Reliable Solid Modelling Using Subdivision Surfaces

Les surfaces de subdivision fournissent une méthode alternative prometteuse dans la modélisation géométrique, et ont des avantages sur la représentation classique de trimmed-NURBS, en particulier dans la modélisation de surfaces lisses par morceaux. Dans ce mémoire, nous considérons le problème des opérations géométriques sur les surfaces de subdivision, avec l'exigence stricte de forme topologique correcte. Puisque ce problème peut être mal conditionné, nous proposons une approche pour la gestion de l'incertitude qui existe dans le calcul géométrique. Nous exigeons l'exactitude des informations topologiques lorsque l'on considère la nature de robustesse du problème des opérations géométriques sur les modèles de solides, et il devient clair que le problème peut être mal conditionné en présence de l'incertitude qui est omniprésente dans les données. Nous proposons donc une approche interactive de gestion de l'incertitude des opérations géométriques, dans le cadre d'un calcul basé sur la norme IEEE arithmétique et la modélisation en surfaces de subdivision. Un algorithme pour le problème planar-cut est alors présenté qui a comme but de satisfaire à l'exigence topologique mentionnée ci-dessus.Subdivision surfaces are a promising alternative method for geometric modelling, and have some important advantages over the classical representation of trimmed-NURBS, especially in modelling piecewise smooth surfaces. In this thesis, we consider the problem of geometric operations on subdivision surfaces with the strict requirement of correct topological form, and since this problem may be ill-conditioned, we propose an approach for managing uncertainty that exists inherently in geometric computation. We take into account the requirement of the correctness of topological information when considering the nature of robustness for the problem of geometric operations on solid models, and it becomes clear that the problem may be ill-conditioned in the presence of uncertainty that is ubiquitous in the data. Starting from this point, we propose an interactive approach of managing uncertainty of geometric operations, in the context of computation using the standard IEEE arithmetic and modelling using a subdivision-surface representation. An algorithm for the planar-cut problem is then presented, which has as its goal the satisfaction of the topological requirement mentioned above

