Numerical analysis of a model of metal solidification, 2D case

The work discusses numerical simulation of metal solidification. The difficulties of calculatibg such processes are that simulation is conducted at several scale levels simultaneously. At present experimental studies cover numerous aspects of metal solidification; however, there is no generalized concept of this process. The model applied in this study is based on the concept of solidified alloy in the form of a porous medium, where propagation of perturbations is described by the Biot equations. In order to describe nucleation, the modified Cahn-Hillard equation is applied. Previously a one-dimensional (1D) numerical model has been developed, its convergence has been demonstrated and a possibility of obtaining of various modes of solidification has been shown at modification of parameters. This work presents the 2D numerical model and examples of calculations. Since a multiscale approach requires high CPU resources, the 2D calculations are based on explicit and explicit-iterative algorithms which are efficiently implemented using a multiprocessor computer.     

Bifurcation and metastability in a new one-dimensional model for martensitic phase transitions

Materials undergoing stress-induced martensitic phase transitions often form complex twinned microstructures with multiple phase boundaries. They also exhibit hysteretic mechanical behavior. We propose and analyze a one-dimensional model for twinning. We consider two elastic bars coupled by a system of continuously distributed linear springs. One of the bars has a two-well nonconvex elastic energy density that models a two-variant martensitic phase. The other bar is linearly elastic and is meant to model the parent austenite phase. Interfacial energy is modeled by a strain-gradient term. Various types of boundary conditions model parameter-dependent loading. A local bifurcation analysis shows that local energy minima (metastable states) often involve a large number of phase boundaries. This is confirmed by the global-bifurcation diagrams obtained numerically. We observe that this microstructure emerges via both sudden (finite) and gradual (infinitesimal) phase nucleation. We propose an energetic argument that predicts hysteresis in overall load-deformation behavior due to metastability of multiple equilibria. A limiting case with zero interfacial energy is treated analytically, yielding global solution diagrams.     

Numerical solutions for the one-dimensional heat-conduction equation using a spreadsheet

We show how to use a spreadsheet to calculate numerical solutions of the one-dimensional time-dependent heat-conduction equation. We find the spreadsheet to be a practical tool for numerical calculations, because the algorithms can be implemented simply and quickly without complicated programming, and the spreadsheet utilities can be used not only for graphics, printing, and file management, but also for advanced mathematical operations.We implement the explicit and the Crank-Nicholson forms of the finite-difference approximations and discuss the geological applications of both methods. We also show how to adjust these two algorithms to a nonhomogeneous lithosphere in which the thermal properties (thermal conductivity, density, and radioactive heat generation) change from the upper crust to the lower crust and to the mantle.The solution is presented in a way that can fit any spreadsheet (Lotus-123, Quattro-Pro, Excel). In addition, a Quattro-Pro program with macros that calculate and display the thermal evolution of the lithosphere after a thermal perturbation is enclosed in an appendix.     

Numerical solution of a phase field model for polycrystallization processes in binary mixtures

We consider the numerical solution of a phase field model for polycrystallization in the solidification of binary mixtures in a domain \( \varOmega \subset \mathbb {R}^2\). The model is based on a free energy in terms of three order parameters: the local orientation \(\varTheta \) of the crystals, the local crystallinity \(\phi \), and the concentration c of one of the components of the binary mixture. The equations of motion are given by an initial-boundary value problem for a coupled system of partial differential equations consisting of a regularized second order total variation flow in \( \varTheta \), an \(L^2\) gradient flow in \(\phi \), and a \(W^{1,2}(\varOmega )^*\) gradient flow in c. Based on an implicit discretization in time by the backward Euler scheme, we suggest a splitting method such that the three semidiscretized equations can be solved separately and prove existence of a solution. As far as the discretization in space is concerned, the fourth order Cahn–Hilliard type equation in c is taken care of by a \(\hbox {C}^0\) Interior Penalty Discontinuous Galerkin approximation which has the advantage that the same finite element space can be used as well for the spatial discretization of the equations in \( \varTheta \) and \( \phi \). The fully discretized equations represent parameter dependent nonlinear algebraic systems with the discrete time as a parameter. They are solved by a predictor corrector continuation strategy featuring an adaptive choice of the time-step. Numerical results illustrate the performance of the suggested numerical method.

     

Analysis of a new algorithm for one-dimensional minimization

Davidon has recently introduced a new approach to optimization using the idea of nonlinear scaling. In this paper we study the algorithm that results when applying his ideas to the one-dimensional case. We show that the algorithm is locally convergent withQ-order equal 2 and compare it with the method of cubic interpolation.     

Numerical solutions of non-linear kinetic equations for a one-dimensional evaporation-condensation problem

Numerical solutions have been obtained for both the nonlinear Boltzmann equation (for two collision laws) and the Krook equation for a one-dimensional evaporation-condensation problem for a range of parameters. Our calculations of rate indicate that the linear prediction underestimates the non-equilibrium hindrance effect due to intermolecular collisions. The Krook evaporation rates are lower than the corresponding Boltzmann results and, thus, differ even more from the linear values. The Boltzmann evaporation rates for a gas of hard spheres are lower than those for Maxwelian molecules at lower values of Knudsen number. The microscopic and macroscopic properties obtained from the Krook solutions differ appreciably from the corresponding Boltzmann results.     

