Progress in Industrial Mathematics at ECMI 2004

Some years ago the national CFD project MEGAFLOW was initiated in Germany, which combined many of the CFD development activities from DLR, universities and aircraft industry. Its goal was the development and validation of a dependable and efficient numerical tool for the aerodynamic simulation of complete aircraft which met the requirements of industrial implementations. The MEGAFLOW software system includes the block-structured Navier-Stokes code FLOWer and the unstructured Navier-Stokes code TAU. Both codes have reached a high level of maturity and they are intensively used by DLR and the German aerospace industry in the design process of new aircraft. Recently, the follow-on project MEGADESIGN was set up which focuses on the development and enhancement of efficient numerical methods for shape design and optimization. This paper highlights recent improvements and enhancements of the software. Its capability to predict viscous flows around complex industrial applications for transport aircraft design is demonstrated. First results concerning shape optimization are presented.

The optimal profile of turbine blades is crucial for the efficiency of modern powerplants. The applied SQP algorithms are based on gradient information.

In this work I present a technique of construction and fast evaluation of a family of cubic polynomials for analytic smoothing and graphical rendering of particles trajectories for flows in a generic geometry. The principal result of the work was implementation and test of a method for interpolation of 3D points by regular parametric curves, and fast and efficient evaluation of these functions for a good resolution of rendering. For this purpose I have used a parallel environment using a multiprocessor cluster architecture. The efficiency of the used method is good, mainly reducing the number of floating-points computations by caching the numerical values of some line-parameter’s powers, and reducing the necessity of communication among processes. This work has been developed for the Research & Development Department of my company for planning advanced customized models of industrial burners.

Optimal path-constrained trajectories of an ISS-based, three-link robot are investigated with a monorail as an additional fourth and prismatic joint. This results in a problem of optimal control for a multiple constrained nonlinear system of differential-algebraic equations. After transformation into minimum coordinates, the only remaining control is the acceleration of the end-effector along the prescribed trajectory, replacing four actuator torques/forces in the original formulation. The simpler structure is achieved at the price of introducing piecewise defined equations of motion, two highly nonlinear control constraints and two state constraints of first order. Switching points between partly linear and fully rotational motion are optimized. Solutions are presented including touch points of the state constraints with the two control constraints being active simultaneously. For the mathematical treatment of those problems, new interior point conditions are derived.

The transmitting spherical reflector antenna (SRA) has a well-known rigorous solution form as a second kind Fredholm system that is well conditioned when truncated to a finite system. The size of such systems for extremely large SRAs require specially designed highly efficient numerical algorithms to make their analysis feasible. Two significant features of the system are that its convolution format admits a computationally rapid implementation of the bi-conjugate gradient method, and at high frequencies, a certain decoupling occurs. These features allow an effective numerical treatment of apertures some thousands of wavelengths.

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.

Surface acoustic wave filters are widely used for frequency filtering in telecommunications. These devices mainly consist of a piezoelectric substrate with periodically arranged electrodes on the surface. The periodic structure of the electrodes subdivides the frequency domain into stop-bands and pass-bands. This means only piezoelectric waves excited at frequencies belonging to the pass-band-region can pass the devices undamped.

The goal of the presented work is the numerical calculation of so-called “dispersion diagrams”, the relation between excitation frequency and a complex propagation parameter. The latter describes damping factor and phase shift per electrode.

The mathematical model is governed by two main issues, the underlying periodic structure and the indefinite coupled field problem due to piezoelectric material equations. Applying Bloch-Floquet theory for infinite periodic geometries yields a unit-cell problem with quasi-periodic boundary conditions. We present two formulations for a frequency-dependent eigenvalue problem describing the dispersion relation.

Reducing the unit-cell problem only to unknowns on the periodic boundary results in a small-sized quadratic eigenvalue problem which is solved by QZ-methods. The second method leads to a large-scaled generalized non-hermitian eigenvalue problem which is solved by Arnoldi methods.

The effect of periodic perturbations in the underlying geometry is confirmed by numerical experiments. Moreover, we present simulations of high frequency SAW- filter structures as used in TV-sets and mobile phones.

Diffraction gratings are often used in optical metrology. When an electromagnetic wave is incident on a grating, the periodicity of the grating causes a multiplicity of diffraction orders. In many metrology applications one needs to know the diffraction efficiency of these orders. Since the period of a grating is often of the same order of magnitude as the wavelength, it is needed to solve Maxwell’s equations rigorously in order to obtain these diffraction efficiencies. Two of those methods are the rigorous coupled-wave analysis (RCWA) and the C method.

In this paper a comparison is made between RCWA and the C method with respect to accuracy and speed. Restrictions are made to one-interface problems, which means that only two media are involved separated by one interface, and only gratings are considered with a periodicity in only one direction.

