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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.

In this paper we are concerned with the elliptic PDEs with highly oscillating coefficients which model the behaviour of composite linear elastic materials. Analytical expressions for the effective coefficients are obtained for the case of layered materials using the theory of homogenisation. Some properties of the homogenised materials are analysed. An efficient algorithm for the numerical determination of the effective coefficients is proposed.

This paper describes the present status of using lattice gauge and ghost field methods for the simulation of on-chip interconnects and integrated passive components at low and high frequencies. Test structures have been developed and characterized in order to confront the simulation techniques with experimental data. The solution method gives results that are in agreement with the measurements.

Blow moulding is an essential stage of manufacturing glass and polymer containers, i.e. bottles or jars. A preform is brought into a mould and subsequently blown into the mould shape to produce the container. Two different problems regarding blow moulding are considered: the forward problem, which consists of determining the mould shape from the preform shape, and the inverse problem, which consists of determining the optimal preform shape corresponding to the designed container shape. This paper is concerned with the constraints on the mould surface and sensitivity to perturbations in the shape for both problems.

In the present paper we address the problem of optimal wall-shape design of a single phase laminar thermosyphon loop. The model takes the buoyancy forces into account via the Boussinesq approximation. We focus our study on showing the effects of wall shape on the flow and on the temperature inside the thermosyphon. To this extend we determine the dependency of the flow rate and the increase in temperature, on the geometrical characteristics of the loop. The geometry considered is a set of axially symmetric corrugated pipes described by a set of parameters; namely the pipe inner radius, the period of the corrugation, the amplitude of the corrugation, and the ratio of expansion and contraction regions of a period of the pipe. The governing equations are solved using the Finite Element Method, in combination with an adaptive mesh refinement technique in order to capture the effects of wall shape. We characterize the effects of the amplitude and of the ratio of expansion and contraction. In particular we show that for a given fixed amplitude it is possible to find an optimal ratio of expansion and contraction that minimizes the temperature inside the thermosyphon. The results show that by adequately choosing the design parameters, the performance of the thermosyphon loop can be improved.

In this paper we address the problem of wall-shape optimization for axially symmetric corrugated pipes. The main objective is, to increase the flow rate in a periodic section of a pipe, by modifying the wall-shape from the traditional cylindrical shape. We tackle this problem by numerically solving the Navier-Stokes equations for the periodic section of the pipe. The numerical model is validated by comparing our numerical results with available experimental data on the pressure drop and friction factor. The wall-shape optimization problem is tackled by considering a family of periodic pipes, in which the wall-shape is characterized by the amplitude, and the ratio between the lengths of expansion and contraction of the periodic section. We first study the effects of varying these parameters and then we show that for small Reynolds numbers the optimal shape turns out to be symmetric, while for large Reynolds numbers, a configuration with a large expansion region, followed by a short contraction region, performs better. The dependency of the optimal ratio on the pressure gradient is studied, and at the same time, we quantify the improvement in terms of flow rate (reduction in friction). Depending on the kind of geometry and the applied pressure gradient, the flow rate can be increased by 8%, for a geometry with small period. In the case of a geometry with large period, the flow rate increases by 35%, for large Reynolds number, and even 120% for small Reynolds numbers.