Numerical analysis of a time allocation model accounting for choice overload

The cost of time has been suggested as a factor associated with the choice overload problem. This term refers to the discomfort or paralysis experienced by individuals when facing a choice within a large set of alternatives, as it has been evidenced in experiments by behavioural and social psychologists. We introduce a rational model of time allocation to analyse how increasing the number of options of a given product may change consumer's allocation of time and in turn affect her welfare. Under some standard assumptions, the numerical analysis of the model reproduces two key experimental findings, namely choice paralysis – i.e. the choice problem is abandoned if the number of options is too large – and choice dissatisfaction – that is, the apparent paradox that increasing the number of considered options beyond certain limit, in turn choosing better, eventually diminishes welfare. The model analysis provides specific threshold values for the occurrence of both phenomena.     

Advanced control of crystallization processes

A key bottleneck in the production of pharmaceuticals and many other products is the formation of crystals from solution. The control of the crystal size distribution can be critically important for efficient downstream operations such as filtration and drying, and product effectiveness (e.g., bioavailability, tablet stability). This paper provides an overview of recent developments in the control of crystallization processes, including activities in sensor technologies, model identification, experimental design, process simulation, robustness analysis, and optimal control.     

On a rough AUSM scheme for a one-dimensional two-phase model

We are interested in exploring Advection Upstream Splitting Method (AUSM) schemes for hyperbolic systems of conservation laws which do not allow any analytical calculation of the Jacobian. For this purpose, we consider a two-phase model which has been used for modeling of unsteady compressible flow of oil and gas in pipes. The model consists of two mass conservation equations, one for each phase, and a common momentum equation. Since no analytical Jacobian can be obtained it is more difficult to use classical schemes such as Roe- and Godunov-type schemes. We propose an AUSM scheme for the current two-phase model obtained through natural generalizations of ideas described in M.-S. Liou [J. Comput Phys 129(2) (1996) 364]. A main feature of AUSM is simplicity and efficiency since it does not require the Jacobian. In particular, we prove that the proposed AUSM type scheme preserves the positivity of scalar quantities such as pressure, fluid densities and volume fractions. This guarantees that the scheme can handle the important and delicate case of transition from two-phase to single-phase flow without introducing negative masses. Many numerical results are included to confirm the accuracy and robustness of the proposed AUSM scheme. In particular, it is demonstrated that the AUSM scheme gives low numerical dissipation at volume fraction contact discontinuities and is able to produce stable and non-oscillatory solutions, also when more complex slip relations are used, that is, when the relative motion of one phase with respect to the other is more or less complicated. This makes the scheme suitable for simulations of many important two-phase flow processes.     

Empirical analysis of anticipatory standardization processes: a case study

The processes followed for developing anticipatory standards such as those for web services are still not well-understood. In spite of the openness of the process, there are few analyses that shed light on the roles that different participants play or the actions they engage in during the development of these standards. We analyze archival documents that capture development of SOAP, a core web service standard. Our analysis shows that participants spend a bulk of their time discussing technical issues, identifying action items, and engaging in discussion to reach consensus. These activities reveal prototypical roles that participants take on such as: Advocate, Architect, Bystander, Critic, Facilitator, Guru, and Procrastinator. Together, the findings support the existence of three clusters in standards development processes: design activities performed by Architects, sense-making activities performed by Critics, and managerial activities performed by Facilitators; along with the important activity of coordinating the work of multiple participants. We discuss implications of our findings and identify opportunities for future work.     

Numerical analysis of rational processes beyond Markov chains

Matrix exponential distributions and rational arrival processes have been proposed as an extension to pure Markov models. The paper presents an approach where these process types are used to describe the timing behavior in quantitative models like queueing networks, stochastic Petri nets or stochastic automata networks. The resulting stochastic process, which is called a rational process, is defined and it is shown that the matrix governing the behavior of the process has a structured representation which allows one to represent the matrix in a very compact form.     

