Discrete Cosserat Rod Kinematics Constructed on the Basis of the Difference Geometry of Framed Curves—Part I: Discrete Cosserat Curves on a Staggered Grid

Abstract

The theory of Cosserat rods provides versatile models to simulate large spatial deformations of slender flexible structures. As the strain measures of the mechanical theory are given in terms of the differential invariants of Cosserat curves, the kinematics of Cosserat rods is closely related to the differential geometry of framed curves. We utilize ideas from the difference geometry of framed curves in Euclidean space to construct the discrete kinematics of Cosserat rod models on a staggered grid, preserving the essential geometric properties independent of the coarseness of the discretization.

Appendices

Appendix A: Geometric Concepts for Frenet Curves

Below we revisit the fundamental concepts of the elementary differential geometry of smooth curves in Euclidean space, as presented in standard texts (e.g.: [20]), from the geometric viewpoint emphasized throughout [12, 13], and outline some essential ideas how to transfer the continuous concepts to the discrete setting, following the ideas of Sauer [54].

1.1 A.1 Geometric Curves in Euclidean Space

We regard geometric curves as simple arcs [64] corresponding to smooth, one-dimensional connected submanifolds, which are either simple arcs diffeomorphic to an interval, or simple loops diffeomorphic to a circle [51]. Thus, the mapping $\mathscr {C}\ni p \mapsto \xi (p) \in \mathbb{R}$ of the points $p$ on a geometric curve $\mathscr {C} \subset \mathscr {E}^{3}$ to their real coordinates$\xi$ is (at least once) differentiable and invertible, and the inverse mapping $\xi \mapsto p(\xi )$ from open intervals in ℝ into $\mathscr {E}^{3}$ provides a local parametrization of the curve. By joining the open intervals of local parametrizations, we obtain a larger one $(a,b) \subset \mathbb{R}$ corresponding to a global parametrization $\phi : [a,b] \to \mathscr {C}$ of the geometric curve, such that $(a,b) \ni \xi \mapsto p = \phi (\xi )$ yields all interior points of $\mathscr {C}$, and the two boundary points of $\mathscr {C}$ are given by $\phi (a)$ and $\phi (b)$. The position vectors $\mathbf {x}(p) \in \mathbb{E}^{3}$ of curve points are then given by a parameter curve$\xi \mapsto \mathbf {r}(\xi ) := \mathbf {x}(\phi ( \xi ))$ in $\mathbb{E}^{3}$.

1.2 A.2 The Derivative of Vector Fields on a Geometric Curve

For a vector field $\mathscr {C}\ni p \mapsto \mathbf {v}(p) \in \mathbb{E}^{3}$ mapping points on a smooth geometric curve to vectors in Euclidean space, the derivative $\mathsf{d}\mathbf {v}_{p}: T_{p} \mathscr {C}\to T_{\mathbf {v}(p)}\mathbb{E}^{3}\simeq \mathbb{E}^{3}$ is a linear mapping of tangent vectors (see [51] §1, or [4], Chap. 5 §34 Sect. 4). For the tangent vectors $\mathbf {r}'(\xi ) = \| \mathbf {r}'(\xi ) \| \mathbf {t}(p(\xi ))$ of the parameter curve $\mathbf {r}(\xi ) = \mathbf {x}(p(\xi ))$ induced by a (local) parametrization $\xi \mapsto p(\xi )$ of the curve and, by the usual abuse of notation, $\mathbf {v}(\xi ) \equiv \mathbf {v}(p(\xi ))$, the identities $\mathsf{d}\mathbf {v}_{p} [\mathbf {r}'(\xi )] = \| \mathbf {r}'( \xi ) \| \mathsf{d}\mathbf {v}_{p} [\mathbf {t}(p(\xi ))] = \mathbf {v}'(\xi )$ define the operation of $\mathsf{d}\mathbf {v}_{p}$ on tangent vectors. In particular, $\mathsf{d}\mathbf {v}_{p} [\mathbf {t}(p)] = \mathbf {v}'(s)$ for arc length parametrization, and $\mathsf{d}\mathbf {x}_{p} [\mathbf {t}(p)] \equiv \mathbf {t}(p)$.

For $\mathbf {v}(p) \in S^{2}$ the derivative $\mathsf{d}\mathbf {v}_{p}: T _{p}\mathscr {C}\to T_{\mathbf {v}(p)} S^{2}$ maps tangent vectors to the tangent plane $T_{\mathbf {v}(p)}S^{2}$ to $S^{2}$ orthogonal to $\mathbf {v}(p)$, such that $\langle \mathbf {v}(p), \mathsf{d}\mathbf {v}_{p} [ \mathbf {t}(p)] \rangle = 0$ holds. This explains briefly the geometric meaning of the derivative $\mathsf{d}\mathbf {a}_{p}^{(k)}$ of the directors of a moving frame $\mathsf {R}(p) = \mathbf {a}^{(k)}(p) \otimes \mathbf {e} _{k}$ on a geometric curve, as well as the derivative $\mathsf{d}\mathsf {R}_{p}$ of a moving frame field $\mathscr {C}\ni p \mapsto \mathsf {R}(p) \in \mathsf{SO(3)}$, and the derivative $\mathsf{d} \hat{\mathsf {q}}_{p}$ of a rotational quaternion field $\mathscr {C} \ni p \mapsto \hat{\mathsf {q}}(p) \in S^{3} \subset \mathbb{H}$ on $\mathscr {C}$. If $\mathsf {R}(p) = \mathfrak{E}(\hat{\mathsf {q}}(p))$, then the derivatives of the $\mathsf{SO(3)}$ frame and the rotational quaternion are related by $\mathsf{d}\mathsf {R}_{p} = \nabla _{S^{3}} \mathfrak{E}(\hat{\mathsf {q}}(p)) \cdot \mathsf{d}\hat{\mathsf {q}}_{p}$ via the (tangential) gradient $\nabla _{S^{3}}\mathfrak{E}$ of the Euler map on $S^{3}$.

The notation $\mathsf{d}\mathbf {x}_{p} = \mathsf {R}(p) \cdot \pmb{\varGamma }(p)$ and $\mathsf{d}\mathbf {a}_{p}^{(k)} = \pmb{\kappa }(p) \times \mathbf {a}^{(k)}(p)$ of the derivative equations of a moving frame, or the equivalent versions $\mathsf{d}\mathsf {R}_{p} = \mathsf {R}(p) \cdot \tilde{\mathsf {K}}(p)$ or $\mathsf{d}\hat{\mathsf {q}}_{p} = \frac{1}{2} \pmb{\kappa }(p) \circ \hat{\mathsf {q}}(p) = \frac{1}{2} \hat{\mathsf {q}}(p) \circ \mathbf {K}(p)$ of the generalized Frenet equations, emphasizes their geometric meaning, independent of any particular parametrization of a geometric curve.

