SUMMARYIn this paper, an approach for three-dimensional frictionless contact based on a dual mortar formulation and using a primal-dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two-dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi-smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis.

The present work focuses on geometrically exact finite elements for highly slender beams. It aims at the proposal of novel formulations of Kirchhoff-Love type, a detailed review of existing formulations of Kirchhoff-Love and Simo-Reissner type as well as a careful evaluation and comparison of the proposed and existing formulations. Two different rotation interpolation schemes with strong or weak Kirchhoff constraint enforcement, respectively, as well as two different choices of nodal triad parametrizations in terms of rotation or tangent vectors are proposed. The combination of these schemes leads to four novel finite element variants, all of them based on a C 1 -continuous Hermite interpolation of the beam centerline. Essential requirements such as representability of general 3D, large-deformation, dynamic problems involving slender beams with arbitrary initial curvatures and anisotropic cross-section shapes, preservation of objectivity and path-independence, consistent convergence orders, avoidance of locking effects as well as conservation of energy and momentum by the employed spatial discretization schemes, but also a range of practically relevant secondary aspects will be investigated analytically and verified numerically for the different formulations. It will be shown that the geometrically exact Kirchhoff-Love beam elements proposed in this work are the first ones of this type that fulfill all these essential requirements. On the contrary, Simo-Reissner type formulations fulfilling these requirements can be found in the literature very well. However, it will be argued that the shear-free Kirchhoff-Love formulations can provide considerable numerical advantages such as lower spatial discretization error level, improved performance of time integration schemes as well as linear and nonlinear solvers or smooth geometry representation as compared to shear-deformable Simo-Reissner formulations when applied to highly slender beams. Concretely, several representative numerical test cases confirm that the proposed Kirchhoff-Love formulations exhibit a lower discretization error level as well as a considerably improved nonlinear solver performance in the range of high beam slenderness ratios as compared to two representative Simo-Reissner element formulations from the literature.

SUMMARYIn recent years, nonconforming domain decomposition techniques and, in particular, the mortar method have become popular in developing new contact algorithms. Here, we present an approach for 2D frictionless multibody contact based on a mortar formulation and using a primal-dual active set strategy for contact constraint enforcement. We consider linear and higher-order (quadratic) interpolations throughout this work. So-called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non-penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. The necessity of such a linearization in order to obtain a consistent Newton scheme is demonstrated. By further interpreting the active set search as a semi-smooth Newton method, contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation contact problems. Numerical examples illustrate the efficiency of our method and the high quality of results.

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The objective of this work is the development of a novel finite element formulation describing the contact behavior of slender beams in complex 3D contact configurations involving arbitrary beam-to-beam orientations. It is shown by means of a mathematically concise investigation of well-known beam contact models based on point-wise contact forces that these formulations fail to describe a considerable range of contact configurations, which are, however, likely to occur in complex unstructured systems of thin fibers. In contrary, the formulation proposed here models mechanical contact interaction of slender continua by means of distributed line forces, a procedure that is shown to be applicable for any geometrical contact configuration. The proposed formulation is based on a Gauss-point-to-segment type contact discretization and a penalty regularization of the contact constraint. Additionally, theoretical considerations concerning alternative mortar type contact discretizations and constraint enforcement by means of Lagrange multipliers are made. However, based on detailed theoretical and numerical investigations of these different variants, the penalty-based Gauss-point-to-segment formulation is suggested as the most promising and suitable approach for beam-to-beam contact. This formulation is supplemented by a consistently linearized integration interval segmentation that avoids numerical integration across strong discontinuities. In combination with a smoothed contact force law and the employed C 1 -continuous beam element formulation, this procedure drastically reduces the numerical integration error, an essential prerequisite for optimal spatial convergence rates. The resulting line-to-line contact algorithm is supplemented by contact contributions of the beam endpoints, which represent boundary minima of the minimal distance problem underlying the contact formulation. Finally, a series of numerical test cases is analyzed in order to investigate the accuracy and consistency of the proposed formulation regarding integration error, spatial convergence behavior and resulting contact force distributions. For one of these test cases, an analytical solution based on the Kirchhoff theory of thin rods is derived, which can serve as valuable benchmark for the proposed model but also for future beam-to-beam contact formulations. In addition to these examples, two real-world applications are presented in order to verify the robustness of the proposed formulation when applied to practically relevant problems.