Polynomial shape functions on the logarithmic space: the LogFE method

Abstract

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...

Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

“…By elementary calculations, it can be shown that, for identical shape functions N I,r,q (ξ ) and initial configuration x 0 (ξ , η), the results of the action of g ξ := exp (ψ 1 (u)(ξ )) andg ξ := exp (ψ 2 (u)(ξ )) on the respective embeddings i 1 resp. i 2 of the initial configuration result in the same current configuration [25]. That is, we have…”

“…Following the approach outlined in [24] and [25], we intend to present a formulation that includes, in addition to rotations and dilatations, the translations at the nodes of the configuration in a future publication.…”

Co‐rotational extension of the Logarithmic finite element method

“…In the Logarithmic finite element (LogFE) method, a novel finite element approach proposed by the authors [1,2], shape functions are defined on the Lie algebra of the deformation function, i.e.…”

The Logarithmic finite element method in a multigrid setting

“…With a suitable formulation, however, it is possible to ensure commutativity at relevant points on the border of a finite element, which is a precondition for achieving the desired linear relationship between such field variables and the set of global degrees of freedom. We refer to our previous contributions for details on the explicit formulation of the Logarithmic finite element method, including an implementation for a planar Bernoulli beam element [1,2].…”

“…The Logarithmic finite element ("LogFE") method [1][2][3] provides a smooth interpolation of the field variables on the coarse grid. It thus minimizes spurious high-frequency components on the fine grid and can avoid locking phenomena associated with the use of linear interpolation.…”