Progress in Industrial Mathematics at ECMI 2002

This paper deals with some examples of new mathematical problems which were formulated during the mathematical modeling of real-world applications. Mostly we investigate numerical algorithms for solving systems of PDEs.

Alan Tayler, one of the founders of ECMI, influenced our work in various ways. For instance, the intensive cooperation between Milano and Linz in the field of polymer crystallization, see, e.g., [BCE99], was triggered by ECMI. Furthermore, Alan and his colleagues at Oxford, especially Hilary and John Ockendon, educated us in the theory of asymptotic and analytical methods for partial differential equations. In our opinion, a combination of these techniques with numerics will become more and more crucial for solving complex industrial problems. We hope to be able to illustrate this by the industrially motivated parameter identification problems treated in this paper.

Challenge of improving efficiency of recovery of hydrocarbons, crude oil and gas, from natural petroleum reservoirs, is an important and urgent global problem facing mankind. Many branches of science and technology contribute to its solution. Nowadays, computer modeling becomes a key element of integration of all multifarious knowledge and decision-making tool. Here, an effort is made to present basic mathematical models of enhanced oil recovery and some mathematical problems emerging from their study.

The paper provides an overview of business modelling techniques — both at language and tool level. The basic elements of some most popular business modelling languages are briefly outlined, including the GRAPES-BM language developed by IMCS, University of Latvia. The paper presents also the basic principles, how metamodel based generic modelling techniques can be used for supporting several modelling notations simultaneously.

To develop the path that goes from the clinical experience to the laboratories and back to the clinic, research in cancer modelling need the establishment of strong interactions among different branches of science and a genuinely multidisciplinary approach. In fact, starting from very practical situations this path passes through progressive abstractions and simplification steps to gain insight into the complex phenomena occurring during tumour evolution and growth and this involves different research areas (see Fig. 1).

Autonomous linear problems produce a purely oscillatory (i.e., with a zero temporal mean) response under a purely oscillatory excitation. Nonlinearity and/or oscillatory coefficients introduce steady (or slowly varying) terms, just because the product of in-phase purely oscillatory terms gives a non-zero mean, as in, e.g.,sin2wt = − 1/2 cos 2wt + 2 = purely oscillatory + steady. These non-oscillatory contributions may have an overall, unexpected effect on the slowly varying dynamics. A well known example is the stabilization of the upper equilibrium of a simple pendulum by high frequency vibration of the support [1], first uncovered by Stephenson [2]. A similar effect is even more striking in infinite-dimensional dynamical systems. If a glass of water is placed in an inverted position, then the liquid falls down quite rapidly, as everyday experience shows. This system exhibits a steady state with the free surface perfectly horizontal, which is of course highly unstable due to gravity. This is the simplest example of the well known instability named after Lord Rayleigh [3] and G.I. Taylor [4], which is always present when a light fluid is accelerated towards a denser one, as in accelerated fronts appearing in combustion [5], astrophysics [6], plasma physics [7], and inertial confinement fusion [7]. As in the simple pendulum, the static solution in the inverted glass can be stabilized by high frequency, vertical vibration of the container, as shown by Wolf [8,9]; see also [10] for a weakly nonlinear analysis of this effect in large aspect ratio containers.

The motion of flexible fibres suspended in an incompressible flow is of interest to researchers in various fields and in particular in textile manufacturing. This work deals with the coupled problem of a single fibre moving in the flow field. An elegant way to describe their interactions might well be to embed the fibre’s motion within a meshless flow concept using the ideas of the partition of unity method (PUM). The construction of PUM spaces that depend on the flow test spaces permits the adoption of the flow resolution and a simplification of the external forces acting on the fibre. The stability and accuracy of the PUM depend on the quality of the fibre cover: a filter algorithm is described that improves this quality. Simulations are presented to demonstrate the applicability of this method but the fact that the approximation spaces need to be modified at each time step can lead to a considerable increase in computational time.

A mathematical model presented in this paper describes the separation of pourable materials into two components. We prove that there is a quantitative convergence of each grain-size class at every stage of separation and obtain a calculation formula for finding the degree of fractional extraction for each narrow class.

One of the mathematical problems in the simulation of large electrical circuits is the solution of high-dimensional linear equation systems. While these systems are usually solved by direct methods iterative solvers are likely to outperform these if the problem dimensions are large enough. Though it is often possible to reduce the system size by utilizing the hierarchical design of electrical circuits some important problems do not lend themselves easily to this approach, and the demand for efficient solvers is ever-increasing.

