Progress in Industrial Mathematics at ECMI 2014

We present a customized system for vehicle tracking and classification. The main purpose of the system is tracking the vehicles in order to understand lane changes, gates transits and other behaviors useful for traffic analysis. The classification of the vehicles into two classes (short vehicles vs. tall vehicles) is also performed for electronic truck-tolling as well as to optimize the performances of the tracker module. The whole system has been developed through a data driven approach based on video sequences acquired by QFree. (Q-Free (www.q-free.com) is a global supplier of solutions and products for Road User Charging and Advanced Transportation Management having applications mainly within electronic toll collection for road financing, congestion charging, truck-tolling, law enforcement and parking/access control.) The sequences are acquired by wide angle cameras from the top of the road and are preprocessed in order to obtain a normalized, low-resolution representation of the scene where the distance between neighboring pixels is constant in the real world. The sequences exhibit high variability in terms of lighting changes, contrast changes and distortion. We assume that the vehicle detection is performed by an external module for plate recognition.

Iris segmentation is driven by three different quality factors: accuracy, usability and speed. Unfortunately the deeply analysis of the literature shows that the greatest efforts of the researchers mainly focus on accuracy and speed. Proposed solutions, in fact, do not meet the usability requirement since they are based on specific optimizations related to the operating context and they impose binding conditions on the sensors to be used for the acquisition of periocular images. This paper tries to fill this gap by introducing an innovative iris segmentation technique that can be used in unconstrained environments, under non-ideal imaging conditions and, above all, that does not require any interaction for adaptation to different operating conditions. Experimental results, carried out on challenging databases, demonstrate that the high usability of the proposed solution does not penalize segmentation accuracy which, in some respects, outperforms that of the leading approaches in the literature.

Reading is an activity which takes place widely on the web: almost all newspapers have his own digital version on the internet and there are even a lot of magazines only on the web. In such a scenario, Computer Vision can offer a useful set of tools that can help web editors to improve the quality of the provided service. One of these tools is here presented: given a webpage of a newspaper or journal, the proposed framework localizes news items remotely clicked by users, giving the bounding box of the content of an article in its relative homepage. The tool is hence able to track an article in the page in which is contained at any time during the day: such an information is very useful for web editors to understand the trend of the published items and to rearrange the contents of the homepage accordingly.

The paper demonstrates when the Wind Atlas Analysis and Application Program (WAsP) is comparable to Computational Fluid Dynamics (CFD) in order to use the WAsP wind prediction later for time consuming CFD simulations. Three different numerical methods (WAsP, RANS, LES) for observation of wind flow over the hills are described and compared with the wind-tunnel experiment. The paper shows that WAsP provides reasonably realistic results for the flow over the commonly found in nature shallow hills.

Ensemble Prediction System (EPS) is the approach used in present day weather predictions to estimate the uncertainty of predictions. Along with the main prediction an ensemble of simulations is launched with perturbed initial values. Recently, the EPS with simultaneous parameter estimation approach (EPPES) has been proposed to tune model parameters online, without additional computational costs, by perturbing the parameter values and monitoring the respective performances. The key point of EPPES is the estimation of the parameter covariance by sequentially updating the covariance as hyperparameters by aid of importance weights. Here, we study the Differential Evolution (DE) optimization approach as a new way to solve the problem as a stochastic optimization task. We show that the convergence is improved using DE, especially in case when initial values of model parameters are far enough from the true ones.

We develop a Monte Carlo algorithm to price an Asian option whose underlying price is driven by a jump diffusion process. By conditioning on the number of jumps, we characterise the underlying asset process as lognormally distributed from which a control variate for the generic Monte Carlo algorithm is derived. Numeric results confirm that the control variate method is an effective variance reduction method.

In this paper we propose an explicit finite-difference scheme to solve the American option pricing problem. It is based on front-fixing transformation that involves unknown free boundary to the equation. The proposed stable and consistent numerical scheme preserves positivity and monotonicity of the solution in accordance with the behavior of the exact solution. Numerical examples and comparison with other methods are included. This technique can be applied to some types of two-asset options after reducing the dimension. In the paper the front-fixing method is applied to exchange option pricing.

In order to overcome the drawbacks of assuming deterministic volatility coefficients in the standard LIBOR market models, several extensions of LIBOR models to incorporate stochastic volatilities have been proposed. The efficient calibration to market data of these more complex models becomes a relevant target in practice. The main objective of the present work is to efficiently calibrate some recent SABR/LIBOR market models to real market prices of caplets and swaptions. For the calibration we propose a parallelized version of the simulated annealing algorithm for multi-GPUs. The numerical results clearly illustrate the advantages of using the proposed multi-GPUs tools when applied to real market data and popular SABR/LIBOR models.

A Backward Stochastic Differential Equation (BSDE) is a stochastic differential equation for which a terminal condition has been specified. In Ruijter and Oosterlee (A Fourier-cosine method for an efficient computation of solutions to BSDEs, 2013) a Fourier-cosine method to solve BSDEs is developed. This technique is known as BCOS method and consists of the approximation of the BSDE’s solution backwards in time by the use of the COS method developed in Fang and Oosterlee (SIAM J Sci Comput 31(2):826–848, 2008) to compute the conditional expectations that rise after the discretization by means of a θ-method for the time-integration.In this work, the methodology is extended to the case in which there are more than one source of uncertainty or the terminal condition depends on more than one process, allowing the pricing of derivatives contracts such as rainbow options. The extension of the BCOS technique can be done taking into account some ideas developed in Ruijter and Oosterlee (SIAM J Sci Comput 34(5):B642–B671, 2012). We present some results concerning to derivatives on two processes without jumps. We also apply our extended method to solve the BSDEs that rise with the use of quadratic hedging techniques for pricing in incomplete markets without or with jumps (Lim, Math Oper Res 29(1):132–161, 2004; Lim, SIAM J Sci Comput 44(5):1893–1922, 2005). Problems in which the randomness of the terminal condition depends not only on the risky asset but also on the insurance risk or the counterparty default risk can be introduced in this framework (Delong, Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. Springer, London, 2013).

The Fichera theory was first proposed in 1960 by Gaetano Fichera and later developed by Olejnik and Radkevič in 1973. It turned out to be very useful for establishing the well-posedness of initial boundary value problems for parabolic partial differential equations degenerating to hyperbolic ones at the boundary.In this paper we outline the application of the Fichera theory to interest rates models of Cox-Ingersoll-Ross (CIR) and Chan-Karolyi-Longstaff-Sanders (CKLS) type. For the one-factor CIR model the obtained results are consistent with the corresponding Feller condition.

It is well known that the correlation between financial products, financial institutions, e.g., plays an essential role in pricing and evaluation of derivatives. Using a constant or deterministic correlation may lead to correlation risk, since market observations give evidence that the correlation is hardly a deterministic quantity.Here, the approach of Teng et al. (A versatile approach for stochastic correlation using hyperbolic functions. Preprint 13/14. University of Wuppertal, 2013) for modelling the correlation as a hyperbolic function of a stochastic process is generalized to derive stochastic correlation processes (SCP) from a hyperbolic transformation of the modified Ornstein-Uhlenbeck process. We determine a transition density function of this SCP in closed form which could be used easily to calibrate SCP models to historical data.As an example we compute the price of a quantity adjusting option (Quanto) and discuss concisely the effect of considering stochastic correlation on pricing the Quanto.

The aim of this paper is to construct a reliable and efficient finite difference scheme for American option pricing under Bates model. First, we transform the associated partial-integro differential equation for this model into another suitable one without the cross derivative. Thereafter, a finite difference discretization has been used for the partial derivatives while the integral part is discretized using the four-points open type formula. The obtained finite difference scheme is solved using PSOR method. Several examples are included showing the advantage of the proposed approach.

Finite difference methods (FDM) have been developed and optimized in a technology context that has radically changed. When FDMs became a standard it used to be that memory was a scarce resource and that algorithms were either memory or compute bound. As a consequence traditional FDMs have been designed to minimize the number of operations and the memory footprint given a certain level of accuracy. In this paper we describe how the potential of GPU computing can be exploited to rethink the way FDM are implemented in the context of financial applications.

PDE pricing methods such as backward and forward induction are typically implemented as unconditionally marginally stable algorithms in double precision for individual transactions. In this paper, we reconsider this strategy and argue that optimal GPU implementations should be based on a quite different strategy involving higher level BLAS routines. We argue that it is advantageous to use conditionally strongly stable algorithms in single precision and to price concurrently sub-portfolios of similar transactions. To support these operator algebraic methods, we propose some BLAS extensions. CUDA implementations of our extensions turn out to be significantly faster than implementations based on standard cuBLAS. The key to the performance gain of our implementation is in the efficient utilization of the memory system of the new GPU architecture.

In the coming years offshore wind energy will be one of the most promising areas in the renewable power generation field. Achieving the optimum design of floating platforms requires a rigorous analysis chain to establish the response of the whole platform under different scenarios. With this aim, we have developed a software package that automatically analyzes the feasibility of a floating structure. The structure of the platform is defined according to a very general set of parameters, allowing us to consider a wide range of designs. The package calls some commercial applications and some own codes, to complete the analysis process. Returned results include the hydrostatic equilibrium position, hydrodynamic pressure, RAOs (response-amplitude operators), material costs and static stresses.

Since 2007, a group of Spanish mathematicians has been promoting mathematical knowledge transfer through the Ingenio Mathematica (i-MATH) Project’s ‘Consulting Platform’. The outcome of this groundwork has been the creation of the national network, math-in. In this paper we will briefly present its aims, management, main activities and workspace.