Computational topology for approximations of knots

[EN] The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding:Hausdorff distance, anda sum of total curvature and derivative.High degree Bézier curves are often used as smooth representations, where computational efficiency is a practical concern. Subdivision can produce PL approximations for a given B\'ezier curve, fulfilling the above two conditions. The primary contributions are:       (i) a priori bounds on the number of subdivision iterations sufficient to achieve a PL approximation that is ambient isotopic to the original B\'ezier curve, and       (ii) improved iteration bounds over those previously established. The first, two authors acknowledge, with appreciation, partial support from NSF Grants 1053077 and 0923158 and also from IBM. The findings presented are the responsibility of these authors, not of the funding programs.Li, J.; Peters, TJ.; Jordan, KE. (2014). Computational topology for approximations of knots. Applied General Topology. 15(2):203-220. doi:http://dx.doi.org/10.4995/agt.2014.2281.SWORD203220152Amenta, N., Peters, T. J., & Russell, A. C. (2003). Computational topology: ambient isotopic approximation of 2-manifolds. Theoretical Computer Science, 305(1-3), 3-15. doi:10.1016/s0304-3975(02)00691-6L. E. Andersson, S. M. Dorney, T. J. Peters and N. F. Stewart, Polyhedral perturbations that preserve topological form, CAGD 12, no. 8 (1995), 785-799.Burr, M., Choi, S. W., Galehouse, B., & Yap, C. K. (2012). Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves. Journal of Symbolic Computation, 47(2), 131-152. doi:10.1016/j.jsc.2011.08.021Chazal, F., & Cohen-Steiner, D. (2005). A condition for isotopic approximation. Graphical Models, 67(5), 390-404. doi:10.1016/j.gmod.2005.01.005W. Cho, T. Maekawa and N. M. Patrikalakis, Topologically reliable approximation in terms of homeomorphism of composite Bézier curves, CAGD 13 (1996), 497-520.Denne, E., & Sullivan, J. M. (2008). Convergence and Isotopy Type for Graphs of Finite Total Curvature. Discrete Differential Geometry, 163-174. doi:10.1007/978-3-7643-8621-4_8G. E. Farin, Curves and surfaces for computer-aided geometric design: A practical code, Academic Press, Inc., 1996.Hirsch, M. W. (1976). Differential Topology. Graduate Texts in Mathematics. doi:10.1007/978-1-4684-9449-5J. Li, T. J. Peters, D. Marsh and K. E. Jordan, Computational topology counterexamples with 3D visualization of Bézier curves, Applied General Topology 13, no. 2 (2012), 115-134.Lin, L., & Yap, C. (2011). Adaptive Isotopic Approximation of Nonsingular Curves: the Parameterizability and Nonlocal Isotopy Approach. Discrete & Computational Geometry, 45(4), 760-795. doi:10.1007/s00454-011-9345-9T. Maekawa, N. M. Patrikalakis, T. Sakkalis and G. Yu, Analysis and applications of pipe surfaces, CAGD 15, no. 5 (1998), 437-458.Milnor, J. W. (1950). On the Total Curvature of Knots. The Annals of Mathematics, 52(2), 248. doi:10.2307/1969467G. Monge, Application de l'analyse à la géométrie, Bachelier, Paris, 1850.Moore, E. L. F., Peters, T. J., & Roulier, J. A. (2007). Preserving computational topology by subdivision of quadratic and cubic Bézier curves. Computing, 79(2-4), 317-323. doi:10.1007/s00607-006-0208-9G. Morin and R. Goldman, On the smooth convergence of subdivision and degree elevation for Bézier curves, CAGD 18 (2001), 657-666.J. Munkres, Topology, Prentice Hall, 2nd edition, 1999.D. Nairn, J. Peters and D. Lutterkort, Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon, CAGD 16 (1999), 613-63.Reid, M., & Szendroi, B. (2005). Geometry and Topology. doi:10.1017/cbo978051180751

Automated CAD Model Generation for Structural Optimisation

Computer-aided design (CAD) models play a crucial role in the design, manufacturing and maintenance of products. Therefore, the mesh-based finite element descriptions common in structural optimisation must be first translated into CAD models. Currently, this translation either can be performed semi-manually or fails to reserve the structural optimality found by the structural optimisation due to the intrinsic difference in geometric representation between finite element mesh and CAD model. This thesis propose a fully automated and topologically accurate approach to synthesise structurally sound parametric CAD models from topology-optimised finite element models to fill the long-existing gap between structural optimisation and CAD systems. This approach successfully preserves the optimal structural performance during the mesh-CAD conversion. The solution provided in this thesis is to first convert the topology-optimised structure into a spatial frame structure and then to regenerate it in a CAD system using standard constructive solid geometry (CSG) operations. The obtained parametric CAD models are compact, that is, have as few as possible geometric parameters, which makes them ideal for editing and further processing within a CAD system. The critical task of converting the topology-optimised structure into an optimal spatial frame structure is accomplished in several steps. The first step is to generate a one-voxel-wide voxel chain model from the topology-optimised voxel model using a topology-preserving skeletonisation algorithm from digital topology. The undirected graph defined by the voxel chain model yields a spatial frame structure after processing it with the proposed graph algorithms. Subsequently, the cross-sections and layout of the frame members are optimised to recover its optimality, which may have been compromised during the conversion process. At last, the obtained frame structure is generated in a CAD system by repeatedly combining primitive solids, like cylinders and spheres, using boolean operations. The resulting solid model is a boundary representation (B-Rep) consisting of trimmed non-uniform rational B-spline (NURBS) curves and surfaces. The numerical studies in this thesis clarify that the converted spatial frame structures are with equivalent structural performance. Moreover, CAD models generated from the spatial frame structures have significantly fewer geometric degree of freedom compared to the topology-optimised structures. Though the numerical studies use topology-optimised structures as input and compact CSG models as output, this thesis also provides the way to extend the proposed generation process to taking other optimised meshes and producing outputs of various geometric representations. This offers a wide range of possible applications and brings new thoughts to industrial design and manufacturing.Chinese Scholarship Counci

Modelling and Animation using Partial Differential Equations. Geometric modelling and computer animation of virtual characters using elliptic partial differential equations.

