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We propose a novel technique for low-frequency approximations of boundary-value problems. In contrast to the standard Ritz-Galerkin approach, the shape functions are given on the logarithmic space of the deformation function. The method is able to capture the complex interplay of the rotational and the translational components of the deformation. It is likely to be particularly efficient in the context of multigrid algorithms.We propose a novel technique -the Logarithmic finite element, or "LogFE" formulation -that significantly reduces the number of degrees of freedom (d.o.f.) necessary to obtain reasonably accurate low-frequency approximations of boundary-value problems, while minimizing spurious high-frequency components in the approximate solution. The method is likely to be particularly efficient as a coarse grid correction step in the context of a multigrid algorithm.In contrast to the standard Ritz-Galerkin approach, the d.o.f. are given as coefficients of shape functions on a Lie algebra, allowing to reduce the number of d.o.f. without incurring the locking phenomena associated with linear shape functions. In contrast to existing methods, the interpolation function, which is being constructed on the logarithmic space, combines both rotational and translational components. Unlike many existing geometrically exact formulations, e.g.[1], we do not identify positions or, more generally, sets of field variables that may include rotations, with elements of a Lie group. Instead, we are searching for a deformation function that transforms the given initial configuration so as to achieve a quasi-static equilibrium. Choosing appropriate shape functions defined on the Lie algebra is of crucial importance for the performance of the method. To the knowledge of the authors, existing formulations, e.g. [2, 3], for geometrically exact beam models rely on a strictly linear interpolation between the d.o.f. given at the respective nodes and thus do not make use of possible additional shape functions defined on the Lie algebra. As a result, the curvature along the neutral axis remains constant on each beam element. The LogFE formulation is not limited by this restriction and thus makes it possible to further reduce the number of nodal degrees of freedom.In this contribution, we focus on the use of the LogFE formulation in the case of a Bernoulli-type beam and limit the d.o.f. to rotations and dilatations at the nodes. We are preparing a more general exposition of the LogFE formulation for publication in the near future, and intend to include translations at the nodes of the structure in future research.

We extend the Logarithmic finite element method, a novel finite element approach for solving boundary-value problems proposed in [1], to a complete set of degrees of freedom, i.e. translational and rotational degrees of freedom in three dimensions. In contrast to the standard Ritz-Galerkin formulation, the shape functions are given on the logarithmic space of the deformation function. Unlike existing formulations based on Lie groups, they may include polynomial functions of arbitrary degree. The method focuses on reducing the low-frequency components in the error, while minimizing spurious high-frequency deformations, a characteristic that is particularly advantageous in the context of a multigrid algorithm, in which the method may be used to construct an approximation for the coarse grid. MotivationIn [1], we have proposed a novel technique in finite element analysis -the Logarithmic finite element, or "LogFE" method -that significantly reduces the number of degrees of freedom (d.o.f.) necessary to obtain reasonably accurate low-frequency approximations of boundary-value problems, while minimizing spurious high-frequency components in the approximate solution.In contrast to the standard Ritz-Galerkin approach, the d.o.f. are given as coefficients of shape functions on a Lie algebra, allowing to reduce the number of d.o.f. without incurring the locking phenomena associated with linear shape functions. In contrast to existing methods, the interpolation function, which is being constructed on the logarithmic space, combines both rotational and translational components. Unlike existing geometrically exact formulations, e.g.[2], we do not identify positions or, more generally, sets of field variables that may include rotations, with elements of a Lie group. Instead, we formulate a deformation function that transforms the given initial configuration. This deformation function provides the kinematics for different finite element algorithms.To the knowledge of the authors, existing formulations, e.g. [3,4], for geometrically exact beam models rely on a strictly linear interpolation between the d.o.f. given at the respective nodes and thus do not make use of polynomial shape functions of higher degree on the Lie algebra. As a result, the curvature along the neutral axis is being assumed to remain constant across each beam element. The LogFE method is not limited by this restriction.We extend the finite element formulation presented in [1] in two important ways: we move from a two-dimensional to a three-dimensional formulation, and we extend the model by including translational degrees of freedom at the nodes. The latter extension also allows us to conceptualize the standard Ritz-Galerkin approach, in which shape functions are based on translations, as a special case of the Logarithmic finite element method. FormulationIn [1], we have embedded the initial configuration, x 0 , and the current configuration, x, into the homogenized plane R 2 ×{ 1 } ∼ = C×{ 1 }. In order to accommodate translations as well as rotations, wh...

We propose a novel finite element formulation that significantly reduces the number of degrees of freedom necessary to obtain reasonably accurate approximations of the low-frequency component of the deformation in boundary-value problems. In contrast to the standard Ritz-Galerkin approach, the shape functions are defined on a Lie algebra-the logarithmic space-of the deformation function. We construct a deformation function based on an interpolation of transformations at the nodes of the finite element. In the case of the geometrically exact planar Bernoulli beam element presented in this work, these transformation functions at the nodes are given as rotations. However, due to an intrinsic coupling between rotational and translational components of the deformation function, the formulation provides for a good approximation of the deflection of the beam, as well as of the resultant forces and moments. As both the translational and the rotational components of the deformation function are defined on the logarithmic space, we propose to refer to the novel approach as the "Logarithmic finite element method", or "LogFE" method.

Mesh generation is an important step in many numerical methods. The authors present a novel approach to mesh generation, based on algebraic graph theory, that can be used to systematically construct configurations exhibiting multiple hierarchies and complex symmetry characteristics. The mesh is expressed as a hierarchically symmetric graph that preserves both the hierarchical structure as well as the symmetry characteristics of such configurations. MotivationMesh generation is an important step in many numerical methods. The characteristics of a mesh that can be successfully used to solve a given numerical problem, however, depend on the problem at hand. As a result, a large number of different approaches to mesh generation exist, and researchers with different backgrounds may use quite different notations and concepts [1]. Often, the algorithms that are being used to generate meshes are given in terms of rather elementary operations. In particular, the operations given in such algorithms generally are not formally related to the objects that are being modified or constructed.Algebraic graph theory can be used to express the steps involved in mesh generation in a formal way, as operations of an algebraic structure. This allows for a bottom-up approach to mesh generation, in which meshes are expressed as graphs, and unary as well as binary operations on such graphs are being used to construct new graphs from partial graphs, eventually resulting in a mesh that represents the complete configuration.Existing graph-theoretic approaches on mesh generation mainly focus on the symmetry characteristics of configurations, rather than on hierarchy levels [2], or are based on undirected graphs as their basic objects [3].The algorithm presented in this paper provides an efficient method for the systematic generation of meshes representing configurations that exhibit multiple hierarchy and symmetry characteristics. Those characteristics are being preserved in the resulting graph. The conceptualization of a mesh as an element of an algebraic structure based on directed graphs is the main feature of this approach.This method also allows to exploit the hierarchy and symmetry characteristics of a given configuration in subsequent steps of various numerical methods, e.g. finite element calculations, a possible subject of further research. Meshes and graphsA directed graph G can be identified with ordered pairs (V, A) of a set of nodes V and a set of arcs A. An arc a, A a (d(a) , r(a)), can be represented by an ordered pair of nodes, and connects the initial node d(a) ∈ V to the terminal node r(a) ∈ V. The opposite graph G * of a directed graph G can be obtained by taking the opposite of each arc of G, which is given by a *