A numerical study of domain wall relocation during slow voltage switching is presented for doped semiconductor superlattices. Unusual relocation scenarios are found and interpreted according to previous theory.

A nonlocal (quantum) drift-diffusion equation for the electric field and the electron density is derived from a Wigner-Poisson equation modelling quantum vertical transport in strongly coupled semiconductor superlattices, by using a consistent Chapman-Enskog procedure. Numerical solutions for a device consisting of a n-doped superlattice placed in a

n

+

-

n

-

n

+

diode under a constant voltage bias are presented and compared with those obtained by using a semiclassical approximation.

Some propositions for approximation of the controllability and observability gramians for nonlinear systems are presented. This enables a balancing type model reduction to be performed for nonlinear systems in the same manner as for linear systems.

The distribution of electromagnetic fields, forces and source term of temperature induced by an alternating axially-symmetric system of electric current in a cylinder of a finite length with 6 electrodes has been investigated and calculated in [2, 1].

In this paper the three-phase alternating current with phase shift 120 degree is fed to every of 9 discreate circular conductors-electrodes, which are placed on the internal wall of the cylinder. The motion of electrolyte and temperature distribution in a cylinder has been calculated in dependence of the arrangement of electrodes.

Numerical aspects for solving of certain problem arising in gyrotron theory are discussed. Particularly, finite-difference schemes using quasistationarization and method of lines were applied and the relevant results analyzed.

A deterministic solution method for the coupled Boltzmann-Poisson system regarding spatially two-dimensional problems is presented. The method is based on a discontinuous piecewise polynomial approximation of the carrier distribution function. The conduction band of silicon is modelled by a non-parabolic six-valley model. In particular, we applied the multicell method to simulate the transients of a silicon MESFET. The results are compared to Monte Carlo simulations.

Vector Fitting

(VF) is an iterative technique to construct rational approximations based on multiple frequency domain samples, introduced by Gustavsen and Semlyen [1, 3]. VF is nowadays widely investigated and used in the

Power Systems

and

Microwave Engineering

communities. Numerical experiments show that VF has favorable convergence properties. However, so far, no theoretical proof for its convergence, or conditions to guarantee convergence, have been published. This paper gives a description of a general iterative Least-Squares framework for rational approximation and shows that VF fits into this framework.

Krylov subspace methods are well-known for their nice properties, but they have to be implemented with care. In this article the mathematical consequences encountered during implementation of Krylov subspace methods in an existing layout-simulator are discussed. Briey, the representation in a circuit is visited and two methods to avoid parts of the redundancy are drawn.

Several iterative methods have been tested in nonlinear DC analysis of industrial electronic circuits.

The efficient implementation of the FDTD algorithm in C, particularly the data types and nested loops required, is discussed. The different constructs were run on four computer platforms indicating significant performance improvement with proper implementation. The extent of the improvement depends on the data type, compiler and computer used.

Finite element approximation in space and Crank-Nicolson approximation in time are used to model incompressible creeping flow of molten glass with temperature dependent viscosity. Iso-P triangle elements and second degree approximation of temperature and velocity fields are applied. Localized thermal behavior is captured with adaptively refined unstructured mesh.

The simulation of circuits including signals with widely separated time scales can easily become very time-consuming. To avoid this, a multidimensional signal model was developed. The resulting system of network equations can be solved very efficiently by a method of characteristics. We investigate the applicability of this method to circuits including digital signal structures. Moreover, systems given in linear-implicit form are solved using the multidimensional approach.

We present a modification of a well-known mathematical model based on the Rigorous Coupled-Wave Analysis (RCWA) that can be used to solve optical diffraction problems on periodic structures (both 1-D and 2-D gratings with approximated layer-structure). The algorithm calculates the reflected and transmitted field which in turn determine the diffraction efficiencies for all reflected and transmitted orders.

Results created with a Matlab implementation of the modified RCWA algorithm (MSolver) show excellent overlap with other published and measured data.

One of the fields of engineering science in which numerical simulation is playing a role of increasing importance is the design of piezoelectric transducers. Efficient techniques to solve the forward problem of computing the mechanical displacements and electric potential for a given configuration play a crucial role in the design itself, but also in the related problem of identifying the correct material parameters. In this paper we consider the iterative solution of linear systems arising from a Finite-Element discretisation of the piezoelectric forward problem with the Generalised Minimal Residual method in combination with incomplete LU decomposition and inexact block diagonal preconditioning.

Transport phenomena in a submicron

npn

silicon bipolar junction transistor are described by using an extended hydrodynamic model for the electrons, combined with a solution of the drift-diffusion model for the holes. Under suitable scaling assumptions, the above model reduces to the energy transport model, or to the Navier-Stokes-Fourier model, in which all the transport coefficients are now explicitly determined. The validity of the constitutive equations is investigated by using Monte Carlo simulations.

