid redistribution, as in the IGR method. We demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude.

On the intersection of sets of incoming and outgoing waves

In many applications noisy signals are measured. These signals have to be filtered and, sometimes, their derivative has to be computed.In this paper a method for filtering the signals and computing the derivatives is presented. This method is based on expansion onto transformed Legendre polynomials.Numerical examples demonstrate the efficacy of the method as well as the theoretical estimates.

High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors

Abstract : A key difficulty in isothermal, isobaric chemical vapor infiltration is the long processing times that are typically required. With this in mind, it is important to minimize infiltration times. This optimization problem is addressed here, using a relatively simple model for dilute gases. The results provide useful asymptotic expressions for the minimum time and corresponding conditions. These approximations are quantitatively accurate for most cases of interest, where relatively uniform infiltration is required. They also provide useful quantitative insight in cases where less uniformity is required. The effects of homogeneous nucleation were also investigated. This does not affect the governing equations for infiltration of a porous body, however, powder formation can restrict the range of permissible infiltration conditions. This was analyzed for the case of carbon infiltration from methane.