In this notation the quantities on both sides are vector (or tensor) valued 1-forms, defined on the tangent bundle $\mathscr {C} \times T\mathscr {C}$ of a geometric curve, and evaluated at $\mathbf {t}(p) \in T_{p}\mathscr {C}$. The vectors $\mathbf {K}(p) \equiv \mathbf {K}_{p}[\mathbf {t}(p)]$, $\pmb{\kappa }(p) \equiv \pmb{\kappa }_{p}[ \mathbf {t}(p)] = \mathsf {R}(p) \cdot \mathbf {K}(p) = \hat{\mathsf {q}}(p) \circ \mathbf {K}(p) \circ \hat{\mathsf {q}}^{\ast }(p)$ and the matrix $\tilde{\mathsf {K}}(p) \equiv \tilde{\mathsf {K}}_{p}[\mathbf {t}(p)]$ result from the 1-forms $\pmb{\kappa }_{p}[\cdot ] = \mathsf {R}(p) \cdot \mathbf {K}_{p}[\cdot ] = \hat{\mathsf {q}}(p) \circ \mathbf {K}_{p}[\cdot ] \circ \hat{\mathsf {q}}^{\ast }(p)$ and $\tilde{\mathsf {K}}_{p}[\cdot ]$. The derivative equations for the vector fields $(\mathbf {n}, \mathbf {b}, \mathbf {t})$ of the Frenet frame can be written in the same manner as $\mathsf{d}\mathbf {x}_{p} = \mathbf {t}(p)$, $\mathsf{d}\mathbf {t}_{p} = \kappa (p) \mathbf {n}(p)$, $\mathsf{d}\mathbf {b}_{p} = -\tau (p) \mathbf {n}(p)$ and $\mathsf{d}\mathbf {n}_{p} = \tau (p) \mathbf {b}(p) - \kappa (p) \mathbf {t}(p)$.

The pull back of the above derivative equations by a local parametrization $\zeta \mapsto p(\zeta )$ yields their well known classical form for parameter curves $\zeta \mapsto \mathbf {r}(\zeta )$ as presented in Sect. 3.1, in general with an additional scaling by $\| \mathbf {r}'(\zeta ) \|$ on both sides as described above, which simplifies for arc length parametrization, as $\| \mathbf {r}'(s) \| = 1$ holds in this case.

For a systematic treatment of moving frames formulated in terms of differential forms within the framework of calculus on manifolds we refer to Blaschke and Reichardt [14] or Cartan [19].

1.3 A.3 Discrete arc Length and Edge Tangent Vectors

The mutual distance of points on a smooth curve can be measured by unbending the curve to a straight line, such that the distance of the same points on the straightened curve equals their Euclidean distance.

This procedure of continuously “unrolling” a smooth curve to the real axis can be understood most easily for the simplified case of a discrete approximation of the curve by a polygonal arc, given by a sequence of curve points $p_{j} = \phi (\xi _{j})$ obtained from a given discretization $a =: \xi _{0} < \xi _{1} < \cdots < \xi _{n} := b$ of the parameter interval with position vectors $\mathbf {r} _{j} := \mathbf {r}(\xi _{j}) \equiv \mathbf {x}(p_{j})$. The corresponding polygonal arc is the piecewise linear curve in $\mathscr {E}^{3}$ defined as the union $\mathscr {P}_{n} [p_{0}, \ldots , p_{n}] := \cup _{j=1} ^{n} [p_{j-1},p_{j}]$ of edges (see Fig. 3)

$$[p_{j-1},p_{j}] := \bigl\{ p \in \mathscr {E}^{3}\mid p = p_{j-1} + \lambda (p_{j} - p_{j-1}) , 0 \leq \lambda \leq 1 \bigr\} , $$

spanned by pairs of adjacent points (also named vertices) $p_{j-1}$ and $p_{j} = p_{j-1} + \mathbf {l}_{j-\frac{1}{2}}$, which are linked by edge vectors$\mathbf {l}_{j-\frac{1}{2}} := p_{j} - p _{j-1} = \mathbf {r}_{j} - \mathbf {r}_{j-1}$ of length $\ell _{j-\frac{1}{2}} := \| \mathbf {l}_{j-\frac{1}{2}} \|$.

Then, the distance of any pair $(p_{k},p_{l})$ of vertices, measured along the path of the polygonal arc $\mathscr {P}_{n}$, is given by the sum $\sum_{j=k+1}^{l} \ell _{j-\frac{1}{2}}$ (with $k < l$) of edge lengths in between, which equals the Euclidean distance of $(p_{k},p _{l})$ if the polygonal arc is straightened out to a line. If the discretization is refined, the polygonal arc $\mathscr {P}_{n}$ approximates the curve $\mathscr {C}$ with increasing accuracy, provided the curve is sufficiently smooth (i.e.: at least Lipschitz continuous).

According to $\int _{\xi _{k}}^{\xi _{l}} \| \mathbf {r}'(\xi ) \| d \xi \approx \sum_{j=k+1}^{l} \ell _{j-\frac{1}{2}}$ the sum of edge lengths provides a discrete approximation of $\int _{\xi _{k}}^{\xi _{l}} \| \mathbf {r}'(\xi ) \| d\xi = \sum_{j=k+1}^{l} \int _{\xi _{j-1}} ^{\xi _{j}} \| \mathbf {r}'(\xi ) \| d\xi$ by evaluating the integrals of $\| \mathbf {r}'(\xi ) \|$ over intervals $[\xi _{j-1}, \xi _{j}]$ of length $h_{j - \frac{1}{2}}$ via the midpoint rule according to the quadrature formula $\int _{\xi _{j-1}}^{\xi _{j}} \| \mathbf {r}'(\xi ) \| d\xi \approx \| \mathbf {r}'(\xi _{j-\frac{1}{2}}) \| h_{j-\frac{1}{2}} + O(h_{j - \frac{1}{2}}^{3})$, with an approximation of $\mathbf {r}'(\xi )$ at the midpoints $\xi _{j - \frac{1}{2}} := \frac{1}{2} (\xi _{j-1} + \xi _{j})$ of the parameter intervals $[\xi _{j-1}, \xi _{j}]$ by a central finite difference as $\mathbf {r}'( \xi _{j-\frac{1}{2}}) \approx (\mathbf {r}_{j} - \mathbf {r}_{j-1}) / h_{j- \frac{1}{2}} + O(h_{j-\frac{1}{2}}^{2})$.