Sportello Matematico per l’Industria Italiana is a project developed by the National Research Council of Italy to build an effective and high-quality network of research groups in Industrial Mathematics in Italy. Here we will recall the objectives and the main actions taken by the project team during its first year of activities.

In this paper we review the recent research done by the authors in Solar Power Tower systems design focusing on the heliostat field problem. We first analyze the basic problem, in which all heliostats have the same size, as commonly addressed in the literature. A brief review of the problem itself and the pattern-free procedure proposed to solve it is given. The algorithm proposed, a greedy-based heuristic procedure, provides a new way to solve the problem different from previous algorithms in the literature. Our methodology consists of a pattern-free heliostat location and therefore it can be easily (even though carefully) adapted to solve other issues such as multi-size or multiple-receiver heliostat field.The multi-size heliostat fields design is also reviewed given the similarities of the problem. This algorithm is tested using two different heliostat sizes. Some ideas about the application of the procedure to more general settings, such as multiple-receiver field, are given as further work.

Order picking consists in retrieving products from storage locations to satisfy independent orders from multiple customers. It is generally recognized as one of the most significant activities in a warehouse (Koster et al, Eur J Oper Res 182(2):481–501, 2007). In fact, order picking accounts up to 50 % (Frazelle, World-class warehousing and material handling. McGraw-Hill, New York, 2001) or even 80 % (Van den Berg, IIE Trans 31(8):751–762, 1999) of the total warehouse operating costs. The critical issue in today’s business environment is to simultaneously reduce the cost and increase the speed of order picking. In this paper, we address the order picking process in one of the Portuguese largest companies in the grocery business. This problem was proposed at the 92nd European Study Group with Industry (ESGI92). In this setting, each operator steers a trolley on the shop floor in order to select items for multiple customers. The objective is to improve their grocery e-commerce and bring it up to the level of the best international practices. In particular, the company wants to improve the routing tasks in order to decrease distances. For this purpose, a mathematical model for a faster open shop picking was developed. In this paper, we describe the problem, our proposed solution as well as some preliminary results and conclusions.

Study groups were first introduced to Ireland in 2008 by MACSI. We present an overview of MACSI study groups focusing on the role study groups play in initiating longer term interactions between industry and academia.

Pricing early-exercise financial options under multi-dimensional stochastic processes is a challenge in the financial sector. For this purpose, the authors in Jain and Oosterlee (The stochastic grid bundling method: efficient pricing of Bermudan options and their Greeks, 2015) proposed a practical simulation-based algorithm called Stochastic Bundling Grid Method (SGBM). SGBM is a Monte Carlo based method for pricing multi-dimensional Bermudan options. The method is based in a combination of dynamic programming, simulation, regression and bundling of paths. In the present work, the SGBM method is taken to the extreme with as a purpose a near-future extension of the method, for example, to Credit Value Adjustment (CVA) calculations. Here, the number of Monte Carlo paths, the problem dimensions, the amount of bundles are increased drastically. As a consequence, the SGBM method becomes significantly more (almost impractically) expensive. Overall, with the increase of the number of bundles, the iterative bundling process used in the original method would take too much computing time. In addition, the algorithm needs a huge storage because many bundles contain many more Monte Carlo paths. In order to make the method affordable, the General-Purpose computing on Graphics Processing Units (GPGPU) paradigm is used to parallelize the algorithm. More specifically, the Nvidia CUDA platform (CUDA webpage: URL http://www.nvidia.com/object/cuda_home_new.html) is chosen to reach this aim, taking advantage of its latest features. Two steps of parallelization are performed, one for the Monte Carlo path simulation and another one for the bundling calculations. Furthermore, a new way to make the bundles is proposed, which is efficient and overcomes the drawbacks caused by the increasing number of bundles and the problem dimensionality.

The finance world, relying more and more on mathematical models, also expects them to be fast, robust and cheap, especially for calibration purposes. The recent revolution in Graphical Processing Units (GPU) and Field-Programmable Gate Array (FPGA) has helped to reduce time and costs but it is the algorithms that ultimately prevail. In this respect, Model Order Reduction (MOR) seems to be especially suited to financial problems as it can reduce extremely computational costs (Achdou and Pironneau, Computational methods for option pricing. SIAM frontiers in applied mathematics, vol 30. Society for Industrial and Applied Mathematics, Philadelphia, 2005). We present two cases when MOR can be extremely useful and how Proper Orthogonal Decomposition (POD) stands out as a valid MOR technique in finance (Volkwein, Proper orthogonal decomposition: theory and reduced-order modelling. Lecture notes, Universität Konstanz, 2013). We show the validity of its application to pricing of basket options, as well as to stochastic volatility models (Heston, Rev Financ Stud 6:327–343, 1993), through the solution of a reduced Black-Scholes PDE. Finally, its computational efficiency when compared with some extensively used numerical methods, as well as some of its limitations are discussed.

In this work we address the inverse problem of reconstructing inclusions and their thermal parameters given temperature measurements at the accessible side of a material. We describe an iterative descent method that combines topological derivative computations to reconstruct the geometry of the defects with gradient iterations to approximate the material parameters. A numerical experiment showing the ability of the method to obtain reasonable reconstructions in a few iterations is presented.

Reduction of computational time in high resolution image reconstruction is essential in basic research and applications as well. This reduction is important for different types of traditional non diffractive tomography in medical diagnosis as well as for applications in nanomaterials research, related to modern technologies. Alternatives to alleviate the computationally intense part of each iteration of iterative methods in tomographic reconstruction have all been based on interpolation over a regular grid in the Fourier domain or in fast nonuniform Fourier transforms. Both approaches speed up substantially the computation of each iteration of classical algorithms, but are not suitable for being used in a large class of more advanced faster algorithms: incremental methods such as OS-EM, BRAMLA or BSREM, among others, cannot benefit from these techniques. The backprojection is a stacking operator, known to be the adjoint of the Radon transform. As a mapping ℬ $$\mathcal{B}$$ , the backprojection can be recast as a convolution operator, in a different coordinate system, which is an improvement in accelerating the computation of ℬ $$\mathcal{B}$$ . In this work, we propose several analytical representations for the operator ℬ $$\mathcal{B}$$ , in order to find a fast algorithm.

Technology for promoting nucleation is important in a number of contexts, for instance degassing carbon dioxide lakes, designing champagne glasses and stout beer widgets. A new design of stout beer widget has recently been proposed which makes use cellulose fibres to initiate foaming in canned stout beers. However, our current scientific understanding of the nucleation of bubbles by cellulose fibres is incomplete, making it impossible to optimise this technology. One particularly poorly understood aspect is the detachment of bubbles from a gas pocket in the fibre. We report experimental and theoretical results towards a model of the detachment based on a model of Rayleigh-Plateau instability including a disjoining pressure.

The modelling of the continuous casting of metals is known to involve the complex interaction of non-isothermal fluid and solid mechanics. However, using asymptotic methods and an earlier numerical result obtained via computational fluid dynamics, we demonstrate how the motion of the liquid metal can be systematically decoupled from the stresses induced in the solidified shell. The resulting asymptotically reduced model can then serve as a computationally efficient module for stress mechanics models that aim to predict segregation and crack formation in the solid metal.

The drip filter coffee market is a multi-billion euro industry. Despite this, although the chemistry of coffee brewing has been investigated in great detail, the physics of the process has received relatively little attention. In order to explain in scientific terms correlations between the coffee quality and the process variables, a physical model is required. In this study, flow through a static, saturated coffee bed, under the influence of a pressure gradient, is described using a double porosity model. The model is parametrised using experimentally obtained data from a cylindrical flow-through cell containing a coffee bed. Mass transfer from the coffee grains to the interstitial water is modelled using two mechanisms; mass transfer from the surface of the grains and mass transfer from the interior (bulk) of the grains. Mass transfer resistances are estimated by fitting experimental data. Initially coffee extraction is dominated by mass transfer from the grain surface, while transfer from the kernel of the grain is the rate limiting mechanism once the surface coffee has been exhausted.

Dry powder inhalers deliver drugs in powdered form to the lungs. The drug is stored within the inhaler bound to an excipient. The drug-excipient conglomerate is broken apart in a vortex chamber by collisions with the walls and other conglomerates. During the initial doses, some drug adheres to the wall of the vortex chamber reducing the amount of drug delivered to the patient. We developed mathematical models for particle-wall adhesion to investigate why drug particles adhere to the wall of the vortex chamber. Two different models are developed to validate our results and a good agreement has been obtained. The first model describes the motion of particles in a turbulent flow field based on Stochastic Differential Equations (SDE). The second model is a continuum model of particle-wall adhesion based on Partial Differential Equations (PDE). This model focuses on the rate at which drug particles are captured by the wall and the time taken for drug particles to fill the wall area. Estimates of magnitudes of adhesive forces suggest that excipient particles do not adhere to the walls, while drug particles bind to the wall due to van der Waals forces when their velocity is below a critical value.

We propose a partial differential equations model for the formation and evolution of a holographic grating in a photopolymer system and use perturbation methods and numerical simulations in order to investigate the dynamical mechanism by which distortions of the illumination pattern arise during recording. The parameters of interest are diffusion and photopolymerization rates as well as exposure time, for which we seek to determine regimes which allow for high fidelity copying.