This work addresses various applications pertaining to the design, modelling and animation of parametric surfaces using elliptic Partial Differential Equations (PDE) which are produced via the PDE method. Compared with traditional surface generation techniques, the PDE method is an effective technique that can represent complex three-dimensional (3D) geometries in terms of a relatively small set of parameters. A PDE-based surface can be produced from a set of pre-configured curves that are used as the boundary conditions to solve a number of PDE. An important advantage of using this method is that most of the information required to define a surface is contained at its boundary. Thus, complex surfaces can be computed using only a small set of design parameters. In order to exploit the advantages of this methodology various applications were developed that vary from the interactive design of aircraft configurations to the animation of facial expressions in a computer-human interaction system that utilizes an artificial intelligence (AI) bot for real time conversation. Additional applications of generating cyclic motions for PDE based human character integrated in a Computer-Aided Design (CAD) package as well as developing techniques to describe a given mesh geometry by a set of boundary conditions, required to evaluate the PDE method, are presented. Each methodology presents a novel approach for interacting with parametric surfaces obtained by the PDE method. This is due to the several advantages this surface generation technique has to offer. Additionally, each application developed in this thesis focuses on a specific target that delivers efficiently various operations in the design, modelling and animation of such surfaces.The project files will not be available online

A framework for hull form reverse engineering and geometry integration into numerical simulations

The thesis presents a ship hull form specific reverse engineering and CAD integration framework. The reverse engineering part proposes three alternative suitable reconstruction approaches namely curves network, direct surface fitting, and triangulated surface reconstruction. The CAD integration part includes surface healing, region identification, and domain preparation strategies which used to adapt the CAD model to downstream application requirements. In general, the developed framework bridges a point cloud and a CAD model obtained from IGES and STL file into downstream applications

Advanced Techniques for Design and Manufacturing in Marine Engineering

Modern engineering design processes are driven by the extensive use of numerical simulations; naval architecture and ocean engineering are no exception. Computational power has been improved over the last few decades; therefore, the integration of different tools such as CAD, FEM, CFD, and CAM has enabled complex modeling and manufacturing problems to be solved in a more feasible way. Classical naval design methodology can take advantage of this integration, giving rise to more robust designs in terms of shape, structural and hydrodynamic performances, and the manufacturing process.This Special Issue invites researchers and engineers from both academia and the industry to publish the latest progress in design and manufacturing techniques in marine engineering and to debate the current issues and future perspectives in this research area. Suitable topics for this issue include, but are not limited to, the following:CAD-based approaches for designing the hull and appendages of sailing and engine-powered boats and comparisons with traditional techniques;Finite element method applications to predict the structural performance of the whole boat or of a portion of it, with particular attention to the modeling of the material used;Embedded measurement systems for structural health monitoring;Determination of hydrodynamic efficiency using experimental, numerical, or semi-empiric methods for displacement and planning hulls;Topology optimization techniques to overcome traditional scantling criteria based on international standards;Applications of additive manufacturing to derive innovative shapes for internal reinforcements or sandwich hull structures

Accurate Real-Time Framework for Complex Pre-defined Cuts in Finite Element Modeling