Inverse problems arise in various fields of science, technology and agriculture where from measurements of state of the system or process it is required to determine a certain typesetting of the causal characteristics. It is known that infrigement of the natural causal relationships can entail incorrectness of the mathematical formulation of inverse problem. Therefore the development of efficient methods for solving such problems allow us to simplify experimental research considerably and to increase the accuracy and reliability of the obtained results due to certain complication of algoritms for processing the experemental data. The problem of the determination of the coefficient of thermal conductivity is among the incorrect inverse problem.

One of the modern areas of applications developed during last years is effective use of electrical energy produced by alternating current in production of heat energy. This process is ecologically clean.The water is weakly electrically conducting medium (electrolyte). Devices based on this principle are developed during last ten years. Compared to classical devices with heating elements, new devices are more compact.

The finite element method is widely used as a numerical method for handling elliptic partial differential equations. There is ongoing research in trying to find good finite element solutions for parabolic problems, which do not use an ODE-solver to evolve in time. Here we present a new method for treating this problem, which allows low regularity in the source term. Such low regularity terms can be found in e.g. electrochemical engineering. The problem under consideration in this article is the following boundary value problem.

A reduction of a recent model of crystallization is presented. After adimensionalization, we solve it numerically. The temperature field exhibits oscillations and the crystallization process exhibits jumps. In fact, there is an advancing front of crystallization outside which the degree of crystallinity is constant. For sufficiently large samples, the model (a PDE and a ODE) can be reduced to a pair of ODEs for the parameters which defines the front.

We examine specific asymptotic and numerical aspects of a mathematical model for determining the dryout point in an LMFBR. By considering a paradigm problem we show that regularisation is essential for the calculation of accurate numerical solutions.

Since the end of the 70s, the oil industry has intensively used probabilistic methods for reservoir modelling. The problems addressed range from estimating the positions of the tops of reservoirs or surfaces limiting geological units based on well information and seismic data, to imaging of the internal architecture of reservoirs. After a brief introduction we will discuss some of the main applications of geostatistics for petroleum. From a technical point of view the tools range from (in most cases linear) estimation, to simulations where the aim is to produce several possible realisations of the variable of interest, conditioned by as much information as possible.

We shall briefly review recent progress in Global Quantitative Uncertainty and Sensitivity Analysis (UA/SA) techniques, relating these to multidimensional global calibration approaches of the “Monte Carlo filtering” type. Global quantitative techniques for Sensitivity Analysis (SA), that are based on the decomposition of the variance of the target model output, have received a considerable boost in recent years, due both to more efficient computational strategies and to a widening of their range of applications. Monte Carlo Filtering, and the GLUE (Generalised Likelihood Uncertainty Estimate) approach that derives from it are also promising tools to use in the presence of structural model uncertainty, as in the case of petroleum engineering that was the focus of the ECMI2002 mini-symposium “Advanced Mathematical Tools for Petroleum System Modelling”.

A mathematical model which describes the functioning of a Goldmann-type applanation tonometer is proposed in order to verify the validity of the Imbert-Fick principle. The spherical axisymmetric elastic equilibrium equations are solved using a Love Stress function. Conclusions are drawn regarding the circumstances under which the Imbert-Fick principle may or may not be valid.

The flow of hydrocarbons in a reservoir is a very complex process involving the interaction of several fluids and rock. Accurate and fast modelling of this complex flow is crucial to optimize reservoir exploitation.We have developed a 3D streamline based reservoir simulator for two phase (water and oil) incompressible flow. In this case, the model simplifies to a non-linear elliptic equation for the pressure, coupled to an equation for the evolution of saturation. The streamline method assumes that displacement along any streamline follows a one-dimensional solution, and that there is no communication among streamlines, so that the 3D saturation equation is decomposed into a set of one-dimensional problems. It takes advantage of the fact that the pressure varies slowly compared to saturation and therefore needs to be updated after several steps in the saturation equation.The decoupling of the pressure equation from the saturation equation speeds up the simulation orders of magnitude with respect to the conventional finite difference method, and this increase in speed is very important in modern reservoir engineering.The method is best suited to problems dominated by displacement as opposed to gravity and capillarity. However, these effects can be taken into account in the model and we will show how to use operator splitting techniques to this end.We will describe the main characteristics of the streamline method, analyze different numerical methods to solve the non-linear hyperbolic equation along streamlines, and present numerical results to show the effects of capillarity in the production of the reservoir.

The current status of electromagnetic simulation capability in the electronics industry is summarised and reviewed: some comments are also made as to what the future might hold for such simulation tools and what probelms need to be solved.