The position vector $\bar{\mathbf {r}}_{j-\frac{1}{2}} := \frac{1}{2} ( \mathbf {r}_{j-1} + \mathbf {r}_{j}) \equiv \mathbf {x}(\bar{p}_{j-\frac{1}{2}})$ of the edge center$\bar{p}_{j-\frac{1}{2}}$, i.e. the midpoint of $[p_{j-1}, p_{j}]$, can likewise be shown to approximate the position vector $\mathbf {r}(\xi _{j-\frac{1}{2}})$ at the midpoint of $[\xi _{j-1}, \xi _{j}]$ with second order accuracy, i.e.: $\mathbf {r}(\xi _{j-\frac{1}{2}}) \approx \bar{\mathbf {r}}_{j-\frac{1}{2}} + O(h_{j-\frac{1}{2}}^{2})$, by adding the Taylor expansions of $\mathbf {r}_{j-1}$ and $\mathbf {r}_{j}$ w.r.t. $\xi = \xi _{j-\frac{1}{2}}$, which results in the approximation $\bar{\mathbf {r}}_{j-\frac{1}{2}} = \mathbf {r}(\xi _{j-\frac{1}{2}}) + h_{j-\frac{1}{2}}^{2} / 4 \mathbf {r}''( \xi _{j-\frac{1}{2}}) + \cdots$ due to the cancellation of the first order terms $\pm h_{j-\frac{1}{2}}/2 \mathbf {r}'(\xi _{j-\frac{1}{2}})$.

The polygonal approximation of a curve (see Fig. 3) naturally leads to the concept of edge based tangent vectors$\mathbf {l}_{j-\frac{1}{2}} / h_{j- \frac{1}{2}} \approx \mathbf {r}'(\xi _{j-\frac{1}{2}})$, with unit length edge tangent vectors $\mathbf {t}_{j-\frac{1}{2}} := \mathbf {l}_{j- \frac{1}{2}} / \ell _{j-\frac{1}{2}}$ located at $\bar{p}_{j- \frac{1}{2}}$. Requiring that consecutive vertices $p_{j}$ and $p_{j+1}$ must not coincide yields the canonical definition of discrete regularity of a polygonal arc, which implies non-zero edge vectors (i.e.: $\| \mathbf {l}_{j+\frac{1}{2}} \| > 0 \Leftrightarrow p_{j} \neq p_{j+1}$) and assures that unit vectors $\mathbf {t}_{j- \frac{1}{2}}$ are well defined for all edges of a discrete regular geometric curve.

Discrete Geometric Curves and Discrete Parameter Curves

The mapping $j \mapsto p_{j}$ of integer indices to vertex points in Euclidean point space $\mathscr {E}^{3}$ may be interpreted as a discrete geometric curve, with its orientation induced by the ordering of the integer indices. This induces a corresponding mapping $j \mapsto \mathbf {r}_{j} \equiv \mathbf {x}(p_{j})$ of integer indices to position vectors in $\mathbb{E}^{3}$.

A discrete parameter curve is defined by the mapping $\xi _{j} \mapsto \mathbf {r}(\xi _{j})$ induced by a discretization of the domain of a smooth parameter curve. The discrete counterpart of arc length parametrisation corresponds to the case $h_{j-\frac{1}{2}} \equiv \ell _{j-\frac{1}{2}}$ of grid constants being equal to edge lengths, with discrete arc length parameters defined as $\varsigma _{j} := \varsigma _{0} + \sum_{i=1}^{j} \ell _{i-\frac{1}{2}}$, marking the vertex positions of the polygonal arc straightened out parallel to the real axis.

Remark 1

In the special case $h_{j-\frac{1}{2}} \equiv 1$ and $\varsigma _{0} \in \mathbb{Z}$ also $\varsigma _{j} \in \mathbb{Z}$, such that the mapping $j \mapsto \mathbf {r}_{j} \equiv \mathbf {x}(p_{j})$ mentioned above may be interpreted as a discrete parametrization of a geometric curve given by $j \mapsto p_{j}$. If additionally $\ell _{j-\frac{1}{2}} \equiv 1$ holds, the mapping $j \mapsto \mathbf {r}_{j}$ corresponds to a discrete arc length parametrization of the discrete geometric curve.

Remark 2

For a rectifiable geometric curve $s \mapsto p(s)$ parametrized by arc length, a discretization of its domain in general does not lead to discrete arc length parametrization, as by construction the distance $s_{j} - s_{j-1} = \Delta s_{j-\frac{1}{2}} \equiv h_{j-\frac{1}{2}}$ of vertices $p_{j} = p(s_{j})$ measured in arc length along the continuous curve is always greater or equal to the respective edge lengths $\ell _{j-\frac{1}{2}} = \| \mathbf {r}_{j} - \mathbf {r}_{j-1} \|$, i.e.: $\Delta s_{j-\frac{1}{2}} / \ell _{j-\frac{1}{2}} \ge 1$, unless the geometric curve is a polygonal arc, and the discrete parameters $s_{j}$ coincide with the vertex position coordinates.

Remark 3

As in general $\Delta s_{j-\frac{1}{2}} = \ell _{j-\frac{1}{2}} + O(h _{j-\frac{1}{2}}^{3})$ holds for arbitrary regular parametrizations, for $s_{0} \equiv \zeta _{0}$ one obtains the relation $s_{j} = \zeta _{j} + \sum_{i=1}^{j} O(h_{i-\frac{1}{2}}^{3})$, i.e.:

For a polygonal arc approximating a smooth geometric curve, the discrete arc length parameters $\zeta _{j}$ approximate the continuous arc length $s(\xi )$ at $s_{j} = s(\xi _{j})$ with second order accuracy.

1.4 A.4 Geometric Notions of Frenet Curvature and Torsion

Any regular geometric curve $\mathscr {C}$ possesses a well defined tangent line$T_{p}\mathscr {C}$ in every point $p \in \mathscr {C}$, as well as a normal plane$N_{p}\mathscr {C}$ defined as the affine plane orthogonal to the tangent line containing the point $p$. All points $q \in T_{p}\mathscr {C}$ are given parametrically in terms of the base point $p$ and either of the two unit tangent vectors $\pm \mathbf {t}(p)$ (depending on the choice of orientation) as $q(\lambda ) = p + \lambda \mathbf {t}(p)$ for $\lambda \in \mathbb{R}$.