PhD ThesisAchieving detailed pre-defined cuts on deformable materials is vitally pivotal for many commercial applications, such as cutting scenes in games and vandalism effects in virtual movies. In these types of applications, the majority of resources are allocated to achieve high-fidelity representations of materials and the virtual environments. In the case of limited computing resources, it is challenging to achieve a convincing cutting effect. On the premise of sacrificing realism effects or computational cost, a considerable amount of research work has been carried out, but the best solution that can be compatible with both cases has not yet been identified. This doctoral dissertation is dedicated to developing a unique framework for representing pre-defined cuts of deformable surface models, which can achieve real-time, detailed cutting while maintaining the realistic physical behaviours. In order to achieve this goal, we have made in-depth explorations from geometric and numerical perspectives. From a geometric perspective, we propose a robust subdivision mechanism that allows users to make arbitrary predetermined cuts on elastic surface models based on the finite element method (FEM). Specifically, after the user separates the elements in an arbitrary manner (i.e., linear or non-linear) on the topological mesh, we then optimise the resulting mesh by regenerating the triangulation within the element based on the Delaunay triangulation principle. The optimisation of regenerated triangles, as a process of refining the ill-shaped elements that have small Aspect Ratio, greatly improves the realism of physical behaviours and guarantees that the refinement process is balanced with real-time requirements. The above subdivision mechanism can improve the visual effect of cutting, but it neglects the fact that elements cannot be perfectly cut through any pre-defined trajectories. The number of ill-shaped elements generated yield a significant impact on the optimisation time: a large number of ill-shaped elements will render the cutting slow or even collapse, and vice versa. Our idea is based on the core observation that the producing of ill-shaped elements is largely attributed to the condition number of the global stiffness matrix. Practically, for a stiffness matrix, a large condition number means that it is almost singular, and the calculation of its inverse or the solution of a system of linear equations are prone to large numerical errors and time-consuming. It motivates us to alleviate the impact of condition number of the global stiffness matrix from the numerical aspects. Specifically, we address this issue in a novel manner by converting the global stiffness matrix into the form of a covariance matrix, in which the number of conditions of the matrix can be reduced by exploiting appropriate matrix normalisation to the eigenvalues. Furthermore, we investigated the efficiency of two different scenarios: an exact square-root normalisation and its approximation based on the Newton-Schulz iteration. Experimental tests of the proposed framework demonstrate that it can successfully reproduce competitive visuals of detailed pre-defined cuts compared with the state-of-the-art method (Manteaux et al. 2015) while obtaining a significant improvement on the FPS, increasing up to 46.49 FPS and 21.93 FPS during and after the cuts, respectively. Also, the new refinement method can stably maintain the average Aspect Ratio of the model mesh after the cuts at less than 3 and the average Area Ratio around 3%. Besides, the proposed two matrix normalisation strategies, including ES-CGM and AS-CGM, have shown the superiority of time efficiency compared with the baseline method (Xin et al. 2018). Specifically, the ES-CGM and AS-CGM methods obtained 5 FPS and 10 FPS higher than the baseline method, respectively. These experimental results strongly support our conclusion which is that this new framework would provide significant benefits when implemented for achieving detailed pre-defined cuts at a real-time rate

Proceedings of the International Workshop on Medical Ultrasound Tomography: 1.- 3. Nov. 2017, Speyer, Germany

Ultrasound Tomography is an emerging technology for medical imaging that is quickly approaching its clinical utility. Research groups around the globe are engaged in research spanning from theory to practical applications. The International Workshop on Medical Ultrasound Tomography (1.-3. November 2017, Speyer, Germany) brought together scientists to exchange their knowledge and discuss new ideas and results in order to boost the research in Ultrasound Tomography

