Contact with friction using the augmented Lagrangian Method: a conditional constrained minimization problem

This work presents a formulation of the contact with friction between elastic bodies. This is a non linear problem due to unilateral constraints (inter-penetration of bodies) and friction. The solution of this problem can be found using optimization concepts, modelling the problem as a constrained minimization problem. The Finite Element Method is used to construct approximation spaces. The minimization problem has the total potential energy of the elastic bodies as the objective function, the non-inter-penetration conditions are represented by inequality constraints, and equality constraints are used to deal with the friction. Due to the presence of two friction conditions (stick and slip), specific equality constraints are present or not according to the current condition. Since the Coulomb friction condition depends on the normal and tangential contact stresses related to the constraints of the problem, it is devised a conditional dependent constrained minimization problem. An Augmented Lagrangian Method for constrained minimization is employed to solve this problem. This method, when applied to a contact problem, presents Lagrange Multipliers which have the physical meaning of contact forces. This fact allows to check the friction condition at each iteration. These concepts make possible to devise a computational scheme which lead to good numerical results.

Covalidation of Integral Transforms and Method of Lines in Nonlinear Convection-Diffusion with Mathematica

The Mathematica system (version 4.0) is employed in the solution of nonlinear difusion and convection-difusion problems, formulated as transient one-dimensional partial diferential equations with potential dependent equation coefficients. The Generalized Integral Transform Technique (GITT) is first implemented for the hybrid numerical-analytical solution of such classes of problems, through the symbolic integral transformation and elimination of the space variable, followed by the utilization of the built-in Mathematica function NDSolve for handling the resulting transformed ODE system. This approach ofers an error-controlled final numerical solution, through the simultaneous control of local errors in this reliable ODE's solver and of the proposed eigenfunction expansion truncation order. For covalidation purposes, the same built-in function NDSolve is employed in the direct solution of these partial diferential equations, as made possible by the algorithms implemented in Mathematica (versions 3.0 and up), based on application of the method of lines. Various numerical experiments are performed and relative merits of each approach are critically pointed out.

Numerical Simulation of the Cavitation in the Hydrodynamic Lubrication of Journal Bearings: a Parallel Algorithm

In this paper we present an algorithm for the numerical simulation of the cavitation in the hydrodynamic lubrication of journal bearings. Despite the fact that this physical process is usually modelled as a free boundary problem, we adopted the equivalent variational inequality formulation. We propose a two-level iterative algorithm, where the outer iteration is associated to the penalty method, used to transform the variational inequality into a variational equation, and the inner iteration is associated to the conjugate gradient method, used to solve the linear system generated by applying the finite element method to the variational equation. This inner part was implemented using the element by element strategy, which is easily parallelized. We analyse the behavior of two physical parameters and discuss some numerical results. Also, we analyse some results related to the performance of a parallel implementation of the algorithm.

An Hp-Adaptive Hierarchical Formulation for the Boundary Element Method Applied to Elasticity in Two Dimensions

This paper presents an HP-Adaptive Procedure with Hierarchical formulation for the Boundary Element Method in 2-D Elasticity problems. Firstly, H, P and HP formulations are defined. Then, the hierarchical concept, which allows a substantial reduction in the dimension of equation system, is introduced. The error estimator used is based on the residual computation over each node inside an element. Finally, the HP strategy is defined and applied to two examples.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Modal testing and numerical modeling of the dynamic properties of layered sheet-steel structure

Recently, due to the increasing total construction and transportation cost and difficulties associated with handling massive structural components or assemblies, there has been increasing financial pressure to reduce structural weight. Furthermore, advances in material technology coupled with continuing advances in design tools and techniques have encouraged engineers to vary and combine materials, offering new opportunities to reduce the weight of mechanical structures. These new lower mass systems, however, are more susceptible to inherent imbalances, a weakness that can result in higher shock and harmonic resonances which leads to poor structural dynamic performances. The objective of this thesis is the modeling of layered sheet steel elements, to accurately predict dynamic performance. During the development of the layered sheet steel model, the numerical modeling approach, the Finite Element Analysis and the Experimental Modal Analysis are applied in building a modal model of the layered sheet steel elements. Furthermore, in view of getting a better understanding of the dynamic behavior of layered sheet steel, several binding methods have been studied to understand and demonstrate how a binding method affects the dynamic behavior of layered sheet steel elements when compared to single homogeneous steel plate. Based on the developed layered sheet steel model, the dynamic behavior of a lightweight wheel structure to be used as the structure for the stator of an outer rotor Direct-Drive Permanent Magnet Synchronous Generator designed for high-power wind turbines is studied.

