Numerical Testing of a New Positivity-Preserving Interpolation Algorithm

An important component of a number of computational modeling algorithms is an interpolation method that preserves the positivity of the function being interpolated. This report describes the numerical testing of a new positivity-preserving algorithm that is designed to be used when interpolating from a solution defined on one grid to different spatial grid. The motivating application is a numerical weather prediction (NWP) code that uses spectral elements as the discretization choice for its dynamics core and Cartesian product meshes for the evaluation of its physics routines. This combination of spectral elements, which use nonuniformly spaced quadrature/collocation points, and uniformly-spaced Cartesian meshes combined with the desire to maintain positivity when moving between these necessitates our work. This new approach is evaluated against several typical algorithms in use on a range of test problems in one or more space dimensions. The results obtained show that the new method is competitive in terms of observed accuracy while at the same time preserving the underlying positivity of the functions being interpolated.Comment: 58 pages, 17 figure

Applied Mathematics and Computational Physics

As faster and more efficient numerical algorithms become available, the understanding of the physics and the mathematical foundation behind these new methods will play an increasingly important role. This Special Issue provides a platform for researchers from both academia and industry to present their novel computational methods that have engineering and physics applications

Fast methods for modelling fluid flow and characterising petroleum reservoirs

This thesis tackles three kinds of computationally efficient methods widely applicable in the fields of engineering, simulation and numerical modelling. First, the Non-Intrusive Reduced Order Modelling (NIROM) is discussed, reframed, generalised and tested. While NIROM is a general methodology, the main focus of this work is to evaluate its potential in the field of reservoir modelling. For this purpose a new method for constructing parameterised NIROMs is proposed and the POD-RBF approach is investigated on a number of representative test cases. A detailed analysis concludes with NIROM not being a viable practical solution at this stage; the underlying issues, their causes and future development the method are discussed in detail. Second, a method for classifying well log data is given. The method is an alternative to typical machine learning (ML) approaches, which up to date have been the only tools utilised for the purpose. Our approach is motivated by (and mitigates a number of) issues with applying ML in practical applications, in particular the lack of explainability. Instead of being a complex surrogate with a large number of degrees of freedom (cf ML), our model consists of the automatically re-scaled training set and a single additional number extracted during the training procedure. The technique proposed is characterised by a case-independent design, very high computational efficiency and relies on an intuitively meaningful operating principle; it also provides additional functionality in comparison with alternatives. It is demonstrated that (out of the box) the method outperforms the vast majority of alternatives on a realistic data set in terms of efficiency and accuracy, even when implemented in serial in an interpreted programming language. Finally, the last part of the thesis addresses the issue of efficient semi-analytical modelling of solid boundaries in Smoothed Particle Hydrodynamics (SPH) simulations. More precisely, this work focuses on the purely technical aspect of efficient evaluation of correction factors governing the boundary effects; the framework utilising their values is already well established. Mathematically, the problem is described as efficiently integrating a spherically symmetric function over its compact spherical support truncated by a surface (or a collection of surfaces) representing a solid boundary (wall). Three types of boundary geometries are considered, namely piecewise-planar, spherical and super-ellipsoid/super-toroid surfaces, with the latter two categories addressed for the first time in the literature. All methods provided are characterised by an arbitrary degree of accuracy and simplicity of implementation, especially in comparison with all to up to date alternatives. A number of representative test cases is studied.Open Acces

Some results on the 1D linear wave equation with van der Pol type nonlinear boundary conditionsand the Korteweg-de Vries-Burgers equation

Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt − wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F (un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics 2015