In radio frequency (RF) application, electric circuits often exhibit multitone signals, where time scales differ by several orders of magnitude. Thus circuit simulation by means of transient analysis becomes inefficient. A multivariate model yields an alternative strategy considering amplitude as well as frequency modulation. Consequently, a warped multirate partial differential algebraic equation (MPDAE) has to be solved using periodic boundary conditions. Thereby, the determination of a local frequency function is crucial for the efficiency of the model. For this purpose, two special choices of continuous phase conditions are applied as additional boundary conditions. Numerical simulations show that these continuous phase conditions identify local frequency functions, which are physically reasonable.

The maximum entropy moment systems of the Boltzmann equation is only solvable with physically unrealistic restrictions on the choice of the macroscopic variables. We show that no such difficulties appear in the semiconductor case if Kane’s dispersion relation is used for the energy band of electrons. As an application the 5-moment model is discussed.

Two main approaches are known for the reduced order modelling of linear time-invariant systems: Krylov subspace based and SVD based approximation methods. Krylov subspace based methods have large scale applicability, but do not have a global error bound. SVD based methods do have a global error bound, but require full space matrix computations and hence have limited large scale applicability. In this paper features and short-comings of both types of methods will be addressed. Furthermore, ideas for improvements will be discussed and the possible application of Jacobi-Davidson style methods such as JDQR and JDQZ for model reduction will be considered.

This paper presents a new Runge-Kutta type integration method that is well-suited for time-domain simulation of oscillators. A unique property of the new method is that its damping characteristics can be controlled by a continuous parameter.

Adaptive stepsize control is used to control the local errors of the numerical solution. For optimization purposes smoother stepsize controllers are wanted, such that the errors and stepsizes also behave smoothly. We consider approaches from digital linear control theory applied to multistep BDF-methods.

The stabilization of a Bunsen flame above the burner rim is simulated using the method of characteristics. Oscillations of the flame front and of its area due to flow oscillations are computed.

A generalized deffinition is given for the time index and a new prototype example is introduced, which serves as a general case for the computation of the time index for a hierarchy of molten carbonate fuel cell models, including a 2D model. The time indices are computed by a new approach using linear integral equations.

The gelation of polymer mixtures under constant cooling rate has been found to be an attractive product structuring mechanism for the food industry. As applications become wider, a predictive method for the process is warranted. To this end, we apply the so-called ‘

S

γ

concept’ in a CFD module for the modeling for microstructure formation of gelling mixtures, where moments of the particle size distribution are evaluated using the local flow conditions as obtained from CFD simulations for the processes considered. The major driving force for these processes is the competition between phase separation, gelation and hydrodynamic phenomena such as break-up and coalescence. Based on theoretical investigations, analytical expressions for the source terms representing the hydrodynamics (break up and coalescence of the droplets) as well as the gelation process were produced. Constitutive models are developed to incorporate the effects of phase separation and gelation on the rheology of the phases. The simulations for different cooling rates clarified the inter-relationships between the competitive mechanisms by depicting the time interval of the domination of each.

We analyze the evolution of a diffusion flame in a turbulent mixing layer. The location of the flame-center is defined by the “stoichiometric” interface. Geometrical properties such as its surface-area, wrinkling and curvature are characterized using an accurate numerical level-set quadrature method. This allows to quantify flame-properties as well as turbulence modulation effects due to coupling between combustion and turbulent transport. We determine the active flame-region which is responsible for the main part of the chemical processing in the flame.

Recently, Burger and Capasso [M

3

AS 11 (2001) 1029–1053] derived a coupled system of partial differential equations to describe non-isothermal crystallization of polymers. The system is based on a spatial averaging of the underlying stochastic birth-and-growth process describing the nucleation and growth of single crystals. Using an appropriate scaling of the original system, we derive a simplified model which only consists of a reaction-diffusion equation with memory for the underlying temperature, such that the degree of crystallization can be explicitly given by a time integration of the temperature-dependent growth and nucleation rate. Numerical simulations indicate that the reduced model shows at least qualitatively the same behavior like the original model.

The aim of this work is to introduce and numerically solve an axisymmetric mathematical model for thermoelectrical simulation of an induction heating furnace.

Thermal explosion of diesel fuel droplets in the presence of thermal radiation is studied. The process is presented in terms of the dynamics of a multi-scale and singularly perturbed system, which is analyzed using the geometrical version of the Method of Integral Manifolds. Analytical estimates of the total ignition delay times in two limiting cases are obtained. The influence of the thermal radiation on the heat transfer and ignition delay time are clarified.

An outline of the Local Defect Correction (LDC) method is given. The method is combined with a procedure to construct an orthogonal, curvilinear fine grid and it is applied to the thermo-diffusive model for laminar flames.