We consider the boundary value problem for a parabolic equation in the form 1 $$\frac{{\partial {\text{u}}}}{{\partial t}} = \frac{1}{{p(x)}}\frac{\partial }{{\partial x}}\left( {p(x)f'(u)\frac{{\partial u}}{{\partial x}}} \right) + F(u),x \in (0,l),t0,$$ 2 $$u(0,x) = {u_0}(x),$$ 3 $$\frac{{\partial u}}{{\partial x}}{|_{x = 0}} = {f_1}\left( {{u_1}} \right),$$ 4 $$\frac{{\partial u}}{{\partial x}}{|_{x = 1}} = {f_2}\left( {{u_2}} \right),$$ where u = u(t,x) is the unknown function, f₁, f₂, F, f are nonlinear functions and f′ (u) > 0, $${u_1} = {u_1}\left( t \right) \equiv u\left( {t,0} \right),{u_2} = {u_2}(t) \equiv u\left( {t,l} \right),f'\left( u \right) \equiv df(u)/du,p(x) \geqslant 0.$$

Multipole expansions can be regarded as a classical analytical method in Electromagnetics [Jac98] [Str41], however they are of considerable interest for numerical procedures as well. Generally, multipole methods allow an efficient calculation of electromagnetic fields, for instance to evaluate the interference between radiating systems. Since each of the multipole terms can be regarded as a mode, the expansion allows a simple physical interpretation. This contribution starts with an introduction into the spherical-multipole analysis of electromagnetic fields. Two applications of multipole expansions will be discussed: First a brief summary of the Fast-Multipole Method will be given, then a procedure is described which shows how the benefits from multipole representations can be exploited to efficiently represent and post-process numerically or asymptotically obtained results.

We consider a system of weakly nonlinear equations with a small positive parameter ε: 1 $${U_t} + A(U){U_x} = \varepsilon B(t,x,\varepsilon t,\varepsilon x,U,{U_x},{U_{xx}},{U_{xxx}},$$ $$U = {({u_1},{u_2}...,{u_n})^T},A(U) = ||{a_{ij}}(U)|{|_{nxn}}.$$

An American option is a contract giving its holder the right to buy (call option) or sell (put option) one unit of an underlying security of value S for a prearranged amount. This right can be exercised at any time prior to the expiration date T. In contrast, a European option can be exercised only at the expiry. Define the amount paid to the holder of an American option at the moment of exercise, the payoff, as Ψ (S, t) ≥ 0; a standard contract is a put option where Ψ = max(K − S, 0) and K is the strike price. The discounted exercise value of the option is Z(t) = Ψ (t) / B(t), where B(t) is the value at time t of $1 invested in a riskless money market account at t = 0. American option valuation can be characterised as an optimal stopping problem. The time 0 value of an American option is given by 1 $$V(0) = \mathop {\sup }\limits_{0\tau T} E\left[ {Z\left( \tau \right)} \right]$$ where the supremum is taken over all the possible stopping times τ less than the expiration date T, and the expectation is taken over the risk-neutral probability density. This is the primal problem.

The Folgar-Tucker equation is used in flow simulations of fiber suspensions to predict fiber orientation depending on the local flow. In this paper, a complete, frame-invariant description of the phase space of this differential equation is presented for the first time.

If a bass loudspeaker is run at high power, owing to the low frequency, the displacements of the membrane and of the air in the reflex tube may exceed a centimeter. Hence, linear acoustics are no longer applicable. Solving the complete Euler equations, however, requires an unreasonably high effort. Therefore, we propose a correction to the linear Helmholtz equation which consists just in a second inhomogeneous Helmholtz equation for the first harmonic and a Poisson equation for the radiation pressure. For medium displacements it allows to predict how the energy inserted into the loudspeaker at a given frequency is distributed between the keynote and the first harmonic (nonlinear distortion). Moreover, it explains how chamfered edges of the reflex tube reduce the excitation of higher harmonics.

In biomedical applications it is often required to determine accurately the absolute pressure acting between a body part such as an arm or foot and a pressure applying part such as a tourniquet cuff or bandage as occurs in tactile sensing, tourniquet applications and in the design of car seats. Bandage compression therapy is the principal treatment for leg ulcers associated with venous disease [Hir98]. Pressure is derived from such sensors by scaling the force by the sensor active area but the effective sensor area varies with applied pressure and with body tissue properties due to the draping of the bandage or cuff over the sensor — the so called ‘hammocking’ effect [Cas01], [OBr02]. Here we present here a model of this effect and compare with experimental results.

