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A space-time finite element formulation for geometrically exact shear-deformable beams

Ivo Steinbrecher, Alexander Humer

arXiv:2610.03285v1cs.CEcs.NAmath.NA

TL;DR

Transient analysis of geometrically exact shear-deformable beams requires a unified treatment of space and time, including cases with moving material boundaries. The paper proposes a mixed space-time finite element formulation and reports convergence to classical time-stepping solutions, objectivity, and application to the sliding spaghetti problem.

  • Problem

    Existing space-time beam formulations focus on fixed material domains, while sliding structures and axially moving continua have boundaries that move relative to material points.

    Source: §2.3 · p. 6

  • Method

    The formulation discretizes space and time together, uses independent translational and angular velocity fields, and interpolates rotations objectively with nodal rotation vectors.

    Source: §1 · p. 1, §5 · pp. 21–22

  • Results

    The formulation converges to classical time-stepping solutions, demonstrates objectivity, and uses time-upwind stabilization to suppress spurious oscillations associated with backward-in-time information propagation.

    Source: §5 · p. 22

  • Takeaways & Limitations

    Time-dependent material domains and non-material boundaries can be represented through space-time geometry without modifying the beam formulation, as illustrated by the sliding spaghetti problem.

    Source: §5 · p. 22

    The approach does not directly treat the dynamics of the beam segment inside the guide; a suitable coordinate transformation would be needed to include it.

    Source: §4.3 · p. 21

Abstract

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We present a space-time finite element formulation for the transient analysis of geometrically exact shear-deformable beams. The formulation treats space and time within a unified finite element framework and employs a mixed approach in which translational and angular velocities are introduced as independent fields. The rotation field is parametrized using nodal rotation (pseudo-)vectors and a suitable objective interpolation of the triad field is employed. The resulting formulation supports generally curved beam structures, including geometric kinks and rigid joints. Furthermore, time-dependent material domains can be represented directly through the geometry of the space-time surface, enabling the treatment of non-material boundaries without modifying the underlying beam formulation. Numerical examples demonstrate convergence to solutions obtained with classical time-stepping schemes, objectivity of the formulation, and the influence of the space-time stabilization. Finally, the sliding spaghetti problem illustrates the applicability of the proposed approach to axially moving continua.

1. Introduction

The paper develops a unified space-time finite element formulation for transient, geometrically exact shear-deformable beams. Its rotation interpolation supports curved structures and joints, while the space-time geometry can represent time-varying material domains.

1. Introduction

  • The formulation discretizes spatial and temporal domains simultaneously, unlike the conventional method-of-lines approach that first discretizes space and then advances the resulting equations in time.Transient beam simulations address vibrations and response of slender components, including problems with large displacements and rotations.Source: §1 · p. 1
  • The paper situates its space-time beam approach among prior rigid-body methods, parabolic-problem IGA using time-upwind stabilization, nonlinear-elasticity formulations, and geometrically exact string methods.In, rotations were parametrized by directors, whose orthonormality is realized through Lagrange multipliers.Source: §1 · p. 2
  • Prior space-time beam formulations treated fixed material domains, leaving time-dependent material domains unexplored for geometrically exact beams.Such domains arise in sliding structures and axially moving continua, where spatial boundaries do not remain tied to fixed material points.Source: §1 · p. 2
  • The proposed mixed formulation uses rotation-vector interpolation of the triad field, preserves objectivity, and accommodates curved beams, geometric kinks, and rigid joints.Translational and angular velocities are independent fields; varying material domains are represented through the reference space-time surface, as illustrated by the sliding spaghetti problem.Source: §1 · p. 2

2. Governing equations

The governing formulation combines geometrically exact beam kinematics and balance laws with a mixed space-time weak form. It represents the beam over a space-time surface, including when the material domain varies with time.

2.1. Large rotations

  • A rotation pseudo-vector parametrizes the triad, and its relative rotation is non-additive; the rotation interpolation must preserve objectivity.The relative rotation vector is not the difference of the two rotation vectors.Source: §2.1 · p. 3

2.2. Geometrically exact beam formulation

  • The beam is modeled by a centerline and cross-section triad, with local linear- and angular-momentum balances closed by a linear relation between stress resultants and conjugate strains.The Simo–Reissner description permits cross sections to remain plane without requiring them to stay perpendicular to the centerline.Source: §2.2 · pp. 4–5

2.3. Space-time formulation

  • For time-varying material domains, the spatial interval has moving material endpoints, while each material point’s reference position remains time-independent.The changing subset of material points is represented directly by the reference space-time surface when boundary motion is prescribed kinematically.Source: §2.3 · p. 6
  • The mixed space-time weak form treats space and time together, imposes initial position, rotation, and velocity conditions, and introduces independent translational and angular velocities to reduce temporal derivatives to first order.The kinematic relations between configuration and velocity fields are enforced weakly; no terminal conditions are prescribed.Source: §2.3 · pp. 6–7

2.3.1. Space-time kinematics

  • The reference space-time surface uses curvilinear coordinates, with tangent directions corresponding to changes in arc length and time; its metric and Jacobian support derivative and area transformations.The temporal tangent is the time direction at fixed material position, while the spatial tangent is the arc-length direction at fixed time.Source: §2.3.1 · p. 8

3. Discretization

The discretization uses surface finite elements for space-time and treats vector fields and rotations with distinct interpolation schemes. Additional choices address objective rotation interpolation, stabilization, mesh generation, and integration.