Anteryon BV produces optical lenses at the wafer scale, which have applications in several areas, the predominant one being in mobile phone cameras. The basic components of a wafer lens are a polymer, which satisfies the optical requirements, and a glass substrate on which this is bonded. During the manufacturing process, the polymer shrinks due to the polymerisation and as a result the final product will not have the designed shape. We have succeeded in modelling this deformation of the rotationally symmetric lens, using the principles of linear elasticity, both for an incompressible and for a more realistic, compressible medium. We have formulated the governing differential equations for the system, under a set of assumptions and used a perturbation technique based on the smallness of the aspect ratio e, to obtain analytical solutions which include the O(e^2) terms, for the displacements of the material points of the lens. These displacements are used to formulate an analytical formula for the resulting lens shape, after a certain amount of shrinkage, given by the parameter zeta has taken place. With the second order terms, the developed model is an improvement on the existing models at Anteryon, which correspond with the zeroth order terms of our model. Using this analytical formula, we also studied and developed analytical formulae for the inverse problem, i.e., what should be the mould shape in order to achieve the target lens shape. We have illustrated the developed model by applying it to a theoretical spherical mould. We have validated the model by comparing it to the measurements of moulds and lenses provided by Anteryon. We have applied the model to measured mould-lens combinations for many positions on the wafer, and for different lens shapes. For a given wafer, we have optimised the parameter zeta by considering the Root Mean Square value of the difference between the model output and the observed lens shape for each combination. The main results are a validation of our model and a method to determine the shrinkage parameter zeta, under specified process conditions. With this value of zeta, and the inverse model, we have a method to predict the ideal mould shape for a target lens shape. In the most ideal case, only one iteration step would be needed to predict this mould shape. Finally, we have also proposed a relation to determine Poisson's ratio for the medium; this result is based on a comparison between the incompressible and compressible models and the measurements.

Most autofocus 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 gradient-based sharpness function for two-dimensional images (2-D setting). Within this setting we are able to take into account the asymmetry of the optical device objective lens (astigmatism aberration). This study provides a useful extension of the analytical observations for one-dimensional images (1-D setting) that have been done before. The gradient-based autofocus method is implemented and demonstrated for the real-world application running in the FEI scanning transmission electron microscope prototype.

The Fourier modal method (FMM) is a method for efficiently solving Maxwell’s equations with periodic boundary conditions. In order to apply the FMM to non-periodic structures, perfectly matched layers need to be placed at the periodic boundaries and the Maxwell equations have to be formulated in terms of a contrast (scattered) field. This reformulation modifies the structure of the resulting linear systems and makes the direct application of available stable recursion algorithms impossible. We adapt the well-known S-matrix algorithm for use with the aperiodic Fourier modal method in contrast field formulation (aFMM-CFF). To this end, stable recursive relations are derived for linear systems with non-homogeneous structure. The stability of the algorithm is confirmed by numerical results.

In the present paper we address the problem of optimal wall-shape design of a single phase laminar thermosyphon loop. The model takes the buoyancy forces into account via the Boussinesq approximation. We focus our study on showing the effects of wall shape on the flow and on the temperature inside the thermosyphon. To this extend we determine the dependency of the flow rate and the increase in temperature, on the geometrical characteristics of the loop. The geometry considered is a set of axially symmetric corrugated pipes described by a set of parameters; namely the pipe inner radius, the period of the corrugation, the amplitude of the corrugation, and the ratio of expansion and contraction regions of a period of the pipe. The governing equations are solved using the Finite Element Method, in combination with an adaptive mesh refinement technique in order to capture the effects of wall shape. We characterize the effects of the amplitude and of the ratio of expansion and contraction. In particular we show that for a given fixed amplitude it is possible to find an optimal ratio of expansion and contraction that minimizes the temperature inside the thermosyphon. The results show that by adequately choosing the design parameters, the performance of the thermosyphon loop can be improved. Keywords: Thermosyphon, Corrugated Pipes, Shape Optimization

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.

This paper extends the area of application of the Fourier modal method from periodic structures to non-periodic ones illuminated under arbitrary angles. This is achieved by placing perfectly matched layers at the lateral boundaries and reformulating the problem in terms of a contrast field.

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. Keywords: Fourier modal method, FMM, aperiodic, aFMM-CFF, perfectly matched layer, PML.

A simultaneous autofocus and two-fold astigmatism correction method for electron microscopy is described. The method uses derivative-free optimization in order to find a global optimum of an image variance, which is an image quality measure. The Nelder-Mead simplex method and the Powell interpolation-based trust-region method are discussed and compared for an application running on a scanning transmission electron microscope.

This paper contains an overview of a self-adjusting multirate method. A simple extension which allows the improvement of the efficiency of the method is introduced. The performance of the extended and the original method is compared for a test problem.