Application and development of numerical methods for the modelling of innovative gas cooled fission reactors

Innovative gas cooled reactors, such as the pebble bed reactor (PBR) and the gas cooled fast reactor (GFR) offer higher efficiency and new application areas for nuclear energy. Numerical methods were applied and developed to analyse the specific features of these reactor types with fully three dimensional calculation models. In the first part of this thesis, discrete element method (DEM) was used for a physically realistic modelling of the packing of fuel pebbles in PBR geometries and methods were developed for utilising the DEM results in subsequent reactor physics and thermal-hydraulics calculations. In the second part, the flow and heat transfer for a single gas cooled fuel rod of a GFR were investigated with computational fluid dynamics (CFD) methods. An in-house DEM implementation was validated and used for packing simulations, in which the effect of several parameters on the resulting average packing density was investigated. The restitution coefficient was found out to have the most significant effect. The results can be utilised in further work to obtain a pebble bed with a specific packing density. The packing structures of selected pebble beds were also analysed in detail and local variations in the packing density were observed, which should be taken into account especially in the reactor core thermal-hydraulic analyses. Two open source DEM codes were used to produce stochastic pebble bed configurations to add realism and improve the accuracy of criticality calculations performed with the Monte Carlo reactor physics code Serpent. Russian ASTRA criticality experiments were calculated. Pebble beds corresponding to the experimental specifications within measurement uncertainties were produced in DEM simulations and successfully exported into the subsequent reactor physics analysis. With the developed approach, two typical issues in Monte Carlo reactor physics calculations of pebble bed geometries were avoided. A novel method was developed and implemented as a MATLAB code to calculate porosities in the cells of a CFD calculation mesh constructed over a pebble bed obtained from DEM simulations. The code was further developed to distribute power and temperature data accurately between discrete based reactor physics and continuum based thermal-hydraulics models to enable coupled reactor core calculations. The developed method was also found useful for analysing sphere packings in general. CFD calculations were performed to investigate the pressure losses and heat transfer in three dimensional air cooled smooth and rib roughened rod geometries, housed inside a hexagonal flow channel representing a sub-channel of a single fuel rod of a GFR. The CFD geometry represented the test section of the L-STAR experimental facility at Karlsruhe Institute of Technology and the calculation results were compared to the corresponding experimental results. Knowledge was gained of the adequacy of various turbulence models and of the modelling requirements and issues related to the specific application. The obtained pressure loss results were in a relatively good agreement with the experimental data. Heat transfer in the smooth rod geometry was somewhat under predicted, which can partly be explained by unaccounted heat losses and uncertainties. In the rib roughened geometry heat transfer was severely under predicted by the used realisable k − epsilon turbulence model. An additional calculation with a v2 − f turbulence model showed significant improvement in the heat transfer results, which is most likely due to the better performance of the model in separated flow problems. Further investigations are suggested before using CFD to make conclusions of the heat transfer performance of rib roughened GFR fuel rod geometries. It is suggested that the viewpoints of numerical modelling are included in the planning of experiments to ease the challenging model construction and simulations and to avoid introducing additional sources of uncertainties. To facilitate the use of advanced calculation approaches, multi-physical aspects in experiments should also be considered and documented in a reasonable detail.

Advanced analysis and design methods for preparative chromatographic separation processes