Entropie‐dominierte Selbstorganisationsprozesse birnenförmiger Teilchensysteme

The ambition to recreate highly complex and functional nanostructures found in living organisms marks one of the pillars of today‘s research in bio- and soft matter physics. Here, self-assembly has evolved into a prominent strategy in nanostructure formation and has proven to be a useful tool for many complex structures. However, it is still a challenge to design and realise particle properties such that they self-organise into a desired target configuration. One of the key design parameters is the shape of the constituent particles. This thesis focuses in particular on the shape sensitivity of liquid crystal phases by addressing the entropically driven colloidal self-assembly of tapered ellipsoids, reminiscent of „pear-shaped“ particles. Therefore, we analyse the formation of the gyroid and of the accompanying bilayer architecture, reported earlier in the so-called pear hard Gaussian overlap (PHGO) approximation, by applying various geometrical tools like Set-Voronoi tessellation and clustering algorithms. Using computational simulations, we also indicate a method to stabilise other bicontinuous structures like the diamond phase. Moreover, we investigate both computationally and theoretically(density functional theory) the influence of minor variations in shape on different pearshaped particle systems, including the stability of the PHGO gyroid phase. We show that the formation of the gyroid is due to small non-additive properties of the PHGO potential. This phase does not form in pears with a „true“ hard pear-shaped potential. Overall our results allow for a better general understanding of necessity and sufficiency of particle shape in regards to colloidal self-assembly processes. Furthermore, the pear-shaped particle system sheds light on a unique collective mechanism to generate bicontinuous phases. It suggests a new alternative pathway which might help us to solve still unknown characteristics and properties of naturally occurring gyroid-like nano- and microstructures.Ein wichtiger Bestandteil der heutigen Forschung in Bio- und Soft Matter Physik besteht daraus, Technologien zu entwickeln, um hoch komplexe und funktionelle Strukturen, die uns aus der Natur bekannt sind, nachzubilden. Hinsichtlich dessen ist vor allem die Methode der Selbstorganisation von Mikro- und Nanoteilchen hervorzuheben, durch die eine Vielzahl verschiedener Strukturen erzeugt werden konnten. Jedoch stehen wir bei diesem Verfahren noch immer vor der Herausforderung, Teilchen mit bestimmten Eigenschaften zu entwerfen, welche die spontane Anordnung der Teilchen in eine gewünschte Struktur bewirken. Einer der wichtigsten Designparameter ist dabei die Form der Bausteinteilchen. In dieser Dissertation konzentrieren wir uns besonders auf die Anfälligkeit von Flüssigkristallphasen bezüglich kleiner Änderungen der Teilchenform und nutzen dabei das Beispiel der Selbstorganisation von Entropie-dominierter Kolloide, die dem Umriss nach verjüngten Ellipsoiden oder "Birnen" ähneln. Mit Hilfe von geometrischen Werkzeugen wie z.B. Set-Voronoi Tessellation oder Cluster-Algorithmen analysieren wir insbesondere die Entstehung der Gyroidphase und der dazugehörigen Bilagenformation, welche bereits in Systemen von harten Birnen, die durch das pear hard Gaussian overlap (PHGO) Potential angenähert werden, entdeckt wurden. Des Weiteren zeigen wir durch Computersimulationen eine Strategie auf, um andere bikontinuierliche Strukturen, wie die Diamentenphase, zu stabilisieren. Schlussendlich betrachten wir sowohl rechnerisch (durch Simulationen) als auch theoretisch (durch Dichtefunktionaltheorie) die Auswirkungen kleiner Abweichungen der Teilchenform auf das Verhalten des kolloiden, birnenförmigen Teilchensystems, inklusive der Stabilität der PHGO Gyroidphase. Wir zeigen, dass die Entstehung des Gyroids auf kleinen nicht-additiven Eigenschaften des PHGO Birnenmodells beruhen. In ''echten'' harten Teilchensystemen entwickelt sich diese Struktur nicht. Insgesamt ermöglichen unsere Ergebnisse einen besseren Einblick auf das Konzept von notwendiger und hinreichender Teilchenform in Selbstorganistationsprozessen. Die birnenförmigen Teilchensysteme geben außerdem Aufschluss über einen ungewöhnlichen, kollektiven Mechanismus, um bikontinuierliche Phasen zu erzeugen. Dies deutet auf einen neuen, alternativen Konstruktionsweg hin, der uns möglicherweise hilft, noch unbekannte Eigenschaften natürlich vorkommender, gyroidähnlicher Nano- und Mikrostrukturen zu erklären

Numerical Simulation of Frictional Contact Problems using Nagata Patches in Surface Smoothing