1.4.1 A.4.1 The Tangent Map and Its Derivative

Qualitatively the geometric notion of curvature reflects the change of the spatial direction of the tangent line while its base point varies along the curve. Quantitatively the total amount of curvature at a curve point $p \in \mathscr {C}$ is measured by Frenet curvature$\kappa (p) := \| \mathsf{d}\mathbf {t}_{p} \|$, definable as the norm of the derivative $\mathsf{d}\mathbf {t}_{p}: T_{p} \mathscr {C}\to T_{\mathbf {t}(p)} S^{2}$ of the tangent map$\mathscr {C}\ni p \mapsto \mathbf {t}(p) \in S^{2}$ taking curve points to tangent vectors.

As the base point $p \in \mathscr {C}$ slides on the curve, the unit tangent vector $\mathbf {t}(p)$ traces a curve on the unit sphere $S^{2}$. As the differential arc length on the latter curve is given by $\mathsf{d}\theta (s) = \| \mathbf {t}'(s)\| ds = \kappa (s) ds$, one may obtain the curvature $\kappa (p)$ by considering the quotient of the length of the curve segment joining $\mathbf {t}(p)$ and $\mathbf {t}(q)$ on $S^{2}$ to the distance $\Delta s_{pq} := s(p) - s(q)$ of the two curve points $p, q \in \mathscr {C}$ measured in arc length in the limit $q \to p \Leftrightarrow \Delta s_{pq} \to 0$, i.e.: $\kappa (p) = \lim_{q \to p} \int _{s(q)}^{s(p)} \mathsf{d}\theta (s) / \Delta s_{pq}$.

The shortest (geodesic) path on $S^{2}$ that connects $\mathbf {t}(p)$ and $\mathbf {t}(q)$ is a circular arc of length $\theta _{pq} := \arccos (\langle \mathbf {t}(p),\mathbf {t}(q) \rangle ) \in [0,\pi ]$, equal to the (non-negative) angle enclosed by both tangent vectors, as sketched in Fig. 2. While by construction $\theta _{pq} \le \int _{s(q)}^{s(p)} \mathsf{d}\theta (s)$, it can be shown that $\theta _{pq} / \int _{s(q)}^{s(p)} \mathsf{d}\theta (s) = \mathscr {O}(1)$ holds asymptotically for $q \to p$, such that the curvature at a point $p$ may likewise be obtained by considering the limit $\kappa (p) = \lim_{q \to p} \theta _{pq} / \Delta s_{pq}$.

Both the enclosed tangent angle $\theta _{pq}$, measuring the geodesic distance of unit tangent vectors $\mathbf {t}(p)$ and $\mathbf {t}(q)$ on $S^{2}$, as well as the arc length distance $\Delta s_{pq} = s(p) - s(q)$ of their base points, are invariant quantities that are independent of any parametrization, possess an elementary geometric interpretation and thus provide a proper starting point to construct a discrete version of Frenet curvature for polygonal arcs.

Definition 4.3 of discrete Frenet curvature$\kappa _{j}$ of a polygonal arc at the vertex $p_{j}$ by the formula (3) given in Sect. 4.2 results from considering pairs $\mathbf {t}(p_{j \pm \frac{1}{2}})$ of tangent vectors of a smooth geometric curve at points $p_{j \pm \frac{1}{2}} = p(s _{j \pm \frac{1}{2}})$ located on the curve at half arc length distance from $p_{j} = p(s_{j})$ to $p_{j \pm 1}$. Replacing the angle between $\mathbf {t}(p_{j \pm \frac{1}{2}})$ by the angle $\theta _{j}$ between edge tangent vectors $\mathbf {t}_{j \pm \frac{1}{2}}$ and the arc length distance of curve points $p_{j \pm \frac{1}{2}}$ by the discrete arc length distance $\Delta \varsigma _{j}$ of edge centers $\bar{p} _{j \pm \frac{1}{2}}$ amounts to a second order accurate approximation of the respective quantities. Therefore (3) provides a consistent, invariant and perfectly geometric discretization of continuous Frenet curvature that approximates $\kappa (p_{j})$, in the case of an equidistant discretization with second order accuracy.

1.4.2 A.4.2 Frenet Frame, Curvature Binormal and Geometric Torsion

As $T_{\mathbf {t}(p)} S^{2} \simeq N_{p}\mathscr {C}$, the derivative of the tangent map can be interpreted as a linear mapping $\mathsf{d}\mathbf {t} _{p}: T_{p}\mathscr {C}\to N_{p}\mathscr {C}$ of tangent vectors to vectors in the normal plane. Due to linearity, this mapping is uniquely defined by its value at the base vector $\mathbf {t}(p)$ of $T_{p} \mathscr {C}$, given by $\mathsf{d}\mathbf {t}_{p} [\mathbf {t}(p)] = \kappa (p) \mathbf {n}(p)$ in all points $p\in \mathscr {C}$ where $\kappa (p) > 0$, by means of a decomposition of the image into its direction given by the principal normal$\mathbf {n}(p)$ and its modulus $\kappa (p)$. The osculating plane at $p$ is the affine plane spanned by the orthonormal pair $\{\mathbf {t}(p), \mathbf {n}(p)\}$ of vectors. It is well defined at all points of positive Frenet curvature, with the binormal vector $\mathbf {b}(p) := \mathbf {t}(p) \times \mathbf {n}(p)$ corresponding to one of the two possible orientations of its normal direction.

The curvature binormal$\kappa (p) \mathbf {b}(p) = \mathbf {t}(p) \times \mathsf{d}\mathbf {t}_{p}$ encodes information about the differential rotation of the tangent direction around the binormal axis $\mathbf {b}(p)$ by an amount $\kappa (p)$. Analogously, geometric torsion$\tau (p)$ is the curvature quantity that measures the differential rotation of the osculating plane (as well as the normal plane) around the tangent axis $\mathbf {t}(p)$ according to the identity $\tau (p) \mathbf {t}(p) = \mathbf {b}(p) \times \mathsf{d}\mathbf {b}_{p}$. Different from the Frenet curvature, the geometric torsion $\tau (p)$ is a signed quantity: With $\mathbf {t}(p)$ fixed by the choice of the orientation of the curve, the sign of $\tau (p)$ can be either negative or positive, as differential rotations of $N_{p}\mathscr {C}$ around the $\mathbf {t}(p)$ direction can be either clockwise or counter-clockwise.