Preparative liquid chromatography is one of the most selective separation techniques in the fine chemical, pharmaceutical, and food industries. Several process concepts have been developed and applied for improving the performance of classical batch chromatography. The most powerful approaches include various single-column recycling schemes, counter-current and cross-current multi-column setups, and hybrid processes where chromatography is coupled with other unit operations such as crystallization, chemical reactor, and/or solvent removal unit. To fully utilize the potential of stand-alone and integrated chromatographic processes, efficient methods for selecting the best process alternative as well as optimal operating conditions are needed. In this thesis, a unified method is developed for analysis and design of the following singlecolumn fixed bed processes and corresponding cross-current schemes: (1) batch chromatography, (2) batch chromatography with an integrated solvent removal unit, (3) mixed-recycle steady state recycling chromatography (SSR), and (4) mixed-recycle steady state recycling chromatography with solvent removal from fresh feed, recycle fraction, or column feed (SSR–SR). The method is based on the equilibrium theory of chromatography with an assumption of negligible mass transfer resistance and axial dispersion. The design criteria are given in general, dimensionless form that is formally analogous to that applied widely in the so called triangle theory of counter-current multi-column chromatography. Analytical design equations are derived for binary systems that follow competitive Langmuir adsorption isotherm model. For this purpose, the existing analytic solution of the ideal model of chromatography for binary Langmuir mixtures is completed by deriving missing explicit equations for the height and location of the pure first component shock in the case of a small feed pulse. It is thus shown that the entire chromatographic cycle at the column outlet can be expressed in closed-form. The developed design method allows predicting the feasible range of operating parameters that lead to desired product purities. It can be applied for the calculation of first estimates of optimal operating conditions, the analysis of process robustness, and the early-stage evaluation of different process alternatives. The design method is utilized to analyse the possibility to enhance the performance of conventional SSR chromatography by integrating it with a solvent removal unit. It is shown that the amount of fresh feed processed during a chromatographic cycle and thus the productivity of SSR process can be improved by removing solvent. The maximum solvent removal capacity depends on the location of the solvent removal unit and the physical solvent removal constraints, such as solubility, viscosity, and/or osmotic pressure limits. Usually, the most flexible option is to remove solvent from the column feed. Applicability of the equilibrium design for real, non-ideal separation problems is evaluated by means of numerical simulations. Due to assumption of infinite column efficiency, the developed design method is most applicable for high performance systems where thermodynamic effects are predominant, while significant deviations are observed under highly non-ideal conditions. The findings based on the equilibrium theory are applied to develop a shortcut approach for the design of chromatographic separation processes under strongly non-ideal conditions with significant dispersive effects. The method is based on a simple procedure applied to a single conventional chromatogram. Applicability of the approach for the design of batch and counter-current simulated moving bed processes is evaluated with case studies. It is shown that the shortcut approach works the better the higher the column efficiency and the lower the purity constraints are.

The effect of calculators on the numerical problem-solving skills and attitudes of primary students towards mathematics /

The purpose of this study was to determine the effect that calculators have on the attitudes and numerical problem-solving skills of primary students. The sample used for this research was one of convenience. The sample consisted of two grade 3 classes within the York Region District School Board. The students in the experimental group used calculators for this problem-solving unit. The students in the control group completed the same numerical problem-solving unit without the use of calculators. The pretest-posttest control group design was used for this study. All students involved in this study completed a computational pretest and an attitude pretest. At the end of the study, the students completed a computational posttest. Five students from the experimental group and five students from the control group received their posttests in the form of a taped interview. At the end of the unit, all students completed the attitude scale that they had received before the numerical problem-solving unit once again. Data for qualitative analysis included anecdotal observations, journal entries, and transcribed interviews. The constant comparative method was used to analyze the qualitative data. A t test was also performed on the data to determine whether there were changes in test and attitude scores between the control and experimental group. Overall, the findings of this study support the hypothesis that calculators improve the attitudes of primary students toward mathematics. Also, there is some evidence to suggest that calculators improve the computational skills of grade 3 students.

Quantum Monte Carlo : some theoretical and numerical studies

In Part I, theoretical derivations for Variational Monte Carlo calculations are compared with results from a numerical calculation of He; both indicate that minimization of the ratio estimate of Evar , denoted EMC ' provides different optimal variational parameters than does minimization of the variance of E MC • Similar derivations for Diffusion Monte Carlo calculations provide a theoretical justification for empirical observations made by other workers. In Part II, Importance sampling in prolate spheroidal coordinates allows Monte Carlo calculations to be made of E for the vdW molecule var He2' using a simplifying partitioning of the Hamiltonian and both an HF-SCF and an explicitly correlated wavefunction. Improvements are suggested which would permit the extension of the computational precision to the point where an estimate of the interaction energy could be made~

The effectiveness of explicit strategy instruction on the top level structure of a text

Thesis (M.Ed.)-- Brock University, 1995.

Developing a numerical inverse-theory-based extraction of orientation-dependent relaxation rates from partially- relaxed spectra

Second-rank tensor interactions, such as quadrupolar interactions between the spin- 1 deuterium nuclei and the electric field gradients created by chemical bonds, are affected by rapid random molecular motions that modulate the orientation of the molecule with respect to the external magnetic field. In biological and model membrane systems, where a distribution of dynamically averaged anisotropies (quadrupolar splittings, chemical shift anisotropies, etc.) is present and where, in addition, various parts of the sample may undergo a partial magnetic alignment, the numerical analysis of the resulting Nuclear Magnetic Resonance (NMR) spectra is a mathematically ill-posed problem. However, numerical methods (de-Pakeing, Tikhonov regularization) exist that allow for a simultaneous determination of both the anisotropy and orientational distributions. An additional complication arises when relaxation is taken into account. This work presents a method of obtaining the orientation dependence of the relaxation rates that can be used for the analysis of the molecular motions on a broad range of time scales. An arbitrary set of exponential decay rates is described by a three-term truncated Legendre polynomial expansion in the orientation dependence, as appropriate for a second-rank tensor interaction, and a linear approximation to the individual decay rates is made. Thus a severe numerical instability caused by the presence of noise in the experimental data is avoided. At the same time, enough flexibility in the inversion algorithm is retained to achieve a meaningful mapping from raw experimental data to a set of intermediate, model-free

Subliminal Exposure to Empathy Primes: A New Method to Reduce Intergroup Bias?