This volume contains the full papers accepted for presentation at the ECCOMAS Thematic Conference on Multibody Dynamics 2015 held in the Barcelona School of Industrial Engineering, Universitat Politècnica de Catalunya, on June 29 - July 2, 2015. The ECCOMAS Thematic Conference on Multibody Dynamics is an international meeting held once every two years in a European country. Continuing the very successful series of past conferences that have been organized in Lisbon (2003), Madrid (2005), Milan (2007), Warsaw (2009), Brussels (2011) and Zagreb (2013); this edition will once again serve as a meeting point for the international researchers, scientists and experts from academia, research laboratories and industry working in the area of multibody dynamics. Applications are related to many fields of contemporary engineering, such as vehicle and railway systems, aeronautical and space vehicles, robotic manipulators, mechatronic and autonomous systems, smart structures, biomechanical systems and nanotechnologies. The topics of the conference include, but are not restricted to: ● Formulations and Numerical Methods ● Efficient Methods and Real-Time Applications ● Flexible Multibody Dynamics ● Contact Dynamics and Constraints ● Multiphysics and Coupled Problems ● Control and Optimization ● Software Development and Computer Technology ● Aerospace and Maritime Applications ● Biomechanics ● Railroad Vehicle Dynamics ● Road Vehicle Dynamics ● Robotics ● Benchmark ProblemsPostprint (published version

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement

Multibody dynamics 2015

This volume contains the full papers accepted for presentation at the ECCOMAS Thematic Conference on Multibody Dynamics 2015 held in the Barcelona School of Industrial Engineering, Universitat Politècnica de Catalunya, on June 29 - July 2, 2015. The ECCOMAS Thematic Conference on Multibody Dynamics is an international meeting held once every two years in a European country. Continuing the very successful series of past conferences that have been organized in Lisbon (2003), Madrid (2005), Milan (2007), Warsaw (2009), Brussels (2011) and Zagreb (2013); this edition will once again serve as a meeting point for the international researchers, scientists and experts from academia, research laboratories and industry working in the area of multibody dynamics. Applications are related to many fields of contemporary engineering, such as vehicle and railway systems, aeronautical and space vehicles, robotic manipulators, mechatronic and autonomous systems, smart structures, biomechanical systems and nanotechnologies. The topics of the conference include, but are not restricted to: Formulations and Numerical Methods, Efficient Methods and Real-Time Applications, Flexible Multibody Dynamics, Contact Dynamics and Constraints, Multiphysics and Coupled Problems, Control and Optimization, Software Development and Computer Technology, Aerospace and Maritime Applications, Biomechanics, Railroad Vehicle Dynamics, Road Vehicle Dynamics, Robotics, Benchmark Problems. The conference is organized by the Department of Mechanical Engineering of the Universitat Politècnica de Catalunya (UPC) in Barcelona. The organizers would like to thank the authors for submitting their contributions, the keynote lecturers for accepting the invitation and for the quality of their talks, the awards and scientific committees for their support to the organization of the conference, and finally the topic organizers for reviewing all extended abstracts and selecting the awards nominees.Postprint (published version

Estimation des forces musculaires du membre supérieur humain par optimisation dynamique en utilisant une méthode directe de tir multiple