If one considers the binormal map$\mathscr {C}\ni p \mapsto \mathbf {b}(p) \in S^{2}$, as sketched in Fig. 2, and quantifies the spatial rotation of the osculating plane in terms of the angular quantity $|\alpha |_{pq} := \arccos (\langle \mathbf {b}(p), \mathbf {b}(q) \rangle ) \in [0,\pi ]$, which measures the length of the circular arc joining $\mathbf {b}(p)$ and $\mathbf {b}(p)$ on $S^{2}$, relative to the arc length distance $\Delta s_{pq}$, the limit $\Delta s_{pq} \to 0$ yields the modulus of geometric torsion according to: $|\tau (p)| = \lim_{q \to p} |\alpha |_{pq} / \Delta s_{pq}$. Geometric torsion as a signed quantity may be obtained as: $\tau (p) = \lim_{q \to p} \alpha _{pq} / \Delta s_{pq} = \lim_{q \to p} \langle \mathbf {t}(p), \mathbf {b}(p) \times \mathbf {b}(q) \rangle / \Delta s_{pq}$. Note that the latter expression may be interpreted as a FD approximation of the identity $\tau (p) = \langle \mathbf {t}(p) , \mathbf {b}(p) \times \mathsf{d}\mathbf {b}_{p} \rangle$, with its geometric interpretation as a differential torsional rotation in terms of $\alpha _{pq} / \Delta s _{pq}$, using a FD approximation $\mathsf{d}\mathbf {b}_{p} \approx [ \mathbf {b}(q) - \mathbf {b}(p)] / \Delta s_{pq}$.

Using this simple FD approximation in the discrete setting, one would obtain Sauer’s definition $\ell _{j+\frac{1}{2}} \tau ^{(S)}_{j+ \frac{1}{2}} = \sin (\alpha _{j+\frac{1}{2}}) = \langle \mathbf {t}_{j+ \frac{1}{2}}, \mathbf {b}_{j} \times \mathbf {b}_{j+1} \rangle$ of discrete geometric curvature. Apart from its low order accuracy, the main drawback of this approximation consists of its lack of monotony over the complete range $(-\pi ,\pi ]$ of admissible torsion angles. Both accuracy as well as monotonicity can be improved by choosing a central FD approximation w.r.t. the edge center $\bar{p}_{j+ \frac{1}{2}}$, utilizing the averaged binormal vector $\bar{\mathbf {b}} _{j+\frac{1}{2}}$ introduced in Definition 4.10, which yields the symmetric, second order accurate FD approximation $\langle \mathbf {t}_{j+\frac{1}{2}} , \bar{\mathbf {b}}_{j+\frac{1}{2}} \times [ \mathbf {b}_{j+1} - \mathbf {b}_{j} ] / \ell _{j+\frac{1}{2}} \rangle = 2 \tan (\alpha _{j+\frac{1}{2}}/2) / \ell _{j+\frac{1}{2}}$ on which the formula (4) for $\tau _{j+\frac{1}{2}} := \alpha _{j+\frac{1}{2}} / \ell _{j+\frac{1}{2}}$ is based.

1.4.3 A.4.3 Parallel Transport Along a Smooth Geometric Curve

The tangent map $\mathscr {C}\ni p \mapsto \mathbf {t}(p) \in S^{2}$ and its derivative $\mathsf{d}\mathbf {t}_{p}: T_{p}\mathscr {C}\to N_{p} \mathscr {C}$ define the parallel transport (PT) in the normal bundle $\mathscr {C}\times N\mathscr {C}$ of a geometric curve in terms of the infinitesimal rotations of an adapted frame $\mathsf {P} = ( \mathbf {P}^{(1)}, \mathbf {P}^{(2)}, \mathbf {t})$ along the curve $\mathscr {C}$ generated by the Darboux vector $\pmb{\kappa }= \mathbf {t} \times \mathsf{d}\mathbf {t}_{p} = \kappa \mathbf {b}$. The characteristic feature of such rotations is identically vanishing geodesic torsion (i.e.: $\varkappa ^{(3)} \equiv 0$). This implies generalized Frenet equations of the normal directors of the form $\mathsf{d}\mathbf {P}^{(1)}_{p} = - \varkappa ^{(2)} \mathbf {t}$, $\mathsf{d}\mathbf {P}^{(2)}_{p} = \varkappa ^{(1)} \mathbf {t}$ with normal and geodesic curvatures $\varkappa ^{( \alpha )} = \langle \kappa \mathbf {b}, \mathbf {P}^{(\alpha )} \rangle \Leftrightarrow \kappa \mathbf {b} = \varkappa ^{(\alpha )} \mathbf {P}^{( \alpha )}$ obtained from the orthogonal decomposition of the curvature binormal, complemented by the Frenet equation $\mathsf{d}\mathbf {t}_{p} = \varkappa ^{(2)}\mathbf {P}^{(1)} - \varkappa ^{(1)}\mathbf {P}^{(2)}$ for the tangent vector. The equations for $\mathbf {P}^{(\alpha )}$ may be rewritten equivalently as $\mathsf{d}\mathbf {P}^{(\alpha )}_{p} = - \langle \mathbf {P}^{(\alpha )}, \mathsf{d}\mathbf {t}_{p} \rangle \mathbf {t}$.

With given $\mathbf {t}(p)$ and $\mathsf{d}\mathbf {t}_{p}$, all solutions of the linear ODE $\mathsf{d}\mathbf {v}_{p} = - \langle \mathbf {v}, \mathsf{d} \mathbf {t}_{p} \rangle \mathbf {t}$ defined for arbitrary vector fields $p \mapsto \mathbf {v}(p)$ on $\mathscr {C}$ have constant $\langle \mathbf {v}(p), \mathbf {t}(p) \rangle$ at all points $p \in \mathscr {C}$ due to $\mathsf{d}\langle \mathbf {v}, \mathbf {t} \rangle \equiv 0$, which implies that if $\mathbf {v}(p) \perp \mathbf {t}(p)$ at some point $p = p_{0}$, then $\mathbf {v}(p) \in N_{p} \mathscr {C}$ for all $p \in \mathscr {C}$. This in turn implies that the scalar product $\langle \mathbf {N}, \mathbf {M} \rangle$ of any two normal vector fields $\mathbf {N}, \mathbf {M} \in N\mathscr {C}$ remains constant as well (i.e.: $\mathsf{d}\langle \mathbf {N}, \mathbf {M} \rangle \equiv 0$). If we choose at some point $p = p_{0}$ an orthonormal pair $(\mathbf {N}_{0}, \mathbf {M}_{0})$ of vectors as initial values satisfying $\langle \mathbf {N}_{0}, \mathbf {t}(p_{0}) \rangle = 0 = \langle \mathbf {M}_{0}, \mathbf {t}(p_{0}) \rangle$ and $\mathbf {N}_{0} \times \mathbf {M}_{0} = \mathbf {t}(p_{0})$, then the linear ODE yields an orthonormal pair $\mathbf {N}(p), \mathbf {M}(p) \in N_{p}\mathscr {C}$ of solutions in the normal bundle $N\mathscr {C}$ corresponding to a parallel frame.