A new T-S fuzzy model predictive control for nonlinear processes

In this paper, a novel fuzzy Generalized Predictive Control (GPC) is proposed for discrete-time nonlinear systems via Takagi-Sugeno system based Kernel Ridge Regression (TS-KRR). The TS-KRR strategy approximates the unknown nonlinear systems by learning the Takagi-Sugeno (TS) fuzzy parameters from the input-output data. Two main steps are required to construct the TS-KRR: the first step is to use a clustering algorithm such as the clustering based Particle Swarm Optimization (PSO) algorithm that separates the input data into clusters and obtains the antecedent TS fuzzy model parameters. In the second step, the consequent TS fuzzy parameters are obtained using a Kernel ridge regression algorithm. Furthermore, the TS based predictive control is created by integrating the TS-KRR into the Generalized Predictive Controller. Next, an adaptive, online, version of TS-KRR is proposed and integrated with the GPC controller resulting an efficient adaptive fuzzy generalized predictive control methodology that can deal with most of the industrial plants and has the ability to deal with disturbances and variations of the model parameters. In the adaptive TS-KRR algorithm, the antecedent parameters are initialized with a simple K-means algorithm and updated using a simple gradient algorithm. Then, the consequent parameters are obtained using the sliding-window Kernel Recursive Least squares (KRLS) algorithm. Finally, two nonlinear systems: A surge tank and Continuous Stirred Tank Reactor (CSTR) systems were used to investigate the performance of the new adaptive TS-KRR GPC controller. Furthermore, the results obtained by the adaptive TS-KRR GPC controller were compared with two other controllers. The numerical results demonstrate the reliability of the proposed adaptive TS-KRR GPC method for discrete-time nonlinear systems.     

Programmable calculator section: Numerical analysis of one-dimensional and cylindrical diffusion systems using a programmable calculator

In this article the calculation of the diffusion coefficient according to the solution of Fick's second law for one-dimensional and cylindrical geometry respectively is treated. The calculation is carried out on the basis of analytical data quoting the time-dependent penetration depth. This is done by means of a curve-fitting method with the help of programmes for a Texas Instruments 59 pocket calculator. The diffusion of Cl⁻ into concrete is discussed exemplarily.     

Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes

This paper deals with the equilibria and stability analysis of the two step anaerobic model initially proposed by [12] to describe the dynamical behavior of an anaerobic fixed-bed wastewater treatment process. In a first part, the model is analyzed: its equilibria and their stability are established considering qualitative properties of the kinetics. In a second part, it is shown that the overloading tolerance (denoted herein OT), a parameter proposed in [9] to monitor anaerobic processes on-line, may not be suitable for monitoring the system and even causes serious problems under certain functioning conditions. Based on the analysis results established in the first part, a modified OT is proposed and evaluated in simulation.     

Numerical solution of a one-dimensional inverse problem by the discontinuous Galerkin method

In this paper, we combine a Tikhonov regularization with a discontinuous Galerkin method to solve an inverse problem in one-dimension. We show that the regularization is simpler than in the case of the inversion using continuous finite elements. We numerically demonstrate that there exist optimal step sizes and polynomial degrees for inversion using the DG method. Numerical results are compared with those obtained by applying the standard finite element method with B-splines as a basis.     

Batch-to-batch model improvement for cooling crystallization

Two batch-to-batch model update strategies for model-based control of batch cooling crystallization are presented. In Iterative Learning Control, a nominal process model is adjusted by a non-parametric, additive correction term which depends on the difference between the measured output and the model prediction in the previous batch. In Iterative Identification Control, the uncertain model parameters are iteratively estimated using the measured batch data. Due to the different nature of the model update, the two algorithms have complementary advantages and disadvantages which are investigated in a simulation study and through experiments performed on a pilot-scale crystallizer.     

Numerical analysis of the mathematical model of hydroelastic oscillations in a curved pipeline

Numerical experiments on computational mechanics of internal waves in pipes were performed. We study selected problems of hydraulic impact and acoustic vibrations. A comparison with the results published in the press was performed. It was established that the considered mathematical model adequately describes all the studied samples and can be applied to the calculation of industrial pipelines.     

Numerical analysis and computing of a non-arbitrage liquidity model with observable parameters for derivatives

This paper deals with the numerical analysis and computing of a nonlinear model of option pricing appearing in illiquid markets with observable parameters for derivatives. A consistent monotone finite difference scheme is proposed and a stability condition on the stepsize discretizations is given.     

Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional processes

This paper presents consistent criteria for evaluating the selection of tuning parameters for an industrial model predictive control of large-scale cross-directional (CD) processes using a two-dimensional (temporal and spatial) frequency analysis technique. The concept of rectangular circulant matrices (RCMs) and their properties are presented. It is shown that large-scale CD processes can be approximated as RCMs and then diagonalized by complex Fourier matrices, allowing analysis in terms of a family of SISO transfer functions across the spatial frequencies. Familiar concepts from control engineering such as bandwidth and stability margin are extended into the two-dimensional frequency domain, providing intuitive measures of closed-loop performance and robustness.     

Development of a maturity model for electronic invoice processes

The digitalization of invoice processes provides a good opportunity for companies to pare down expenses, optimize administrative tasks, and increase efficiency and competitiveness. But the digitalization is limited by a variety of software solutions, legal uncertainties, heterogeneous demands, lack of know-how, and information system infrastructure incompatibilities. A holistic map of electronic invoice processes is mandatory, especially to demonstrate different levels of process integration and optimization. A maturity model puts this into practice and provides companies with a tool to identify their current situation and to derive recommendations to optimize that situation. In this paper, a maturity model for electronic invoice processes will be developed using exploratory data from focus groups. A theoretical approach that is based on a procedure-model for developing maturity models is applied. Four categories (strategy, acceptance, processes & organization, and technology) are identified and enriched by sub-categories. Future research requires the development of detailed maturity metrics.