The Immersed Boundary Method (IBM) is an effective mathematical model and approximation scheme for the discretization of biological systems which involve the interaction of fluids and solids. The Finite Element IBM (FE-IBM) proved to be competitive with respect to the original IBM (based on finite differences and on a suitable approximation of a Dirac delta function) in several aspects: in particular, the position of the solid can be dealt with in a natural way by taking advantage of the underlying variational formulation (thus avoiding the use of the delta function); moreover, the use of finite elements allows for sharp pressure jumps when discontinuous pressure schemes are adopted. Recently [see Boffi et al. (Coupled Problems 2013. Computational Methods for Coupled Problems in Science and Engineering V, Cimne, 2013)], a fully variational approach of the FE-IBM has been introduced, which can be shown to be unconditionally stable with respect to the time discretization. The novelty consists in the treatment of the coupling between the solid and the fluid: in the standard formulation, this is given by a differential equation stating that the velocity of the solid is equal to that of the fluid, while in the new formulation this coupling is imposed in a weak form. A rigorous mathematical analysis shows the stability of the coupling and the unconditional time stability.

The pathogenesis of many blinding diseases, such as diabetic retinopathy, glaucoma or retinopathy of prematurity, is thought to be related to retinal tissue hypoxia. Yet, the mechanisms governing oxygen delivery to the retina are still poorly understood. Since it is not currently possible to disentangle the influence of all the concurring factors in retinal oxygenation during experimental and clinical measurements, mathematical models can serve as virtual laboratories to separately investigate the individual influence of different parameters. In this contribution, we propose a mathematical model which describes the oxygen profile along the whole retinal depth, including sources from blood circulation and tissue metabolic consumption. An analytical solution for the profile is computed and quantitative estimates of the sensitivity of retinal oxygen profiles to changes in geometrical and metabolic parameters of the retinal tissue are provided. In particular, this analysis highlights the important role played by the thickness of the different retinal layers and warns of potential issues when using experimental data across species.

We investigate spectral deferred correction (SDC) methods for time stepping and their interplay with spatio-temporal adaptivity, applied to the solution of the cardiac electro-mechanical coupling model. This model consists of the Monodomain equations, a reaction-diffusion system modeling the cardiac bioelectrical activity, coupled with a quasi-static mechanical model describing the contraction and relaxation of the cardiac muscle. The numerical approximation of the cardiac electro-mechanical coupling is a challenging multiphysics problem, because it exhibits very different spatial and temporal scales. Therefore, spatio-temporal adaptivity is a promising approach to reduce the computational complexity. SDC methods are simple iterative methods for solving collocation systems. We exploit their flexibility for combining them in various ways with spatio-temporal adaptivity. The accuracy and computational complexity of the resulting methods are studied on some numerical examples.

Since the start of the liberalization of energy markets the energy sector has undergone major changes. Energy companies now provide electricity at variable prices and are faced to a competitive market environment. Their trading is subject to risks and uncertainty about future price developments. In this work we introduce a regularized regression approach to forecast Phelix Peak prices in the German electricity market. Additionally we investigate the influence of fundamental price drivers on the forecasting accuracy. Since the problem complexity grows exponentially with the dimension of the feature space, the regression problem suffers from the curse of dimensionality. To cope with this problem we apply the combination technique, which enables us to reduce the complexity while keeping a high approximation accuracy.

Estimating the demand on the low voltage network is essential for the distribution network operator (DNO), who is interested in managing and planning the network. Such concerns are particularly relevant as the UK moves towards a low carbon economy, and the electrification of heating and transport. Furthermore, small to medium enterprises (SMEs) contribute a significant proportion to network demand but are often overlooked. The smart meter roll out will provide greater visibility of the network, but such data may not be readily available to the DNOs. The question arises whether useful information about customer demand can be discerned from limited access to smart meter data? We analyse smart meter data from 196 SMEs so that one may create an energy demand profile based on information which is available without a smart meter. The profile itself comprises of simply two estimates, one for operational power and another for non-operational power. We further improve the profile by clustering the SMEs using a simple Gaussian mixture model. In both cases, the average difference between the actual and predicted operational/non-operational power is less than 0.15 kWh, and clustering reduces the range around this difference. The methods presented here out perform the flat profile (akin to current methods).

In this paper we present a general model of drug release from a delivery device and the subsequent drug transport in biological tissue. Our model consists of a system of partial differential equations describing the solid–liquid mass transfer and diffusion in the device coating as well as the drug transport through the biological tissue via diffusion, convection and reaction. The drug release from the device depends not only on the properties within the coating and the tissue, but also on the coupling of the two layers. In order to take this into account, our model fully couples the two distinct layers through flux and permeable interface conditions. The model has a wide applicability and we point the reader towards some solution methods, noting that simplifications may be made depending on the parameter values in a given system.

Imperfections in manufacturing processes may cause unwanted connections (faults) that are added to the nominal, “golden”, design of an electronic circuit. By fault simulation we simulate all situations: new connections and each with different values for the newly added element. We also consider “opens” (broken connections). During the transient simulation the solution of a faulty circuit is compared to the golden solution of the fault-free circuit. A strategy is developed to efficiently simulate the faulty solutions until their moment of detection. We fully exploit the hierarchical structure of the circuit in the simulation process to bypass parts of the circuit that appear to be unaffected by the fault. Accurate prediction and efficient solution procedures lead to fast fault simulation in which the golden solution and all faulty solutions are calculated over a same time step. Finally, we store a database with detectable deviations for each fault. If such a detectable output “matches” a measurement result of a product that has been returned because of malfunctioning it helps to identify the subcircuit that may contain the real fault.

We consider coupled problems with uncertain parameters modelled as random variables. Due to the largely differing behaviour of subsystems in coupled problems, we introduce a strategy of adjusted grids defined in the parameter domain for resolving the stochastic model. This allows us to adapt quadrature grids to each subsystem. The communication between the different grids requires global approximations of coupling variables in the random space. Since implicit time integration methods are typically included, we investigate dynamic iteration schemes to realise this approach. Numerical results for a thermal-electric test circuit outline the feasibility of the method.

Nanofluids have been hailed as a possible winner in the race to find sufficiently powerful cooling systems for emerging high-power electronic devices. There exist numerous experiments demonstrating nanofluids to have remarkable properties. However, there has been much controversy in the literature with discrepancies between results concerning the heat transfer and thermal conductivity of nanofluids. In this paper we analyse a popular model for nanofluid flow which previously has been employed to demonstrate the improved heat transfer. We find the opposite result and then move on to explain some of the reasons behind the discrepancies.

Biofilms are bacterial aggregates that grow on moist surfaces. Thin homogeneous biofilms naturally formed on the walls of conducts may serve as biosensors, providing information on the status of microsystems (MEMS) without disrupting them. However, uncontrolled biofilm growth may largely disturb the environment they develop in, increasing the drag and clogging the tubes. To ensure controlled biofilm expansion we need to understand the effect of external variables on their structure. We formulate a hybrid model for the computational study of biofilms growing in laminar microflows. Biomass evolves according to stochastic rules for adhesion, erosion and motion, informed by numerical approximations of the flow fields at each stage. The model is tested studying the formation of streamers in three dimensional corner flows, gaining some insight on the effect of external variables on their structure.

Analog circuit sizing has strongly focused on the optimization of nominal performance and of the yield in the past. Recently, more topics in analog sizing have come up. These are Pareto optimization, optimization with discrete parameter values and consideration of aging effects in addition to manufacturing and operating tolerances. This contribution will illustrate these tasks and give problem formulations and solution approaches.

Progressive CMOS technology scaling leads to high increase in statistical variations, whose impact on circuit performances must be taken into account already in the design phase. Reliable surrogate models can replace expensive circuit simulations to statistically characterize the figures of merit of a circuit with a reduced computational effort. We implemented a software framework which allows the automatic generation of surrogate models based on machine learning techniques such as Artificial Neural Networks (ANNs) and Support Vector Machines (SVMs). These methodologies have been used to generate statistical variation aware models for leakages and propagation delays of a set of digital standard cells.

The nano-CMOS technology scaling makes the figures of merit of a circuit, such as performance and power, extremely sensitive to uncontrollable statistical process variation (PV). In this context, multi-objective optimization algorithms and statistical analysis are essential to ensure stable manufacturing and secure high foundry yields. The CAD and Design Services group, part of the IPG R&D in STMicroelectronics, has created a consortium in order to develop, test and implement “Methods for Advanced Multi-objective Optimization for eDFY of Complex Nano-scale Circuits”: the MAnON Project. The contribution presents the industrial and scientific project challenges, the research results, and consequent methodology enhancements and their implementation into a software prototype in order to be usable inside a nanoelectronics industrial design environment.

The production of semiconductor integrated circuits is very complex and expensive. Therefore, it is essential to verify the designed circuits before they are fabricated. Due to the process variations, nanoscale circuits have to be simulated many times during the design flow. This kind of analysis can be very expensive because of their complexity and the high number of simulations. For this reason the semiconductor industry is deeply interested in using less complex but accurate models to speed up time consuming SPICE simulations. This contribution presents a method that creates a compact model, which replaces a semiconductor integrated circuit or sub circuit to significantly reduce the transient simulation time.

In this work we propose an approach that combines a Support Vector learning Machine with a Derivative-Free black box optimization algorithm in order to maximize the yield in the production of electronic circuits. This approach is tested on a circuit provided by ST-Microelectronics, to be employed in consumer electronics. The results of the approach are compared with the results of WiCkeD, a commercial software largely used for integrated circuits analysis.

