A mathematical framework for contact detection between quadric and superquadric surfaces

The calculation of the minimum distance between surfaces plays an important role in computational mechanics, namely, in the study of constrained multibody systems where contact forces take part. In this paper, a general rigid contact detection methodology for non-conformal bodies, described by ellipsoidal and superellipsoidal surfaces, is presented. The mathematical framework relies on simple algebraic and differential geometry, vector calculus, and on the C2 continuous implicit representations of the surfaces. The proposed methodology establishes a set of collinear and orthogonal constraints between vectors defining the contacting surfaces that, allied with loci constraints, which are specific to the type of surface being used, formulate the contact problem. This set of non-linear equations is solved numerically with the Newton-Raphson method with Jacobian matrices calculated analytically. The method outputs the coordinates of the pair of points with common normal vector directions and, consequently, the minimum distance between both surfaces. Contrary to other contact detection methodologies, the proposed mathematical framework does not rely on polygonal-based geometries neither on complex non-linear optimization formulations. Furthermore, the methodology is extendable to other surfaces that are (strictly) convex, interact in a non-conformal fashion, present an implicit representation, and that are at least C2 continuous. Two distinct methods for calculating the tangent and binormal vectors to the implicit surfaces are introduced: (i) a method based on the Householder reflection matrix; and (ii) a method based on a square plate rotation mechanism. The first provides a base of three orthogonal vectors, in which one of them is collinear to the surface normal. For the latter, it is shown that, by means of an analogy to the referred mechanism, at least two non-collinear vectors to the normal vector can be determined. Complementarily, several mathematical and computational aspects, regarding the rigid contact detection methodology, are described. The proposed methodology is applied to several case tests involving the contact between different (super)ellipsoidal contact pairs. Numerical results show that the implemented methodology is highly efficient and accurate for ellipsoids and superellipsoids.Fundação para a Ciência e a Tecnologia (FCT

Investigation of end-stop motion constraint for a wave energy converter

This work develops a design protocol for wave energy converter motion constraint, endstop systems. It applies the protocol by first using a numerical hydrodynamic wave energy converter (WEC) model to obtain preliminary design loads. Following a definitive set of selection criteria, comprehensive design of a system of load-bearing, helical springs is produced. A preliminary design is modeled with finite element analysis, and compared to analytical results. New dynamical collision models are conceived for impact damping systems based on spring-mass and anisotropic surface friction phenomena, by applying the concept observed on the snake ventral skin. Friction and compressive forces are correlated by classical mechanics. Finally, dimensional analysis is applied to yield design parameterization to directly compare the micro and macro influences within these distinct models, resulting in new knowledge on the physical relationships within contact interfaces and a dimensionless mechanical impedance formulationEsta dissertação desenvolve um protocolo para projetos de sistemas de restrição de movimentos associado a um limitador de fim de curso em conversores de energia das ondas. Inicialmente, é aplicado um modelo numérico hidrodinâmico para análise de cargas em um conversor de energia das ondas (WEC). Em seguida, é apresentado um conjunto definitivo de critérios de seleção, para análise de um sistema de molas helicoidais compressivas, para atenuar as forças provocadas pelos movimentos extremos da boia. Um projeto preliminar é modelado com análises de elementos finitos e comparado com os resultados analíticos. Novos tipos de modelos dinâmicos são idealizados para amortecimento do impacto, baseados em molas e no fenômeno de atrito superficial anisotrópico, aplicando o conceito observado na pele ventral de cobras. As forças de atrito e compressivas foram correlacionadas por meio de princípios de mecânica clássica. Finalmente, uma análise adimensional é utilizada para gerar a parametrização do projeto, para comparar diretamente as micro e macro influências entre esses modelos distintos, resultando em novos conhecimentos sobre as relações físicas nas interfaces de contato e uma formulação adimensional de impedância mecânica

Simulation of conforming contact in real-time multibody dynamics using a volumetric force model