This contribution deals with the mathematical modelling of a high temperature molten carbonate fuel cell (MCFC) and serves as a basis for the following three contributions of this mini-symposium. After a motivation and a short introduction into the working principle of the MCFC, the most important equations of the model are presented. This model is applied for optimisation purposes and as a basis for the derivation of reduced models specifically designed for different tasks.

The reduction of exhaust particulate emissions from diesel vehicles is a great upcoming challenge. As a result of their harmful effects, new legislation on diesel vehicles has been introduced throughout the world specifying low emissionlevels. Today, the use of diesel particulate filter (DPF) in addition to engine modifications is the favoured method to fulfil these criteria. The principle of a DPF is based on the accumulation of particles in the alternating open and closed channels of the filter. The pressure drop over the DPF increases with time. This increase is associated with the rise of fuel consumption. For this reason, the deposited filter cake must be occasionally regenerated. To minimise complex and expensive investigations on test benches, a mathematical model has been developed describing the loading and regeneration behaviour of a DPF. The model is integrated in a commercial CFD-Code using user-defined subroutines (UDS). The CFD-Code was used for the calculation of the fluid flow and the particle tracks of different kinds of particles (e.g. soot, additives) in a two-dimensional model of the DPF. Thus, the axial and radial structure of the deposited particles on the filter can be determined. In the UDS models are implemented to calculate the pressure loss, the separation efficiency and the regeneration behaviour. Comparing the simulation results with the results gained experimentally, it can be seen that both sets of data concur. Further development concerning the implementation of a subroutine to describe the long-term behaviour and transport of the deposited particles will be carried out.

A model was applied on experimental data to study the mass transport of oxygen diffusing through the oil phase and the packaging materials as well as the oxidation reactions. A nonlinear system was numerically solved for various combinations of materials, temperatures, and light availability, by adopting a typical Newton method, in conjunction with a multi-step up-winding finite differences scheme. The probability of the packaged olive oil not to reach the end of its shelf life (

P

safe

) and its time evolution, was in very good agreement with the experimental data.

P

safe

was proposed as a reduction indicator for shelf life predictions at “real-life” conditions. Exposure to light at any pattern could significantly stimulate the oxidative degradations, only assisted by elevated temperatures and presence of oxygen. Plastic containers showed particularly higher protective role when oil was stored at light, while glass was the most protective material when oil was stored at dark.

A reduced nonlinear model of a planar molten carbonate fuel cell is presented. The model is derived from a spatially distributed dynamic model of the cell by applying the Karhunen Loève Galerkin procedure. The reduced model is of considerably lower order than the original one and requires much less computation time. The comparison between the two models shows that the reduced model can describe the dynamic of the temperature field with sufficient accuracy and has good extrapolation qualities with respect to changes in the model parameters.

We present the results of the numerical simulation of the first stages of the melting from a side of a gallium slab by adding to the heat transfer and to the melt flow the description of the effects of the deformations of the solid phase. The experiment by Gau and Viskanta in [4] has been considered.

This work presents a dimensionless analysis of mass transport equations in fixed-bed absorbers. Focus is centered in isothermal and incompressible problems, with special attention to nonlinear adsorption and desorption processes that take place at absorbent particles. The general differential-algebraic equation system is expressed in dimensionless form, and the model is particularized into four different formulations. The model is analyzed and used to simulate a standard industrial test efficiently. Formulations are selected depending on the relative importance of the different physical phenomena involved in each part of test.

For studying the impact of a hight pressure vapor on a concrete wall, we propose a stationary 3D homogenized model. We show that the interface evolves as a (shock or rarefaction) wave accordingly with the mobility coefficient values

M

. Moreover, we prove the existence of a finite asymptotical position for the interface when

t

goes to +∞.

We implement a multigrid algorithm to solve the radiative heat transfer equations in glass production. The time, angle and space coordinates are discretized using Crank-Nicolson, discrete-ordinate and Galerkin methods, respectively. Based on the same mesh hierarchy for both heat conduction and radiative transfer, our multigrid algorithm consists on using the Newton-Gmres and Atkinson-Brakhage solvers as smoothers on the coarse meshes.

This paper describes the modelling of the toner behaviour in the development nip of the Océ Direct Imaging print process. The discrete element method is used as the simulation tool for a quantitative description of the system. The interaction rules and the associated parameters are determined for the toner particles and the surfaces of the development rollers. The model is validated with print quality results. It is shown that it is possible to achieve quantitative agreement between DEM simulations and experimental print quality results.

Drying in porous structures is simulated with a 2-D pore network model that accounts for various processes at the pore-scale (mass transfer by advection and diffusion in the gas phase, viscous flow in liquid and gas phases and capillary effects at the gas-liquid interface). We further study the effect of capillarity-driven viscous flow through macroscopic liquid films. It is shown that film flow is a major transport mechanism in drying of porous media, its effect being dominant when capillarity controls the process, which is the case in typical applications.