An optimal control approach based on the adjoint method for the design of a semiconductor device is considered. A consistent energy transport model, free of any fitting parameters, formulated on the basis of the maximum entropy principle (MEP) is used as mathematical model. The robustness of the optimal control approach is verified by a numerical sensitivity analysis, performed by introducing a Gaussian noise in the reference doping profile.

Eddy current method is widely used in practice for quality testing of conducting materials (examples include determination of electrical conductivity, thickness of metal coatings, identification of flaws in a conducting medium). In the present paper a semi-analytical method for solution of direct eddy current problems for the case of a conducting medium of finite size is considered. The method is applied to several eddy current problems with cylindrical symmetry. The following problem is analyzed in detail. Consider a coil with alternating current located above a conducting medium in the form of a circular cylinder (such a model can be used for design of coin validators which are based on the estimation of electrical conductivity of a coin). We assume that the electromagnetic field is exactly zero at a sufficiently large distance from the coil (the distance can be chosen on the basis of the required accuracy of the solution). The solution is constructed using the method of separation of variables which includes two steps where numerical calculations are necessary: (a) computation of complex eigenvalues without good initial guess for the roots and (b) solution of a system of linear algebraic equations. Computations of the change in impedance of the coil for different frequencies with the semi-analytical method are in good agreement with experimental data and results of numerical simulation with finite element method. Solution of other problems with cylindrical symmetry is also discussed (a flaw in the form of a circular cylinder in a conducting half-space or a plate). Such models can be used for the analysis of quality of spot welding (in case of a volumetric flaw) and estimation of the effect of corrosion (for surface flaws).

Given a multiphysics problem with components of different dynamical behaviour reduction-multirate methods start with a model order reduction of the slow part system and apply than a multirate ODE-integration to the whole system. This approach lets us profit as much as possible from properties of the given system related to computational efficiency. In this paper we present the motivation and the idea behind this reduction-multirate approach.

In the last 10 years many 3D numerical schemes have been developed for the study the flow of a mixture of liquid and gas in a pipeline (Frank, Numerical simulation of slug flow regime for an air-water two-phase flow in horizontal pipes. In: The 11th international topical meeting on nuclear reactor thermal-hydraulics (NURETH-11), Avignon, 2005; Vallée et al., Nucl Eng Des 238(3):637–646, 2008; Höhne, Experiments and numerical simulations of horizontal two-phase flow regimes. In: Proceeding of the seventh international conference on CFD in the minerals and process industries, Melbourne, 2009; Bartosiewicz et al., Nucl Eng Des 240(9):2375–2381, 2010) but although they offer a very good accuracy, they are rarely fit for modelling a long pipe, due to the high computational costs. Then one is usually led to consider 1D models, see e.g. the works of Issa and his group (Issa and Kempf, Int J Multiphase Flow 29(1):69–95, 2003). Such models offer much faster simulations than 3D schemes, on the other hand they almost completely miss the dynamics in the transversal direction. Here we present a model able of representing the full 3D dynamics, but with the computational cost typical of 1D simulation. The main feature of our model consists in describing the dynamical variables in the direction transversal to the pipe by means of a family of functions depending on a set of parameters. The model is then solved by a standard finite volume scheme.

The aim of this contribution is to provide a thorough description of a heat pipe. This is a particular type of heat exchanger used in a variety of industrial applications, such as the cooling of electrical devices and solar cells, the temperature equalisation in spacecrafts, or the reduction of local heat gains in reactors and air-conditioning systems. Usually, lumped parameter models are used to study the behaviour of heat pipes and the thermal ranges in which they work optimally. In the following analysis, a quite comprehensive thermo-fluid dynamic model of the liquid/vapour pair operating in a heat pipe is developed. The model, which accounts for several phenomena taking place in this kind of devices, has the purpose of predicting the optimal thermal range of a given heat pipe, and preventing the occurrence of off-design conditions. The present investigation is done by considering a heat pipe working in zero-gravity conditions, to be used for Aerospace applications.

In order to study the dynamics of anisotropic chemotaxis-fluid models, a detailed numerical analysis is established in this paper. To discretize this type of models, a monotone combined scheme is proposed as a compromise between the nonconforming finite elements, enabling in particular the use of general meshes and the discretization of anisotropic diffusion tensors, and between the finite volumes enabling to avoid spurious oscillations in the convection-dominated regime. Moreover, this monotone scheme ensures the discrete maximum principle and therefore the confinement of the density of cells and the positivity of the chemical concentration. Finally, a test is given to illustrate the numerical study.

Most of the models of epidemic propagations do not take into the account the spatial distribution of the individuals. They give only the temporal change of the number of the infected, susceptible and recovered patients. In our presentation we present a spatial epidemic propagation model and give some of its qualitative properties both in the continuous and the finite difference numerical case: boundedness, nonnegativity preservation, the condition of forming epidemic waves. Some of the results are demonstrated on numerical tests.

Front propagation can be studied by two alternative approaches: the level set method and the reaction-diffusion equation. When a front propagates in a random environment it gets a random character and these two approaches can indeed be considered complementary and reconciled. In fact, if the level set contour is randomized accordingly to the probability density function of the front particle displacement, the resulting averaged process emerges to be governed by an evolution equation of the reaction-diffusion type. This approach turns out to be useful to simulate random effects in wildland fire propagation as those due to turbulent heat convection and fire spotting phenomena.

A numerical code that employs a higher-order finite volume scheme on unstructured meshes for approximating enhanced Boussinesq-type equations is presented. The objective of this study is to further investigate wave propagation over complex bathymetries using the developed code and to present an approach for the parallelization of the resulted code, along with preliminary numerical results.

Boussinesq-type equations are used to describe the propagation and transformation of free-surface waves in the nearshore region. The nonlinear and dispersive performance of the equations are determined by tunable parameters. Recently the authors presented conditions on the free parameters under which a Nwogu-type equations would yield bounded eigenspectra (Eskilsson and Engsig-Karup, J Comput Phys 271:261–280, 2014). This leads to a global conditional CFL time-step restriction which is shown to not be affected by the discretisation method and in this sense the CFL condition is tamed to impose a minimal constraint. In this paper we extend the previous study and provide numerical experiments which confirms the theoretical results also is valid in two horizontal dimensions.

In this paper, we investigate the nonlinear properties of Boussinesq models. In particular, we consider the wave shoaling obtained in physical regimes which go from linear to weakly nonlinear, to the wave breaking limit. For a given asymptotic accuracy in terms of dispersion and nonlinearity, we consider two families of models: the first depending on derivatives of the velocity, the second on derivatives of the volume flux. We show that, while linear dispersion and linear shoaling characteristics are strongly dependent on the type of dispersive terms introduced, when approaching the nonlinear regime the only influencing factor is whether the model is in amplitude-velocity of amplitude-flux form. We investigate these two alternative formulations of several known models, and propose a new model with a compact differential form, and the same linear characteristics of the model of Nwogu. The nonlinear shoaling properties of the models are investigated numerically showing that inside one given family, all the models have almost identical behaviour.

In this paper we are interested in a linear elasticity system modeling the presence of a crack inside a volcano. The traction force on this crack induces discontinuities of the displacement field. The computation of the latter is carried out with a finite element method for which the boundary of the crack is taken into account with a fictitious domain approach; It means that the mesh we consider does not fit to the crack. The interest of this approach lies in a framework where the position and the shape of the crack is lead to evolve, and in that case no re-meshing is required.

We propose a finite-difference ghost-point method for the numerical solution of thermo-poroelastic equations. The method is applied to evaluate deformation, gravity and thermomagnetic changes in Campi Flegrei area caused by hydrothermal fluid circulation during an unrest.

The Solfatara-Pisciarelli area represents the most active zone within the Campi Flegrei caldera (CFc) in terms of hydrothermal manifestations and local seismicity. Periodic injections of hot CO₂-rich fluids at the base of a relatively shallow hydrothermal system has already been correlated to ground uplift in a wide range of numerical modelling of the CFc unrests, that highlight a strong correlation between chemical composition of the Solfatara and Pisciarelli fumaroles, seismicity and ground movements. In particular, a new simulation has been realised via the coupling of TOUGH2©and Comsol Multiphysics©. Recent uplift episodes in the in the centre of Pozzuoli Bay have been reconstructed imposing fluid flows in the system as experimentally recorded. Numerical studies, geochemical data and Magnetotelluric (MT) survey have been integrated, to guess the main features of the shallower part of the hydrothermal system of the Solfatara-Pisciarelli area.

This paper considers the path following of unmanned helicopters based on dynamic optimization. We assume that the helicopter is equipped with a flight control system that provides an approximation of its closed-loop dynamics. The task at hand is to compute inputs for this flight control system in order to track a geometrically specified path. A concise problem formulation and a discussion of an efficient implementation are presented. The implementation achieves computation times below the flight duration of the path by exploiting differential flatness properties of parts of the dynamics. Finally, we present quantitative results with respect to convergence and required iterations for a challenging nonlinear path. We show that the proposed optimization based approach is capable of tackling nonlinear path following for unmanned helicopters in an efficient and practicable manner.

Local energy storage and smart energy scheduling can be used to flatten energy profiles with undesirable peaks. Extending a recently developed model to allow controllable loads, we present Centralized and Decentralized Model Predictive Control algorithms to reduce these peaks. Numerical results show that the additional degree of freedom leads to improved performance.

