A comparison of smooth basis constructions for isogeometric analysis

In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modeling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-$C^1$, Analysis-Suitable $G^1$ and the Approximate $C^1$ constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate $C^1$ and Analysis-Suitable $G^1$ converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-$C^1$ and D-Patch provide relatively easy construction on complex geometries. The Almost-$C^1$ method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate $C^1$ and Analysis-Suitable $G^1$ applicable to more complex geometries

G1-smooth Biquintic Approximation of Catmull-Clark Subdivision Surfaces

International audienceIn this paper a construction of a globally G1 family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of Bézier patches, which are bicubic C2 around regular vertices and biquintic G1 around extraordinary vertices (and C1 on their one-rings vertices). Each Bézier point is computed using a locally defined mask around the neighboring mesh vertices. To define G1 conditions, we assign quadratic gluing data around extraordinary vertices that depend solely on their valence and we use degree five patches to satisfy these G1 constraints. We explore the space of possible solutions, considering several projections on the solution space leading to different explicit formulas for the masks. Certain control points are computed by means of degree elevation of the C0 scheme of Loop and Schaefer [22], while for others, explicit masks are deduced by providing closed-form solutions of the G1 conditions, expressed in terms of the masks. We come up with four different schemes and conduct curvature analysis on an extensive benchmark in order to assert the quality of the resulting surfaces and identify the ones that lead to the best result, both visually and numerically. We demonstrate that the resulting surfaces converge quadratically to the CC limit when the mesh is subdivided

Ferramentas numéricas para análise isogeométrica em regime não-linear

Doutoramento em Engenharia MecânicaThe present work deals with the development of robust numerical tools for Isogeometric Analysis suitable for problems of solid mechanics in the nonlinear regime. To that end, a new solid-shell element, based on the Assumed Natural Strain method, is proposed for the analysis of thin shell-like structures. The formulation is extensively validated using a set of well-known benchmark problems available in the literature, in both linear and nonlinear (geometric and material) regimes. It is also proposed an alternative formulation which is focused on the alleviation of the volumetric locking pathology in linear elastic problems. In addition, an introductory study in the field of contact mechanics, in the context of Isogeometric Analysis, is also presented, with special focus on the implementation of a the Point-to-Segment algorithm. All the methodologies presented in the current work were implemented in a in-house code, together with several pre- and post-processing tools. In addition, user subroutines for the commercial software Abaqus were also implemented.O presente trabalho foca-se no desenvolvimento de ferramentas numéricas robustas para problemas não-lineares de mecânica dos sólidos no contexto de Análises Isogeométricas. Com esse intuito, um novo elemento do tipo sólido-casca, baseado no método das Deformações Assumidas, é proposto para a análise de estruturas do tipo casca fina. A formulação proposta é validada recorrendo a um conjunto de problemas-tipo disponíveis na literatura, considerando tanto regimes lineares como não-lineares (geométrico e de material). É ainda apresentada uma formulação alternativa para aliviar o fenómeno de retenção volumétrica para problemas em regime linear elástico. Adicionalmente, é apresentado um estudo introdutório da mecânica Do conta to no contexto de Análises Isogeométricas, dando especial ênfase ao algoritmo de Ponto-para-Segmento. As metodologias apresentadas no presente trabalho foram implementadas num código totalmente desenvolvido durante o de correr do mesmo, juntamente com diversas ferramentas para pré- e pós processamento. Foram ainda implementadas rotinas do utilizador para o software comercial Abaqus