La modélisation musculo-squelettique permet d’estimer les forces internes du corps humain, à savoir, les forces musculaires et articulaires. Ces estimations sont nécessaires pour comprendre l’anatomie fonctionnelle, les mécanismes de blessures ou encore de concevoir des aides techniques à la motricité. Le défi est d’utiliser l’ensemble des données biomécaniques existantes pour prédire des forces internes qui tiennent compte des stratégies neuro-musculo-squelettiques propres à chacun. L’objectif de cette thèse était d’estimer les forces musculaires du membre supérieur humain par optimisation dynamique, en proposant une méthode innovante de suivi simultané des données électromyographiques (EMG) et cinématiques. À cet égard, nos quatre objectifs spécifiques étaient de : (1) résoudre ce problème d’optimisation dynamique en utilisant une méthode directe de tir multiple ; (2) déterminer sa pertinence et sa performance par rapport aux autres algorithmes existants ; (3) valider son applicabilité à des données expérimentales ; et (4) caractériser des techniques d’identification (numériques et expérimentales) des propriétés musculaires, notamment à l’aide d’un ergomètre isocinétique. Nos différentes études ont permis d’établir que, en un temps de calcul raisonnable (~ 1 heure), notre nouvelle méthode de suivi simultané en optimisation dynamique est à-même de reproduire la cinématique attendue avec une précision de l’ordre de 5°. En outre, l'erreur quadratique moyenne sur les forces musculaires a été réduite d’au moins cinq fois avec notre nouvelle méthode, comparativement aux optimisations statique, hybride et dynamique reposant sur des fonctions-objectif de moindres-activations/excitations (erreur sur les forces musculaires de 18,45 ± 12,60 N avec notre nouvelle méthode contre 85,10 ± 116,40 N avec une optimisation hybride faisant le suivi des moments articulaires). Notre algorithme a également montré son efficacité lors de l’identification des propriétés musculaires d’un modèle musculo-squelettique générique : ce faisant, des excitations musculaires avec deux fois moins d’erreurs vis-à-vis de l’EMG expérimental ont été obtenues, comparativement à l’optimisation statique. Finalement, en termes de calibration du modèle musculo-squelettique, nous avons pu établir que la mesure expérimentale du moment articulaire à l’épaule au moyen de l’ergomètre isocinétique est inadéquate, en particulier lors de mouvements de rotation interne/externe de l’épaule. En effet, les composantes en flexion et abduction du moment à l’épaule mesurées par l’ergomètre isocinétique sont significativement sous-estimées (jusqu'à 94,9% par rapport au moment résultant calculé à partir des efforts tridimensionnels à la main et au coude, mesurés par des capteurs de force six axes). Par conséquent, cette thèse a mis en évidence l’importance du suivi simultané de l’EMG et de la cinématique en optimisation dynamique, afin de rendre fiables les estimations de forces musculaires du membre supérieur – notamment, dans les cas de forte co-contraction musculaire. Elle également a permis d’établir des recommandations qui serviront lors de la calibration du modèle à partir de l’ergomètre isocinétique. Notre méthode innovante pourra être appliquée à des populations pathologiques, afin de comprendre la pathomécanique et mieux intervenir auprès des professionnels de la santé et de leurs patients.Musculoskeletal modeling is used to estimate the internal forces of the human body, namely, muscle and joint forces. These estimates are necessary to understand functional anatomy and pathogenesis or to design technical devices supporting the movement. The challenge is to use all existing biomechanical data to predict internal forces that account for the neuro-musculoskeletal strategies of each individual. The purpose of this thesis was to estimate the human upper-limb muscles forces using forward dynamic optimisation. To do so, we proposed an innovative method tracking both electromyographic (EMG) and kinematic data directly into the optimisation objective-function. In this regard, our four specific objectives were: (1) solving the forward-dynamic optimisation problem using a direct multiple shooting method; (2) determining its relevance and performance compared to other existing algorithms in the literature; (3) validating its applicability to experimental data; and (4) characterizing techniques to identify the model muscle properties using the isokinetic dynamometer. In our different studies, we have demonstrated that, in a reasonable computation time (~ 1 hour), our new dynamic-optimisation method is able to predict the joint kinematics with an accuracy of about 5°. In addition, the muscle forces root-mean-square error was reduced by at least five times with our new method compared to static, hybrid, and dynamic optimisations based on least-activations/excitations objective-functions (muscle forces error of 18.45 ± 12.60 N with our new method vs. 85.10 ± 116.40 N with a traditional hybrid optimisation tracking the joint torques). Our new algorithm also proved to be efficient in identifying the muscle properties of a generic musculoskeletal model: in doing so, the error between the optimised muscle excitations and the experimental EMG was two time lower than the one obtained with static optimisation. Finally, regarding the calibration of the musculoskeletal model, we established that the experimental joint torque measurement at the shoulder using the isokinetic dynamometer was not suitable, especially during internal/external rotation movements of the shoulder. In fact, the flexion and abduction components of the shoulder torque measured by the isokinetic dynamometer are significantly underestimated (up to 94.9% compared to the resulting torque calculated from the three-dimensional forces at the hand and at the elbow, measured by six-axis force sensors). Therefore, this thesis has emphasized the importance of tracking both EMG and kinematics in dynamic optimisation, in order to make reliable estimations of the upper-limb muscle forces – specifically when high co-contraction occurs. Besides, recommendations were issued about calibrating the musculoskeletal model from the experimental torques measured with the isokinetic dynamometer. It will be possible to apply our innovative forward-dynamic optimisation method to pathological populations to increase understanding of the pathomechanics of human movement and better assist health professionals and their patients