An abundant literature has demonstrated the benefits of empathy for intergroup relations (e.g., Batson, Chang, Orr, & Rowland, 2002). In addition, empathy has been identified as the mechanism by which various successful prejudice-reduction procedures impact attitudes and behaviour (e.g., Costello & Hodson, 2010). However, standard explicit techniques used in empathy-prejudice research have a number of potential limitations (e.g., resistance; McGregor, 1993). The present project explored an alternative technique, subliminally priming (i.e., outside of awareness) empathy-relevant terms (Study 1), or empathy itself (Study 2). Study 1 compared the effects of exposure to subliminal empathy-relevant primes (e.g., compassion) versus no priming and priming the opposite of empathy (e.g., indifference) on prejudice (i.e., negative attitudes), discrimination (i.e., resource allocation), and helping behaviour (i.e., willingness to empower, directly assist, or expect group change) towards immigrants. Relative to priming the opposite of empathy, participants exposed to primes of empathy-relevant constructs expressed less prejudice and were more willingness to empower immigrants. In addition, the effects were not moderated by individual differences in prejudice-relevant variables (i.e., Disgust Sensitivity, Intergroup Disgust-Sensitivity, Intergroup Anxiety, Social Dominance Orientation, Right-wing Authoritarianism). Study 2 considered a different target category (i.e., Blacks) and attempted to strengthen the effects found by comparing the impact of subliminal empathy primes (relative to no prime or subliminal primes of empathy paired with Blacks) on explicit prejudice towards marginalized groups and Blacks, willingness to help marginalized groups and Blacks, as well as implicit prejudice towards Blacks. In addition, Study 2 considered potential mechanisms for the predicted effects; specifically, general empathy, affective empathy towards Blacks, cognitive empathy towards Blacks, positive mood, and negative mood. Unfortunately, using subliminal empathy primes “backfired”, such that exposure to subliminal empathy primes (relative to no prime) heightened prejudice towards marginalized groups and Blacks, and led to stronger expectations that marginalized groups and Blacks improve their own situation. However, exposure to subliminal primes pairing empathy with Blacks (relative to subliminal empathy primes alone) resulted in less prejudice towards marginalized groups and more willingness to directly assist Blacks, as expected. Interestingly, exposure to subliminal primes of empathy paired with Blacks (vs. empathy alone) resulted in more pro-White bias on the implicit prejudice measure. Study 2 did not find that the potential mediators measured explained the effects found. Overall, the results of the present project do not provide strong support for the use of subliminal empathy primes for improving intergroup relations. In fact, the results of Study 2 suggest that the use of subliminal empathy primes may even backfire. The implications for intergroup research on empathy and priming procedures generally are discussed.

