Robert M. M. Mattheij

Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures

This paper extends the area of application of the Fourier modal method (FMM) from periodic structures to aperiodic ones, in particular for plane-wave illumination at arbitrary angles. This is achieved by placing perfectly matched layers at the lateral sides of the computational domain and reformulating the governing equations in terms of a contrast field that does not contain the incoming field. As a result of the reformulation, the homogeneous system of second-order ordinary differential equations from the original FMM becomes non-homogeneous. Its solution is derived analytically and used in the established FMM framework. The technique is demonstrated on a simple problem of planar scattering of TE-polarized light by a single rectangular line.

Model order reduction for nonlinear IC models

Model order reduction is a mathematical technique to transform nonlinear dynamical models into smaller ones, that are easier to analyze. In this paper we demonstrate how model order reduction can be applied to nonlinear electronic circuits. First we give an introduction to this important topic. For linear time-invariant systems there exist already some well-known techniques, like Truncated Balanced Realization. Afterwards we deal with some typical problems for model order reduction of electronic circuits. Because electronic circuits are highly nonlinear, it is impossible to use the methods for linear systems directly. Three reduction methods, which are suitable for nonlinear differential algebraic equation systems are summarized, the Trajectory piecewise Linear approach, Empirical Balanced Truncation, and the Proper Orthogonal Decomposition. The last two methods have the Galerkin projection in common. Because Galerkin projection does not decrease the evaluation costs of a reduced model, some interpolation techniques are discussed (Missing Point Estimation, and Adapted POD). Finally we show an application of model order reduction to a nonlinear academic model of a diode chain.

A Derivative-Based Fast Autofocus Method in Electron Microscopy

Most automatic focusing methods are based on a sharpness function, which delivers a real-valued estimate of an image quality. In this paper, we study an L2-norm derivative-based sharpness function, which has been used before based on heuristic consideration. We give a more solid mathematical foundation for this function and get a better insight into its analytical properties. Moreover an efficient autofocus method is presented, in which an artificial blur variable plays an important role.We show that for a specific choice of the artificial blur control variable, the function is approximately a quadratic polynomial, which implies that after the recording of at least three images one can find the approximate position of the optimal defocus. This provides the speed improvement in comparison with existing approaches, which usually require recording of more than ten images for autofocus. The new autofocus method is employed for the scanning transmission electron microscopy. To be more specific, it has been implemented in the FEI scanning transmission electron microscope and its performance has been tested as a part of a particle analysis application.

Mathematical modelling of glass forming processes

An important process in glass manufacture is the forming of the product. The forming process takes place at high rate, involves extreme temperatures and is characterised by large deformations. The process can be modelled as a coupled thermodynamical/mechanical problem including the interaction between glass, air and equipment. In this paper a general mathematical model for glass forming is derived, which is specified for different forming processes, in particular pressing and blowing. The model should be able to correctly represent the flow of the glass and the energy exchange during the process. Various modelling aspects are discussed for each process, while several key issues, such as the motion of the plunger and the evolution of the glass-air interfaces, are examined thoroughly. Finally, some examples of process simulations for existing simulation tools are provided.

Discretizations with Dichotomic Stability for Two-Point Boundary Value Problems

For a two-point boundary value problem to be well conditioned, the system of ordinary differential equations must necessarily possess a dichotomic set of fundamental solutions [7], with decaying modes controlled by initial conditions, and growing modes controlled by terminal conditions [10]. It was shown in [3] that it is important for a discretization of such a problem to preserve the dichotomy property, and the implications of this stability criterion were examined in a number of particular cases. Some simple difference schemes were examined for second order ordinary differential equations, and also various discretizations for first order systems, including those obtained by piecewise collocation and implicit Runge-Kutta type formulae. The last two examples were of multistep schemes, of such a form that they could be used in a sequential stepping mode, as would be done in shooting methods, or more generally in multiple shooting.

Orientation identification of the power spectrum

The image Fourier transform is widely used for defocus and astigmatism correction in electron microscopy. The shape of a power spectrum (the square of a modulus of image Fourier transform) is directly related to the three microscope controls, namely, defocus and twofold (two-parameter) astigmatism. We propose a new method for power-spectrum orientation identification. The method is based on the three measures that are related to the microscopes controls. The measures are derived from the mathematical moments of the power spectrum and is tested with the help of a Gaussian benchmark, as well as with the scanning electron microscopy experimental images. The method can be used as an assisting tool for increasing the capabilities of defocus and astigmatism correction a of nonexperienced scanning electron microscopy user, as well as a basis for automated application.

Efficient estimation of the friction factor for forced laminar flow in axially symmetric corrugated pipes

In this paper we present an efficient method for calculating the friction factor for forced laminar flow in arbitrary axially symmetric pipes. The approach is based on an analytic expression for the friction factor, obtained after integrating the Navier-Stokes equations over a segment of the pipe. The friction factor is expressed in terms of surface integrals over the pipe wall, these integrals are then estimated by means of approximate velocity and pressure profiles computed via the method of slow variations. Our method for computing the friction factor is validated by comparing the results, to those obtained using CFD techniques for a set of examples featuring pipes with sinusoidal walls. The amplitude and wavelength parameters are used for describing their influence on the flow, as well as for characterizing the cases in which the method is applicable. Since the approach requires only numerical integration in one dimension, the method proves to be much faster than general CFD simulations, while predicting the friction factor with adequate accuracy.Copyright

A Numerical Method for the Solution of Time‐Harmonic Maxwell Equations for Two‐Dimensional Scatterers

The Fourier modal method (FMM) is a method for efficiently solving Maxwell equations with periodic boundary conditions. In a recent paper [1] the extension of the FMM to non‐periodic structures has been demonstrated for a simple two‐dimensional rectangular scatterer illuminated by TE‐polarized light with a wavevector normal to the third (invariant) dimension. In this paper we present a generalized version of the aperiodic Fourier modal method in contrast‐field formulation (aFMM‐CFF) which allows arbitrary profiles of the scatterer as well as arbitrary angles of incidence of light.

Local defect correction for time-dependent partial differential equations

A Local Defect Correction (LDC) method for solving time-dependent partial differential equations whose solutions have highly localized properties is discussed. We present some properties of the technique. Results of numerical experiments illustrate the accuracy and the efficiency of the method.

An explicit method for solving flows of ODE

This paper is concerned with finding numerical solutions of a flow of ODE solutions. It describes a new method for finding solutions of problems, when the flow field is not given explicitly, using an implicit method, in particular Euler Backward, as well as interpolation. The special aspect is that the method as such is explicit, and resolves the flow with a similar accuracy as Euler Backward, even when the problem is stiff. An analysis is given of both stability and accuracy and extensions to non-autonomous problems are discussed. A number of numerical examples illustrates this analysis.