The drill-strings used in drilling operate under extreme condi-tions, therefore, an accurate understanding of their dynamics is necessary and has attracted much interest. Although a bottom hole assembly (BHA) is to a great ex- tent responsible for the dynamics of the system, the in uence of the drill-pipes has been increasingly neglected by current models. Their dynamics and geometrical behaviour should be better analysed for a deeper understanding of underlying phe- nomena. For example, under stick-slip oscillations, the torque on the drill-string may cause torsional buckling of the drill-pipes, incurring in helical con guration, in which the apparent length is reduced, a ecting the forces at the bit{rock interface. With such behaviour and interactions in mind, this work focuses on elaborating a comprehensive mathematical model to investigate the dynamics of drill-strings, with attention to the drill-pipes section. Firstly, lower dimensional models are used to analyse the stick-slip limit cycle and its limits of existence. Then, a model developed for MEMS is used as a base for a comprehensive model using the formu- lation of Cosserat rods. Relevant boundary conditions are applied and a numerical simulation procedure is established. Simulations are performed for a range of sce- narios under stick-slip occurrence, and the behaviour of the drill-pipes is analysed. Focus is then given to axial vibrations with bit-bounce and the in uence on stick- slip, later to lateral vibrations with whirling motion of the drill-pipes, and nally to helical con gurations, taken by the drill-string under combined torsional, axial and lateral loads, showing the consequent shortening of the drill-string.

In the effort to simulate the biologically inspired continuum robot’s dynamic capabilities, researchers have been faced with the daunting task of simulating—in real-time—the complete three dimensional dynamics of the the “beam-like” structure which includes the three “stiff” degrees-ofreedom transverse and dilational shear. Therefore, researchers have traditionally limited the difficulty of the problem with simplifying assumptions. This study, however, puts forward a solution which makes no simplifying assumptions and trades off only the real-time requirement of the desired solution. The solution is a Finite Difference Time Domain method employing an explicit single step method with cheap right hands sides. The cheap right hand sides are the result of a rather ingenious formulation of the classical beam called the Cosserat rod by, first, the Cosserat brothers and, later, Stuart S. Antman which results in five nonlinear but uncoupled equations that require only multiplication and addition. The method is therefore suitable for hardware implementation thus moving the real-time requirement from a software solution to a hardware solution.

In summary, this thesis focuses on developing an a priori theory for geometric finite element discretizations of a Cosserat rod model, which is derived from incompatible elasticity. This theory will be supported by corresponding numerical experiments to validate the convergence behavior of the proposed method. The main result describes the qualitative behavior of intrinsic H1-errors and L2-errors in terms of the mesh diameter 0 < h ≪ 1 of the approximation scheme. Geometric Finite Element functions uh with its subclasses Geodesic Finite Elements and Projection- based Finite Elements as conforming path-independent and objective discretizations of Cosserat rod configurations were used. Existence, regularity, variational bounds and vector field transport estimates of the Cosserat rod model were derived to ob- tain an intrinsic a-priori theory. In the second part, this thesis concerns the derivation of the Cosserat rod from 3D elasticity featuring prestress together with numerical experiments for microheteroge- neous prestressed materials.

The application of polynomial reproducing methods has been explored in the context of linear and non linear problems. Of specific interest is the application of a recently developed reproducing scheme, referred to as the error reproducing kernel method (ERKM), which uses non-uniform rational B-splines (NURBS) to construct the basis functions, an aspect that potentially helps bring in locall support, convex approximation and variation diminishing properties in the functional approximation. Polynomial reproducing methods have been applied to solve problems coming under the class of a simplified theory called Cosserat theory. Structures such as a rod which have special geometric properties can be modeled with the aid of such simplified theories. It has been observed that the application of mesh-free methods to solve the aforementioned problems has the advantage that large deformations and exact cross-sectional deformations in a rod could be captured exactly by modeling the rod just in one dimension without the problem of distortion of elements or element locking which would have had some effect if the problem were to be solved using mesh based methods. Polynomial reproducing methods have been applied to problems in fracture mechanics to study the propagation of crack in a structure. As it is often desirable to limit the use of the polynomial reproducing methods to some parts of the domain where their unique advantages such as fast convergence, good accuracy, smooth derivatives, and trivial adaptivity are beneficial, a coupling procedure has been adopted with the objective of using the advantages of both FEM and polynomial reproducing methods. Exploration of SMW (Sherman-Morrison-Woodbury) in the context of polynomial reproducing methods has been done which would assist in calculating the inverse of a perturbed matrix (stiffness matrix in our case). This would to a great extent reduce the cost of computation. In this thesis, as a first step attempts have been made to apply Mesh free cosserat theory to one dimensional problems. The idea was to bring out the advantages and limitations of mesh free cosserat theory and then extend it to 2D problems.