The Process Plant Layout (PPL) problem involves decisions concerning the spatial allocation of equipment items and the required connections among them, [3]. The objective of the PPL problem is to determine the optimal spatial allocation of equipment items and the required connections between them. PPL problems have mostly been solved by heuristic rules but in recent years significant research effort has been put on more rigorous methods, mainly based on mathematical programming techniques. The resulting problem is often subsequently discretised in linear form and solved using linear solvers. In this paper, a non-linear approach to the general PPL problem is investigated. A comparative study between different non-linear solvers is carried out and the performance of the solvers is evaluated.

We present a simple mathematical model of fluid flow in a Scraped-Surface Heat Exchanger (SSHE). Specifically we consider steady isothermal flow of a Newtonian fluid around a periodic array of pivoted scraper blades in a channel with one stationary and one moving wall, when there is an applied pressure gradient in a direction perpendicular to the wall motion. The flow is fully three-dimensional, but decomposes naturally into a two-dimensional transverse flow driven by the boundary motion and a longitudinal pressure-driven flow.

The transmission line matrix (TLM) for simulating sound wave propagation in stationary and moving media is presented. TLM is inherently a timedomain approach which does not require solution of a differential equation. TLM and FEM are compared in terms of accuracy and computational complexity. It is concluded that TLM may represent a more efficient alternative to FEM when predicting acoustic fields in stationary media. Furthermore, applicability of TLM to moving media is examined. A TLM model of 2D moving media based on the idea of [3] is introduced.

We develop a lubrication model for the viscous drop spreading with evaporation. It is then solved in the quasi-static case and the numerical method is used in the parameter identification in the application to DNA chips.

We propose and evaluate a method for the recognition of airborne fungi spores. We suggest a similarity-based object-recognition method to identify spores in a digital microscopic image. We do not use the gray values of the case, but the object edges instead. The similarity measure measures the average angle between the vectors of the template and the object. Case generation is done semi-automatically by manually tracing the object, automatic shape alignment, similarity calculation, clustering, and prototype calculation.

Miniaturized robotic manipulators are a key element in future highprecision minimally invasive surgery and telesurgery. This development is supported by the rapidly decreasing size of robotic sensors and actuators. Severe limitations are currently induced by the drives of the micro-joints.

The present paper deals with the optimal control of an advanced six-sectional branched manipulator. Joints are driven by weak, but fast, and strong, but slow, actuators acting in parallel. This results in the new and challenging problem of constrained optimal control of multibody systems subject to rivalling controls. For efficient treatment the differential equations of the state and adjoint variables are recursively defined. Geometric constraints lead to state constraints of second order. The complete problem of optimal control is transferred into a piecewise defined, highly nonlinear multi-point boundary value problem. The numerical solution of the boundary value problem is by the advanced multiple shooting method

JANUS

.

A mathematical and numerical model to predict the non-linear behaviour of concrete as multiphase porous material is proposed. The model can be usefully applied to several practical cases: evaluation of concrete performance in the high temperature range, e.g. during fire, to early stages of maturing of massive concrete structures, to shotcrete in tunnelling, and to durability. All the important phase changes of water and chemical reactions, i.e. adsorption-desorption, condensation-evaporation, and hydration-dehydration, as well as the related heat and mass sources or sinks are considered. Changes of the material properties caused by temperature and pressure changes, concrete damage or carbonation, fresh concrete hardening, as well as coupling between thermal, hygral and mechanical phenomena are taken into account. This model further allows to incorporate sorption hysteresis. Some relevant applications of the model will be shown in this work.

For the modelling of the glass press-blow process level set functions are used. Special difficulties arise due to velocity gradients in the domain. A reinitialisation procedure for unstructured triangular meshes is adapted to these difficulties and is applied.

We consider a model for laser surface remelting, a process to improve the surface quality of steel components. The mathematical model consists of the two-dimensional heat equation for temperature and an ordinary differential equation for the liquid phase. The equations are coupled via source terms. We study the efficient numerical simulation using adaptive grids, which are especially well-suited for problems with moving heat sources. To account for the local high activity due to the heat source, we introduce local uniform grids and couple the solutions on the global coarse and local fine grids using the local defect correction (LDC) technique.

We consider minimizing the mass of an injection moulding machine, fulfilling certain constraints. The deformation of its frame is described by the plain stress state equations for linear elasticity. The minimization problem is a nonlinear constrained one. When the design parameters change, then also the shape will change. Generating a new finite element mesh for each single shape leads to a non-differentiable objective. Here we deform the mesh elastically.

This work deals with the motion of a long slender elastic fiber in a turbulent flow. Neglecting the fiber effect on the turbulence, a centered differentiable Gaussian field is derived for the randomly fluctuating component of the flow velocity. The construction of the initial double-velocity correlation tensor is hereby based on the

κε

model and Kolmogorov’s universal equilibrium theory. Its dynamic is described by Taylor’s hypothesis of frozen turbulence. Using an empirical drag coefficient, the developed fluctuation field leads to a correlated stochastic force that can numerically be treated as white noise with flow dependent amplitude.