In this work, results obtained by the flight control simulations of a prototype of hexarotor Unmanned Aerial Vehicle (UAV) are shown. The mathematical model and control of the hexacopter airframe are presented. To stabilize the entire system, Linear Quadratic Regulator (LQR) control is used in such a way to set both Proportional Derivative (PD) and Proportional Integral Derivative (PID) controls. Particle Swarm Optimization has been used to set the optimal coefficient matrices of the LQR control algorithm. The simulations are performed to show how LQR tuned PD and PID controllers lead to zero the error of the position along gravity acceleration direction, stop the rotation of UAV around body axes and stabilize the hexarotor. Moreover, the obtained LQR-PD and LQR-PID controllers have been tested by comparing the response to impulse disturbances of the nonlinear dynamical system with the response of the linearized one.

During the last years, alternative drive technologies, for example electrically powered vehicles (EV), have gained more and more attention, mainly caused by an increasing awareness of the impact of CO2 emissions on climate change and by the limitation of fossil fuels. However, these technologies currently come with new challenges due to limited lithium ion battery storage density and high battery costs which lead to a considerably reduced range in comparison to conventional internal combustion engine powered vehicles. For this reason, it is desirable to increase the vehicle range without enlarging the battery. When the route and the road slope are known in advance, it is possible to vary the vehicles velocity within certain limits in order to reduce the overall drivetrain energy consumption. This may either result in an increased range or, alternatively, in larger energy reserves for comfort functions such as air conditioning. In this presentation, we formulate the challenge of range extension as a multiobjective optimal control problem. We then apply different numerical methods to calculate the so-called Pareto set of optimal compromises for the drivetrain power profile with respect to the two concurrent objectives battery state of charge and mean velocity. In order to numerically solve the optimal control problem by means of a direct method, a time discretization of the drivetrain power profile is necessary. In combination with a vehicle dynamics simulation model, the optimal control problem is transformed into a high dimensional nonlinear optimization problem. For the approximation of the Pareto set, two different optimization algorithms implemented in the software package GAIO are used. The first one yields a global optimal solution by applying a set-oriented subdivision technique to parameter space. By construction, this technique is limited to coarse discretizations of the drivetrain power profile. In contrast, the second technique, which is based on an image space continuation method, is more suitable when the number of parameters is large while the number of objectives is less than five. We compare the solutions of the two algorithms and study the influence of different discretizations on the quality of the solutions. A MATLAB/Simulink model is used to describe the dynamics of an EV. It is based on a drivetrain efficiency map and considers vehicle properties such as rolling friction and air drag, as well as environmental conditions like slope and ambient temperature. The vehicle model takes into account the traction battery too, enabling an exact prediction of the batterys response to power requests of drivetrain and auxiliary loads, including state of charge.

In this paper we present a reduced order method for the solution of parametrized Stokes equations in domain composed by an arbitrary number of predefined shapes. The novelty of the proposed approach is the possibility to use a small set of precomputed bases to solve Stokes equations in very different computational domains, defined by combining one or more reference geometries. The selection of the basis functions is performed through an optimization greedy algorithm.

Uncertainties—more precisely bounded and stochastic disturbances—play a major role in control and estimation tasks in general. Examples for bounded uncertainty are lack of knowledge about specific parameters and manufacturing tolerances. Moreover, stochastic disturbances have a large influence on dynamic systems, especially on sensor measurements. These issues make it difficult to control a system such that robustness and stability are guaranteed if system parameters are not exactly known and system states cannot be measured with high accuracy due to process and measurement noise. Sliding mode techniques are known for their robustness, so that an extension of classical approaches is presented that accounts for uncertainties and estimates non-measurable states as well as unknown parameters.

Interval-based sliding mode controllers can be used efficiently for a robust stabilization of systems with bounded uncertainty. The real-time implementation of these procedures makes use of software libraries that provide functionalities for interval analysis and algorithmic differentiation. This paper gives an overview of possible extensions of such control procedures for the reliable stabilization of the thermal behavior of high-temperature solid oxide fuel cell systems. During the real-time stabilization, limitations of the range of state and control variables are treated by constraints implemented in a barrier Lyapunov function approach.

In this paper we propose a discrete time sliding mode congestion controller for a single virtual circuit in connection-oriented communication networks. The circuit is characterized by the non-negligible propagation delay, the maximum link capacity and unknown, time-varying data loss rate. The proposed controller generates non-negative and limited transmission rates, ensures upper bounded queue length in the bottleneck link buffer and may guarantee full utilization of the link capacity. In order to ensure fast reaction to the unpredictable data loss and unknown changes of the available bandwidth, the controller employs the dead-beat sliding hyperplane. However, straightforward application of the dead-beat paradigm could lead to unacceptably big transmission rates. Therefore, the controller is designed using the concept of the reaching law, which helps to attenuate the excessive magnitude of control signal at the beginning of the transmission process.

In this work we present a semi-classical modeling and simulation approach for ultra-narrow channels that has been implemented as part of the Vienna Schrödinger-Poisson (VSP) simulation framework (Baumgartner, J Comput Electron 12:701–721, 2013; http://www.globaltcad.com/en/products/vsp.html (2014)) over the past few years. Our research has been driven by two goals: maintaining high physical accuracy of the models while producing a computationally efficient and flexible simulation code.

Semiconductor spintronics is promising, because it allows creating microelectronic elements which are smaller and consume less energy than present charge-based devices. Silicon is the main element of modern charge-based electronics, thus, understanding the peculiarities of spin propagation in silicon is the key for designing novel devices. We investigate the electron momentum and the spin relaxation in thin (001) oriented SOI films using a k ⋅ p-based approach with spin degree of freedom properly included. We demonstrate that shear strain routinely used to enhance the electron mobility can boost the spin lifetime by an order of magnitude.

The existence and uniqueness of the electron transport Wigner equation solution, determined by boundary conditions, is analyzed in terms of the Neumann series expansion of the integral form of the equation, obtained with the help of Newton’s trajectories. For understanding of the peculiarities of Wigner-quantum electron transport in semiconductor structures such mathematical issues can not be separated from the physical attributes of the solution. In the presented analysis these two sides of the problem mutually interplay.The problem is first formulated from a physical point of view, where the stationary solution is considered as the long time limit of the general evolution problem posed by both initial and boundary conditions. The proof of convergence relies on the assumption for reasonable local conditions which may be specified for the kernel and on the fact that the Neumann series expansion corresponds to an integral equation of Volterra type with respect to the time variable.

Cutting edge semiconductor devices for power electronic applications, such as Phase Control Thyristors (PCTs) or Bimode Insulated Gate Transistor (BIGTs), present large area and complex 3D geometry, thus requiring full scale 3D models for their simulation. Moreover, sensitivity to temperature variations and complex loading conditions call for mixed mode simulation of distributed devices coupled to external controlling circuits. In this work, we describe a strategy for coupled simulation of 3D devices and lumped circuit networks, with particular emphasis on efficient iterative solution strategies for nonlinear equations. The algorithm presented is tested on a p-i-n power diode, for which quasi-static on-state and transient switching (reverse recovery) simulations are performed.

A hydrodynamical model for the charge and the heat transport in graphene is presented. The state variables are moments of the electron, hole and phonon distribution functions, and their evolution equations are derived from the respective Boltzmann equations by integration. The closure of the system is obtained by means of the maximum entropy principle and all the main scattering mechanisms are taken into account. Numerical simulations are presented in the case of a suspended graphene monolayer.

In this contribution, which is based on the results published in Barletti (J Math Phys 55:083303, 2014) and Barletti et al. (Tr Inst Mat 11:11–29, 2014), we apply the maximum entropy closure technique in order to derive equations of hydrodynamic type for a system of particles with spin-orbit interaction, with particular focus on the case of electrons on a graphene sheet.

The aim of this work is to use a numerical scheme based on the discontinuous Galerkin method for finding deterministic (non stochastic) solutions of the electron Boltzmann transport equation in graphene. The same methods has been already successfully applied to a more conventional semiconductor material like Si (Cheng et al., Comput Methods Appl Mech Eng 198(37–40):3130–3150, 2009; Cheng et al., Boletin de la Sociedad Espanola de Matematica Aplicada 54:47–64, 2011). A n-type doping or equivalently a high value of the Fermi potential is considered. Therefore we neglect the inter band scatterings but retain all the main electron-phonon scatterings. Simulations in graphene nano-ribbons are presented and discussed.

We show that in a semiconductor superlattice with long scattering times, damping of Bloch oscillations due to scattering is so small that nonlinearities may compensate it and Bloch oscillations persist even in the hydrodynamic regime. In order to demonstrate this, we propose a Boltzmann-Poisson transport model of miniband superlattices with inelastic collisions and we derive by singular perturbation methods hydrodynamic equations for electron density, electric field, and the complex amplitude of the Bloch oscillations. Numerical solutions of these equations show stable Bloch oscillations with spatially inhomogeneous field, charge, current density, and energy density profiles. These Bloch oscillations disappear as scattering times become sufficiently short. For sufficiently low lattice temperatures (70 K), Bloch and Gunn type oscillations mediated by electric field, current, and energy domains coexist for a range of voltages. For larger lattice temperatures (300 K), there are only Bloch oscillations with stationary amplitude and electric field profiles.