This thesis presents a numerical method for the simulation of fluid-structure interaction (FSI) problems on high-performance computers. The proposed method is specifically tailored to interactions between Newtonian fluids and a large number of slender viscoelastic structures, the latter being modeled as Cosserat rods. From a numerical point of view, such kind of FSI requires special techniques to reach numerical stability. When using a partitioned fluid-structure coupling approach this is usually achieved by an iterative procedure, which drastically increases the computational effort. In the present work, an alternative coupling approach is developed based on an immersed boundary method (IBM). It is unconditionally stable and exempt from any global iteration between the fluid part and the structure part. The proposed FSI solver is employed to simulate the flow over a dense layer of vegetation elements, usually designated as canopy flow. The abstracted canopy model used in the simulation consists of 800 strip-shaped blades, which is the largest canopy-resolving simulation of this type done so far. To gain a deeper understanding of the physics of aquatic canopy flows the simulation data obtained are analyzed, e.g., concerning the existence and shape of coherent structures.

[pt] Uma fase crítica do processo de obtenção do petróleo é a perfuração do solo para o acesso ao reservatório. Um dos problemas, em particular, é compreender o comportamento dinâmico da coluna de perfuração durante o processo de perfuração diante de diversos fatores como a interação broca-rocha, choques da coluna de perfuração contra a parede do poço, estratégias de controle da velocidade angular de operação e outros fatores. Uma etapa fundamental para lidar com este problema é a representação do sistema dinâmico para caracterizar a coluna de perfuração, isto é, o modelo matemático que representará a resposta dinâmica da estrutura diante dos carregamentos. Neste contexto, este trabalho abordará o problema da dinâmica de uma coluna de perfuração através de um modelo matemático baseado na teoria de Cosserat, que resultará em um sistema de seis equações diferenciais parciais que descrevem a resposta dinâmica de uma estrutura unidimensional, inserida no espaço euclidiano tridimensional, em termos das variáveis de deslocamento linear da curva e angular das seções. O modelo é capaz de descrever uma dinâmica não-linear, incluindo flexão, torsão, extensão e cisalhamento. Inicialmente, o sistema de EDPs é resolvido na forma quase estática, satisfazendo as condições de contorno, utilizando o método de Perturbação Regular. As soluções aproximadas são utilizadas como funções base para implementação no método de Elementos Finitos. Estas funções base são conhecidas como elemento de Cosserat Modificado (Modfied Cosserat Rod Element - MCRE). Verifica-se a limitação destas funções base para problemas que não envolvam grandes deslocamentos, não sendo adequadas para o problema proposto. Diante deste fato, o sistema de EDPs é escrito na forma fraca e resolvido por um software comercial de análise de Elementos Finitos considerando as condições de contorno, o modelo de interação broca-rocha, a estratégia de controle da velocidade angular e eventuais contatos da coluna contra a parede do poço. O modelo proposto produziu resultados que estão de acordo com a literatura e se mostrou capaz de lidar com grandes deslocamentos. / [en] A critical step in the oil exploration process is drilling the soil for access to the petroleum reservoir. One of the problems is understanding the dynamic behavior of the drill string during the drilling process in the face of various factors such as drill bit-rock interaction, drill string shocks against the well wall, angular velocity control strategies and other factors. A key part of dealing with this problem is the representation of the dynamic system to characterize the drill string, e.g., the mathematical model that will represent the dynamical response of the structure when facing different types of loads. In this context, this work will address the problem of the dynamics of a drill string using a mathematical model based on Cosserat theory that will result in a system of six partial differential equations that describe the dynamic response of a one-dimensional structure, inserted in three-dimensional Euclidean space, in terms of the linear displacement variables of the curve and angular displacement of the cross sections. The model is able to describe nonlinear dynamics, including flexure, torsion, extension and shear. Initially, the system of partial differential equations is solved in a quasi-static sense, satisfying the boundary conditions, using the Regular Perturbation method. The approximate solutions are used as shape functions for implementation in the Finite Element method. These shape functions are known as Modified Cosserat Rod Element (MCRE). It is verified that these shape functions are restricted to problems that do not involve large displacements and for this reason they are not suitable for the proposed problem. Given this fact, the system of partial differential equations is written in a weak form and solved by a commercial software based on Finite Element analysis, considering the boundary conditions, the drill bit-rock interaction model, the angular velocity control strategy and for any string contacts against the well wall. The proposed model produced results that are in agreement with the literature and is capable of dealing with large displacements.