Tese de doutoramento em Engenharia Mecânica, na especialidade de Tecnologias de Produção, apresentada ao Departamento de Engenharia Mecânica da Faculdade de Ciências e Tecnologia da Universidade de CoimbraAll movements in the world involve contact and friction, from walking to car driving. The contact mechanics has application in many engineering problems, including the connection of structural members by bolts or screws, design of gears and bearings, sheet metal or bulk forming, rolling contact of car tyres, crash analysis of structures, as well as prosthetics in biomedical engineering. Due to the nonlinear and non-smooth nature of contact mechanics (contact area is not known a priori), such problems are currently solved using the finite element method within the field of computational contact mechanics. However, most of the commercial finite element software packages presently available are not entirely capable to solve frictional contact problems, demanding for efficient and robust methods. Therefore, the main objective of this study is the development of algorithms and numerical methods to apply in the numerical simulation of 3D frictional contact problems between bodies undergoing large deformations. The presented original developments are implemented in the in-house finite element code DD3IMP. The formulation of quasi-static frictional contact problems between bodies undergoing large deformations is firstly presented in the framework of the continuum mechanics, following the classical scheme used in solid mechanics. The kinematic description of the deformable bodies is presented adopting an updated Lagrangian formulation. The mechanical behaviour of the bodies is described by an elastoplastic constitutive law in conjunction with an associated flow rule, allowing to model a wide variety of contact problems arising in industrial applications. The frictional contact between the bodies is established by means of two conditions: the principle of impenetrability and the Coulomb’s friction law, both imposed to the contact interface. The augmented Lagrangian method is applied for solving the constrained minimization incremental problem resulting from the frictional contact inequalities, yielding a mixed functional involving both displacements and contact forces. The spatial discretization of the bodies is performed with isoparametric solid finite elements, while the discretization of the contact interface is carried out using the classical Node-to-Segment technique, preventing the slave nodes from penetrating on the master surface. The geometrical part of the contact elements, defined by a slave node and the closest master segment, is created by the contact search algorithm based on the selection of the closest point on the master surface, defined by the normal projection of the slave node. In the particular case of contact between a deformable body and a rigid obstacle, the master rigid surface can be described by smooth parameterizations typically used in CAD models. However, in the general case of contact between deformable bodies, the spatial discretization of both bodies with low order finite elements yields a piecewise bilinear representation of the master surface. This is the central source of problems in solving contact problems involving large sliding, since it leads to the discontinuity of the surface normal vector field. Thus, a surface smoothing procedure based on the Nagata patch interpolation is proposed to describe the master contact surfaces, which led to the development of the Node-to-Nagata contact element. The accuracy of the surface smoothing method using Nagata patches is evaluated by means of simple geometries. The nodal normal vectors required for the Nagata interpolation are evaluated from the CAD geometry in case of rigid master surfaces, while in case of deformable bodies they are approximated using the weighted average of the normal vectors of the neighbouring facets. The residual vectors and tangent matrices of the contact elements are derived coherently with the surface smoothing approach, allowing to obtain quadratic convergence rate in the generalized Newton method used for solving the nonlinear system of equations. The developed surface smoothing method and corresponding contact elements are validated through standard numerical examples with known analytical or semi-analytical solutions. More advanced frictional contact problems are studied, covering the contact of a deformable body with rigid obstacles and the contact between deformable bodies, including self-contact phenomena. The smoothing of the master surface improves the robustness of the computational methods and reduces strongly the non-physical oscillations in the contact force introduced by the traditional faceted description of the contact surface. The presented results are compared with numerical solutions obtained by other authors and experimental results, demonstrating the accuracy and performance