The Logarithmic finite element method: Shape functions defined on the logarithmic space

C. Schröppel, J. Wackerfuß

In most finite element methods based on the Ritz-Galerkin approach, the degrees of freedom are endowed with a straightforward physical meaning, and shape functions are defined on the basis of individual degrees of freedom. By contrast, the Logarithmic finite element method, or LogFE method defines degrees of freedom on the logarithmic space, or the Lie algebra associated with linear transformations (i.e. translations, rotations, dilatations and possibly transformations involving stretch or shear components) on the configuration space in the physical domain.

Figure 1 illustrates how the deformation function g, which depends on the value ξ attained by the embedded parameterization, transforms the initial configuration (of a two-dimensional beam, in this case), resulting in the current configuration. It is this deformation function g which is obtained, by applying the exponential function, from the shape functions defined on the Lie algebra.

Figure 1: Material body, parameterization, and configurations in the LogFE method.

The shape functions therefore become multi-dimensional embeddings in the physical domains on the respective finite elements, generally spanning the dimensions associated with multiple degrees of freedom. On the Lie algebra, which is, by its mathematical construction, a vector space, rotations can be represented as translations, and it is this aspect which makes this approach particularly interesting for finite-element calculations that involve the modelling of rotations.

In established finite element methods, translations and rotations are assigned different degrees of freedom (d.o.f.). Often, the components of the interpolant between two or more nodes belonging to a given finite element are then constructed separately on the basis the individual degrees of freedom, thus eliminating the interplay between translations and rotations that can be observed in the physical reality. Well-known deficiencies of many of these formulations, such as lack of geometric invariance, path dependence and insufficient accuracy in coarse meshes, have been traced back to the characteristics of the various methods proposed for the interpolation of the rotational variables by G. Jelenić, M. A. Crisfield and I. Romero, among others.

The LogFE approach aims to avoid these difficulties by introducing a natural coupling between translational and rotational components of the deformation. Insofar as this natural coupling largely captures the physically non-linear components of the physical interaction, it can thus enlarge the domain of the configuration space which can be modelled with sufficient accuracy without resorting to techniques that induce various drawbacks associated with the existing approaches.

The LogFE approach allows to significantly reduce the number of degrees of freedom, while keeping a high degree of accuracy with regard to the low-frequency part of the deformation (In this context, “low-frequency” refers to those spatial displacements in a quasi-static setting that are composed of eigenvectors associated with small eigenvalues). In addition, we observe that spurious influences on the high-frequency part are minimal. Therefore, the LogFE approach can be used as a coarse grid approximation method within a multigrid framework.

Figure 2: Three-dimensional beam, subjected to given boundary conditions and kinematically modeled by the Logarithmic finite element method. The entire beam is modeled by one single finite element only.

As part of the work of the MISMO research group, we have implemented a LogFE formulation of a Kirchhoff beam, in order to prove the basic characteristics an applicability of the method (see Figure 2). We are currently implementing a multigrid algorithm for the simulation of super carbon nanotubes (SCNTs) based on the LogFE approach. In this context, we represent each carbon nanotube junction by a node endowed with several d.o.f. on the logarithmic scale. We apply the concept of finite elements not only on the fine, but also on the coarse grid, thus making it possible to describe the mechanical properties not only of a complete SCNT configuration, but also of its components, in a concise way, and to reveal the scaling laws associated with the self-similarities that characterize super carbon nanotubes. 


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