The time evolution of power systems is modeled by a system of differential and algebraic equations. The variables involved in the system may exhibit different time scales. In standard numerical time integration methods the most active variables impose the time step for the whole system. We present a strategy, which allows the use of different, local time steps over the variables. The partitioning of the components of the system in different classes of activity is performed automatically and is based on the topology of the power system.

Automatic focusing methods are based on an image quality measure, which is a realvalued estimation of an image’s sharpness. In this paper we study L_1- or L_2-norm derivative-based image quality measures. For a bench mark case these measures turn out to be quadratic, which implies that after obtaining of at least three images one can find the position of the optimal defocus. The resulting autofocus method is demonstrated for a reference scanning transmission electron microscopy application. Keywords: Electron microscopy · Autofocus · Linear image formation · Image quality measure.

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 which does not contain the incoming field. Due to 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.

Multirate time-integration methods [3–5] appear to be attractive for initial value problems for DAEs with latency or multirate behaviour. Latency means that parts of the circuit are constant or slowly time-varying during a certain time interval, while multirate behaviour means that some variables are slowly time-varying compared to other variables. In both cases, it would be attractive to integrate these slow parts with a larger timestep than the other parts. This saves the computational workload while the accuracy is preserved. A nice property of multirate is that it does not use any linear structure, in contrast to MOR, but only a relaxation concept. If the coupling is sufficiently monitored and the partitioning is well chosen, multirate can be very efficient.

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’s controls, namely defocus and two-fold (two-parameter) astigmatism. In this paper the new method for power spectrum orientation identification is proposed. The method is based on the three measures which are related to the microscope’s controls. The measures are derived from the mathematical moments of the power spectrum. The method 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 non-experienced scanning electron microscopy user, as well as a basis for automated application.

A new simultaneous autofocus and two-fold astigmatism correction method is proposed for High Angle Annular Dark Field Scanning Transmission Electron Microscopy (HAADF-STEM). The method makes use of a modification of an image variance, which has already been used before as an image quality measure for different types of microscopy. In this paper we describe numerical simulations based on a classical HAADF-STEM linear image formation model showing that the modified variance reaches it's maximum for Scherzer focus and zero astigmatism. In order to find this maximum in a three-parameter space we employ the well-known Nelder-Mead simplex optimization algorithm. The method is implemented and tested on a FEI Tecnai F20.It successfully finds the optimal defocus and zero astigmatism with the time and accuracy, compared with the human operator. The method is iterative, and finding the optimal defocus and zero astigmatism requires obtaining typically 20-50 images.

Composites are often exposed to harsh loading conditions. This may lead to crack formation and propagation. In this paper an algorithm is described to predict the path of pre-existing cracks in homogeneous materials, based on an incremental approach. The approach is shown to also deal with the highly heterogeneous behaviour of periodically distributed composites.

A one-dimensional mathematical model is derived for a three-stage pulse-tube refrigerator (PTR) that is based on the conservation laws and the ideal gas law. The three-stage PTR is regarded as three separate single-stage PTRs that are coupled via proper junction conditions. At the junctions there are six fluid flow possibilities each defining its own boundary conditions for the adjacent domains. Each single stage cools down the gas in the regenerator to a lower temperature such that the system reaches its lowest temperature at the cold end of the third stage. The velocity and pressure amplitudes are decreasing towards the higher stages and there is an essential phase difference between them at different positions. The system of coupled PTRs is solved simultaneously first for the temperatures and then for the velocities and the regenerator pressures. The final result is a robust and accurate simulation tool for the analysis of multi-stage PTR performance.

Mc 2009. Microscopy Conference, Graz, Austria. 30 August - 4 September 2009. First Joint Meeting Of Dreiländertagung And Multinational Congress On Microscopy.

The interaction between premixed flames and acoustic perturbations of the gas velocity leads to combustion noise. In order to design noise-free combustion devices, one needs to understand the detailed mechanism by which the combustion noise is produced. The conical Bunsen flame is an excellent model for theoretical studies of the combustion noise.