Remark 4

The canonical linear ODE $\mathsf{d}\mathbf {P}_{p} = - \langle \mathbf {P}, \mathsf{d}\mathbf {t}_{p} \rangle \mathbf {t}$ of PT in the normal bundle $\mathscr {C}\times N\mathscr {C}$ provides a ‘bottom up’ approach to construct the Frenet equations for the normal directors of parallel frames from its characteristic geometric conservation properties, i.e.: $\langle \mathbf {P}, \mathbf {t} \rangle \equiv 0$, $\| \mathbf {P} \| = 1$ and $\langle \mathbf {P}, \mathbf {Q} \rangle = \mathrm{const}$. for any solution pair $(\mathbf {P}, \mathbf {Q}) \in N\mathscr {C}$. These properties are (by construction) built into discrete parallel transport (DPT) introduced in Sect. 6.2.1 in terms of rotations $\mathsf {B}_{j}$ connecting adjacent tangent vectors $\mathbf {t}_{j\pm \frac{1}{2}}$.

Appendix B: The Stored Energy of Elastic Cosserat Rods and Internal Constraints

While kinematic boundary conditions have been discussed for different variants of discrete framed curves in the Sects. 4.4.1, 5.6 and 6.6 of this article, the subject of internal constraints, which is likewise a kinematical topic important for the construction of discrete rod models, has been mentioned only briefly so far, mostly within the context of the adaption condition$\mathbf {a}^{(3)} \equiv \mathbf {t} \Leftrightarrow \pmb{\varGamma }\equiv \mathbf {e}_{3}$ that restricts general Cosserat curves (assumed to be parametrized by arc length) to ribbons.

Although the concepts of an inextensible centerline or unshearable cross sections are standard topics of the mechanical theory of rods, they are not part of the differential geometry of framed curves, as the latter is concerned with the geometrical properties of individual configurations, whereas the mechanical concepts describe restrictions on configurations that are deformed relative to a reference state and thus involve properties of pairs of configurations rather than individual ones.

A discussion of the stored energy of an elastic rod, as given below in the updated excerpt from the respective sections of the article [48], provides a proper framework to address the topic of such internal constraints. We refer to the article [42] for a detailed exposition including numerical aspects.

2.1 B.5 The Elastic Energy of a Cosserat Rod

Static equilibria of deformed elastic structures can be computed as minima of their elastic energy, subject to the assumed boundary conditions. To model slender flexible structures as elastic Cosserat rods, one need to specify a corresponding elastic energy function. For linear elastic material behaviour, the elastic (stored) energy function of a 3D body is a quadratic form of its Green–Lagrange strain tensor $\mathsf {E} = \frac{1}{2} (\mathsf {F}^{T} \cdot \mathsf {F} - \mathsf {I})$, where $\mathsf {F} = \mathsf{d}\varPhi (\mathbf {X})$ is the deformation gradient computed as the derivative of the positions $\mathbf {x} = \varPhi (\mathbf {X})$ of the material points in the deformed body volume w.r.t. their positions $\mathbf {X}$ in the undeformed body (see [31, 52] for details). The deformation gradient and Green–Lagrange strain tensor for the deformed configurations $\mathbf {x} = \mathbf {r}(s) + \xi _{\alpha } \mathbf {a}^{(\alpha )}(s)$ of a Cosserat rod w.r.t. its undeformed reference configuration $\mathbf {X} = \mathbf {r}_{0}(s) + \xi _{\alpha } \mathbf {a}_{0}^{(\alpha )}(s)$ given by a smooth regular curve $\mathbf {r}_{0}(s)$ parametrized by arc length and its adapted frame $\mathsf {R}_{0}(s) = \mathbf {a}_{0}^{(k)}(s) \otimes \mathbf {e}_{k}$ with $\mathbf {r}_{0}'(s) = \mathbf {a}_{0}^{(k)}(s)$, are obtained [46] as functions of the differences $\mathbf {K}(s) - \mathbf {K}_{0}(s)$ and $\pmb{\varGamma }(s) - \pmb{\varGamma }_{0}$ (with $\pmb{\varGamma }_{0} = (0,0,1)^{T}$) of the invariants of the framed curve in their deformed and undeformed configurations.

For slender rod geometries, one may always assume that the local strains remain small, although the deformations of the rod configuration correspond to large (finite) rotations and displacements in space. In this case one may approximate the exact expression for $\mathsf {E}$ by taking only the leading order terms in the differences of the invariants into account. The resulting approximated energy density can then be integrated analytically over the cross section coordinates $(\xi _{1},\xi _{2})$ in closed form, which finally yields [46] the elastic energy $\mathscr {W}_{el}$ of a Cosserat rod as a quadratic functional in the differences $\mathbf {K} - \mathbf {K}_{0}$ and $\pmb{\varGamma }- \pmb{\varGamma }_{0}$ of the invariants, given by the sum $\mathscr {W}_{el} = \mathscr {W}_{es} + \mathscr {W}_{bt}$ of the two integrals

$$\begin{aligned} \mathscr {W}_{es} =& \frac{1}{2} \int _{0}^{L} ds [\mathit{EA}] \bigl( \varGamma ^{(3)}(s) - 1 \bigr)^{2} + [ \mathit{GA}_{\alpha }] \varGamma ^{(\alpha )}(s)^{2} , \end{aligned}$$

(46)

$$\begin{aligned} \mathscr {W}_{bt} =& \frac{1}{2} \int _{0}^{L} ds [\mathit{EI}_{\alpha }] \bigl( K^{(\alpha )}(s) - K_{0}^{(\alpha )}(s) \bigr) ^{2} + [\mathit{GJ}] \bigl( K^{(3)}(s) - K_{0}^{(3)}(s) \bigr)^{2} . \end{aligned}$$

(47)

The first term (46) represents the elastic energy related to rod deformations by longitudinal extension combined with transverse shearing, the second term (47) accounts for the elastic energy stored in bending and torsional deformations of the rod. The parameters $[\mathit{EA}]$, $[\mathit{GA}_{\alpha }]$, $[\mathit{EI} _{\alpha }]$ and $[\mathit{GJ}]$ quantify the effective stiffness properties of the local cross sections of the rod related to the respective deformation modes. They may be constants, or vary along the rod as functions of $s$. In the simple case of a homogeneous and isotropic material characterized by the elastic moduli $E$ and $G$, they are given as products of the moduli and geometric quantities (i.e.: area $A$, area moments $I_{\alpha }$, polar moment $J$) of the cross section.

