Simulation of linear elastic structural elements using the Petrov–Galerkin finite element method

In this contribution, it is demonstrated that the mesh sensitivity of linear elastic Reissner–Mindlin finite-element plate formulations can be significantly reduced by using a Petrov–Galerkin-based approach. In contrast to the usual Bubnov–Galerkin method, Petrov–Galerkin methods are generally characterized by the fact that the test function and the trial function are approximated using different shape functions. To provide an overview, established Petrov–Galerkin methods for 2D solid elements, which have been shown to reduce mesh sensitivity, are reviewed first. It is then investigated whether a suitable Petrov– Galerkin plate formulation can be developed. In this context, it is demonstrated that a full Petrov–Galerkin method leads to problems in the treatment of transverse shear locking. However, the proposed partial Petrov–Galerkin method shows the desired mesh-insensitive behavior

Director-based IGA beam elements for sliding contact problems

The geometrically exact beam theory is one of the most prominent non-linear beam models. It can be used to simulate aerial runways or pantograph-catenaries, where a sliding contact condition between two or more beams is used. A smooth discretization of at least C1continuity is needed to not introduce any unphysical kinks. This can be achieved using the isogeometric analysis, which we apply to a director-based formulation of the geometrically exact beam. For a stable time integration scheme we use an energy-momentum conserving scheme. Using the notion of the discrete gradient, an energy-momentum conserving algorithm is constructed, including the case of sliding contact between beams

Extension of the enhanced assumed strain method based on the structure of polyconvex strain‐energy functions

In this work, two well-known approaches for mixed finite elements are combined to render three novel classes of elements. First, the widely used enhanced assumed strain (EAS) method is considered. Its key idea is to enhance a compatible kinematic field with an incompatible part. The second concept is a framework for mixed elements inspired by polyconvex strain-energy functions, in which the deformation gradient, its cofactor and determinant are three principal kinematic fields. The key idea for the novel elements is to treat enhancement of those three fields separately. This approach leads to a plethora of novel enhancement strategies and promising mixed finite elements. Some key properties of the newly proposed mixed approaches are that they are based on a Hu-Washizu type variational functional, fulfill the patch test, are frame-invariant, can be constructed completely locking free and show no spurious hourglassing in elasticity. Furthermore, they give additional insight into the mechanisms of standard EAS elements. Extensive numerical investigations are performed to assess the elements\u27 behavior in elastic and elasto-plastic simulations

Open issues on the eas method and mesh distortion insensitive locking-free low-order unsymmetric eas elements

One of the most popular mixed finite elements is the enhanced assumed strain (EAS) approach. However, despite numerous advantages there are still some open issues. Three of the most important, namely robustness in nonlinear simulations, hourglassing instabilities and sensitivity to mesh distortion, are discussed in the present contribution. Furthermore, we propose a novel Petrov-Galerkin based EAS method. It is shown that three conditions have to be fulfilled to construct elements that are exact for a specific displacement mode regardless of mesh distortion. The so constructed novel element is lockingfree, exact for bending problems, insensitive to mesh distortion and has improved coarse mesh accuracy

Mesh distortion insensitive and locking-free Petrov–Galerkin low-order EAS elements for linear elasticity

One of the most successful mixed finite element methods in solid mechanics is the enhanced assumed strain (EAS) method developed by Simo and Rifai in 1990. However, one major drawback of EAS elements is the highly mesh dependent accuracy. In fact, it can be shown that not only EAS elements, but every finite element with a symmetric stiffness matrix must either fail the patch test or be sensitive to mesh distortion in bending problems (higher order displacement modes) if the shape of the element is arbitrary. This theorem was established by MacNeal in 1992. In the present work we propose a novel Petrov–Galerkin approach for the EAS method, which is equivalent to the standard EAS method in case of regular meshes. However, in case of distorted meshes, it allows to overcome the mesh-distortion sensitivity without loosing other advantages of the EAS method. Three design conditions established in this work facilitate the construction of the element which does not only fulfill the patch test but is also exact in many bending problems regardless of mesh distortion and has an exceptionally high coarse mesh accuracy. Consequently, high quality demands on mesh topology might be relaxed

Structure-preserving space-time discretization of large-strain thermo-viscoelasticity in the framework of GENERIC

Large‐strain thermo‐viscoelasticity is described in the framework of GENERIC. To this end, a new material representation of the inelastic part of the dissipative bracket is proposed. The bracket form of GENERIC generates the governing equations for large‐strain thermo‐viscoelasticity including the nonlinear evolution law for the internal variables associated with inelastic deformations. The GENERIC formalism facilitates the free choice of the thermodynamic variable. In particular, one may choose (i) the internal energy density, (ii) the entropy density, or (iii) the absolute temperature as the thermodynamic variable. A mixed finite element method is proposed for the discretization in space which preserves the GENERIC form of the resulting semi‐discrete evolution equations. The GENERIC‐consistent space discretization makes possible the design of structure‐preserving time‐stepping schemes. The mid‐point type discretization in time yields three alternative schemes. Depending on the specific choice of the thermodynamic variable, these schemes are shown to be partially structure‐preserving. In addition to that, it is shown that a slight modification of the mid‐point type schemes yields fully structure‐preserving schemes. In particular, three alternative energy‐momentum‐entropy consistent schemes are devised associated with the specific choice of the thermodynamic variable. Numerical investigations are presented which confirm the theoretical findings and shed light on the numerical stability of the newly developed schemes

Inverse Dynamics of Geometrically Exact Beams

This paper is concerned with the inverse dynamics of flexible mechanical systems whose motion is governed by quasi-linear hyperbolic partial differential equations. Problems that appear by applying classical solution strategies to the problem at hand, e.g. integrating the problem at hand sequentially in space and time will be adressed in this work. Motivated by the hyperbolic structure of the underlying initial boundary value problem, two methods that are based on a simultaneous space-time integration will be presented. Special emphasize will be given to the phenomena of wave propagation within geometrically exact beams and its relevance regarding the inverse dynamics problem

Controlling nonlinear elastic systems in structural dynamics

This contribution deals with the feedforward control of continuous mechanical systems. After introducing a general formulation of such problems and adressing the limitations of the commonly used semi-discrete method, two numerical methods are presented that resolve these limitations