Dual Phase steel (DP steel) has shown high potential for automotive and other applications, due to its remarkable combined properties of high strength and good formability. The mechanical properties of the material are strictly related to the spatial distribution of the two steel phases, ferrite and martensite, and with their stochastic geometry. Unfortunately the experimental costs to obtain images of sections of steel samples are very high, so that one important industrial problem is to reduce the required number of 2D sections in order to either reconstruct the 3D geometry of the material, or to simulate realistic ones. In this work we will present a germ-grain statistical model which can be used for a best fitting of the main geometric characteristics of the martensite phase. The parameters of the model are estimated on the basis of morphological characteristics of the images of about 150 tomographic sections taken from a real sample. After optimization or tuning of the relevant parameters, the statistical model can then be used to identify the minimum number of sections of the sample which are needed to estimate the parameters in a reliable way.

We introduce a model to compute the annual performance of a heliostat field. We take into account topography, tracking errors, and the position and intensity of the sun. An approach is introduced, which improves on the otherwise expensive pairwise comparison to calculate shading and blocking. Because the computational time is reduced significantly, the presented implementation is sufficiently fast to allow for heliostat field layout optimization within a couple of hours. The optimization is executed via a genetic algorithm, which optimizes the heliostat positioning parameters as well as other design parameters, e.g. receiver tilt angle. A novel approach is used to reduce the search domain. Because the search domain delivers several local optima with comparable values of the objective function, the objective function is augmented. We use smoothing functionals to disperse the local optima. A field layout is optimized on a hilly ground in South Africa, with additional constraints on the heliostat positions.

In this paper an overview on modelling techniques and numerical methods applied to problems in water network simulation is given. The considered applications cover river alarm systems (Rentrop and Steinebach, Surv Math Ind 6:245–265, 1997), water level forecast methods (Steinebach and Wilke, J CIWEM 14(1):39–44, 2000) up to sewer and water supply networks (Steinebach et al., Mathematical Optimization of Water Networks Martin. Springer, Basel, 2012).The hyperbolic modelling equations are derived from mass and momentum conservation laws. A typical example are the well known Saint-Venant equations. For their numerical solution a conservative semi-discretisation in space by finite differences is proposed. A new well-balanced space discretisation scheme is presented which improves the local Lax-Friedrichs approach applied so far. Higher order discretisations are achieved by WENO methods (Kurganov and Levy, SIAM J Sci Comput 22(4):1461–1488, 2000).Together with appropriate boundary and coupling conditions this method of lines approach leads to an index-one DAE system. Efficient solution of the DAE system is the topic of Jax and Steinebach (ROW methods adapted to network simulation for fluid flow, in preparation).

This work deals with the model order reduction (MOR) of a nonlinear-parametric system of partial differential equations (PDEs). Applying a semidiscretization in space and replacing the nonlinearities by introducing new state variables, we set up quadratic-linear differential algebraic systems (QLDAE) and use a Krylov-subspace MOR. The approach is investigated for gas pipeline modeling.

Simulating free-surface and pressurised flow is important to many fields of application, especially in network approaches. Modelling equations to describe flow behaviour arising in these problems are often expressed by one-dimensional formulations of the hyperbolic shallow water equations. One established approach to realise their numerical computation is the method of lines based on semi-discretisation in space (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001; Steinebach and Weiner, Appl Numer Math 62:1567–1578, 2012; Steinebach et al., Modeling and numerical simulation of pipe flow problems in water supply systems. In: Martin, Klamroth, et al. (eds) Mathematical optimization of water networks. International series of numerical mathematics, vol 162, pp 3–15. Springer, Basel, 2012). It leads to index-one DAE systems as algebraic constraints are required to realise coupling and boundary conditions of single reaches.Linearly implicit ROW schemes proved to be effective to solve these DAE systems (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001). However, under certain conditions an extended partial explicit time-integration of the shallow water equations could be worthwhile to save computational effort. To restrict implicit solution by ROW schemes to stiff components while using explicit solution by RK methods for remaining terms, we adapt ROW method ROS34PRW (Rang, J Comput Appl Math 262:105–114, 2014) to an AMF and IMEX combining approach (Hundsdorfer and Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations. Springer, Berlin, Heidelberg, New York, 2003). Applied to first test problems regarding open channel flow, efficiency is analysed with respect to flow behaviour. Results prove to be advantageous especially concerning dynamical flow.

Parametric modeling as well as parametric model order reduction (PMOR) of parametric systems are being widely researched in many micro- and nano-electrical(-mechanical) problems as well as in coupled micro- and nano-electro-thermal problems. We propose an adaptive technique for automatically implementing PMOR, so as to automatically construct the reduced-order models. The adaptive technique is based on a posteriori error estimation and is realized through a greedy algorithm which uses the error estimation as a stopping criteria.

We present an extended analytic formulation for the determination of the temperature distribution along a bond-wire within a package in order to extract the maximum allowable current to not exceed a specific temperature. The closed-form formula involves the essential physical parameters that define a package, i.e., moulding compound material and dimensions, bond-wire characteristics, etc. This is very important if one wants to assess the influence of (randomly distributed) parameter variations on the current capacity of the wire by means of uncertainty quantification methods.

This paper presents a new solution method to deal with thermal effects in power designs. The new ingredients are: (1) the treatment of the electric and thermal fields are done fully self-consistent, (2) the dealing with (fragments of) the transistor fingers by using table models.

The project nanoCOPS (http://www.fp7-nanocops.eu) is a collaborative research project within the FP7-ICT research program funded by the European Union. The consortium comprises experts in mathematics and electrical engineering from seven universities (BU Wuppertal, HU Berlin, Brno UT, TU Darmstadt, FH OÖ Hagenberg, U Greifswald, KU Leuven), one research institute (MPI Magdeburg), two industrial partners (NXP Semiconductors Netherlands, ON Semiconductor Belgium) and two SMEs (MAGWEL—Belgium, ACCO Semiconductor—France).We present an overview of the project subjects addressing the “bottlenecks” in the currently-available infrastructure for nanoelectronic design and simulation. In particular, we discuss the issues of an electro-thermal-stress coupled simulation for Power-MOS device design and of simulation approaches for transceiver designs at high carrier frequencies and baseband waveforms such as OFDM (Orthogonal Frequency Division Multiplex).

In this paper we present a meshfree Lagrangian particle method for the simulation of dynamic wetting phenomena. The essence of dynamic wetting is that the contact angle between the interface of the immiscible fluids and the solid surface is a dynamic quantity. The dynamic contact angle is modeled as a boundary condition. The two-phase immiscible flow is described by the incompressible Navier-Stokes equations in combination with the continuous surface tension force model. The phases are distinguished by assigning colors to the particles, and the normal vector and curvature of the interface are computed from this color function. Chorin’s pressure projection method is used to solve the model equations in a meshfree framework. A two-phase Couette flow is considered, with a capillary bridge spanning the distance between the two walls. The details of the numerical methods can be found in Tiwari and Kuhnert (J Comput Appl Math 203:376–386, 2007), Tiwari et al. (Numerical simulation of wetting phenomena by a meshfree particle method. J Comput Appl Math 292:469–485, 2016. It is shown that the numerical results reproduce the employed empirical law for the dynamic contact angle.

The convergence for a co-simulation method is commonly based on an error recursion. Usually the contraction condition itself is obtained by some estimations (standard theory). This paper takes a closer look at the coupling structure of co-simulation model for a simple electric circuit. It is shown that with standard theory for our example no contraction could be inferred. However, co-simulation converges. By a detailed analysis, we can prove convergence in this case.

In this report we present the approximation of an infinite horizon optimal control problem for the evolutive Navier-Stokes system. The method is based on a model reduction technique, using a POD approximation, coupled with a Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function of the corresponding control problem for the reduced system. Although the approximation schemes available for the HJB are shown to be convergent for any dimension, in practice we need to restrict the dimension to rather small numbers and this limitation affects the accuracy of the POD approximation. We will present numerical tests for the control of the time-dependent Navier-Stokes system in two-dimensional spatial domains to illustrate our approach and to show the effectiveness of the method.

We present a validated fully three-dimensional simulation of a vertical-slot fish pass with GPUSPH, a high-performance CUDA implementation of the Smoothed Particles Hydrodynamics (SPH) numerical method for free-surface flows. The GPUSPH results are compared to flow velocity and water level measurement from a laboratory model with the same geometry. The results show good agreement between the numerical simulations and the experimental data.

A system of differential equations describing the behavior of a single ball in an elementary cell of the twisting-ball information display is considered. Nonlinear ordinary differential equations describe the ball shift and rotation. For efficient practical implementation of the display, the optimal values of ball and cell parameters are needed. To obtain these, the computer simulations were realized. Results of numerical experiments of modeling the balls with different physical parameters are presented. The numerical experiments show that the movement of the elementary particle of the twisting-ball display is extremely sensitive to the physical parameters of the balls but there exist nearly optimal combinations of these parameters. In this case the ball rotation intends towards some complete rotation cycle: if control voltage changes its polarity, the ball rotates nearly 180∘ and exposes right, black or white, size to the observer and the display works as expected.

We show an application of the Smoothed Particle Hydrodynamics (SPH) method to the simulation of a fully three-dimensional dam-break with floating objects. The simulation is done using GPUSPH, an implementation of the SPH method in CUDA which has been recently extended including support for fully coupled fluid/solid interaction. Boundary conditions are computed using the unified semi-analytical model proposed by Ferrand et al. SPH is also used to compute the total force and torque acting on the floating objects, which are then used to integrate the motion of the objects.