3. Discretization

  • The space-time domain is discretized with four-noded quad4 or nine-noded quad9 surface elements, with separate treatments for vector-space fields and orientations in SO3.The section also covers stabilization, global-system assembly, mesh generation, and numerical integration.Source: §3 · p. 9

3.1. Discretization of vector space quantities

  • Centerline position, velocity, and cross-section material angular velocity use standard isoparametric interpolation with the same shape functions as the space-time mesh.The same interpolation is used for variations of these primal fields.Source: §3.1 · p. 10

3.2. Discretization of rotations

  • Nodal total rotation vectors define local rotation vectors, which are interpolated and transformed through an intermediate triad to obtain the orientation field.A Petrov–Galerkin interpolation of nodal spin vectors avoids rotation-dependent interpolation functions that would need reevaluation and consistent linearization.Source: §3.2 · pp. 10–11

3.2.1. Construction of the intermediate triad

  • For elements without a numbering-invariant node, the intermediate triad is formed from the normalized average of nodal unit quaternions, yielding an objective, numbering-independent interpolation.Using one node’s triad is simpler but is suitable only when the element has a node invariant to local numbering.Source: §3.2.1 · p. 11

3.3. Stabilization

  • Residual-based time-upwind stabilization adds a temporal upwind contribution to suppress spurious modes interpreted as nonphysical information propagating backward in time.It is applied to all mixed-formulation test functions as an additional Petrov–Galerkin modification.Source: §3.3 · pp. 11–12

3.4. Global system of equations

  • The global nonlinear system follows by assembling the discretized weak form and factoring out independent variations; rotation-vector increments are updated additively and remapped to keep rotation angles in [0, π].The global tangent is obtained by consistent linearization of the residual, with Jacobians evaluated using automatic differentiation.Source: §3.4 · p. 12

3.5. Space-time mesh generation

  • The tensor-product mesh connects compatible spatial beam meshes at discrete time levels, allowing material points to differ between levels while requiring unchanged topology and node count.This permits time-varying material domains; the space-time module in BeamMe generates meshes for generally curved three-dimensional beams.Source: §3.5 · pp. 12–13

3.6. Numerical integration

  • Tensor-product Gaussian quadrature permits independent integration orders along the beam and second parameter directions, with reduced beam-direction integration for internal forces to alleviate locking.Inertia and external-load contributions use full integration in both directions.Source: §3.6 · p. 13

4. Numerical examples

The examples test agreement with classical time-stepping schemes, convergence and objectivity, then extend the formulation to rigid joints and time-dependent material domains. They also show that stabilization affects convergence and spurious backward-in-time oscillations.

4.1. Curved beam

  • Curved-beam tip displacements and kinetic, elastic, and total energies closely match the classical CJ-line2 results.The comparison uses the same spatial discretization and temporal resolution.Source: §4.1 · p. 14, Figure 5 · p. 15, Figure 6 · p. 16

4.1.1. Convergence for uniform refinement

  • Second-order ST-quad4 and third-order ST-quad9 convergence are observed; ST-quad9 has the lowest error levels, while CJ-line3 remains second-order.ST-quad9 recovers third-order convergence for θ≥0.01; θ≤0.001 reduces its convergence rate and increases its error.Source: §4.1.1 · pp. 16–17, Figure 7 · p. 16, Figure 8 · p. 17

4.1.2. Effect of space-time stabilization

  • During the unloaded rest phase, smaller θ produces larger spurious displacement oscillations; stabilization damps these backward-traveling modes.The authors use θ=0.1 in subsequent simulations and note that the choice is problem-dependent.Source: §4.1.2 · p. 17

4.1.3. Objectivity of the space-time formulation

  • After back-transformation, tip displacements from rigidly rotated initial configurations agree up to nonlinear solver tolerance, confirming objectivity.The absolute tip displacements do not coincide before back-transformation.Source: §4.1.3 · pp. 17–18, Table 1 · p. 18

4.2. Free-floating L-shaped beam

  • For the free-floating L-shaped beam, joint displacement history agrees very well with CJ-line3, demonstrating accurate dynamics for beams with rigid joints.The comparison uses the same spatial discretization and 1000 time steps for CJ-line3.Source: §4.2 · p. 19, Figure 12 · p. 20

4.3. Sliding spaghetti problem

  • In the sliding-spaghetti problem, the beam-tip trajectory agrees excellently with SBF as the exposed beam shortens and its oscillation frequency increases.The space-time mesh represents the changing material domain directly; the guide motion is prescribed kinematically.Source: §4.3 · p. 21

5. Conclusion

The paper presents a mixed space-time formulation for transient analysis of geometrically exact shear-deformable beams. Numerical examples demonstrate convergence to classical time-stepping solutions, objectivity, and direct representation of non-material boundaries.

5. Conclusion

  • Independent translational and angular velocity fields reduce the highest temporal derivatives to first order and allow velocity-level initial conditions to be imposed directly.Finite rotations use nodal rotation (pseudo-)vectors.Source: §5 · p. 21
  • Figure 14 shows deformed beam configurations over the sliding-spaghetti motion and compares the beam-tip trajectory with the Sliding Beam Formulation.Source: Figure 14 · p. 22
  • Objective rotation interpolation makes the formulation applicable to generally curved beams, including geometric kinks and rigid joints.Source: §5 · p. 22
  • The formulation converges to the same solutions as classical time-stepping approaches, and numerical investigations demonstrate objectivity and the role of time-upwind stabilization in suppressing spurious backward-time oscillations.The sliding-spaghetti example demonstrates that non-material boundaries can be represented through the space-time surface geometry without modifying the underlying beam formulation.Source: §5 · p. 22

CRediT authorship contribution statement

Ivo Steinbrecher and Alexander Humer share contributions in methodology, analysis, software, validation, data curation, investigation, and writing; Steinbrecher is additionally credited with conceptualization and visualization.

CRediT authorship contribution statement

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