The conventional material from which silos are usually constructed is steel, and the existing codes and standards on these structures reect the design criteria appropriate for an isotropic material. This paper deals with the design optimisation of the silos made from composite materials. The purpose of the present study is to perform the design optimisation of cylindrical composite silos loaded with the unsymmetrical external pressure caused by the action of wind. The design methodology is outlined, and the effectiveness of the optimisation is demonstrated using a particular example. In this case, the resultant optimised design produced a 29% saving in wall thickness, and thus material cost, in comparison with the non-optimised wall thickness.

In this paper, we perform a linear stability analysis using normal modes on the two-dimensional system of two superposed fluids confined between two infinite plates in the presence of a large temperature gradient. The movement of the fluids is characterized by a combination of inertial and buoyancy forces, thus we are dealing with a mixed convection problem. The results of the linear stability analysis show that for large wave numbers, the small amplitude waves travel with the interface velocity.

The question of interest in the present study is the inverse problem for high precision glass forming, i.e. ‘How to design the mould and the temperature regime so that at the very end of the forming process we will get at room temperature a prescribed glass geometry with a precision in the order of the Micron?’ The aim is to eliminate from the manufacturing process the costly and time-consuming post-processing when the final shape does not conform precisely to the desired one.

A nonlinear first-order PDE describing the displacement of a glass surface subject to solid particle erosion is presented. The analytical solution is derived by means of the method of characteristics. Alternatively, the Engquist-Osher scheme is applied to compute a numerical solution.

This article discusses mathematical outlines of the numerical project combining particle method with radiation models in order to simulate glass cooling process. Its initial part gives a sketch of the particle Finite Pointset Method (FPM) [1], the next one debriefs the radiation models considered to implement in the method framework and the final one presents some preliminary, qualitative results of current research.

We introduce a novel multiscale approach for reservoir simulation as an alternative to industry-standard upscaling methods. In our approach, reservoir pressure and total velocity is computed separately from the fluid transport. Pressure is computed on a coarse grid using a multiscale mixed-finite element method that gives a mass-conserving velocities on a fine subgrid. The fluid transport is computed using streamlines on the underlying fine geogrid.

In the seventies of the previous century, the reinsurance treaty ECOMOR used to enjoy some limited popularity. However, since then the treaty has been largely neglected by most reinsurers, partly because of its technical complexity. In this paper, we give results pertaining to asymptotic properties of this reinsurance form. In particular, we formulate asymptotic estimates for the tail of the distribution of the ECOMOR-quantity. Furthermore, we give its weak laws.

We present an accurate numerical solution for the discrete Black-Scholes equation with only a few grid points. European and American option problems with deterministic discrete dividend modelled by a jump condition at the exdividend date are solved. Fourth order finite differences are employed, as well as a grid stretching in space and a Lagrange interpolation at the ex-dividend date.

Semi-Lagrange time integration is used with the finite difference method to provide accurate stable prices for Asian options, with or without early exercise. These are combined with coordinate transformations for computational efficiency and compared with published results.

The derivation of the risk neutral probabilities in a binary tree, in the presence of uncertainty on the underlying asset moves, boils down to the solution of dual fuzzy linear systems. The issue has previously been addressed and different solutions to the dual systems have been found. The aim of this paper is to apply a methodology which leads to a unique solution for the dual systems.

We present an effective way to estimate liquidity risk.

We focus on the inter-related roles of scale and heterogeneity of porous medium properties for fluid flow and contaminant transport in isothermal and non-isothermal multiphase systems across a range of scales. Multiscale network and macro-scale continuum models, and detailed laboratory experiments are used to support the investigation. We demonstrate the critical role of scale in determining the dominant forces in a porous medium system, the importance of heterogeneity across a range of scales, and the dominant role of block heterogeneities on macro-scale fluid flow and non-isothermal contaminant remediation. We give special attention to the numerical approximations of pressure-saturation-conductivity relations in heterogeneous systems, and we show the effects of interface approximation schemes on both the ability to resolve phenomena of concern and on the efficiency of the numerical simulator.

In this paper we analyse the unsteady expansion and contraction of a long, two-dimensional bubble confined between superheated or subcooled parallel plates, whose motion is driven by mass transfer between the liquid and the vapour.

Semi-Lagrangian techniques are proposed for animating water waves in realistic events. The two-dimensional shallow water equations are considered to model the motion of water flow and a second order time marching procedure which combines the characteristic method with a finite differencing discretization is used to integrate the model. Numerical results are carried out on a squared pool without and with obstacles. The obtained results show that our algorithm is robust, stable and highly accurate.