The Heston model is a well established model for the description of the stock price dynamics. In reference [DRA02] the authors discovered a solution for long time behaviour that seems to be very interesting due the good agreement with empirical data from the New York stock exchange (NYSE). Based on this result we took the calculated stationary standardized probability density distribution and compared it with our empirical data from the German stock exchange. We use the German tick-by-tick data of the stock index DAX and its stocks from May 1996 until December 2001 [Kur02]. We calculated the probability density distributions for different time lags and compared them with the theoretical solution for the long time behaviour of the Heston model developed in [DRA02].

Electrorheological fluids (ERF) belong to the class of so-called smart materials. The viscosity of these fluids is continuously controllable. Thus ERF-devices are excellent interfaces between electronic control units and mechanical systems [5]. The application within vehicle suspensions exploits the controllability of high frequencies and forces over a wide range. In this paper we will focus on control issues rather than on ERF damper modeling.

The Explicit Jump Immersed Interface Method reduces the irregular domain problem with non-grid aligned boundaries to solving a sequence of problems in a rectangular parallelepiped on a Cartesian grid using standard central finite differences. Each subproblem is solved using a Fast Fourier Transform based fast solver.The resulting method is second order convergent for the displacements in the maximum norm as the grid is refined. It makes the method attractive for applications where information about the local displacements, stresses and strains is needed, like optimal shape design and others.

A model for the dewatering of pulp suspension in the twin-blade forming papermaking process is formulated and analysed. The slenderness of the geometry permits reduction to a one-dimensional problem, which can be rewritten in the form of a highly non-linear second-order ODE. Analysis of its asymptotic structure up- and downstream indicates a strategy for computing solutions numerically. Subsequent results indicate that at industrially realistic suction pressures, the pressure within the pulp suspension will be lower than the surrounding ambient pressure, suggesting model breakdown as air is entrained.

The propagation of excitation waves through a cluster of insulin-secreting β-cells (a pancreatic islet of Langerhans) is modelled, and the results are related to recent image analysis experiments.

Different approaches to the mathematical modelling of the cardiovascular system are discussed. The compartment model is used as a basis for construction of a simplified model, which can be useful in the investigation of the role of regulation mechanisms on the partition of the blood volume between the systemic and pulmonary circulations.

We describe a fluid mechanics model that has been constructed in order to allow anunderstanding of the drawing of microstructured optical fibres, or ‘holey fibres’, to be gained, and furtherour ability to predict and control the final fibre geometry. The effects of fibre rotation are included in the model. Predictions are made by solving the final model numerically.

The influence of mesoscopic surface heterogeneity on the dynamics of the heterogeneous catalytical reactions is investigated. It is shown that in the case of the CO oxidation on modified Pt(111) local defects may assist nucleation of new stable non-equilibrium state with low or high reaction rate. The same defects under appropriate conditions can stabilize the metastable state of high reactivity due to the pinning-effect. In the case of the NO+CO reduction on Pt(100) the inhomogeneity of catalyst surface can induce the regime of chemical turbulence which gives the high reaction rate.

In this paper are discussed mathematical models for the liquid film generated by impinging jets. These models describe only the film shape under special assumptions about processes. Attention is stressed on the interaction of the liquid film with some obstacle. The idea is to generalize existing models and to investigate qualitative behavior of liquid film using numerical experiments. G.I. Taylor [TaI59]-[TaIII59] found that the liquid film generated by impinging jets is very sensitive to properties of the wire which was used as an obstacle. The aim of this presentation is to propose a modification of the Taylor’s model, which allows to simulate the film shape in cases when the angle between jets is different from 180°. Numerical results obtained by discussed models give two different shapes of the liquid film similar as in Taylors experiments. These two shapes depend on the regime: either droplets are produced close to the obstacle or not. The difference between two regimes becomes larger if the angle between jets decreases. Existence of such two regimes can be very essential for some applications of impinging jets, if the generated liquid film can have a contact with obstacles.

The railway track level, also in unloaded state, deviates from the theoretical plane form and suffers from geometrical irregularities determined by the actual equilibrium position of the rails and the track under the action of gravity forces. In the paper the excitation effect of the vertical geometrical irregularities of the track level is analysed by a simple hybrid model consisting of a beam with an initial irregularity function, and a wheel, rolling along the rail, modelled by a lumped parameter mass supported on a viscoelastic Hertzian spring representing the wheel/rail contact, and subjected to the action of a constant load. The model can serve as a base for further investigations into the vertical dynamic interactions between the irregular track and lumped parameter vehicle models. The steady-state solution to the problem above is given in an analytical way, generalizing the results obtained in the case of a periodic initial shape [zz00].