Programa Oficial de Doutoramento en Enxeñaría Naval e Industrial . 5015V01[Abstract] Simulation is a tool on the rise, especially in the industrial field. The usage of simulators grants the ability of studying, predicting and improving the behavior of a system, as well as designing a new one. In the case of mechanical processes simulators, the characterization of contacts and collisions between the different elements at play is one of the key factors to achieve a realistic simulation. If, furthermore, the simulator is designed to interact with machines or people, the need of real-time execution is imposed. Usually, these requirements produce a conflict of interest, since more complex algorithms demand larger execution times. Furthermore, all this is worsened by some application’s need of conforming contact simulation, this is, complex contacts where the size of the contact footprint is not negligible compared to the size of the bodies in collision. This work studies two methods suitable for conforming contact simulation and their possibilities to be used in real-time simulators are discussed.[Resumo] A simulación é unha ferramenta en auxe, especialmente no ámbito industrial. O emprego de simuladores otorga a capacidade de estudar, predecir e mellora-lo comportamento dun sistema, así como de deseñar un novo. No caso dos simuladores de procesos mecánicos, a caracterización do contacto e das colisións entre os diferentes elementos en xogo é un dos factores clave para conseguir unha simulación fidedigna. Se, ademáis, ésta está deseñada para interactuar con máquinas ou persoas, imponse a necesidade de que a execución da simulación sexa en tempo real. Xeralmente, estos requerimentos producen un conflicto de intereses, xa que algoritmos máis complexos esixen tempos de execución máis amplos. Ademáis, todo isto vese perxudicado pola necesidade dalgunhas aplicacións de simular contactos conformes, isto é, contactos complexos nos que o tamaño da pegada de contacto non é desprezable en comparación ó tamaño dos corpos en colisión. Neste traballo estúdianse dous métodos adecuados para simular contactos conformes e debátense as súas posibilidades para ser aplicados en simuladores en tempo real.[Resumen] La simulación es una herramienta en auge, especialmente en el ámbito industrial. El empleo de simuladores otorga la capacidad de estudiar, predecir y mejorar el comportamiento de un sistema, así como de diseñar uno nuevo. En el caso de los simuladores de procesos mecánicos, la caracterización del contacto y las colisiones entre los diferentes elementos en juego es uno de los factores clave para conseguir una simulación fidedigna. Si, además, ésta está diseñada para interactuar con máquinas o personas, se impone la necesidad de que la ejecución de la simulación sea en tiempo real. Generalmente, estos requerimientos producen un conflicto de intereses, ya que algoritmos más complejos exigen tiempos de ejecución más amplios. Además, todo esto se ve perjudicado por la necesidad de algunas aplicaciones de simular contactos conformes, esto es, contactos complejos en los que el tamaño de la huella de contacto no es despreciable en comparación al tamaño de los cuerpos en colisión. En este trabajo se estudian dos métodos adecuados para simular contactos conformes y se debaten sus posibilidades para ser aplicados en simuladores en tiempo real

Animating jellyfish through numerical simulation and symmetry exploitation

This thesis presents an automatic animation system for jellyfish that is based on a physical simulation of the organism and its surrounding fluid. Our goal is to explore the unusual style of locomotion, namely jet propulsion, which is utilized by jellyfish. The organism achieves this propulsion by contracting its body, expelling water, and propelling itself forward. The organism then expands again to refill itself with water for a subsequent stroke. We endeavor to model the thrust achieved by the jellyfish, and also the evolution of the organism's geometric configuration. We restrict our discussion of locomotion to fully grown adult jellyfish, and we restrict our study of locomotion to the resonant gait, which is the organism's most active mode of locomotion, and is characterized by a regular contraction rate that is near one of the creature's resonant frequencies. We also consider only species that are axially symmetric, and thus are able to reduce the dimensionality of our model. We can approximate the full 3D geometry of a jellyfish by simulating a 2D slice of the organism. This model reduction yields plausible results at a lower computational cost. From the 2D simulation, we extrapolate to a full 3D model. To prevent our extrapolated model from being artificially smooth, we give the final shape more variation by adding noise to the 3D geometry. This noise is inspired by empirical data of real jellyfish, and also by work with continuous noise functions from the graphics community. Our 2D simulations are done numerically with ideas from the field of computational fluid dynamics. Specifically, we simulate the elastic volume of the jellyfish with a spring-mass system, and we simulate the surrounding fluid using the semi-Lagrangian method. To couple the particle-based elastic representation with the grid-based fluid representation, we use the immersed boundary method. We find this combination of methods to be a very efficient means of simulating the 2D slice with a minimal compromise in physical accuracy