Models based on a filtered Poisson process are used for the flow of a river. The aim is to forecast the next peak value of the flow, given that another peak was observed not too long ago. The most realistic model is the one when the time between the successive peaks does

not

have an exponential distribution, as it is often assumed. An application to the Delaware River, in the USA, is presented.

The robust Parallel Finite Element Method examined in [5] and [4]. It is an element-wise parallel iterative solution method based on a Red-Black domain decomposition. Convection-diffusion problems are solved in an optimal order for a method which makes use of not more than local communication. For the parallellism, the recent paper [8] shows that a near perfect load-balance can be obtained for two-dimensional problems. This paper proves that one of the conditions which is sufficient in the two-dimensional case, unexpectedly is not so for the three-dimensional case.

A mathematical model for the flow and solidification of a thin liquid film is briefly described. Typical results for ice accretion due to incoming rain droplets on a at surface and aerofoil are shown.

A Finite Element Modified Method of Characteristics (FEMMOC) is proposed for numerical solution of the two-dimensional shallow water equations. The method is formulated and implemented for mean flow and hydraulics in the strait of Gibraltar. Preliminary results presented in this work show that the FEMMOC is able to provide stable, accurate and efficient solutions.

We discuss an algorithm for convection-diffusion equations with high activity areas which combines the Local Defect Correction technique with high order compact finite difference schemes.

Since steady-state nonlinear fluid sloshing in moving tanks is caused by a finite set of natural modes, approximate solutions of the original free boundary value problem can be found from a system of nonlinear ordinary differential equations (modal system) coupling time dependent amplitudes of these leading modes. We focus on two-dimensional flows in a rectangular tank. We present an extensive literature survey and examine bifurcations of periodic (steady-state) solutions of a single-dominant modal system derived by [1].

Failures in repairable systems are often described by means of renewal or non-homogeneous Poisson processes, depending upon the repair policy. In the former case repairs bring the system reliability back to its initial value, whereas in the latter they restore the same reliability the system had just before the failure. We focus on the latter process, illustrating some properties and applications, mainly in a Bayesian framework.

We present new efficient schemes of Rosenbrock’s type for numerical solution of differential-algebraic stiff systems. For these schemes, we develop an algorithm for accuracy control.

It is well known that discontinuities in pipe networks give reflections to pressure waves that can be analysed to find the time delay between the original signal and the reflected one. A leak in a pipe will also give a reflection point, though possibly a more diffuse one. It is a reasonably straightforward task (using, say, a cross correlation) to measure the time delay of the first reflection, but more complicated methods are required to extract data about further reflections from, for example, the end of the pipe which has a leak in it.

Cepstrum techniques were used to find the common pipe lengths in the network. Latterly, this has been used in conjunction with wavelet analysis to filter the data. Finally, continuous wavelets are being used. These help to explain many of the results that have previously been produced. These were conducted on both real (experimental) and modelled networks.

In this paper we compare several different strategies for robust design when the experiment is carried out via a computer simulator.

We consider oil-water flow in porous media, with a dynamic capillary pressure relation. This leads to a pseudo-parabolic degenerate regularisation of the Buckley-Leverett (BL) equation. It is known that linear pseudo-parabolic regularisations of BL lead to shock solutions that do not satisfy the Oleinik condition. In this note we analyse the existence of travelling wave solutions that violate the Oleinik condition, taking special care of the degeneracy of the problem.

Electronic products tend to be economically outdated before their technical end-of-life has been reached. The ability to analyze and predict the (remaining) technical life of a product would make it possible either to re-use sub-assemblies in the manufacture process of new products, or to design products for which the technical and economical life match. This requires models to predict and monitor performance degradation profiles. In this paper we report on designed experiments to obtain such models. We show how wavelet analysis can be used to extract features from electrical signals. These features are analyzed using the Analysis of Variance in order to establish relations between these features and performance degradation.

The preparation of simulation models from CAD models is still a difficult task since shape changes are often required to adapt a component or a mechanical system to the hypotheses and specifications of the simulation task. Detail removal or idealization operations are among the current treatments performed during the preparation of simulation models. In this paper we introduce the concept of simplification features, which allows a user to improve the efficiency of the analysis model generation process. As a result, form feature semantics and simulation data are attached to a polyhedral model during the preparation phase to ease the Finite Element(FE) details identification and removal as well as to maintain the consistency between a CAD model and its associated F.E models.

The effects of discretisation parameters on the performance of a space-time VMS FEM are investigated. A moving-wave solution of the one-dimensional viscous Burgers equation is used to limit the influence of SGS modelling errors. Factors influencing the magnitude of the implicit SGS model are discussed.