In the past few years Maxwell’s fish-eye lens has been subject to intense investigation in the context of transformation optics, mainly spurred by the possibility to create perfect imaging without the need to resort to negative refraction, one of the outstanding—but difficult to implement—properties of metamaterials. Here we extend this discussion to an acoustical fish-eye constructed in (2+1)D spacetime. The underlying acoustic wave is governed by a homogeneous spherical Helmholtz equation, which is shown to emerge from a variational principle in inherently covariant manner. The formal analytical solutions of the acoustic potential are derived.

The post-Newtonian terms included in the Frequency Difference of Arrival equation derived here by means of Synge’s world-function are considered to estimate their contribution in the precise Geolocation of passive radio transmitters at rest on the earth surface. Four of these terms are kinematical and the other two are gravitational. The kinematical terms account for the velocities of the radio transmitter and the receivers with respect to the Earth Centered Inertial reference frame, as well as for the relative velocities of the transmitter with respect to the receivers. The other two account for the gravitational attraction of an spherical earth on the receivers. The gravitational time delay has been taken into account to derive these terms.

Different satellite configurations are considered to show by numerical simulations the influence of the post-Newtonian corrections for the standard locations of radio transmitters by the Time Difference of Arrival method to the solutions of the Newtonian Frequency Difference of Arrival equations. The satellites considered are Low, Mid and Geostationary Earth Orbit satellites in a number never smaller than five. The radio transmitters are supposed to be passive and are placed either on the earth surface or in space.

Synge’s equations for time-like geodesics in terms of Fermi coordinates are used to derive post-Newtonian equations for the relative motion of satellites in coplanar circular near orbits about the earth. The reference frame, co-moving with the base satellite, is assumed to be a Fermi frame, that is, inertial guided. The resulting system is autonomous, linear, and reduces to the equation of the geodesic deviation for nearby satellites. Hence, it can be used by some Acquisition, Pointing, and Tracking systems to increase the accuracy presently reached in locating passive radio-transmitters.

Deterioration of copper and bronze artifacts is one of the main concerns for people working in cultural heritage. In particular a significant effort has been devoted to study the corrosion due to environmental conditions, such as temperature, moisture and the concentration of pollutants. We introduce a mathematical model able to describe the corrosion effects on a copper layer, which is subject to deposition of SO₂. The present model is based on a partial differential equation system with a double free boundary for monitoring and detecting copper corrosion products (mainly brochantite and cuprite).We assume to have a copper sample on which is formed a non protective oxide layer (Cu₂O), and, over this layer, a corrosion product (brochantite) grows. We aim to create a new approach to forecasting corrosion behavior without the necessity of an extensive use of laboratory testing using chemical-physical technologies, while taking into account the main chemical reactions. Although the model was kept simple, just describing the main reaction and transport processes involved, the mathematical simulations and the related model calibration are in agreement with the laboratory experiments.

Solutions of partial differential equations (PDEs) arising in science and industrial applications often undergo large variations occurring over small parts of the domain. Resolving steep gradients and oscillations properly is a numerical challenge. The idea of the r-refinement (moving mesh) is to improve the approximation quality—while keeping the computational effort—by redistributing a fixed number of grid points in areas of the domain where they are needed. In this work we develop a general moving mesh framework for 1d PDEs that is based on three parameterization layers representing referential, computational and desired parameters. Numerical results are shown for two different strategies that are applied to a fiber spinning process.

We present a method for the computational construction of virtual non-woven materials in the textile industry. The underlying model is a surrogate model for the lay-down process of a single fibre described by stochastic differential equations. In particular, we illustrate a computational method of constructing a virtual non-woven material from thousands of single fibres. Furthermore, we show a way of identifying contact points between the fibres. These contact points play an essential role in the corresponding fibre network, which is the basis for virtual material testing.

The mechanical properties of nonwoven materials are investigated and optimized. On the micro scale these fabrics are modeled as network of beams. Here linear Timoshenko beams and geometrically exact beams are compared in simple tension tests. Then a homogenization scheme is used to calculate effective material tensors of Kirchhoff-Love plates, where periodically repeatable representative volume elements containing fiber networks define the micro structure. Finally, these effective properties are optimized by changing the shape of the underlying network.

In non-woven manufacturing thousands of slender fibers are swirled by air flows before they lay down to form a web. The fiber-fluid interactions have a crucial influence on the quality of the final product. For the purpose of an efficient and fast computation of the multi-scale, two-way coupled interaction problem, we investigate classical homogenization strategies and a new continuum approach for very long fibers suspended in a fluid flow. We compare the results with Direct Numerical Simulation (DNS) and Immersed Boundary Methods for academic examples.

The dynamics of viscous jets in electrospinning processes varies from drop forming, whipping to coiling depending on the parameter regime. To investigate the practically relevant whipping regime more closely we use an asymptotic Cosserat rod model that is given by a stiff boundary value problem of ordinary differential equations. For the efficient simulation of the six-parametric problem we present a numerical approach that is based on a continuation-collocation method. On top of an implicit Runge-Kutta discretization of fourth order, suitable initial guesses and global convergence of the applied Newton method are achieved by a recursive continuation strategy. The numerical results are very convincing, they show the jet characteristics observed in the experiments.

Electrospinning is commonly used to produce very fine polymeric fibers. In this technique, a conducting liquid is pumped from an electrified needle into a surrounding dielectric media and the meniscus formed exhibits a conical shape, known as Taylor cone, due to the balance of electrical and surface tension forces. If the needle electrical potential is sufficiently high, the very strong electric field generated at the cone apex cannot be balanced by surface tension and a very thin jet is issued which eventually develops lateral instabilities that are responsible of additional stretching. In this work, we use a theoretical model that describes the kinematic of the midline of the jet, its radius and convective velocity from an Eulerian framework. Balances of mass, linear and angular momentum applied to a slice of the jet, as well as viscous law for stretching, bending and torsion describe the dynamics (nonlinear PDE in time and arclength of the midline). Capillary and electric forces are included in the momentum balance. If periodic orbits are explored, the time dependence of the PDE disappears when the motion is considered with respect to a frame rotating with the jet. One obtains a boundary value problem of ODEs with the frequency as a free parameter. This model is also suitable for describing other kinds of instabilities, such as the axisymmetric one which takes place in drop formation (dripping regime, electrospray).

In an airlay process thousands of fibers are distributed by a turbulent air stream to produce a nonwoven. We present models and numerical strategies in order to simulate the dynamics of the fibers until they are laid down to a conveyor belt. In particular, we focus on the effect of the turbulent air flow onto the fibers and their contact with walls. The simulation results of the laydown can be used further, e.g., as input for fiber laydown models in nonwoven production processes.

Individualized dose adaptation usually requires Therapeutic Drug Monitoring (TDM) based on patient blood samplings. However this invasive approach, generally accompanied with discomfort and cost, is not always justified since it may occur that the resulting dose adaptation does not significantly differ in a population whose individuals share similar characteristics. Inspired by the principle of maximum likelihood, we propose a probabilistic approach, based on population-pharmacokinetic modeling and simulation, to evaluate the therapeutic performance of a dosing regimen in terms of dose and time. Two types of therapeutic indicators, time-based and concentration-based, are suggested to assess quantitatively different drug regimens with the aim to identify the optimal one. For the population under investigation, our results identified a stable and robust optimal regimen and determined critical times including toxicity. Moreover, for a same therapeutic target, our approach enables to identify more than one corresponding regimen, giving thus a great flexibility in clinical practice.

Adaptive Algebraic Multigrid (or Multilevel) Methods (αAMG) are introduced to improve robustness and efficiency of classical algebraic multigrid methods in dealing with problems where no a priori knowledge or assumptions on the near-null kernel of the underlined matrix are available. Recently we proposed an adaptive (bootstrap) AMG method, αAMG, aimed to obtain a composite solver with a desired convergence rate. Each new multigrid component relies on a current (general) smooth vector and exploits pairwise aggregation based on weighted matching in a matrix graph to define a new automatic, general-purpose coarsening process, which we refer to as “the compatible weighted matching”. In this work, we present results that broaden the applicability of our method to different finite element discretizations of elliptic PDEs. In particular, we consider systems arising from displacement methods in linear elasticity problems and saddle-point systems that appear in the application of the mixed method to Darcy problems.

Cheddar cheese undergoes a number of biochemical changes during ripening. These processes were modelled with differential equations in a project at MISG2013 (the 2013 mathematics-in-industry study group) at Queensland University of Technology, Australia. Models could aid in the prediction of cheese quality from initial measurements. The model is presented and the effect of small changes in initial conditions is explored.

Stochastic volatility modelling is of fundamental importance in financial risk management. Among the most popular existing models in the literature are the Heston and the CEV stochastic models. Each of these models has some advantages that the other one lacks. For example, the CEV model and the Heston model have different relative properties concerning the leverage as well as the smile effects. In this work we deal with the hybrid stochastic volatility model that is based on the CEV and the Heston models combined. This alternative model is expected to perform better than any of the two previously mentioned models in terms of dealing with both the leverage and the smile effects. We deal with the pricing and hedging problems for European options. We first find the set of equivalent martingale measures (E.M.M.). The market is found to be incomplete within this framework since there are infinitely many of E.M.M. We then find the targeted E.M.M. by minimizing the entropy. Using Ito calculus and risk-neutral method enable us to find the partial differential equation (P.D.E.) corresponding to the option price. Moreover, we use Clark-Ocone formula to obtain a hedging strategy that minimizes the distance between the payoff and the value of the hedged portfolio at the maturity. This hedging strategy is among the most efficient available strategies.