Modélisation du carnet d'ordres limites et prévision de séries temporelles

Le contenu de cette thèse est divisé de la façon suivante. Après un premier chapitre d’introduction, le Chapitre 2 est consacré à introduire aussi simplement que possible certaines des théories qui seront utilisées dans les deux premiers articles. Dans un premier temps, nous discuterons des points importants pour la construction de l’intégrale stochastique par rapport aux semimartingales avec paramètre spatial. Ensuite, nous décrirons les principaux résultats de la théorie de l’évaluation en monde neutre au risque et, finalement, nous donnerons une brève description d’une méthode d’optimisation connue sous le nom de dualité. Les Chapitres 3 et 4 traitent de la modélisation de l’illiquidité et font l’objet de deux articles. Le premier propose un modèle en temps continu pour la structure et le comportement du carnet d’ordres limites. Le comportement du portefeuille d’un investisseur utilisant des ordres de marché est déduit et des conditions permettant d’éliminer les possibilités d’arbitrages sont données. Grâce à la formule d’Itô généralisée il est aussi possible d’écrire la valeur du portefeuille comme une équation différentielle stochastique. Un exemple complet de modèle de marché est présenté de même qu’une méthode de calibrage. Dans le deuxième article, écrit en collaboration avec Bruno Rémillard, nous proposons un modèle similaire mais cette fois-ci en temps discret. La question de tarification des produits dérivés est étudiée et des solutions pour le prix des options européennes de vente et d’achat sont données sous forme explicite. Des conditions spécifiques à ce modèle qui permettent d’éliminer l’arbitrage sont aussi données. Grâce à la méthode duale, nous montrons qu’il est aussi possible d’écrire le prix des options européennes comme un problème d’optimisation d’une espérance sur en ensemble de mesures de probabilité. Le Chapitre 5 contient le troisième article de la thèse et porte sur un sujet différent. Dans cet article, aussi écrit en collaboration avec Bruno Rémillard, nous proposons une méthode de prévision des séries temporelles basée sur les copules multivariées. Afin de mieux comprendre le gain en performance que donne cette méthode, nous étudions à l’aide d’expériences numériques l’effet de la force et la structure de dépendance sur les prévisions. Puisque les copules permettent d’isoler la structure de dépendance et les distributions marginales, nous étudions l’impact de différentes distributions marginales sur la performance des prévisions. Finalement, nous étudions aussi l’effet des erreurs d’estimation sur la performance des prévisions. Dans tous les cas, nous comparons la performance des prévisions en utilisant des prévisions provenant d’une série bivariée et d’une série univariée, ce qui permet d’illustrer l’avantage de cette méthode. Dans un intérêt plus pratique, nous présentons une application complète sur des données financières.

Development of Free Space Method for Material Characterisation and GPR Study & A Case Study on Biological Samples

The thesis mainly focuses on material characterization in different environments: freely available samples taken in planar fonn, biological samples available in small quantities and buried objects.Free space method, finds many applications in the fields of industry, medicine and communication. As it is a non-contact method, it can be employed for monitoring the electrical properties of materials moving through a conveyor belt in real time. Also, measurement on such systems at high temperature is possible. NID theory can be applied to the characterization of thin films. Dielectric properties of thin films deposited on any dielectric substrate can be determined. ln chemical industry, the stages of a chemical reaction can be monitored online. Online monitoring will be more efficient as it saves time and avoids risk of sample collection.Dielectric contrast is one of the main factors, which decides the detectability of a system. lt could be noted that the two dielectric objects of same dielectric constant 3.2 (s, of plastic mine) placed in a medium of dielectric constant 2.56 (er of sand) could even be detected employing the time domain analysis of the reflected signal. This type of detection finds strategic importance as it provides solution to the problem of clearance of non-metallic mines. The demining of these mines using the conventional techniques had been proved futile. The studies on the detection of voids and leakage in pipes find many applications.The determined electrical properties of tissues can be used for numerical modeling of cells, microwave imaging, SAR test etc. All these techniques need the accurate determination of dielectric constant. ln the modem world, the use of cellular and other wireless communication systems is booming up. At the same time people are concemed about the hazardous effects of microwaves on living cells. The effect is usually studied on human phantom models. The construction of the models requires the knowledge of the dielectric parameters of the various body tissues. lt is in this context that the present study gains significance. The case study on biological samples shows that the properties of normal and infected body tissues are different. Even though the change in the dielectric properties of infected samples from that of normal one may not be a clear evidence of an ailment, it is an indication of some disorder.ln medical field, the free space method may be adapted for imaging the biological samples. This method can also be used in wireless technology. Evaluation of electrical properties and attenuation of obstacles in the path of RF waves can be done using free waves. An intelligent system for controlling the power output or frequency depending on the feed back values of the attenuation may be developed.The simulation employed in GPR can be extended for the exploration of the effects due to the factors such as the different proportion of water content in the soil, the level and roughness of the soil etc on the reflected signal. This may find applications in geological explorations. ln the detection of mines, a state-of-the art technique for scanning and imaging an active mine field can be developed using GPR. The probing antenna can be attached to a robotic arm capable of three degrees of rotation and the whole detecting system can be housed in a military vehicle. In industry, a system based on the GPR principle can be developed for monitoring liquid or gas through a pipe, as pipe with and without the sample gives different reflection responses. lt may also be implemented for the online monitoring of different stages of extraction and purification of crude petroleum in a plant.Since biological samples show fluctuation in the dielectric nature with time and other physiological conditions, more investigation in this direction should be done. The infected cells at various stages of advancement and the normal cells should be analysed. The results from these comparative studies can be utilized for the detection of the onset of such diseases. Studying the properties of infected tissues at different stages, the threshold of detectability of infected cells can be determined.