2.2 B.6 Unshearable and Inextensible Rods

A Cosserat rod is unshearable if its cross sections remain orthogonal to its centerline tangents in all configurations, such that the internal constraints $\varGamma ^{(\alpha )}(s) = \langle \mathbf {a}^{( \alpha )}(s) , \mathbf {r}'(s) \rangle \equiv 0$ are fulfilled, which in turn implies that $\mathbf {a}^{(3)}(s) \equiv \mathbf {t}(s) = \mathbf {r}'(s) / \| \mathbf {r}'(s) \|$ holds. In this case $\varGamma ^{(3)}(s) = \| \mathbf {r}'(s) \|$, and the energy integral (46) reduces to the extensional energy$\mathscr {W}_{\mathrm{ext}} := \frac{1}{2} \int _{0}^{L} ds [\mathit{EA}] (\| \mathbf {r}'(s) \| - 1)^{2}$ of a Kirchhoff rod with extensible centerline.

If the centerline is assumed to be inextensible, its arc length $s$ remains the same in all configurations, such that the internal constraint $\| \mathbf {r}'(s) \| = 1 \Leftrightarrow \mathbf {r}'(s) = \mathbf {t}(s)$ is always fulfilled. The inextensibility condition is often assumed to hold in addition to the assumption of unshearable cross sections, which can be expressed by the vector constraint $\mathbf {r}'(s) \equiv \mathbf {a}^{(3)}(s) \Leftrightarrow \mathsf {R}^{T}(s) \cdot \mathbf {r}'(s) = \pmb{\varGamma }(s) \equiv \mathbf {e}_{3}$ that characterizes the configurations of inextensible Kirchhoff rods.

Within the Lagrangian formalism constraints given by equations $g(\ldots ) = 0$ are enforced by introducing Lagrange multipliers$\lambda$ as additional variables into the model by adding product terms $\lambda g(\ldots )$ to the Lagrangian. In the case of an inextensible Kirchhoff rod, the corresponding additive extension of the Lagrangian is given by the integrals $\int _{0}^{L} ds \langle \pmb{\lambda }(s), \mathbf {r}'(s) - \mathbf {a} ^{(3)}(s) \rangle = \int _{0}^{L} ds \langle \pmb{\varLambda }(s), \pmb{\varGamma }(s) - \mathbf {e}_{3} \rangle$ with vector Lagrange multipliers $\pmb{\lambda }(s) = \mathsf {R}(s) \cdot \pmb{\varLambda }(s)$ enforcing the combined constraints of unshearable cross sections and an inextensible centerline equivalently in their spatial and material form.

For inextensible Kirchhoff rods, the vector Lagrange multiplier $\pmb{\lambda }(s)$ takes the role of the sectional force vector $\mathbf {f}(s) = [\mathit{GA}_{\alpha }] \varGamma ^{(\alpha )}(s) \mathbf {a}^{( \alpha )}(s) + [\mathit{EA}] ( \varGamma ^{(3)}(s) - 1 ) \mathbf {a}^{(3)}(s)$ of an elastic Cosserat rod obtained from the energy term (46), both vanishing identically due to the internal constraints. For extensible Kirchhoff rods the tangential part $\mathbf {f}_{t}(s) = [\mathit{EA}] ( \varGamma ^{(3)}(s) - 1 ) \mathbf {t}(s)$ of the sectional force vector is obtained from a constitutive relation, while the transverse force components are replaced by Lagrange multipliers $\varLambda ^{(\alpha )}(s)$ enforcing the constraints $\varGamma ^{(\alpha )}(s) \equiv 0$ via the integral $\int _{0}^{L} ds \varLambda ^{(\alpha )}(s) \varGamma ^{(\alpha )}(s)$ added to the Lagrangian.

2.3 B.7 Discrete Elastic Energy of Quaternionic Cosserat Rods

The integrals (46) and (47) can be approximated by suitable quadrature rules, making use of the discrete curvatures $\{ K_{j}^{(k)} \}^{k=1,2,3}_{j=1,\ldots ,n-1}$ and extensional and shear strains $\{ \varGamma _{j-1/2}^{(k)} \}^{k=1,2,3}_{j=1,\ldots ,n}$ according to the approach described in detail in [41].

We start with a discretization $0 =: s_{0} < s_{1} < \cdots < s_{n} := L$ of the interval domain $[0,L]$ of the arc length parameter $s$ of the reference curve $\mathbf {r}_{0}(s)$ into subintervals $[s_{j-1},s_{j}]$ of length $h_{j-1/2} := s_{j} - s_{j-1}$. The distance between interval midpoints $s_{j \pm 1/2} = \frac{1}{2} (s_{j} + s_{j \pm 1})$ is given by $\Delta s_{j} := s_{j+1/2} - s_{j-1/2} = \bar{h}_{j}$, where $\bar{h}_{j} := \frac{1}{2} (h_{j-1/2} + h_{j+1/2})$ is the average of the grid constants $h_{j \pm 1/2}$ adjacent to $s_{j}$.