In this paper, we consider a nonlinear Control Volume Finite Element (CVFE) scheme to solve an anisotropic degenerate Keller-Segel model over general meshes. This scheme, whose construction is based on the Godunov scheme to approximate the degenerate diffusion fluxes provided by the conforming finite element reconstruction on a primal triangular mesh and on a nonclassical upwind finite volume mesh to approximate the other terms over a dual mesh, ensures the discrete maximum principle whatever the anisotropy of the problem and without any restriction on the transmissibility coefficients. Numerical experiment is provided with full anisotropic and heterogeneous diffusion tensors over general mesh.

ATM Replenishment has become a widely studied popular problem in the modern age of human-machine interaction, due to several reasons. This paper presents a solution that is a two-part system. The first part or the analytics section is capable of providing very highly accurate, hourly forecasts of withdrawals from an ATM, for which past data is available. The Machine Learning algorithm used for obtaining forecasts, M5P, is based on a decision tree approach that reinforces various characteristics empirically found on withdrawal patterns in ATMs. The second and more important section formulates a simple mixed binary, goal programming problem. The weights are decided by the bank at the beginning of each period, and is particularly advantageous in decision flexibility terms. This is done specially keeping in mind the ever-changing operating budgets and customer service goals. In terms of hard numbers, this work describes a system which generates daily schedules with an error in withdrawal forecast per month (non-absolute addition) as low as 0.7 % at a correlation coefficient of 0.92.

There are numerous methods for shadow identification in satellite imagery using variety of means of image processing and pattern recognition. We, however, suggest to detect shadow artifacts using a straight geometrical approach with the help of digital elevation models (DEM). To demonstrate the pipeline of shadow detection process we use Landsat imagery and SRTM DEM database as the sources of images and matlab as software to run computational operations. Landsat provides a huge amount of images covering the surface of our planet. The SRTM collection covers nearly all terrain areas, thus excluding oceans, seas and other zero elevation areas. Both databases are widely used for image processing in various projects working with geographical data. The computational tools of matlab in their turn help to easily create functions and scripts for image processing in a relatively simple and fast way.

Lattice Quantum Chrome Dynamics (Lattice QCD) is a gauge theory formulated on a highly dimensional grid or lattice of points in space and time. It aims at determining observables such as the mass of elementary particles as accurate as possible, with computational costs as low as possible at the same time. Thus high performance computing tools are inevitable, as well as the construction of HPSC hardware tailored to the needs of Lattice QCD. In the Hybrid Monte Carlo (HMC) approach (Duane et al., Phys. Lett. B, 195:216, 1987 http://dx.doi.org/10.1016/0370-2693(87)91197-X), Monte Carlo simulations involving a molecular dynamics step in its core are performed, which yield physical values provided with their statistical errors.In this talk we concentrate on the Wilson Flow, a system of differential equations defined on the Lie group SU(3). The Wilson Flow can be used, e.g., to determine the physical lattice spacing which influences the result of the HMC simulations. We focus on tailored Runge-Kutta Lie group integration methods with step size prediction. The numerical results confirm that our strategy is able to reduce the statistical errors of the simulation.

There is a growing technology-driven interest in using external influences to move or shape small quantities of liquids, a process that is referred to as microfluidic actuation. The use of electrical, rather than mechanical, forces to achieve this actuation is convenient, because the resultant devices contain no moving parts. In this work we consider a sessile drop of an incompressible liquid with a high conductivity resting on the lower substrate inside a parallel-plate capacitor subjected to a relatively low frequency A.C. field. With the application of an electric field the drop deforms into a new static shape where the apex of the drop rises towards the upper electrode in order to balance the Maxwell electric stresses, surface tension and hydrostatic pressure on the interface. From experimental, numerical and asymptotic approaches we determine a predictive equation for the deformation as a function of initial contact angle and drop width, surface tension and applied voltage.

Main result of this article is demonstrating the weak global in time well posedness result for the equations governing fiber suspension flows for sufficiently small initial data under mild assumptions about the nonlinear equation for fiber orientation dynamics and the constitutive law, thus extending the previous local in time results. The required estimate of growth of the H2 norm is granted if the L∞ norm of fiber orientation state variables remains bounded. This is the case for fiber orientation tensors.

We prove global existence of a weak solution to the angiogenesis model proposed by A. Tosin, D. Ambrosi, L. Preziosi in Bull. Math. Biol. (2006) 7, 1819-1836. The model consists of compressible Navier-Stokes equations coupled with a reaction-diffusion equation describing the concentration of a chemical solution responsible of endothelial cells migration and blood vessels formation.Proofs are based on the control of the entropy associated to the hyperbolic equation of conservation mass and the adaptation of the results of P.L. Lions dealing with compressible fluids which are inevitable for all models dealing with compressible Navier-Stokes equations.We use the vanishing artificial viscosity method to prove existence of solutions, the main difficulty for passing to the limit is the lack of compactness due to hyperbolic equation which usually induces resonance phenomenon. This is overcome by using the concept of the compactness of effective viscous pressure combined with suitable renormalized solutions to the hyperbolic equation of mass conservation.

We present a new high-order compact scheme for the multi-dimensional Black-Scholes model with application to European Put options on a basket of two underlying assets. The scheme is second-order accurate in time and fourth-order accurate in space. Numerical examples confirm that a standard second-order finite difference scheme is significantly outperformed.

Bioventing is a technology used to abate the presence of pollutants in the subsoil. Microorganisms biodegrade the pollutant but the biochemical reaction requires oxygen and so an airflow is induced in the subsoil by means of injection and/or extraction wells.Costs, final result and decontamination time are reliant on contaminant type, soil permeability and several other factors, but oxygen subsoil concentration plays a very important role. For this reason a rational choice of well locations and flow rates is required.The mathematical definition of the optimal design problem will be set-up starting from a simplified mathematical model describing the bioventing system.A formal definition of decontaminated subsoil will be given and the set of system control variables will be identified. Optimization strategies such as cost minimization and time optimization will be mathematically described.

The aim of this project is to develop a virtual modelling tool which can be used to construct optimal design of power transmission lines and cables. They should meet the latest power transmission network technical and economical requirements. The mathematical model is based on a general heat conduction equation describing the diffusion, convection and radiation processes. We take into account a linear dependence of the resistance on temperature. The velocity of convective transport of the heat in air regions is obtained by solving a coupled thermoconvection problem including the heat conduction problem and a standard Navier-Stokes model of the flow in air. The changes of material coefficients in soil due to influence of heating are taken by solving a simplified mass balance equation for flows in porous media. The FVM is used to solve the obtained system of differential equations. Discretization of the domain is done by applying “aCute” mesh generator, which is a modification of the well-known Triangle mesh generator. The discrete schemes are implemented by using the OpenFOAM tool. Parallel versions of basic algorithms are also investigated. Results of computational experiments of simulation of real industrial underground cables are presented.

In this paper we consider 2D stationary boundary value problems for the system of magnetohydrodynamic (MHD) equations and the heat transfer equation. The viscous electrically conducting incompressible liquid moves between infinite cylinders with square or round sections placed periodically. We also consider similar 2D MHD channel flow with periodically placed obstacles on the channel walls. We analyse the 2D forced and free MHD convection flow and temperature around cylinders and obstacles in homogeneous external magnetic field. The cylinders, obstacles and walls of the channel with constant temperature are heated. The distributions of electromagnetic fields, forces, velocity and temperature fields have been calculated using the method of finite differences.The goal of such investigation is to obtain the distributions of stream function, temperature, velocity and the vortex formation in the plane of the cross-section between the cylinders and obstacles as function of the external magnetic field and of the direction of the gravitation.

The problem discussed in this paper is detecting the shape of an unknown object in a 2-dimensional static electric field. For simplicity, the problem is defined in a partially rectangular domain, where on a part of the boundary the potential and/or its normal derivative are known. On the other part of the boundary the boundary curve is unknown, and this curve is to be determined. The unknown part of the boundary curve describes the shape of the unknown object.The problem is defined in the complex plane by an analytic function w = f(z) = u(x, y) + iv(x, y) with the potential u as its real part. Then the inverse function is given as f−1(w) = x(u, v) + iy(u, v), where the functions x and y are harmonic in a rectangle with an unknown boundary condition on one boundary. The alternating-field technique is used to solve the unknown boundary condition.

The application of the SDRE-State Dependent Ricatti Equation method for the tracking control of a nonholonomic mobile robot is presented in this work. The proposed control law minimize the quadratic cost functional consisting of tracking errors and control efforts. The numerical simulations demonstrate the efficacy of the control method applied to track the linear and circular trajectory reference robot.

Among the components needed in photonic integrated circuits, dielectric waveguides and small footprint ring resonators play a key role for many applications and require sophisticated electromagnetic analysis and design. In this work, we present an accurate vectorial mode solver based on the finite element method. Considering a general nonreciprocal permittivity tensor, the proposed method allows us to investigate important cases of practical interest. To compute the electromagnetic modes, the Rayleigh-Ritz functional is derived for the non-self adjoint case, it is discretized using the node elements and the penalty function is added to remove the spurious solutions. Although the use of the penalty function is well known for the waveguide problem, it has been introduced for the first time (to the best of our knowledge) in the ring resonator modal analysis. The resulting quadratic eigenvalue problem is linearized and solved in terms of the propagation constant for a given frequency (i.e., γ-formulation). Unlike the earlier developed mode solvers, our approach allows us to precisely compute both forward and backward propagating modes in the nonreciprocal case. Moreover, it avoids time-consuming iterations and preserves matrix sparsity, ensuring high accuracy and computational efficiency.