of the implemented algorithms for highly nonlinear problems.Todos os movimentos no mundo envolvem contato e atrito, desde andar até conduzir um carro. A mecânica do contacto tem aplicação em muitos problemas de engenharia, incluindo a ligação de elementos estruturais com parafusos, projeto de engrenagens e rolamentos, estampagem ou forjamento, contato entre os pneus e a estrada, colisão de estruturas, bem como o desenvolvimento de próteses em engenharia biomédica. Devido à natureza não-linear e não-suave da mecânica do contato (área de contato desconhecida a priori), tais problemas são atualmente resolvidos usando o método dos elementos finitos no domínio da mecânica do contato computacional. No entanto, a maioria dos programas comerciais de elementos finitos atualmente disponíveis não é totalmente capaz de resolver problemas de contato com atrito, exigindo métodos numéricos mais eficientes e robustos. Portanto, o principal objetivo deste estudo é o desenvolvimento de algoritmos e métodos numéricos para aplicar na simulação numérica de problemas de contato com atrito entre corpos envolvendo grandes deformações. Os desenvolvimentos apresentados são implementados no programa de elementos finitos DD3IMP. A formulação quasi-estática de problemas de contato com atrito entre corpos deformáveis envolvendo grandes deformações é primeiramente apresentada no âmbito da mecânica dos meios contínuos, seguindo o método clássico usado em mecânica dos sólidos. A descrição cinemática dos corpos deformáveis é apresentada adotando uma formulação Lagrangeana reatualizada. O comportamento mecânico dos corpos é descrito por uma lei constitutiva elastoplástica em conjunto com uma lei de plasticidade associada, permitindo modelar uma grande variedade de problemas de contacto envolvidos em aplicações industriais. O contacto com atrito entre os corpos é definido por duas condições: o princípio da impenetrabilidade e a lei de atrito de Coulomb, ambas impostas na interface de contato. O método do Lagrangeano aumentado é aplicado na resolução do problema de minimização com restrições resultantes das condições de contato e atrito, produzindo uma formulação mista envolvendo deslocamentos e forças de contato. A discretização espacial dos corpos é realizada com elementos finitos sólidos isoparamétricos, enquanto a discretização da interface de contacto é realizado utilizando a técnica Node-to-Segment, impedindo os nós slave de penetrar na superfície master. A parte geométrica do elemento de contacto, definida por um nó slave e o segmento master mais próximo, é criada pelo algoritmo de deteção de contacto com base na seleção do ponto mais próximo na superfície master, obtido pela projeção normal do nó slave. No caso particular de contato entre um corpo deformável e um obstáculo rígido, a superfície rígida master pode ser descrita por parametrizações normalmente utilizadas em modelos CAD. No entanto, no caso geral de contato entre corpos deformáveis, a discretização espacial dos corpos com elementos finitos lineares origina uma representação da superfície master por facetas. Esta é a principal fonte de problemas na resolução de problemas de contato envolvendo grandes escorregamentos, uma vez que a distribuição dos vetor normais à superfície é descontínua. Assim, é proposto um método de suavização para descrever as superfícies de contacto master baseado na interpolação Nagata, que conduziu ao desenvolvimento do elemento de contacto Node-to-Nagata. A precisão do método de suavização das superfícies é avaliada através de geometrias simples. Os vetores normais nodais necessários para a interpolação Nagata são avaliados a partir da geometria CAD no caso de superfícies rígidas, enquanto no caso de corpos deformáveis são aproximados utilizando a média ponderada dos vetores normais das facetas vizinhas. Tanto os vetores de segundo membro como as matrizes residuais tangentes dos elementos de contacto são obtidas de forma coerente com o método de suavização da superfície, permitindo obter convergência quadrática no método de Newton generalizado, o qual é utilizado para resolver o sistema de equações não lineares. O método de suavização das superfícies e os elementos de contacto desenvolvidos são validados através de exemplos com soluções analíticas ou semi-analíticas conhecidas. Também são estudados outros problemas de contato mais complexos, incluindo o contato de um corpo deformável com obstáculos rígidos e o contato entre corpos deformáveis, contemplando fenómenos de auto-contato. A suavização da superfície master melhora a robustez dos métodos computacionais e reduz fortemente as oscilações na força de contato, associadas à descrição facetada da superfície de contato. Os resultados são comparados com soluções numéricas de outros autores e com resultados experimentais, demonstrando a precisão e o desempenho dos algoritmos implementados para problemas fortemente não-lineares.Fundação para a Ciência e Tecnologia - SFRH/BD/69140/201