We present a continuous wavelet analysis of count data with time-varying intensities. The objective is to extract intervals with significant intensities from background intervals. This includes the precise starting point of the significant interval, its exact duration and the (average) level of intensity. We allow multiple change points in the intensity curve, without specifying the number of change points in advance. We extend the classical (discretised) continuous Haar wavelet analysis towards an unbalanced (i.e., asymmetric) version. This additional degree of freedom allows more powerful detection. Locations of intensity change points are identified as persistent local maxima in the wavelet analysis at the successive scales. We illustrate the approach with simulations on low intensity data. Although the method is presented here in the context of Poisson (count) data, most ideas (apart from the specific Poisson normalization) apply for the detection of multiple change points in other circumstances (such as additive Gaussian noise) as well.

One way to increase the robustness of soft sensors is to use ensembles of symbolic regression-based predictors. Ensembles can increase the robustness because it gives a more consistent estimate of the output, it allows the derivation a measure of confidence and it can be used for problem detection. In this paper we will demonstrate the robust soft sensor by using ensembles of symbolic regression-based predictors in an industrial application.

Large area uniform plasmas are essential in microelectronics processing. Motivated by this application, a macroscopic model is proposed as a framework for investigating the occurrence of instabilities in high frequency plasma discharges for parallel plate geometries. This paper will concentrate on the formation of stationary, spatially inhomogeneous patterns.

A simple mathematical model for the motion of a pipeline bundle being towed using the Controlled Depth Tow Method (CDTM) is constructed and analysed. When the forces exerted by the sea on the bundle are neglected the model predicts that the bundle is neutrally stable and that its motion involves two different timescales. When these forces are not neglected the model predicts that the bundle will always be stable if the tension in the bundle at its downstream end is sufficiently large.

In the design workflow, CAD models of complex components include more and more details. A transformation of such models into Finite Element (F.E.) models often generates a much too large number of elements to be used directly. Generally, the removal of shape details or idealization operations are required to prepare F.E. models. These modifications must preserve the analysis result and the user must control the process in order to ensure sufficient accuracy of the F.E. results. In accordance to the analysis problems, the simplification process generates different appropriate F.E. models. In this paper, we present different operators and criteria to prepare analysis models from CAD models.

In viscoelastic pipes, where the material properties depends on a complex bulk modulus as well as on a complex shear modulus, the sound field within the fluid is affected. Therefore, the dispersion of flexural waves occurs in the pipe, while the speed of flexural waves decreases due to the coupled fluid mass. Coupling between the pipe wall and the fluid also decreases the sound speed in the fluid. Likewise, the speed of sound in fluid is frequency-dependent, just as the group velocity of bending waves depends on the frequency. Wavelet transform of non-stationary sound signals was used to identify the frequency-dependent fluid sound speed. Measurement and analysis of non-stationary signals with the use of time-frequency method provides a view to frequency dependent transfer characteristics of fluid-pipe coupled system. The results also showed that, in the case of propagating small disturbances (such as acoustic waves), the pipe wall inertia has a minor influence on the wave propagation characteristics. The elastic reaction of the wall to expansion of the cross section greatly exceeds the inertial reactions.

In many science and engineering problems, one observes smooth behaviour on macroscopic space and time scales. However, sometimes only a microscopic evolution law is known. In such cases, one can approximate the macroscopic time evolution by performing appropriately initialized simulations of the available microscopic model in small portions of the space-time domain. This coarse-grained time-stepper can be used to perform time-stepper based numerical bifurcation analysis. We discuss our recent results concerning the accuracy of the proposed methods.

Molecular dynamics simulations are typically very costly. We investigate whether optimal prediction, a method to approximate the mean solution of a large system of ordinary differential equations by a smaller system, can in principle be applied to speed up computations. A one-dimensional, solely classical model problem, describing some aspects of coating a copper layer onto a silicon crystal, is considered. Asymptotic methods are employed to approximate the high-dimensional conditional expectations, which arise in optimal prediction. Results of a comparison of the thus derived smaller system with the original system are shown.

The chart surface approach, a variational grid generation method for surface grids, is applied to CAD models describing ship hulls and propellers.

Strongly damped mechanical systems arise, for example, in vehicle dynamics and in modelling joints in biomechanics. Standard explicit integrators become unstable unless very small time steps are chosen. We are interested in the numerical solution of such systems with step sizes that are independent of the damping parameter. The smooth motion of the mechanical system is expanded in terms of solutions of differential-algebraic systems of index 2. These results hold for analytical solutions as well as for numerical solutions of suitable methods such as Radau collocation. In the border case of big damping constants it turns out that the error of numerical solutions of the strongly damped mechanical system is bounded by errors for the differential algebraic systems.

This paper deals with Shape Memory Alloy (SMA) actuators for mechatronic applications. A mathematical model on the macroscopic level is discussed and a computational framework is introduced. The latter makes use of the method of lines and results in a system of differential-algebraic equations in time. Some first simulation results for a 1D wire in the isothermal case illustrate the approach.