A novel musculoskeletal joint modelling for orthopaedic applications

This thesis was submitted for the degree of Docter of Philosophy and awarded by Brunel University.The objective of the work carried out in this thesis was to develop analytical and computational tools to model and investigate musculoskeletal human joints. It was recognised that the FEA was used by many researchers in modelling human musculoskeletal motion, loading and stresses. However the continuum mechanics played only a minor role in determining the articular joint motion, and its value was questionable. This is firstly due to the computational cost and secondly due to its impracticality for this application. On the other hand, there isn’t any suitable software for precise articular joint motion analysis to deal with the local joint stresses or non standard joints. The main requirement in orthopaedics field is to develop a modeller software (and its associated theories) to model anatomic joint as it is, without any simplification with respect to joint surface morphology and material properties of surrounding tissues. So that the proposed modeller can be used for evaluating and diagnosing different joint abnormalities but furthermore form the basis for performing implant insertion and analysis of the artificial joints. The work which is presented in this thesis is a new frame work and has been developed for human anatomic joint analysis which describes the joint in terms of its surface geometry and surrounding musculoskeletal tissues. In achieving such a framework several contributions were made to the 6DOF linear and nonlinear joint modelling, the mathematical definition of joint stiffness, tissue path finding and wrapping and the contact with collision analysis. In 6DOF linear joint modelling, the contribution is the development of joint stiffness and damping matrices. This modelling approach is suitable for the linear range of tissue stiffness and damping properties. This is the first of its kind and it gives a firm analytical basis for investigating joints with surrounding tissue and the cartilage. The 6DOF nonlinear joint modelling is a new scheme which is described for modelling the motion of multi bodies joined by non-linear stiffness and contact elements. The proposed method requires no matrix assembly for the stiffness and damping elements or mass elements. The novelty in the nonlinear modelling, relates to the overall algorithmic approach and handling local non-linearity by procedural means. The mathematical definition of joint stiffness is also a new proposal which is based on the mathematical definition of stiffness between two bodies. Based on the joint stiffness matrix properties, number of joint stiffness invariants was obtained analytically such as the centre of stiffness, the principal translational stiffnesses, and the principal rotational stiffnesses. In corresponding to these principal stiffnesses, their principal axes have been also obtained. Altogether, a joint is assessed by six principal axes and six principal stiffnesses and its centre of stiffness. These formulations are new and show that a joint can be described in terms of inherent stiffness properties. It is expected that these will be better in characterising a joint in comparison to laxity based characterisation. The development of tissue path finding and wrapping algorithms are also introduced as new approaches. The musculoskeletal tissue wrapping involves calculating the shortest distance between two points on a meshed surface. A new heuristic algorithm was proposed. The heuristic is based on minimising the accumulative divergence from the straight line between two points on the surface and the direction of travel on the surface (i.e. bone). In contact and collision based development, the novel algorithm has been proposed that detects possible colliding points on the motion trajectory by redefining the distance as a two dimensional measure along the velocity approach vector and perpendicular to this vector. The perpendicular distance determines if there are potentially colliding points, and the distance along the velocity determines how close they are. The closest pair among the potentially colliding points gives the “time to collision”. The algorithm can eliminate the “fly pass” situation where very close points may not collide because of the direction of their relative velocity. All these developed algorithms and modelling theories, have been encompassed in the developed prototype software in order to simulate the anatomic joint articulations through modelling formulations developed. The software platform provides a capability for analysing joints as 6DOF joints based on anatomic joint surfaces. The software is highly interactive and driven by well structured database, designed to be highly flexible for the future developments. Particularly, two case studies are carried out in this thesis in order to generate results relating to all the proposed elements of the study. The results obtained from the case studies show good agreement with previously published results or model based results obtained from Lifemod software, whenever comparison was possible. In some cases the comparison was not possible because there were no equivalent results; the results were supported by other indicators. The modelling based results were also supported by experiments performed in the Brunel Orthopaedic Research and Learning Centre

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”