2.3.1 B.7.4 Discrete Bending and Torsion Energy

The discrete curvatures $K_{j}^{(k)}$ are vertex based quantities, such that a discrete approximation of $\mathscr {W}_{es}$ can be obtained from the energy integral (47) by (non-equidistant) trapezoidal quadrature. The pull back of the curvatures originally defined w.r.t. discrete arc length to the reference configuration implies a rescaling by the factor $\Delta \varsigma _{j} / \Delta s_{j} \approx \| \mathbf {r}'(s_{j}) \|$, i.e.: $\bar{K}_{j}^{(k)} := (\Delta \varsigma _{j} / \Delta s_{j}) K_{j}^{(k)}$, equivalent to the definition

$$ \bar{K}_{j}^{(k)} := \langle \bar{ \mathbf {K}}_{j} , \mathbf {e}_{k} \rangle , \quad \bar{\mathbf {K}}_{j} := (\Delta s_{j})^{-1} 2 \log {\hat{\mathsf {W}} _{j}} = (\vartheta _{j} / \Delta s_{j}) \hat{\mathbf {U}}_{j} = \bar{K}_{j}^{(k)} \mathbf {e}_{k} $$

(48)

of discrete pulled back material curvatures. The discrete approximation of $\mathscr {W}_{bt}$ can then be written in compactly as

$$ \mathscr {W}_{bt} \approx \mathscr {W}^{(D)}_{bt} := \frac{1}{2} \sum _{j=0}^{n} \Delta s_{j} \langle \Delta \bar{ \mathbf {K}}_{j}, \mathsf {C}_{bt} \cdot \Delta \bar{\mathbf {K}}_{j} \rangle , $$

(49)

with the curvature differences $\Delta \bar{\mathbf {K}}_{j} := \bar{ \mathbf {K}}_{j} - \mathbf {K_{0}}_{j}$ between the deformed and reference configurations, the diagonal matrix $\mathsf {C}_{bt} := \mathrm{diag}([\mathit{EI} _{1}], [\mathit{EI}_{2}], [\mathit{GJ}])$ of bending and torsional stiffness parameters, and the grid constants of the half-edges near the boundary defined as $\Delta s_{0} := h_{1/2}/2$ and $\Delta s_{n} := h_{n-1/2}/2$.

2.3.2 B.7.5 Discrete Extensional and Shear Energy and Discrete Internal Constraints

As the discrete extensional and shear strains $\varGamma ^{(k)}_{j-1/2}$ are edge based quantities, an approximation of the energy integral (46) by midpoint quadrature is the natural choice to obtain a discrete version of $\mathscr {W}_{es}$. The pull back of the strains to the reference configuration is obtained by a rescaling with the factors $\ell _{j-1/2}/h_{j-1/2} \approx \| \mathbf {r}'(s_{j-1/2}) \|$, according to

$$ \bar{\varGamma }^{(k)}_{j-1/2} := \frac{\ell _{j-1/2}}{h_{j-1/2}} \varGamma ^{(k)}_{j-1/2} = \bigl\langle \mathbf {l}_{j-1/2} , \mathbf {a}^{(k)} _{j-1/2} \bigr\rangle / h_{j-1/2} = \langle \bar{\pmb{\varGamma }}_{j-1/2}, \mathbf {e}_{k} \rangle , $$

(50)

where $\bar{\pmb{\varGamma }}_{j-1/2} := \hat{\mathsf {q}}^{\ast }_{j-1/2} \circ (\mathbf {l}_{j-1/2}/h_{j-1/2}) \circ \hat{\mathsf {q}}_{j-1/2}$ is the material vector obtained from rotating the discrete edge tangent vector $\mathbf {l}_{j-1/2}/h_{j-1/2} \approx \mathbf {r}'(s_{j-1/2})$ back to the local frame. The discrete approximation of $\mathscr {W}_{es}$ can be written in compact form as

$$ \mathscr {W}_{es} \approx \mathscr {W}^{(D)}_{es} := \frac{1}{2} \sum _{j=1}^{n} h_{j-1/2} \langle \Delta \bar{ \pmb{\varGamma }}_{j-1/2}, \mathsf {C}_{es} \cdot \Delta \bar{ \pmb{\varGamma }}_{j-1/2} \rangle , $$

(51)

where $\Delta \bar{\pmb{\varGamma }}_{j-1/2} := \bar{\pmb{\varGamma }}_{j-1/2} - \pmb{\varGamma }_{0}$ with $\pmb{\varGamma }_{0} = (0,0,1)^{T}$, and the shear and extensional stiffness parameters collected in the diagonal matrix $\mathsf {C}_{es} := \mathrm{diag}([\mathit{GA}_{1}], [\mathit{GA}_{2}], [\mathit{EA}])$.

The condition of vanishing discrete transverse shear strains $\bar{\varGamma }^{(\alpha )}_{j-1/2} \equiv 0$ implies $\bar{\varGamma } ^{(3)}_{j-1/2} = \ell _{j-1/2}/h_{j-1/2}$, such that (50) reduces to the discrete extensional energy

$$\mathscr {W}^{(D)}_{\mathrm{ext}} := \frac{1}{2} \sum _{j=1}^{n} h_{j-1/2} [ \mathit{EA}] ( \ell _{j-1/2}/h_{j-1/2} -1 )^{2} $$

of a discrete extensible Kirchhoff rod model [45], which approximates its continuum counterpart given by $\mathscr {W}_{\mathrm{ext}} := \frac{1}{2} \int _{0}^{L} ds [\mathit{EA}] (\| \mathbf {r}'(s) \| - 1)^{2}$. Additionally imposing the inextensibility condition$\ell _{j-1/2} \equiv h_{j-1/2}$ on the edges implies $\mathscr {W}^{(D)}_{\mathrm{ext}} \equiv 0 \equiv \mathscr {W} ^{(D)}_{es}$.

The discrete Lagrangian multipliers $\varLambda ^{(\alpha )}_{j-1/2}$ enforcing the constraints $\bar{\varGamma }^{(\alpha )}_{j-1/2} \equiv 0$ are introduced into the rod model by adding the sum term $\sum_{j=1} ^{n} h_{j-1/2} \varLambda ^{(\alpha )}_{j-1/2} \bar{\varGamma }^{( \alpha )}_{j-1/2}$, which approximates the corresponding integral $\int _{0}^{L} ds \varLambda ^{(\alpha )}(s) \varGamma ^{(\alpha )}(s)$ by midpoint quadrature, to the discretized Lagrangian (see [41, 42] for details). The inextensibility constraint on the discrete centerline is analogously enforced by the Lagrange multipliers $\varLambda ^{(3)}_{j-1/2}$, with the sum term $\sum_{j=1}^{n} \varLambda ^{(3)}_{j-1/2} ( \ell _{j-1/2} - h_{j-1/2} ) \approx \int _{0}^{L} ds \varLambda ^{(3)}(s) ( \| \mathbf {r}'(s) \| - 1 )$ replacing the (identically vanishing) discrete extensional energy in the Lagrangian.